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Tps3 Programmer`s Guide Peter B. Andrews Dan

1. EXT REW0 Aoo 2 V L Ax t LEAF3 Agop 2 V L Ax with complete mating L3 L4 The full expansion tree with the two lemmas has the form CONJI Va N CONJT IM P18 VA N Lv N lemma lemma CON J8 theorem L N lemma lemmas The lemma etrees correspond to the proofs of the lemmas and the lemma etrees correspond to the lemmas that can be used to show the theorem The value of the lemmas slot in the expansion proof has a value such as EXT LEMMA 1 EXT LEMMA 2 indicating that there are two lemmas nei ther of which depend on lemmas of their own The jform after dissolving the connections corresponding to the proofs of the lemmas is L14 FORALL q79 A x18 OR L15 L16 q 9 LAMBDA x FALSEHOOD OR q 9 Al L5 L7 A x 15 A x 15 FORALL q 7 OR OR L9 L10 L6 L8 q gt 7 LAMBDA x A x OR FALSEHOOD OR q 7 Al A x715 A x715 L1 P LAMBDA x A x OR FALSEHOOD L2 PA 132 13 MATING SEARCH Here we can find a complete mating by connecting L1 L9 L2 L10 corresponding to the mating of L1 and L2 that we wanted to make and L5 L6 L7 L8 corresponding to the proof that Ax Ax V L and A really are extensionally the same Given this complete expansion proof we can merge the proof and trans late to a natural deduction proof During the merging process the first two lemmas are recognized as unused The translation to natural deduction de pends on the
2. and are satisfied by any partial identities for 0 and a Then we extend 0 and a in stages First whenever y is a selected variable corresponding to a bound variable x in the original wff we let y z if this is legal Next whenever y is a selected variable associated with a bound variable x we try to send both of these to the same pretty no superscript variable if this is possible The function prettify identify free bound is used to try to send a free and a bound to the same pretty variable Third we check for bound variables z which occur in a subwff of the form Azi Azn M Ay An where A is a free or bound variable such A are stored in the prop erties bound try to equate free and bound try to equate bound In this case we try to send A and z to the same pretty variable if possible see the functions prettify identify free bound and prettify identify bound bound Finally we choose the rest of 0 and the rest of using the functions get best alt free name and get best alt bound name This completes the computation of 0 and a so we call rename all vars in etree to actually do the 160 14 MERGING renaming Finally remove unnecessary ab rews eliminates a rewrite nodes where the shallow does not change this may happen since we renamed variables prettify process vars in etree This collects the frees and bounds in the etree into the variables fixed frees fixed bounds frees to rename and b
3. In our example the least pre fixed property is Vu Vz Aula SwA vw V z 0 gt vz D Vz u zDd vz 186 16 SET VARIABLES and the induction property is Vp Vw pw D p Sw A p0 D Yz ut x gt pz The first thing we do is choose a new selected variable p4 p Let P 1fpre be Vz ulz SwApu V z 0 gt p gl P be Vz ulz D pz I indhyp be Vw p w D pt Sw A pt0 and Ip be Va ul x D p x We construct the positive ftree lf f as EXP Least Pre fixed Point Lp IMP va N lffi P7 LEAF1 P and the negative ftree IndF as SEL Induction Principle Lp IMP x N IndF I LEAF2 I where lf f and IndF are constructed by make clos setvar ind negf Since Py and Ig are a equal we can mate LEAF 1 and LEAF2 make clos setvar ind negf 1 We start with two wffs wff1 of the form C1 A AC wff2 of the form YZ Di V V DaD D pz where n is the number of constraints Let be new selected variables and LEAF 1 be a negative literal with shallow for mula pa We return two values a negative ftree of the form SEL VZ D1 Z V V Da Z gt pA a N Fo and a positive ftree PF with shallow formula wff1 where P F gt and N F are constructed by make clos setvar ind negf 2 3 SET CONSTRAINTS 187 In our example wff1 is Vw p w D pt Sw A pt0 and wff2 is Vz dulz SwApu V z 0 gt pt2 We choose a new selected variable z and create a ne
4. Chapter 2 TPS Structures 1 TPS Modules 1 1 The Tps3 Module Structure 1 2 Defining a New Module 2 Categories ili i oo o NNTB BWNNNFR EH a a ee a CONNDFPKBWWNYNNMNTTWOCAO YO 21 21 21 21 22 iv CONTENTS 3 Contexts 4 Flavors Chapter 3 Top Levels 1 Defining a Top Level 2 Command Interpreters Chapter 4 MExpr s 1 Defining MExpr s 2 Argument Types 2 1 List Types 2 2 Consed Types Chapter 5 Representing Well formed formulae 1 Types 2 Terminal Objects of the Syntax 3 Explanation of Properties 4 Non terminal Objects of the Syntax 5 Binders in TPS 5 1 An example How to See the Wf Representations 6 Flavors and Labels of Gwffs 6 1 Representation 6 2 Using Labels 6 3 Inheritance and Subflavors 6 4 Examples Chapter 6 Printing and Reading Well formed formulas 1 Parsing 2 Printing of formulas 2 1 The Basics 2 2 Prefix and Infix 2 3 Parameters and Flags 2 4 Functions available 2 5 Styles and Fonts 2 6 More about Functions 3 Pretty Printing of Formulas 3 1 Parameters and Flags 3 2 Creating the PPlist 3 3 Printing the PPlist 3 4 Pretty Printing Functions 3 5 JForms and Descr JForms 3 6 Some Functions 4 How to speed up pretty printing a bit 4 1 Static and Dynamic Parameters 4 2 A grand solution and why it fails 4 3 A modest solution and why it works 23 23 25 25 26 27 27 29 31 31 33 33 33 36 38 38 40 41 41 42 42 43 45 45 46 4
5. Ws Let gap qj max ej for 1 lt j lt q if e jk 1 lt k lt yj is non empty otherwise let gap q max L L lt qj V In j L qn Then maximize gap giving equal weight to all 1 lt j lt q We Minimize b min ej max dik with an alternative similar to Ws in case any of the sets is empty Wy Minimize t min p max q with an alternative similar to Ws in case any of the sets is empty 3 Updating the support structure Part of the execution of a rule application is updating the plan structure this is one of the reasons why with every rule there comes a description of how the plan structure should be updated Below we will give a description of what the function UPDATE PLAN is supposed to accomplish even in cases when the rule is used in a way different from the defaults Again we assume the UPDATE PLAN is called as in update plan default exist default new where default exist is p di1 dig Pp dpi dpzp default new is q1 11 1y dq q1 Cqyg Recall that there may be variables appearing in place of a line number The following restrictions should be noted e For each plan support pair pi dj diz in default exrist there is at most one occurrence of a variable among dj1 dizx e Any occurrence of a variable in default exist is unique When UPDATE PLAN is called all arguments are filled in that is each place is occupied eithe
6. and Co v is Yw v w D v S w The inversion principle is the familiar statement Vz wz gt z 0 v 3w fx S w A wul The induction principle is the equally familiar statement Vp p0 A Vw pw gt p S w D Vw vw D pu The inversion principle follows from the induction principle by instantiating p with Az x 0 v dw z S w A vw but it is clearly easier if we need the inversion principle to have it in the lemma rather than needing to prove it by instantiating the new set variable p In the general case in which some constraints contain a I v with v free in v we need machinery to show that there is a solution v satisfying the constraints the inversion principle and the induction principle This machinery is provided by the Knaster Tarski Fixed Point Theorem 3 1 Knaster Tarski Fixed Point Theorem The Knaster Tarski Fixed Point Theorem states that monotone set functions have fixed points There are also versions showing there are least and greatest fixed points in fact there are a lattice of such fixed points Definitions Suppose K g A gt A for a power set o A A pre fixed point of K is a set v such that K v C v A post fixed point of K is a set v such that v C K v A fixed point of K is a set v satisfying K v v Knaster Tarski Fixed Point Theorem Suppose K g A gt A for a power set A Further suppose K is monotone function in the sense that for every
7. fined otherwise TPS3 will not realize they are supposed to be the same As an example the flag RULEP MAINFN can be given the value RULEP SIMPLE Since RULEP SIMPLE is defined in the CORE package it must be exported in core exp so that when it is inputted the symbol CORE RULEP SIMPLE is interned not USER RULEP SIMPLE Of course this presumes that the USER package uses the CORE package which it always does in TPS3 2 3 Implementation specific differences Not all Lisp implementa tions are alike This is particularly true in the areas of Common Lisp which are intentionally unspecified including things like how the Lisp top level works how file pathnames are represented how the user exits the Lisp or saves a core image For this reason certain TPS3 source files contain and directives We try to keep the number of these files to a minimum so that when porting to new implementations work is minimized When using and you should try to use as specific a feature of the implementation as possible but avoid using the machine type unless that is the reason you have to make a change For example the feature allegro v3 1 is probably better than allegro as I have found out to my dismay when Allegro 4 0 came out Look at the lisp variable features to find what features that version of lisp recognizes A few examples of features are listed below allegro Allegro Common Lisp clisp Gnu Common Lisp cmu CMU Common L
8. java TpsStart For example lisp I tps3 dxl javainterface cd javafiles usr bin java TpsStart other big rightOffset 10 nopopups 245 246 22 RUNNING TPS WITH AN INTERFACE which tells To send these command line arguments to java TpsStart they should For example java TpsStart big instructs Java to use the big fonts and x2 and x4 instruct Java to multiply the size of the fonts by 2 or 4 respectively The extra argument nopopups will provide an alternative to popup prompts Another way to use the java interface is to start TPS3 as usual then use the command JAVAWIN This requires the flag JAVA COMM to be set appropriately When TPs3 is started in javaservice mode it uses the rest of the com mand line arguments to start the Java interface and creates sockets connect ing TPs3 to the Java interface Two processes are spawned one to receive input from the Java interface either from a prompt or from a menu item selection and another to actually run TPS3 commands The rest of the description does not particularly depend on the Java interface so I will simply refer to the interface and attempt to emphasize that such an interface could be implemented in a variety of ways The code to receive input from the interface is written in external interface lisp It listens to the socket stream and collects characters into strings separated by null characters ASCII 0 There are a few possib
9. the command to print the current wff If no subformula satisfying predicate is found EDSEARCH will return NIL For example defedop fb alias find binder result gt execute mhelp Find the first binder left to right edwff argname gwff defwffop find binder argtypes gwff resulttype edcommand argnames gwff arghelp gwff mhelp Find the first binder left to right defun find binder gwff edsearch gwff function boundwff p 6 Examples Consider the following examples 2s taken from the code 7th July 1994 6 EXAMPLES 81 defedop ib alias instantiate binder result gt edwff edwff argname bdwff defwffop instantiate binder argtypes gwff gwff resulttype gwff argnames term bdwff arghelp term bound wff applicable p lambda term bdwff and ae bd wff p bdwff type equal gar bdwff term mhelp Instantiate a top level universal or existential binder with a term defun instantiate binder term bdwff cond label q bdwff apply label bdwff instantiate binder term bdwff lsymbol q bdwff throwfail Cannot instantiate bdwff gwff a logical symbol boundwff q bdwff cond ae bd wff p bdwff substitute l term var term caar bdwff cdr bdwff t throwfail Instantiate only existential or universal quantifiers t not cdar bdwff fsym t throwfail Cannot instantiate an application defedop subst alias sub
10. 127 127 128 CONTENTS 9 After a Mating is Found 10 How MIN QUANT ETREE Works 11 Lemmas in Expansion Proofs 12 Extensional Expansion Dags 13 Printing Chapter 14 Merging 1 Applying Substitutions and Merging Duplicate Expansions 2 Detecting Unneeded Nodes 3 Modify Dual Rewrites 4 Prune Unmated Branches 5 Subst Skol Terms 6 Remove Leibniz 7 Raise Lambda Nodes 8 Cleanup Etree 9 Prettify 10 Merging Extensional Expansion Proofs Chapter 15 Unification 1 Data Structures Computing Head Normal Form Control Structure First Order Unification Subsumption Checking Notes Oo PR wh Chapter 16 Set Variables 1 Primitive Substitutions 2 Using Unification to Compute Setsubs 3 Set Constraints 3 1 Knaster Tarski Fixed Point Theorem 3 2 Tracing Through An Example Chapter 17 Tactics and Tacticals 1 Overview 2 Syntax for Tactics and Tacticals 3 Tacticals 4 Using Tactics 4 1 Implementation of tactics and tacticals Chapter 18 Proof Translations 1 Data Structures 2 EProofs to Nproofs 3 NProofs to Eproofs 3 1 Chad s Nat Etree 3 1 1 Normal Deductions 3 1 2 Annotations of the Assertions in a Proof 128 129 129 132 133 135 136 137 139 141 141 141 150 154 157 162 163 163 163 163 163 163 165 167 167 167 168 171 174 193 193 195 198 199 199 203 203 203 204 205 206 206 viii CONTENTS 3 1 3 Some Nonstandard ND Rules 3 1 4 Equality Rules 3 1 5 A Se
11. 138 14 MERGING should use the mating provided and not the eager mate everything be cause our mating might not be that big In fact Dan wrote such a SPANS that uses a variant of PROP MSEARCH and the time used in X5207 by SPANS went from 1 second to about 3 sec Unfortunately PROP MSEARCH or rather PROP FIND CHEAPEST PATH appears to have the empty disjunction causes confusion bug Try MS90 3 on the formula falsehood implies A More drastic changes were tougher to implement There were a few suggestions e What this is doing is a lot of duplicated effort so perhaps it would be possible to cache some results This would be pretty space intensive e g THM131 has an astronomical number of vpaths when it begins merging It turned out that attempts to make SPANS better by caching the results were pretty silly because the way it is invoked you can t tell the difference between sets of arguments The differences are made by changing the status of various lower level expansion nodes So that attempt was abandoned e Perhaps it would be possible to check the paths which the suspect node was on It s not clear how to do this e Of course it might be possible to avoid some of the calls to spans in the first place though possibly not with MS90 3 but even elim inating half would only save 3 days in the wolf goat cabbage prob lem without changing what it does In the end the solution used was as follows when path focuse
12. AUTO GENERATE HYPS Flag 106 BACKUP LIB DIR Flag 233 BASIC PROP TAC 208 BETA 115 binder Syntax 35 bktrack limit 117 bogus slot 24 BOOLEAN Argument Type 67 bound must avoid 160 161 bound try to equate bound 160 bound try to equate to free 160 254 BOUNDWFF Q Function 72 BRACKETS Parameter 50 ByteStream Data 88 bytestream tty Function 83 call system Function 6 ceb nat etr lisp File 205 ceb nat etree Function 205 ceb nat seq lisp File 205 ceb proof to natree Function 205 CFONT Property 49 cfont lisp File 49 CFontTable Parameter 62 cgraph 117 char Data 52 check conn Function 126 check etree structure Function 114 check etree structure break Function 114 check shallow formula Function 156 check shallow formulas Function 141 148 149 CJFORM Command 127 CLEANUP Command 109 cleanup all expansions Function 154 156 cleanup etree Function 186 154 cleanup expansion Function 154 cleanup leibniz expansions Function 141 143 147 156 cleanup rewrite node Function 154 CLEANUP SAME Flag 109 clist 173 Cols Data 56 COMMAND 246 command interpreters auto lisp File 26 command interpreters core lisp File 26 components 111 115 116 compound tacl defn Syntax 196 compound tactic Syntax 195 CONCEPT Style 48 CONCEPT S Style 48 conn unif p Function 126 connect socket Function 6 connection graph 127 connection
13. Cambridge MA 1977 IJCAI W W Bledsoe A maximal method for set variables in automatic theorem proving In J E Hayes Donald Michie and L I Mikulich editors Machine Intelligence 9 pages 53 100 Ellis Harwood Ltd Chichester and John Wiley amp Sons 1979 251 252 Ble83 Bro04 Fel86 Gen69 GMW79 MR93 Pfe87 BIBLIOGRAPHY W W Bledsoe Using examples to generate instantiations of set variables In Proceedings of the Eighth International Joint Conference on Artificial Intelli gence pages 892 901 Karlsruhe Germany 1983 IJCAI Chad E Brown Set Comprehension in Church s Type Theory PhD thesis Department of Mathematical Sciences Carnegie Mellon University 2004 Amy P Felty Using extended tactics to do proof transformations Techni cal Report MS CIS 86 89 Department of Computer and Information Science University of Pennsylvania 1986 Gerhard Gentzen The Collected Papers of Gerhard Gentzen North Holland Publishing Co Amsterdam 1969 Edited by M E Szabo Michael J Gordon Arthur J Milner and Christopher P Wadsworth Edin burgh LCF A Mechanised Logic of Computation volume 78 of Lecture Notes in Computer Science Springer Verlag 1979 Neil V Murray and Erik Rosenthal Dissolution Making paths vanish Journal of the ACM 40 3 504 535 July 1993 Frank Pfenning Proof Transformations in Higher Order Logic PhD thesis Carnegie Mellon University 1987 156 pp
14. IMPLIES AND x lt I gt FORALL IMPLIES R lt OI gt x lt I gt P lt OI gt x lt I gt x lt I gt FORALL IMPLIES NOT Q lt OI gt x lt I gt R lt OI gt x lt I gt x lt I gt FORALL OR P lt OI gt x lt I gt Q lt OI gt x lt I gt FLAVOR WEAK And another way lt 2 gt getwff subtype gwff p x2106 IMPLIES AND x lt I gt FORALL IMPLIES R lt OI gt x lt I gt P lt OI gt x lt I gt x lt I gt FORALL IMPLIES NOT Q lt OI gt x lt I gt R lt OI gt x lt I gt x lt I gt FORALL OR P lt OI gt x lt I gt Q lt OI gt x lt I gt And finally a way that only works at type O the 0 below is a zero not a capital O lt 3 gt get gwff0 x2106 CIMPLIES AND x lt I gt FORALL IMPLIES R lt OI gt x lt I gt P lt OI gt x lt I gt x lt I gt FORALL IMPLIES NOT Q lt OI gt x lt I gt R lt OI gt x lt I gt x lt I gt FORALL OR P lt OI gt x lt I gt Q lt OI gt x lt I gt 6 FLAVORS AND LABELS OF GWFFS 41 6 Flavors and Labels of Gwffs It is sometimes desirable to be able to endow a gwff with additional properties For example one may wish to be able to refer to a gwff by a short tag or to specify that a particular gwff is actually a node in an expansion tree For this purpose TPS3 provides the facility of labels and flavors see page 23 A label is an object which as far as TPS3 i
15. Suppose M is B z N for some binder B By induction on N and TU z we have an N satisfying N a N for each x T a x is free for x in N a z is free for z in N for each y T U z A y is free for y in N RO N gY N Let M be Ba z a z z N Note that M a M since a z is free for z in N Suppose we have x I To check if a x is free for x in M we need to know if x occurs free in N in the scope of a binder for a x By the induction hypothesis we know such a binder cannot be in N itself so the binder would have to be the outermost binder of M the one for a z That is we must have a x a z In such a case we need to ensure that x does not occur free at all in M Now condition ensures us that M is not of the form B z N where x bound in context occurs free in N So x cannot occur free in the a equivalent M So a x is free for x in M Suppose z T We need to show 6 z is free for z in M But this is immediate since z does not occur free in B a z a z z N Suppose we have y T where y is not z We need to show 6 y is free for y in M So we need to show no free occurence of y in M occurs withen the scope of a binder for 0 y By the induction hypothesis we know no such binder occurs in N So the binder would have to be the outermost binder of M the one for a z That is we must have a z y In this case we ne
16. function is called All this is also available in an interactive mode where the user is prompted for argu ments after he has been told what the defaults are and which alternatives are open The way all this has been implemented is through MExpr s which constitute special functional objects analogous to Expr s or FExpr s in LISP Every TPs3 command should be an MExpr so that the facilities of TPs3 top level can be utilized 1 Defining MExpr s Mexprs are special functional objects that are recognized by the top level of Tps3 They can be defined with the defmexpr macro which has a number of optional arguments The general format is indicate optional arguments defmexpr it name ArgTypes it type1 it type2 ArgNames it name1 it name2 ArgHelp it help1 it help2 DefaultFns it fnspec1 it fnspec2 EnterFns it fnspect it fnspec2 MainFns it fnspeci it fnspec2 CloseFns it fnspec1 it fnspec2 Print Command it boolean Dont Restore it boolean MHelp it comment There are actually two other possible entries Wffop Typelist and WffArgTypes these are only used in mexprs which are generated automatically by the Rules package In the following a function specification is either a symbol naming a function or an expression of the form Lambda arglist body We also assume that the main function which is to perform
17. graph are represented as numbers so sets of links are just ordered lists of numbers and one can efficiently test for subsets Given a new partial mating M Tps3 just looks at all the entries in the failure record to see if any of them are subsets of M 2 Operations on Expansion Trees 2 1 Deepening The code for deepening an expansion tree is in the file etrees wffops lisp The idea behind deepening an expansion tree is to find the leaves then destructively replace the leaves with a new node depending on the structure of the shallow formula and the setting of many many flags The key function to try to understand is deepen leaf node real Suppose A is the shallow wff of the leaf in question Here is a quick outline of what this function does 1 If A can be A reduced create a rewrite node with a leaf of the reduced form as its child 2 If A is B then create a negation node with a leaf of B as its child except sometimes when ADD TRUTH is set to T 3 If A is BAC then create an econjunction node with leaves for B and C as children 4 If A is BV C then create an edisjunction node with leaves for B and C as children 5 If A is BDC then create an implication node with leaves for B and C as children 6 If Ais B C then create a rewrite node rewriting the equivalence either as a conjunction of implications or disjunction of conjunc tions depending on the values of MIN QUANTIFIER SCOPE and REWRITE EQUIVS and the parit
18. if BRACKETS is NIL 3 PRETTY PRINTING OF FORMULAS 51 Other than that the functions are identical In the above example we would get Vz Pax V qo if Brackets T and Vx Po t V qo if Brackets NIL Both PrintWffPlain and PrintWffScope call PrintWfE where the real work of distinguishing the different kinds of formulas and symbols is being done The distinction between PrintWffPlain and PrintWff is only made for the sake of pretty printing see 3 4 At an even lower level is the function actually a macro PCALL which determines the appropriate way to print a particular symbol in the current style and prints an error if the relevant function is undefined PCALL actually applies to printing functions rather than characters so each function will have a different definition for different styles For example in style scribe the print symbol function is called PP SYMBOL SCRIBE whereas in style xterm it s called PP SYMBOL XTERM Examine the plists of SCRIBE and XTERM to verify this if you like 3 Pretty Printing of Formulas The most commonly used way of printing formulas such as lines or plans is to pretty print them This is a feature quite similar to the way LISP pretty prints functions Formulas which are too long to fit on one line of the current output device are broken at the main connective and printed in several lines The main difference to the LISP pretty printing is that we have to consider infix operators The
19. in string format or NIL which is used for arguments which are not gwffs WFFOP TYPE specifies a type in string format which is the type of the result the wffop or NIL if the result is not a gwff WFFOP TYPELIST A list of type symbols which are to be considered type variables in the definition of the wffop REPLACES The wffop being defined is to replace some previously de fined wffop This is used extremely rarely PRINT OP This is set to T for printing operations which are usu ally defined using the macro DEFPRTOP which sets this property automatically By default this property has value NIL MULTIPLE RECURSION seems to be set to T for most tests of equality and NIL everywhere else I m not entirely sure what it s for Here are some example which may shed more light onto the subject defwffop substitute l term var argtypes gwff gvar gwff resulttype gwff argnames term var inwff arghelp term var inwff wffargtypes A A B TERM and VAR are of type A wffop type B INWFF and result of type B wffop typelist A B where A and B may be any types mhelp defwffop lexpd argtypes gvar gwff gwff occ list resulttype gwff argnames var term inwff occurs arghelp lambda variable term to be extracted contracted form occurrences to be extracted wffargtypes A A B NIL TERM and VAR are of type A INWFF is of type B OCCURS is not wffop type B a gwff result
20. instantiate definitions inwff lambda abbsym chkarg eq abbsym chkarg gabbr defwffop instantiate all argtypes gwff symbollist resulttype gwff argnames inwff exceptions arghelp inwff exceptions defaultfns lambda amp rest rest mapcar lambda argdefault arg if eq arg argdefault arg NIL rest mhelp Instantiate all definitions except the ones specified in the second argument defun instantiate all inwff exceptions instantiate definitions inwff lambda abbsym chkarg not memq abbsym chkarg exceptions defwffop instantiate 1 argtypes gwff resulttype gwff argnames inwff arghelp inwff mhelp Instantiate the first abbreviation left to right defun instantiate 1 inwff let oneflag nil declare special oneflag instantiate definitions inwff lambda abbsym chkarg declare ignore abbsym chkarg special oneflag progi not oneflag setq oneflag t nil defwffrec instantiate definitions argnames inwff chkfn chkarg defun instantiate definitions inwff chkfn chkarg 3 EXAMPLE OF PLAYING WITH A JFORM IN THE EDITOR 77 cond label q inwff apply label inwff instantiate definitions inwff chkfn chkarg lsymbol q inwff cond or logconst q inwff propsym q inwff pmpropsym q inwff inwff pmabbrev q inwff if funcall chkfn get inwff stands for chkarg get pmdefn inwff inwff abbrev q
21. since there is only one literal Similarly NEGF is NEGF Z make clos setvar lemma negf 9 Usually this function is given two wffs which are a equal A positive leaf and a negative leaf are created mated and returned Another case is when set variable occurs embedded in the literal of the constraint In this case the first wff wff1 is of the form wa VE wT D lua T A Alw where the second wff wff2 is A ut In this case we create a positive ftree of the form EXP Ju Nz w Z gt ul z A Alw ou Lut CONJ Via N SEL YT wi T D u z LEAF3 A u La IMP L N LEAF1 uta LEAF ua and a negative leaf with shallow formula A u which we mate to LEAF3 Also LEAF 1 and LEAF2 are mated These two ftrees are returned In our example the literal is LF2 The formulas are both ut w so we create Kuug as a positive leaf and NEGF F as a negative leaf The two are mated and returned e make clos setvar ind negf This function is called with the least pre fixed point property formula 1fwff the induction property formula indwff and the list of constraints paths The goal is to prove the induction property from the pre fixed point property by constructing a negative ftree IndF for the induction property a positive ftree Lf f for the least pre fixed point property and a complete set of connections be tween nodes in them The dynamic variable 1ff is set to lf f IndF is returned
22. usually modest in efficiency but within weeks you will have no idea how they work and others will be even more mystified Those who follow you in your task will curse and despise 1 2 1 INTRODUCTION you consequently your cute programs will probably be completely rewritten anyway 3 Don t panic See section 9 for more minor guidelines 2 Tps3 Conventions 2 1 Filenames The extension of a filename should indicate what it contains lisp for Lisp source code exp for export statements vide infra rules for deduction rule definitions defirule statements mss for Scribe formatted documentation ter for TRXformatted documentation vpw for vpwindow output work for work files prf for proofs Filenames should be descriptive of their contents without being too long For example functions lisp would be a stupid name because from the name no one would know what its purpose was If you have several related files it is a good idea to give them a common prefix so that it is clear just from their names that they are related 2 2 Lisp packages and export files TPs3 creates and uses several different Lisp packages If you don t know what I mean by Lisp package read the chapter on packages in Common Lisp the Language 2nd ed These packages are created when TPS3 is compiled or built by make package forms in the files tps build lisp and tps compile lisp The package structure is set up so that common functions are p
23. 6 A conc This constructs two positive ftrees PF with shallow 0 C gt A and a positive ftree PF with shallow D We return EXP C A y PF and PF Again the constraint gt v 0 has an empty Y so we proceed directly to make clos setvar ind negf 5 with p 0 and z 0 to get the two positive ftrees For the constraint wlv w gt vt S w C is Vw p w D pt Swt wt 0 w w by definition make clos setvar ind negf 5 then constructs two positive ftrees PF with shallow formula p w D p S w and PF with shallow formula z Sw A p w We return EXP Yw pt wt D p Sw W PF and PF make clos setvar ind negf 5 This function starts with the negative leaf LEAF1 leaf constructed in make clos setvar ind negf 1 This function compares the arguments in the shal low formula p b1 bn of LEAF1 leaf and the given for mula p a1 an conc We recursively compare the arguments Suppose we have a negative leaf leaf with shallow formula p a1 ai 1 bi bn If b and a are syntactically equal then these correspond to some z WV Otherwise there should be an equation at the beginning of wff2 That is either wff2 is b aj or bi ai A D The function make ftree subst creates a posi tive ftree with shallow b a a negative leaf LEAF 2 with 190 16 SET VARIABLES shallow formula p a1 bj41 bn and some new con nections to add to the conne
24. 9 After a Mating is Found Here is the sequence of events for pfd 1 An expansion proof has been found A record probably called DUP RECORD of indices for duplicated variables leaves and con nections is maintained by the search process The unification tree associated with the mating has a record of the substitutions for variables 11 LEMMAS IN EXPANSION PROOFS 129 2 Construct jform with final substitutions applied This uses all copies of variables needed for the final mating Duplicate expansion tree from the jform Attach expansion terms to the expansion tree Call propositional search to reconstruct the mating Reorganize mating in ms88 form Merge expansion proof Translate expansion proof DDoS CONDOR W Nites Tiel lee Maat iat email 10 How MIN QUANT ETREE Works After a proof is found TPs3 constructs an expansion proof tree The implementation of flag MIN QUANT ETREE consists of the following steps 1 TPs3 searches through the expansion proof tree to find if there are primsubs which are not in minimized scope form If TPs3 finds some it goes to step 2 First TPS3 transform all the primsubs into their minimized scope forms In order to make sure that the expansion proof tree is still a correct one TPS3 has to modify it This is done by calling two functions namely one step mqe bd and one step mqe infix Now TPS3 goes to step 3 Since the expansion proof tree is still a correct one TPS3 can use a pr
25. Describe VPForm SignAtoms and Literals are described by Describe VPAtom and Describe VPLit respectively The virtual printing functions used for this process are FlatSym and FlatWff FlatSym takes an arbitrary LISP identifier as an argument and returns a pair gencharlist length for this identifier FlatWff takes a wff as argument and returns a PPlist for it The main function which then prints the Descr JForm is Print VPForm It takes the line of the Descr JForm which should be printed as an additional argument On lower levels SPrintAplicn and SPrintPPlist print an aplicn or a PPlist much in the same fashion SPrintAplicn and SPrintPPlist do except that PRINC takes the role of PRINC and TYO This is necessary from the way the actual output is handled If the vertical path diagram does not fit on one page several temporary files are opened and each file contains the information for one of the pages This means that the characters have to be counted and a new file to be selected as the current ouput file whenever 4 HOW TO SPEED UP PRETTY PRINTING A BIT 57 the character count exceeds the global parameter VPFPage The counting as well as the change of the current output file is done by the function PRINC The argument has to be either a LISP atom in which case it will be PRINC ed or a single element list in which case this element will be TYO ed 4 How to speed up pretty printing a bit Pretty printing in TPS or ETPS i
26. Execute the following command at the Unix prompt tar rvhf deuvname 9 A list of file names should pass by on the screen Just watch until you get a new Unix prompt 10 To check the tape enter tar tvf devname The same list of file names should pass by These are the names of the files which are now on the tape If there were already files on the tape you will see all of them listed as well 11 If all is well call the operator and tell them that you are done with the tape and that they can dismount it Then execute exit at the Unix prompt to give up control of the tape drive and log out as usual 12 Go back to the operator s room and pick up the tape To make a tar archive onto a Sun cartridge tape 1 Take the tape to the CS operator and ask her to put it on the machine O GP That is a Sun3 running Mach Go back to a terminal and log in on any machine Again you want to be in the directory home theorem project dist 2 Now you need to make sure that you can write to the O s tape drive You want to check the owner of the file o dev rst8 and make sure it s rfsd If not call the CS operator and ask them to assign it to rfsd so that you can make the tar tape 3 Now execute the following command at the Unix prompt tar cvhf o dev rst8 4 A list of file names should pass by on the screen It will be very slow 5 After you get a new Unix prompt wait a few seconds for the tape to rewind then chec
27. FIRST ORDER MODE MS are often dynamically set by the code This flag for example is removed from the list of flags along with any flag such as RIGHTMARGIN that is never relevant for automatic search The code does attempt to recognize when flags are dynamically bound in a certain context and take this into consideration A user is given the option of computing this flag relevancy information when calling UPDATE RELEVANT or SHOW RELEVANCE PATHS An other option is to load the relevance information from a file Such a file could have been created in a previous TPS3 session using SAVE FLAG RELEVANCY INFO CHAPTER 10 The Monitor The monitor is designed to be called during automatic proof searches its basic operation is described in the User Manual There are three basic steps required to write a new monitor function which are described below using the monitor function monitor check as an example More examples are in the file monitor lisp 1 The Defmonitor Command The command defmonitor behaves just like defmexpr the only difference being that the function it defines does not appear in the list when the user types This command will be called by the user before the search is begun and should be able to accept any required parameters or to calculate them from globally accessible variables at the time the command is called So for example the defmonitor part of monitor check looks like this defmonitor monitor check ar
28. INFIX to find it is 5 and then PUTPROP OR 5 INFIX Unfortunately prefix operators like negation do not currently have a binding strength associated with them and will always be associated to the left This has to be kept in mind when formulas are typed in Definitions can be infix or prefix and the same rules hold for them There are flags which control whether a definition or its instantiation will be printed Similarly logical atoms can appear as names or as values or both In general the appearance of a formula and in particular of a definition very much depends on which output device is used See section 2 5 for more de tail but remember that this only affects the way the primitive or defined symbols appear but not how the formula is assembled from its parts 2 3 Parameters and Flags The flags listed below are global param eters which can be set by the user to control the way formulas are printed These settings can be overridden if specific commands are given PrintTypes T causes all types to be printed If a typed symbol occurs more than once only the first occurrence will have a type symbol unless the same symbol name appears in the same formula with a different type NIL suppresses type symbols PrintDepth This is a parameter which determines how deep the recursion which prints the formula will go Subformulas located at a lower level will simply be replaced by an amp A PrintDepth of 0 means that everyth
29. It helps to give edops and wffops different names the name of a wffop should be longer and more descriptive than the name of the edop for which it is an alias 3 Example of Playing with a Jform in the Editor lt Ed9 gt sub x2115 78 7 WELL FORMED FORMULAE OPERATORS lt Ed10 gt neg lt Ed13 gt cjform AND FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 FORALL u lt I gt EXISTS v lt I gt OR LIT2 AND LIT3 LIT4 FORALL w lt I gt OR LIT5 LIT6 EXISTS u lt I gt FORALL v lt I gt OR LIT7 LIT8 lt Ed14 gt edwff AND FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 FORALL u lt I gt EXISTS v lt I gt OR LIT2 AND LIT3 LIT4 FORALL w lt I gt OR LIT5 LIT6 EXISTS u lt I gt FORALL v lt I gt OR LIT7 LIT8 lt Ed15 gt setq aa edwff AND FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 FORALL u lt I gt EXISTS v lt I gt OR LIT2 AND LIT3 LIT4 FORALL w lt I gt OR LIT5 LIT6 EXISTS u lt I gt FORALL v lt I gt OR LIT7 LIT8 lt Ed1i6 gt aa AND FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 FORALL u lt I gt EXISTS v lt I gt OR LIT2 AND LIT3 LIT4 FORALL w lt I gt OR LIT5 LIT6 EXISTS u lt I gt FORALL v lt I gt OR LIT7 LIT8 lt Ed17 gt auto jform parent aa NIL vp lt Ed19 gt setq bb auto conj
30. SPANNING CLIST PATH Function 137 SPANS Function 137 138 special exp File 5 SPrintPPlist Function 55 stack space 7 stacksize 8 stats 117 status 112 status userid Function 6 statuses 117 stringdt Function 14 stringdtl Function 14 strip exp vars Function 137 strip exp vars for etree Function 137 STYLE Flag 48 50 61 67 246 Style Parameter 48 SUBST 117 subst skol terms Function 136 141 subst skol terms main Function 141 subst vars for params Function 136 substitute in etree Function 137 substitution list 117 SUBSUMPTION CHECK Flag 163 SUBSUMPTION DEPTH Flag 168 165 SUBSUMPTION NODES Flag 163 support 101 swap space 7 8 symmetry 117 symmetry holder 117 tacl defn Syntax 196 tacmode 194 tacmode Flag 199 tactic defn Syntax 195 tactic exp Syntax 195 tactic mode Syntax 195 tactic use Syntax 195 TACTIC VERBOSE Flag 16 tactic verbose Flag 194 199 tacuse Flag 194 199 TAGS File 12 terms 115 TEX Style 49 TEXNAME Property 49 TEXPROOF Command 64 THROWFAIL Function 67 TLIST MExpr 21 token Syntax 195 tops20 lisp File 6 14 tps mss File 49 tps3 save Function 5 6 tps3 save lisp File 243 tps3 patch File 7 tps3 sys File 7 TpsBigFonts java File 248 tpsjobs File 16 tpsjobs done File 1 16 17 TpsSmallFonts java File 248 TpsWin java File 247 248 TREAT HLINES AS DLINES Fla
31. T apply Pfenning s algorithm for removing the Leibniz equality nodes for substitution of equality nodes remove leibniz nodes 4 Try to replace selected parameters by the actual bound variables subst vars for params not always possible because of restric tion that a parameter should appear at most once 5 Raise lambda rewrite nodes so that in natural deduction the lambda normalization occurs as soon as possible raise lambda nodes 6 Clean up the etree cleanup etree For each expansion term in the tree a Lambda normalize it b Minimize the superscripts on bound variables c Make a new expansion with the new term d Deepen the new expansion like the original but removing un necessary lambda norm steps e Remove the original expansion Begin natural deduction proof using current eproof e Set up planned line a Use shallow wff of the current eproof s etree 203 204 18 PROOF TRANSLATIONS b Give it the tree as value for its NODE property c Give it the current eproof s mating list of pairs of node names as its MATING property Call use tactic with tactic desired a Each line in the proof will correspond to a node in the etree the natural deduction proof is stored in the variable dproof b Here is an important property which should remain invariant during the translation process It should always be the case that the line mating of the planned line is a p acceptable mat ing for the etree that one could cons
32. a fo single symbol That works only because consists of only one charac ter def logconst type 00 38 5 REPRESENTING WELL FORMED FORMULAE printnotype t prefix 6 mhelp Negation 4 Non terminal Objects of the Syntax Isymbol logconst propsym pmpropsym abbrev pmabbrev lsymbol roughly corresponds to what was called hatom for Huet atom from Huet s unification algorithm in the old representation Generalized WFFs guff Isymbol Cpropsym binder gwff Generalized binder gwffl gwff2 where cdr type gwff1 type guff2 label 5 Binders in TPS In the discussion about the internal representation of wffs the issue of binders has been neglected so far Currently TPS allows three binders A V J plus some buggy fragments of support for the 4 binder Since most binders are inherently polymorphic there is only one kind of binder Notice that the definition is formulated such that a binder may have a definition but need not In order to determine the type of a bound wff the type of the scope of the binder must be matched against the type stored in the SCOPE TYPE property Also the type of the bound variable must match the type in the VAR TYPE property These matches are performed keeping in mind that all types in the TYPELIST property are considered to be variables Then the bindings established during the match are used to construct the type of the whole bound wff
33. a non NIL TypeConst property For example 0 and I def typeconst o mhelp The type of truth values typevariable Type Variable An identifier with a non NIL TypeVar property It is the parsers responsibility to give the TypeVar property to types not previously encountered typesymbol typeconstant typevariable typesymbol typesym bol 2 Terminal Objects of the Syntax Before going into detail about the terminal objects of the syntax some general remarks about type polymorphism in TPS3 are needed TpPs3 supports polymorphic objects like C subset which is a relation that may hold between sets of any type It must be understood however that the parser completely eliminates this ambiguity of types when actually reading a wff In a given wff every proper subwff has a type Therefore there is a class of objects with polymorphic type which never appear in a wif but nevertheless may be typed by the user The instances of those polymorphic abbreviations or polymorphic proper symbols inside the formula will refer however to those polymorphic primitive symbols or polymorphic abbreviating symbols For reasons of efficiency binders are handled slightly differently Binders are also polymorphic in the sense that a certain binder say V may bind variables of any type The case of binder however is slightly different from that of polymorphic abbreviations since a binder is not a proper subwff Binders there
34. after primsub 119 banned conns list 118 features 3 5 hacked rewrites list 118 139 ho banned conns list 118 instantiated defs list 118 instantiated eqs list 118 leibniz var list 119 rew unsubst exps 119 standard output 14 246 unsubst exp vars 119 big 245 bottomOffset 245 javainterface 245 maxChars 245 nopopups 245 rightOffset 245 screenx 245 screeny 245 x2 245 x4 245 ETPS 4 TPS 4 Syntax 38 SPrintAplicn Function 56 SPrintPPlist Function 56 VPForm Function 56 PRINC Function 56 57 Atom Data 55 JForm Data 55 Literal Data 55 SignAtom Data 55 Var Data 55 typeconstant Type 33 typesymbol Type 33 typevariable Type 33 name 27 29 MARKATOM Data 53 abbrev Syntax 34 Index 253 ABBREV Q Function 72 accept socket conn Function 6 active mating 136 ADD EXT LEMMAS Command 130 ADD HYPS Command 97 ADD TRUTH 116 ADD TRUTH Flag 116 120 136 ALL 126 all banned 116 117 allow nonleaf conns 126 ALLOW NONLEAF CONNS Flag 126 ALLSCOPEFLAG Flag 63 AllScopeFlag Flag 47 48 AndHeight Parameter 62 aplicn Data 52 aplicnlist Data 52 APPLY LABEL Function 70 APPLY MATCH Flag 90 apply thm 146 Function 141 148 149 ascnumber Data 52 ASSEMBLE RULE Command 15 ASSERT Command 97 ASSERT LEMMAS Flag 182 ATOMVALFLAG Flag 63 AtomValFlag Flag 47 48 ATTACH DUP INFO TO NATREE Function 215
35. and also a current list of all module known to TPs3 There is a partial order of modules in TPS3 One whole set of modules called BARE is distinguished from the others All files in the module BARE and all of its submodules must always be present in a TPS3 core image When TPs3 is built from Lisp some of the files in the BARE module can not be loaded with a module loading command since it has not been defined Thus even though every file for TPs3 belongs to a proper module not all modules are loaded the same way because of the bootstrapping problem Another quirk should be mentioned here A module called WFFS defines the basic operations of wffs The modules WFF PRINT and WFF PARSE de pend on WFFS The module WFFS however cannot exist alone the modules WFF PRINT and WFF PARSE must be present also even though this fact can not be deduced from the module structure 1 2 Defining a New Module To define a new module for TPS3 use the DEFMODULE macro Its format is defmodule it name needed modules it module it module macro files it mfile it mfile files it file it file mhelp it help string needed modules These are all modules that must be loaded for the module name to work Because of the transitive structure of mod ules only the direct predecessors of the new module need to be listed 21 22 2 TPS STRUCTURES macro files These are the files the compiler needs before it can
36. are used by the REPORT facility As general conven tions when the template form is a variable use the same name for the template name e g DPROOF If the template form is STATUS statusfn use statusfn as the template name e g DATE for STATUS DATE or USERID for STATUS USERID signal hook an optional function to be called whenever the the event is signalled This should not to the writing of the informa tion but may be used to do something else If the function does a THROWFAIL the calling SIGNAL EVENT will return NIL which means failure of the event The arguments of the function should be the same as EVENT ARGS write when one of IMMEDIATE NEVER or an integer n which means to write after an implementation depended period of n At the moment this will write whenever the number of inputs n EVENT CYCLE where EVENT CYCLE is a global variable say 5 write file the name of the global FLAG with the filename of the file for the message to be appended to write hook an optional function to be called whenever a number gt 0 of events are written Its first argument is the file it will write to if the write hook returns Its second argument is the list of evaluated templates to be written If an event is to be written immediately this will always be a list of length 1 mhelp The mhelp string for the event Remember that an event is ignored until INIT EVENTS or INIT EVENT event has been called 1 2 Signalling E
37. bool gt is t the formula is the residue of a formula in the first sequent e If lt bool gt is nil the formula is the residue of a formula in the second sequent e lt nat gt is a natural number indicating the position of the formula in the first or second sequent For another example given two cut free derivations of B C 3 A A D 224 18 PROOF TRANSLATIONS and E A F gt G with mix formulas might return e a cut free derivation of C F E gt G D positive indices t 1 nil 2 nil 0 and negative indices nil 0 t 2 e or a cut free derivation of B B gt G positive indices t 0 t 0 and negative indices nil 0 e or a cut free derivation of F E gt D D positive indices nil 2 nil 0 and negative indices t 2 t 2 In general we start with two derivations D Tri Ay and Do To Ao and two lists i4 ip and j jz of natural numbers The natural num bers give us the positions of the mix formulas in A and Ig Let A be a list A A and T be a list B Bm The mix formulas are Aj and B These formulas should have a common reduct with respect to A reduction expanding abbreviations and expansion equalities using either extensionality or Leibniz Mix elimination returns a cut free derivation of r gt a and two lists of indl and indl of indices indicating the preimage of the formulas
38. clearly demonstrates that a grand solution is infeasible 4 3 A modest solution and why it works Everything would work out fine if we could limit the number of dynamic parameters This can be achieved very simply by restricting ourselves to saving information about symbols only and not about labels in general Of the various dynamic parameters only one survives this cut Do I put a type on this identifier is the only question that can be solved from the context only This simplification also reduces the number of static parame ters For example AllScopeFlag is irrelevant to the printing of symbols wffs without proper subwffs However care must be taken when the appearance of identifiers is changed We will return to this problem later in the section about other issues 4 4 Implementation All printing requests go through the function PWFF When PWFF is entered all static parameters have their final value 4 HOW TO SPEED UP PRETTY PRINTING A BIT 59 Inside PWFF we will set two more special global variables Hash Notype and Hash Type Hash Type and Hash Notype will have as value of the name of the prop erty which contains the symbol s pplist When constructing the pplist for the given wff the first pass during pretty printing it is checked whether symbols have the appropriate property If yes the symbol itself stands for a pplist We are thus modifying the recursive definition of pplist If not the pplist will be
39. compile any of the files in the module It is generally a good idea to make a file with all the macro definitions e g argument types fla vors of labels etc and separate it from the functions commands etc in the module This means clearer program structure but also minimal overhead for the compiler files These are the rest of the files in the module When the module is loaded first the macro files are loaded then the files The new module should also be added into defpck lisp at an appropriate point and should be added into whichever of tps build lisp tps compile lisp etps build lisp and etps compile lisp are appropriate these files are in the same directory as the Makefile not the main TPS directory 2 Categories TPs3 categories are in a sense data types A category is a way to char acterize a set of similar objects which have properties of the same types use the same auxiliary functions are acted on by the same functions etc Categories are orthogonal to the package module structure i e a cat egory may have members which are defined in many different packages and modules Categories group objects by functionality how they behave whereas packages and modules group objects by purpose why they exist Categories are defined using the defcategory macro For example the definition of the category of TPs3 top levels is defcategory toplevel define deftoplevel properties top prompt fn singlefn command int
40. consider a familiar example in which we have the two constraints gt v0 and w v w gt v S w The minimal solution to this is the least set containing 0 and closed under the function S Since we have the full power of higher order logic we can define such a solution by AaVp p0 A Yz v z D v S z D pe It should be clear at this point that the terms defining these solutions can become quite large Making such instantiations can be prohibitive in theorem proving because we may need to perform unification with the solution in some other part of the problem This suggests it may be simpler to use the 170 16 SET VARIABLES instantiation to prove there is a set satisfying the conditions we want Then using this lemma the set that exists is represented by a selected variable a very small term Another motivation for using such lemmas is that we gain more control over what properties of the sets are included in the lemma See section 11 for more on the implementation of expansion proofs using lemmas Suppose we have a general set of minimal constraints Consider what properties of v an existence lemma should include The most obvious is the condition C v vaT v gt v ti where UW w Of course as mentioned above C ATT so we certainly need to include more conditions Two other conditions are inversion princi ples and inductive principles An inversion principle for the set of constraints would be of the for
41. copy and so on These dupli cations are preserved during backtracking in case they are needed again later we use universal dup mark to remember how many of them are really there e The procedure for expanding the etree after ms90 3 finishes a search has to be completely replaced The old procedure relied on the fact that the names of skolem constants never changed and so it was possible to attach all of the expansion terms to the jform and then duplicate and deepen the etree while applying the appropriate substitutions The names of selections do change this makes the substitutions incorrect because they will contain the names of old selections So we use ms90 3 mating and dup record to duplicate the etree directly and then add the correct connections to it using ADD CONN This procedure is probably still buggy 4 Checking Acyclicity of the Dependency Relation 4 1 The Dependency Relation An expansion proof is given by an expansion tree and a mating One of the conditions on the expansion tree 4 CHECKING ACYCLICITY OF THE DEPENDENCY RELATION 123 is that the so called dependency relation is acyclic irreflexive For an expansion tree Q let Sg be the set of selected variables in Q and let Og be the set of occurrences of expansion terms The following definition can be found in Miller s thesis Definition Let Q be an expansion tree Let lt 2 be the binary relation on Og such that t lt Q s if there exists y Sg so
42. current eproof seealso eproof current eproof in order to decide whether to call ms or ms propositional if there are no free variables in current eproof ms propositional is called 7 1 Sunil s Propositional Theorem Prover The original files for Sunil s fast propositional calculus theorem prover are in home theorem project tps variants si prop qload these files go into the editor and make the edwff the example you wish to run Within the editor test runs the program using edwff as argument When it is done test1 shows the mating it found Most of the code in the above directory is now a permanent part of TPs3 the function prop msearch can be called from the mate top level and will display a correct mating for propositional jforms The way to call it is auto prop msearch auto cr eproof jform the latter function is the internal name for CJFORM For some reason the propositional theorem prover is never used except to reconstruct the mating after a path focused duplication procedure has found it 8 Control Structure and Interface to Unification The non path focused duplication npfd search procedures have a con nection graph but the pfd procedures do not the latter just apply simpl to decide whether literals may be mated Sunil s disjunction heuristic see below is implemented for pfd search procedures but not for npfd The non path focused duplication search procedures break a jform with top level disjunct
43. cut free derivation of r1 2 gt A1 Ag This is usually done by a recursive algorithm on the two derivations where at each step either one of the derivations gets smaller or the cut formula gets smaller while the derivations may get much larger Since we have an explicit merge contraction rule we have the following problematic case Suppose D ends with an application of merge cut D3 Ty gt A A Ai Ty gt A Ay where A is the cut formula Also suppose A is the principal formula of Do The most natural way to handle this case is to first perform cut elimination on D3 and Dg giving a cut free derivation Ds ri T2 A Ay Ao merge Then we could call cut elimination again with D5 and D2 to obtain a cut free derivation De Vt I2 Tp rap Ai Ao A2 With some applications of focus and merge to shuffle and merge the formu las we would have the cut free derivation of D7 r T2 gt Aj A2 we desire However the derivation Ds will in general be bigger than Ds while the cut formula A remains the same So this recursive call to cut elimination may not terminate even with a first order derivation 4 CUT ELIMINATION 223 We can get around this problem by performing mix elimination as did Gentzen Gen69 instead of cut elimination Because we care about the or der of formulas in the sequent mix elimination is complicated The main functions are ftree seq mix elim 1 and ftree seq mix elim pri
44. detected immediately 2 Tps3 CONVENTIONS 3 There is a special export file called special erp This file contains export statements for symbols which may already exist in certain Lisp implementa tions For example some implementations already contain a symbol EXIT while others do not Why is this a problem Because if the CORE packages uses a package from an implementation e g Allegro s EXCL package and that package already exports the symbol EXIT so that EXIT is imported by CORE then an error will result if we try to export EXIT from the CORE package i e you can t export a symbol from a package other than its home package special exp uses the standard and macros to specify in which implementations such symbols should be exported from the CORE package Generally these nuisance symbols are found by trial and error when first porting TPs3 to a Lisp implementation and some symbols may have to be moved from core exp to special exp Another symbol in t CORE that can cause problems with some Lisps is date Note that when TPs3 starts up the USER soon to be COMMON LISP USER as the changes in Common Lisp suggested by the X3J13 committee are im plemented is the value of the variable package What this means is that any symbols typed in by the TPs3 user will be interned in the USER package Thus any symbols that could be inputted by the user as say a flag value should be exported from the package in which they were de
45. different LISP packages If you are doing this think about why it is nec essary Perhaps it is better to export the symbols or rethink the package structure e Symbols should be exported before files containing them are com piled Otherwise you stand the risk of having those symbol occurrences interned in the wrong package e Lisp macros can be very useful but it is easy to overuse them It can be very difficult to debug code that uses many macros and because there is no guarantee that macros will not be expanded when code is 9 WRITING NEW CODE WITHOUT MAKING A NUISANCE OF YOURSELF 15 loaded and they are always expanded when compiled modifying a macro means recompiling every file in which it appears which is quite a nuisance e There are a multitude of functions in TPS3 so one must be careful not to inadvertently redefine a function or macro With the Lisp function APROPOS you can check to see whether a function name is already being used Use the TAGS table See section 4 above e Try not to re invent the wheel look in all the likely places to see if some of the code you need has already been written If your new construct is similar to an existing one use grep i in the directory afs andrew mcs math TPS lisp to find and examine all uses of the existing construct e Remember that rules of inference should be written as rules files and compiled with the ASSEMBLE RULE command if you modify the lisp files directly you run
46. either D or D ends with a logical rule and the principal formula is not a mix formula then we can recursively call mix elimination on the premisses and imitate the rule For example if D is Dil Dio Ty gt A Any T12 gt B A12 Ti i2 gt AA B Ar Ais the function ftree seq invert position list uses the positions of the mix formulas in A AA B 1 Ate to find the positions of the mix formulas in Aj and Aj Using this we can call mix elimination twice to obtain D3 T3 gt A3 4 CUT ELIMINATION 227 and Da V4 7c A4 The function ftree seq mix elim imitate rule finishes this case First ftree seq bring to front uses structural rules to bring the residuals of A and B to the front of A3 and A4 so we have Ds Ts gt A AG and D4 r4 gt B A At this point we apply the A rule D3 Di T gt A A T gt B A A T3 4 gt A B A A and compute the indices of the residuals PRINCIPAL The final and most important case is when both mix formulas are principal This is handled by the function ftree seq mix elim principal In each of the cases above one of the derivations gets smaller in the recursive call The case when both mix formulas are principal is complicated by the need to perform several recursive calls requiring us to adjust and compose the indices as we go along The cases for each connective and quantifier are described next e T l This cannot happen because the only
47. file functions xterm bold font xterm normal font and pptyox use codes to switch between bold and normal font There is currently some confusion about how one switches to bold font The code in xterm lisp switches to bold by sending in ASCII lt ESC gt 5 m However it appears the official ANSI code for switching to bold is lt ESC gt 1 m while lt ESC gt 5 m is for blinking text see http members tripod com oldboard assembly ansi_codes html1 ESC n n nm 11 MISCELLANEOUS 19 Set Graphics Rendition is used to set attributes as well as foreground and background colors If multiple parameters are used they are executed in sequence and the effects are cumulative n is one of the following attributes or colors All attributes off Bold Dim Underline Blink Reverse Video Invisible ON oOFPFNF O The working hypothesis of Chad at the moment is that the impleme nations of xterm we have used for TPS3 render blinking text as bold so that switching to blink is the same as switching to bold In 2005 while using xterm version X Org 6 7 0 192 for TPS3 the symbols were not dis playing as symbols Instead what should have been symbols were blinking normal text This can be fixed by changing the codes in xterm lisp to send lt ESC gt 1 minstead of lt ESC gt 5 m It is possible there was good reason why the original programmer Dan Nesmith used blink instead of bold Instead of explicitly making th
48. function for flavor name which will create a structure of type flavor name To access the attributes of a label which is a symbol use get since all attributes will be on the symbol s property list The attributes of a label which is a structure of type flavor name can be accessed by using the standard Lisp accessing functions for structures Thus if one of the label s attributes is represents the attribute can be accessed by calling the function flavor name represents Flavors can be redefined or modified at any time This may be done if for example one wished to extend a flavor s definition into a Lisp package which was not always loaded Merely put another defflavor statement into the code You need only put the new or changed properties in the redefinition If however you wish to change the attributes of a flavor which is a structure you should put in all of the attributes you desire not just the new ones and be sure to declare any included flavor as well Note it is possible to change a flavor which uses symbols as labels into one which uses structures but if you fail to redefine code which depends on property lists the program will be smashed to flinders 6 3 Inheritance and Subflavors Some flavors may be similar in many ways in fact some flavors may be more specialized versions of other flavors One may wish a new flavor s labels to be operated upon by most wffops in the same way as an existing flavor s labels this
49. function which given a goal to be accomplished will return a list of new goals along with a procedure by which the original goal can be achieved given that the new goals are first achieved Tactics also may fail that is they may not be applicable to the goal with which they are invoked Tacticals operate upon tactics in much the same way that functionals operate upon functions By the use of tacticals one may create a tactic that repeatedly carries out a single tactic or composes two or more tactics This allows one to combine many small tactics into a large tactic which represents a general strategy for solving goals As implemented in TPS3 a tactic is a function which takes a goal as an argument and returns four values a list of new goals a message which tells what the tactic did or didn t do a token indicating what the tactic did and a validation which is a lambda expression which takes as many arguments as the number of new goals and which given solutions for the new goals combines the solutions into a solution for the original goal It is 193 194 17 TACTICS AND TACTICALS possible that the validation is used nowhere in the code and that it should be phased out Consider this example Suppose we are trying to define tactics which will convert an arithmetic expression in infix form to one in prefix form and evaluate it One tactic might if given a goal of the form A B where A and B are themselves arithmetic expres
50. general structure of the functions doing the pretty printing allows future changes to the way printing in general is done without making changes to the pretty printer Whenever a formula is to be pretty printed the usual printing functions as described above are called but instead of printing the characters they will be appended to a list Later this list is used to actually output the characters after the decision where to break the formula has been made From this structure it is clear that all the parameters and flags controlling the appearance of a formula on the several printing devices still work in the way described before There are however a few additional flags which determine how subformulas will be arranged within a line 3 1 Parameters and Flags As new flags particularly for pretty printing we have PPWffag T means that formulas will usually be pretty printed This is the default value NIL means that formulas never will be pretty printed unless the command is given explicitly LocalLeftFlag T will cause the left hand side of an infix expres sion to be aligned with the operator and not with the right hand side 52 6 PRINTING AND READING WELL FORMED FORMULAS NIL is the default and prints left and right hand side of an infix expression with the same indentation FilLineFlag T will try to fill a line as much as possible before starting a new one This only makes a difference for associative infix operators
51. idea works regardless of the arity of the set type of u The functions make knaster tarski lemma make knaster tarski negf make knaster tarski leastfp lemma make knaster tarski leastfp negf make knaster tarski gfp lemma make knaster tarski gfp negf and oth ers with similar names construct an ftree representation of an expansion proof of these different versions of the Knaster Tarski Theorem for a given set type A proof of the Knaster Tarski Theorem for the set type of u can be used as a lemma to show a set existence lemma for u which includes the constraint properties the inversion principle and the induction principle THM2 in TPs3 is a version of the Knaster Tarksi Theorem When we apply the Knaster Tarski theorem we need to instantiate K with a set function we know is monotone This can be ensured syntactically Definition We can define a set of terms positive and negative with respect to u by induction e ut is positive with respect to u so long as u does not occur free in t e P V P gt is positive negative with respect to u if both P and P are e P A Po is positive negative with respect to u if both P and P are e P DP is positive negative with respect to u if P is negative pos itive with respect to u and P is positive negative with respect to u e P is positive negative with respect to u if P is negative positive with respect to u e VxP is positive negative with respect to
52. in T and A T and indl are lists of the same length For each B in T there is a corresponding index b j where either b is t and j is the position of B in T or b is nil and j j1 j is the position of B in T Similarly A and indl have the same length For each A in A there is a corresponding index a i where either a is t and i i1 7 is the position of A in Ag or a is nil and i is the position of A in Ag The main function which attempts to eliminate all cuts from a derivation is ftree seq cut elim This is a simple recursive algorithm which first elimi nates cuts from premisses then either imitates the rule or if the rule was cut uses mix elimination to obtain a cut free derivation from the cut free derivations of the premisses That is given two derivations D r ES A Ay 4 CUT ELIMINATION 225 and Do A To gt A gt we call mix elimination with these derivations and two lists of positions 0 and 0 Mix elimination returns a cut free derivation of A and two lists of indices Using the indices and the focus weaken and merge rules we can shuffle the formulas in I and A to obtain a derivation of r1 r2 gt Aj Ao This final shuffling is performed by ftree seq mix elim finish 4 3 The Mix Elimination Algorithm The mix elimination algo rithm works by recursion on the two given derivations Suppose we are given two cut free derivations D Tri 7 Ai and Do To gt A2 and two l
53. inwff if funcall chkfn inwff chkarg get defn inwff inwff boundwff q inwff if and anyabbrev q binding inwff funcall chkfn binding inwff chkarg get def binder binding inwff bindvar inwff gdr inwff cons car inwff instantiate definitions gdr inwff chkfn chkarg t let mewcar instantiate definitions car inwff chkfn chkarg if and lambda bd p newcar not lambda bd p car inwff lcontr cons newcar instantiate definitions cdr inwff chkfn chkarg cons newcar instantiate definitions cdr inwff chkfn chkarg 2 The formula editor The formula editor is in many ways very similar to the top level of TPs3 The main difference is that we have an entity called current w or edwff which can be operated on All the regular top level commands can still be executed but we can now also call any wffop directly If we want the wffop to act on the edwff we can specify EDWFF which is a legal gwff inside the editor This process is made even easier through the introduction of edops An edop is very similar to a wffop but it ties into the structure of the editor in two very important ways One argument can be singled out so that it will always be the edwff and secondly the edop will specify what happens to the result of the operations which is often the new edwff This is particularly useful for operations which take one argument and return one wff as a value like lambda normalization
54. lib file of zero length in which case the file is deleted Objects which are loaded by the user are re defined as TPS3 objects library objects of type MODE or MODE1 become TPs3 modes whereas gwffs and abbreviations each become both theorems of type library and abbreviations Notice that this blurs the distinction between a gwff and an abbreviation Users are allowed to redefine TPs3 theorems of type library and their corresponding abbreviations theorems of other types may not be redefined this is to prevent users from accidentally overwriting standard abbreviations with their own library definitions Library definitions are parsed every time they are written and this in volves re loading all of the needed objects Since the needed objects are often abbreviations this will frequently result in their redefinition and so LIsPwill generate warning messages If a large file is being re parsed this can take a long time and produce a huge number of warnings 1 Converting TPTP Problems to Tps3 library items Every needed function or flag is set in library2 lisp The utility is made of two library commands INSERT TPTP to insert one TPTP problem into Tps3 and INSERT TPTP to automatically call insert tptp on an entire directory These two commands act on tps files which are generated 233 234 19 LIBRARY using the TPTP2X utility One flag INSERT LIB DIR defines the output directory for the newly created items Please note that a
55. list goal IDTAC succeed Lambda x x mhelp Tactical which always succeeds returns its goal unchanged The following is an example of a compound tactical then and orelse are tacticals deftactical then defn tac lambda taci tac2 then tac1 then orelse tac2 idtac idtac mhelp THEN tactici tactic2 will first apply tactic1 if it fails then failure is returned otherwise tactic2 is applied to each resulting goal If tactic2 fails on any of these goals 198 17 TACTICS AND TACTICALS then the new goals obtained as a result of applying tactici are returned otherwise the new goals obtained as the result of applying both tactic1 and tactic2 are returned 3 Tacticals There are several tacticals available Many of them are taken directly from GMW79 After the name of each tactical is given an example of how it is used followed by a description of the behavior of the tactical when called with goal as its goal The newgoals and validation returned are described only when the tactical succeeds 1 idtac idtac 2 3 NS SS NS Returns goal lambda x x failtac failtac Returns failure call call command Executes command as if it were entered at top level of TPs3 This is used only for side effects Returns goal lambda x x orelse orelse tacticl tactic2 tacticN If N 0 return failure else apply tactic1 to goal If this fails call orelse
56. modified format tps file is used in order to prevent conflit with other objects inside of tps The original file uses const def axiom and thm as functions they have been replaced by const insert def insert axiom insert and thm insert The principal issue when converting TPTP problems into TPS items is to avoid using already defined objects For this reason every inserted item is suffixed usually with the name of the destination library file e g one becomes one ALG2684 More information about how to process this conversion are available in the User s guide CHAPTER 20 Teaching Records 1 Events in TPS3 The primary purpose of events in TPS3 is to collect information about the usage of the system That includes support of features such as automatic grading of exercises and keeping statistics on the application of inference rules Events once defined and initialized can be signalled from anywhere in TpPs3 Settings of flags ordinarily collected into modes control if when and where signalled events are recorded Siganlling of events can be suppressed by changing the values of the flags in subject EVENTS Notice that this of course only suppresses the signalling not the events themselves In ETPS a basic set of events is predefined and the events are signalled automatically whenever appropriate If these events are then recorded de pends on your ETPS profile T
57. module is compiled loaded the macro files are compiled loaded first Also there is code for loading the macro files for a module without loading the other files When adding a file foo lisp to a module designate it as a macro file if many of the other files in that module use structures macros or variables defined in foo lisp In addition to the files it contains the definition of a TPS3 module also specifies the other modules which must also be present when it is used The TPs3 module structure breaks up the source files into chunks each which has some particular purpose or purposes Then to build a version of TPs3 which has certain capabilities one need only load the modules required This is how the files tps build lisp and tps compile lisp specify TPS3 is to be built Note that etps build lisp and etps compile lisp just load fewer modules than the build compile files for Tps3 Likewise there are grader compile lisp and grader build lisp files for building a Grader core image By using the module mechanism a module may be modified by adding deleting or modifying its constituent files and other users don t have to know all they need to know is what the module provides Functions such as LOAD MODULE are provided to load modules making sure that any modules they require are also loaded 2 5 File format In general programmers should use only lower case Why For two reasons It is easier to read and in case sensitive operat
58. not blank xstartData 3 2 1 1 1 2 3 This character is drawn using 7 rectangles starting from x coordinates 3 2 1 1 1 2 3 resp 2 ADDING A NEW SYMBOL 249 ystartData 10 9 8 7 6 5 4 These are the y coordinates of the 7 rectangles widthData 4 1 1 4 1 1 4 These are the widths of the 7 rectangles heightData 1 1 1 1 1 1 1 These are the heights of the 7 rectangles Since all heights are 1 the epsilon character is drawn by drawing 7 horizontal lines ABBO0 ABI 96 And89 And01 BA98 BBP93 BBSS08 BF 93 Bis99 Ble77 Ble79 Bibliography Peter B Andrews Matthew Bishop and Chad E Brown System description Tps A theorem proving system for type theory In David McAllester editor Proceedings of the 17th International Conference on Automated Deduction vol ume 1831 of Lecture Notes in Artificial Intelligence pages 164 169 Pittsburgh PA USA 2000 Springer Verlag http dx doi org 10 1007 10721959_11 Peter B Andrews Matthew Bishop Sunil Issar Dan Nesmith Frank Pfenning and Hongwei Xi TPS A theorem proving system for classical type theory Journal of Automated Reasoning 16 321 353 1996 Reprinted in BBSSO8 http dx doi org 10 1007 BF00252180 Peter B Andrews On connections and higher order logic Journal of Automated Reasoning 5 257 291 1989 Reprinted in BBSS08 Peter B Andrews Classical type theory In Alan Robinson and Andrei Voro
59. of the ETPS manual to see the effect of the following lines in the file home theorem project doc etps system mss seealso Primary Quitting Other tt EXIT seealso Primary Quitting Other tt END PRFW seealso Primary Quitting Other tt OK indexentry key Quitting entry Quitting 5 Other commands in the manuals Any other Scribe commands may be used in the manuals for example we use the typewriter font given by t for command names and the italic font given by i for file names In the TFXversions of the manuals one uses the corresponding TEXcommands We also have TPS in Scribe and TPS in T X to print the string Tps3 and HTPS in Scribe to do the same in headers 6 Converting Scribe to ATEX documentation The aime of this section is to provide helpful information on how to program a new documentation device 6 1 The laterdoc lisp file This file was written as an equivalent to scrdoc lisp Functions and macros are essentially equivalent Nevertheless while the Scribe documentation system contains several calls to special func tion such as scribe one fn which are internal properties of special objects e g tactics latex doc lisp contains every ATpX specific help formatting function Some of these functions uses the tex style in the TPs3 system which is described in defter lisp As the style properties were thought to be used with Scribe some of them cannot be easily translated in ATE
60. say x or x then the new variable will be of the form x or z get best alt free name Returns the nicest legal alternative for a free y using get best alt name and prettify free legal p get best alt bound name Returns the nicest legal alternative for a bound z using get best alt name and prettify bound legal p e rename all vars in etree This corresponds to R on etrees e scope problem p Checks to make sure the new var and old var are either both free in context or were both bound by the same binder If not return T This will cause rename all vars in wff to throw a failure indicating a bug in PRETTIFY rename all vars in wff This corresponds to R on wffs described above We do check to make sure we are avoiding variable capture 162 14 MERGING If a variable capture does occur there is a bug in PRETTIFY and a failure is thrown 10 Merging Extensional Expansion Proofs The code for merging extensional expansion proofs is completely differ ent than the corresponding code for merging expansion proofs Essentially we translate from an open dag ext exp open dag to a ground dag ext exp dag via the function eeod to eed node see ext exp open dags lisp This translation process deletes any unnecessary parts of the extensional expan sion proof Prettify for extensional expansion proofs is performed by ext exp dag prettify in ext exp dags lisp The code is similar to the prettify code in mating merge eq lisp CHAPTER 1
61. should print the external representation of its only argument It can expect this argument to be of the correct type The value returned is ignored A PrintFn may signal an error if printing is not possible e g if the current style does not have a representation of the given data type TestFn Here fnspec should return NIL if its argument is not of type name and return something not NIL otherwise Short Prompt boolean is only used in otl typ lisp but I can t work out what for MHelp This is an optional documentation and is accessed during MHelp or after a while the user supplies command arguments interactively property value Pairs like this allow for more information about the type For example deftype anything getfn lambda anything anything testfin lambda anything declare ignore anything t printfn princ mhelp Any legal LISP object deftype integert getfin testfn testfin and integerp integer gt integer 1 printfn princ mhelp A nonnegative integer deftype boolean getfn cond boolean t t nil testfin or eq boolean t eq boolean nil printfn if boolean princ t princ nil mhelp A Boolean value NIL for false T for true No TestFn or PrintFn is allowed to have any side effects since they may be called arbitrarily often No GetFn needs to expect as an argument since defaults are now figured out elsewhere This avoids confl
62. system This is highly discouraged b When you want to create new nodes or change some nodes in the current topnode make the corresponding changes in the attribute statuses of current eproof which is a hash table Don t forget this otherwise your new commands won t work Actually according to an old email from Dan Nesmith about status the status of etrees is not stored in the status slot of an etree Instead the status of an etree is in a hash table associated with the current eproof So update statuses depends on the value of current eproof The same email contains information on the predecessor slot For refer ence here is the email To Peter Andrews K GP CS CMU EDU Cc issar K GP CS CMU EDU hwxi K GP CS CMU EDU mbishop K GP CS CMU EDU Subject Re STATUS and PREDECESSOR In Reply To Your message of Fri 25 Sep 92 16 05 44 EDT lt 1992 9 25 20 0 41 Peter Andrews K GP CS CMU EDU gt Date Tue 29 Sep 92 16 34 36 0100 Message Id lt 29531 717780876 js sfbslc10 cs uni sb de gt From Dan Nesmith lt nesmith cs uni sb de gt Your message dated Fri 25 Sep 92 16 05 44 EDT gt Can you please explain what the slots STATUS and PREDECESSOR gt in the structure current topnode are for Of course maybe we should gt have a meeting with Sunil to have a discussion about this Sorry it took so long to reply At first glance I thought this was something to do with unification current topnode is a variable whose v
63. than one type the type of each proper must somehow be encoded in its name TPS3 does this by appending the type and replaced by lt and gt respectively to the print name of the symbol pmpropsym Polymorphic Proper Symbols These are just like propsym except that they also have a STANDS FOR property which is the polymorphic primitive symbol pmprsym this polymorphic proper symbol was constructed from Note that this particular instance of the polymorphic primitive symbol always has a specific given type For example IOTA lt I lt OI gt is a pmpropsym after parsing y QQy def pmpropsym iota type A OA typelist A printnotype t fo single symbol iota mhelp Description operator abbrev Abbreviations 2 TERMINAL OBJECTS OF THE SYNTAX 35 For example EQUIV This is separate category from poly morphic abbreviations only for reasons of efficiency An abbrevi ation could be thought of as a polymorphic abbreviation with an empty list of type variables For example def abbrev equiv type 000 printnotype t fo single symbol equiv infix 2 defn 000 pmabbrev Polymorphic Abbreviations For example SUBSET lt 0 lt 0A gt lt 0A SUBSET lt 0 lt 0B gt lt OB are poly morphic abbreviations after parsing A f12 0a SUBSET B or RO f12 obb a SUBSET R b For example def abbrev subset type O OA OA typelist A printnotype t infix 8 fo s
64. that y is selected in a node dominated by t and y is free in s The transitive closure lt Q of lt Q is called the dependency relation While the relation above is defined in terms of expansion terms the way TPS actually searches for a proof is as follows Expansion variables are used in place of expansion terms and TPS finds instantiations for these vari ables These instantiations arise from two sources pre processing usually giving set variable instantiations and unification The instantiations made during pre processing may contain expansion variables which will later be instantiated by unification During the mating search unification constructs substitutions for expan sion variables and checks the acyclicity condition for those substitutions To be sound TPS should include the substitutions made in pre processing as well Until now TPS did not include these substitutions in the check This did not cause a problem with soundness because the omitted substitutions all PRIMSUBS GENSUBS did not contain any selected variables The most obvious way to ensure soundness is to include all the sub stitutions in the acyclicity check not only those arising from unification However since the instantiations made before search begins will not change during search we should be able to find a more efficient method The idea is to start with an expansion tree obtained after pre processing i e after instantiations have been made and the resu
65. the command has n arguments Then the phrases in the above definition have the following meaning name This is the name of the MExpr as called by the user 27 4 MEXPR S ArgTypes This is a list which must have as many elements as the function arguments i e n typel type2 typen have to be valid types which means that they have to have a non NIL ArgType prop erty Each argument supplied by the user on the command line will be processed first by the corresponding GetFn In case an FExpr is to be called each element of the argument list is presupposed to be of the same type This type is specified in parentheses If ArgTypes is omitted the function has no arguments ArgHelp This has to be a list of length n Each element is a string describing the argument or NIL These quick helps for arguments can be accessed via the when being prompted for the argument value For an FExpr there should be only one string DefaultFns The fnspecs declared in this place are being processed in a left to right order where the result of one fnspec is passed on to next A fnspec can signal an error a THROW with a FAIL label if the arguments seem to be contradictory e g if a planned line and a term is supplied for a P rule but the term does not appear in the proof but it can count on the arguments being of the correct type and in internal representation In detail each default fnspec must be either a symbol denoting a function of n a
66. the main file for the manual on lt topic gt one of char etps facilities grader prog teacher and user See the TPs3 User Manual for additional information When new facilities are added to ETPS copy the information about them from the automatically produced facilities mss and facilities tex into the appropriate ETPS mss tex file 4 Indexing in the Manuals The basic mechanisms are in home theorem project doc lib index lib and home theorem project doc lib indexcat mss Note the comment on the use of IndexCategory in the former file In the TRXversion of the Program mer s Guide there are indexing commands defined which mimic the role of the corresponding Scribe commands indexotherDIY TAC in the text on page lt pagenumber gt puts DIY TAC lt pagenumber gt into the index index X WORD in the text on page lt pagenumber gt puts WORD Y lt pagenumber gt into the index Example indexcommandDO GRADES in the text on page lt pagenumber gt puts DO GRADES System Command lt pagenumber gt into the index Here is a partial list of of possible values for X and Y where the complete list is in home theorem project doc lib indexcat mss 6 CONVERTING SCRIBE TO ATEX DOCUMENTATION 87 X command gives Y System Command X edop gives Y Editor Command X flag gives Y flag X function gives Y function X style gives Y style X mexpr gives Y mexpr See quitting in the index
67. the risk of having your modifications accidentally overwritten by future users e When modifying copies of existing files prior to installing them rename the file temporarily for example preface the filename with your initials so that if you compile your new code it won t overwrite the existing compiled file e Don t install code until you ve tested it After installation keep backup copies of the old files in the home theorem project old source directory on gtps and change their extensions from lisp to lisp to 1997 jan 3 or whatever Delete all Emacs backup files from the main lisp directory Compile new code using the CMU Common Lisp version of TPS3 since that compiler is fussier than most e Try to make sure that online documentation is included in all user functions argument types etc that you define Also you should at the very least put comments in your code better yet write some documentation for the manuals Note that online help can be asso ciated with any symbol using the definfo command e Ifanew subject has been created and this subject contains flags im portant for automatic search the function mode flagging lisp should be updated The code starts as defun mode mode let already set nil unless eq mode gettype tps mode maint quiet dolist subject IMPORTANT MATING SEARCH MS88 UNIFICATION PRIMSUBS MTREE TOP The list in the dolist contains all the subjects important for aut
68. tiy th P a A a Pi Alt 2 Pa Alta Let us use the notation Q CUT to indicate this CUT node inside the larger tree Q Assuming we do not have a problem with acyclicity the CUT node 220 18 PROOF TRANSLATIONS should reduce as follows MERGE t1 a Q CUT1 Q CUT2 where CUT is CUT1 Va 4 xN C P t A t Pi A t and CUT 2 is CUT2 L N C SEL Vz A z EXP Yx A x La tof N th P A a P gt A te Py A tn However this leads to a loop as in the following simple example Consider the following proof of Va Vy Ao ty D Aaah Ab b using the lemma Vz Aow ZZ 1 1 Ev Sev Ue Aou LY Hyp 2 1 F Vy Aou 2 Y UI z 1 3 1 H Aoy 2 2 UI z 2 4 1 E Vz Aou 2 2 UGen z 3 5 1 F Ao a UI a 4 6 1 E Agu b b UI b 4 7 1 F Ao a a Ab b Conj 5 6 8 F VaVyAocy D gt Aa aA Ab b Deduct 7 4 CUT ELIMINATION 221 When translating this proof we obtain a tree Q with two cut nodes CUT1 4 4 N R SEL1 EXP1 Le La P c Ace P Aaa and CUT2 x 4 N R SEL2 EX P2 ed b P3 d Add Py Abb corresponding to the two applications of the lemma in line 4 Contracting CUTI gives MERGE VA XN a c Q CUT3 Q CUT4 where CUT3 and CUT4 are CUT3 x 4 N a c R Pi a Aaa Py Aaa and CUT4 x 4 N R SEL3 EX P3 Le P c Ace Both CUT3 and CUT4 are easy to eliminate CUT3 can be replaced by a c R and CUT4 can be replaced by R1 with appropriate changes to the complete mating
69. to Value Value resp If a parameter of the following list is not included in the call of the function its global value will be assumed Possible parameters with their range and the section they are explained in are PrintTypes T NIL 2 3 PrintDepth 0 1 2 3 AllScopeFlag T NIL 2 3 AtomValFlag T NIL 2 3 PPWfllag T NIL 3 1 LocalLeftFlag T NIL 3 1 FilLineFlag T NIL 3 1 FlushLeftFlag T NIL 3 1 Leftmargin 1 Rightmargin 2 6 Rightmargin 1 2 2 6 Style XTERM SCRIBE CONCEPT GENERIC SAIL TEX 2 5 2 5 Styles and Fonts TPs3 can work with a variety of different out put devices producing special characters like V or A where possible and spelling them out as FORALL and AND where not Details of how to produce output files for various purposes are in the ETPS and User s Manuals At no point does the user actually make a commitment whether to work with special characters or not since she can easily switch back and forth The internal representation is completely independent of these switches in the external representation A few commands such as VPForm and VPDiag have an argument Style which specifies the style in which a file is produced Furthermore there is a flag STYLE which TPs3 will use in the absence of any other indication as to the appropriate form of output Along with the style the user can usually specify an appropriate line length by using the LEFTMARGIN and RIGHTMARGIN flags Some com mands most no
70. u if P is or if x is the same variable as u 3 SET CONSTRAINTS 173 e 4xP is positive negative with respect to u if P is or if x is the same variable as u Proposition Suppose P is positive with respect to u Then AuAZP u Z represents a monotone function of u Proof We can prove this by induction on P O The function mon fn negf generates an expansion proof that a particular AuAZP u Z is monotone where P is positive with respect to u This function returns an ftree and pushes new connections on the special variable clist We will solve some constraints by building a monotone function K from the constraints Some constraints do not directly give a monotone function of the set variable In these cases we may want to find a best approximating Definition Given a function F o A gt A let Kp p A gt 9 A and KF A gt A be defined by Kp u z Vw ju Cw D z F w and KF u z 3w w Cu A z F w Proposition Kpr and K are monotone set functions For all u Kp u C F u C KF u Futhermore if L p A gt p A is a montone set function and forall u L u C F u then forall u L u C Kp u Similarly if R g A gt A is a montone set function and forall u F u C R u then forall u K u C R u So Kp and K are the best monotone upper and lower aproximations of F Proof Suppose u C v and z Kp u We need to show that z F w for every w D v Given such a w apply the de
71. value of the flag ASSERT LEMMAS If ASSERT LEMMAS is set to T then the expansion proof translates into two natural deduction proofs The first is a proof of the lemma The second is a proof of the theorem using the lemma The name of the natural deduction proof of the lemma is given by the corresponding symbol in the value of the lemmas slot For example the name could be EXT LEMMA 2 In the proof of the theorem there would be a line justified by Assert EXT LEMMA 2 If ASSERT LEMMAS is NIL the expansion proof translates to a single natu ral deduction proof First the outline 5 FV tol Ago TV L At DAal AxV1 A PLAN4 6 F Vzal Ago t VL Az gt Ax ArV1 A Same as 5 104 F Po o 0t Bie Aolo zvl gt PA PLAN2 is formed then these gaps are filled in using the corresponding parts of the expansion proof Extensionality examples can of this form can be proven automatically using DIY if the flag USE EXT LEMMAS is set to T 12 Extensional Expansion Dags Extensional expansion dags are a generalization of expansion trees which represent proofs in extensional type theory see Chad E Brown s thesis Bro04 There are four lisp structures used to represent extensional expansion dags ext exp dag ext exp arc ext exp open dag and ext exp open arc These structures are defined in ext exp dag macros lisp The intention is that ext exp dag ext exp arc structures represent nodes arcs in ground dags with no expansion
72. var structure is the type of its variable and all exp vars are gwffs Errors of the form Wff operation lt wffop gt cannot be applied to labels of flavor lt label gt are almost always caused by attempting to use a wffop on a flavor for which the corresponding property 24 2 TPS STRUCTURES is undefined for example if we deleted the lines above and recompiled TPS any attempt to find the type of an expansion variable would result in the error Wff operation TYPE cannot be applied to labels of flavor EXP VAR Flavors that are defined within TPS will also have the slot bogus slot this slot is tested for by TPS to confirm that the flavor was defined by TPS but the contents of this slot are never examined This means that there is always one empty slot in each node of an expansion tree or jform which the programmer can use to store information while a program is being tested whereas if you define a new slot you have to recompile all instances of a structure which can be a nuisance Obviously once the new code is working you should define a new slot change all references to bogus slot and recompile TPS For examples of flavors of gwffs see page 41 CHAPTER 3 Top Levels 1 Defining a Top Level Top levels are a TPS3 category whose definition is given in section 2 For an example let s look at the editor top level deftoplevel ed top top prompt fn ed top prompt command interpreter ed command interpreter print ed
73. variables and should only be constructively manipulated On the other hand ext exp open dag ext exp open arc structures represent nodes arcs may contain expansion variables and can be destructively ma nipulated e g via substitution The global variable ext exp dag verbose causes the structures to be printed with a huge amount of verbosity and should be set to NIL unless debugging Similarly setting the global vari able ext exp dag debug to T causes a lot of extra sanity checking to aid debugging The EXT MATE top level can be used to manipulate extensional expan sion dags The code implementing the EXT MATE is in the file ext mate top lisp 13 PRINTING 133 The automatic search procedures ms03 7 and ms04 2 use extensional expansion dags The code for ms03 7 is in ext search lisp The code for ms04 2 is in ms04 search_lisp 13 Printing CHAPTER 14 Merging Once a complete mating is found we enter a merging process The merging process performs the following steps some of which are described in more detail in separate sections Note merging still contains bugs although not very many If a cor rect mating is merged and produces a translation error or a message of the form The formula is not provable as there s no connection on the follow ing path lt path gt then it s likely that a bug in merging is the culprit Within merging the routines for REMOVE LEIBNIZ CLEANUP ETREE and PRETTIFY are the most likely caus
74. w The corresponding selected variable w is used to create K uut nee as EXP Aw Sw Sw uu v Sw 0 1 w Ka LF3LF2 where K unt with NEGF make clos setvar lemma negf 7 The arguments that are sent to this function are those from the main literal which were not the first argument corresponding to the selected variables generated in make clos setvar lemma negf 2 These arguments correspond to a conjunction of reflexive equations If there is more than one literal in the constraint more conjuncts are formed when make clos setvar lemma negf 8 is called The first constraint LF1 has no extra literals so when this function is called K uuh with shallow wff 0 0 is simply constructed as REFL REW 0 0 TRUE The second constraint LF3 LF2 does have the extra literal LF2 When this is called K uu LF Which shallow wff Sw Sw A ul w is constructed as by make clos setvar lemma negf 7 along CONJ L REFL REW Kuu 4 TRUE where K uu is constructed by make clos setvar lemma negf 8 3 SET CONSTRAINTS 185 e make clos setvar lemma negf 8 Given the left side T of a constraint a negative ftree K uuk and positive ftree N EGF are constructed along with connections between them These are constructed as conjuncts for each literal in I using make clos setvar lemma negf 9 to construct the children of the con juncts In our example K uur is simply K wun
75. way T can be positive principal in Do is if the last step is focus weaken or merge which were handled above Similarly if L is negative principal in D1 then it must end with a focus weaken or merge e REW D is Dis Ty gt A gt B A BD A Ai _ _ REW Tri gt A B Ai Do is Dai AD B A B gt A T2 gt v Snel eS RNIN 1 A B T Ag First we recursively call mix elimination for all the unexpanded formulas A B with D and D2 smaller than D2 giving D3 AD B B D A T gt As 228 18 PROOF TRANSLATIONS Next recursively call mix elimination for all the unexpanded for mulas A B with D1 smaller than D1 and D2 giving D4 T4 gt AD B A BD A Ay Finally we can call mix elimination for the two occurrences of the smaller mix formula AD B B D A with D4 and D3 giving D5 Ts gt As Of course with if we weigh more than A and D we can say A D B A B D A is smaller than A B So this case does not cause a problem with termination The hard part is tracing the residuals to return the proper indices Each formula in C in Ts is either the residual of some C in T4 or C in 13 If the preimage is in T4 then C is a residual of a C in Ty or Pg Once we compute the preimage of the preimage we have the proper index The following diagram is helpful when trying to compute preim ages Diy Do D Dai elim m D4 D3 elim Ds EX P SEL
76. we obtain H U Sp C Lp or equivalently H C Lp and Sp C Lp Here we would like to make Lp as small as possible the fewer hypotheses we used the stronger the statement result Alternatively the second line can again be viewed as a constraint on H when matching an existent pline This leads to the following algorithm for determining set of hypothesis 1 Let Dexist be the set of dlines which exist in the current proof Then set ower La a Sald Deris AHE Ha Also let Lq be the strong default for the hypotheses of line d for each d Derists 2 Let Pexise be the set of plines which exist in the current proof Then set upper Lplp Pexist AH Hp Also let Lp be the strong default for the hypotheses of line p for each pE Pesists 3 If not Hiower C Hupper the application of the inference rule is illegal Do a THROWFAIL with proper message 108 12 ETPS 4 If both Hiower and Hupper are undefined empty intersection or union respectively do not fill in any further defaults 5 If exactly one of Hiower and Hupper is undefined let Hiower upper or vice versa 6 For non existent dlines d we let La HupperUSa If AUTO GENERATE HYPS is T make Lg the strong default for that argument otherwise just a regular default 7 For non existent plines p we let Lp HiowerUSp If AUTO GENERATE HYPS is T make Lg the strong default for that argument otherwise just a regular default This algorithm is
77. we replace it with Bi Ai Bi Ai Finally we sort the list lexicographically by the pairs of hash numbers and store it in the subsumed slot In future if we return to this node we can just read off the hash function without recalculating it Now TPs3 compares the dotted pairs of numbers from the hash functions of the new and old node If those for the new node are equal to or a superset of those for the old node then we need to do some more detailed checking This is the point at which TPs3 prints a if UNIFY VERBOSE is not SILENT Otherwise we know there is no subsumption and proceed to the next node If there is still a possibility of subsumption the next thing to do is to enumerate all the ways in which the old node might be considered a subset of the new one If we are lucky each dotted pair of numbers in a given node will be different from each other and from all other dotted pairs at that node and there will only be one way in which this could happen If we aren t so lucky if there are several disagreement pairs that get the same pair of hash numbers or if there is a disagreement pair where the hash numbers for both wffs are the same there may be multiple ways to think about For each possible way we output two disagreement pair lists which will be the entire old node and that subset of the new node to which it might correspond ordered so that the nth element of one is supposed to compare to the nth element of the
78. we will call inher itance of properties In addition one may wish a new flavor to actually be a subtype in Lisp terms of an existing flavor and have the attributes of the existing flavor s labels be included in the attributes of the new flavor s labels this we will call inclusion of attributes The defflavor form allows either or both types of sharing to be used Inheritance of properties is signalled in the defflavor by a form such as inherit properties existing flavor existing flavorN This will cause the properties in the wffop hash table of the existing flavors to be placed into the wffop hash table of the new flavor If any conflict of prop erties occurs e g if existing flavorI and existing flavorJ I lt J have a prop erty with the same name then the value which existing flavorJ has for that property will be the one inherited by the new flavor A new flavor may inherit properties from any number of existing flavors In contrast attributes may be included from only one other flavor This can be done by using the form include ezisting flavor The existing fla vor must be a flavor whose instances are structures and the new flavor s 6 FLAVORS AND LABELS OF GWFFS 43 instances will also be structures whose slots include the attributes of the existing flavor Thus the same accessing functions for those slots will work on labels of both flavors To define default values for those slots add them to the include form a
79. which is part of the pplist or simply recomputed every time the identifier is printed The first solution would require significantly more permanently occupied list space the second solution would take more time during each printing Notice that the time required for the printing is not that long since the identifier will have to be printed only during the actual printing phase not during the virtual printing phase The length is already known through the symbols property list It therefore seems to be much better only to save the printing length of the identifier Another issue arises when we allow that the printing appearance of identifiers be changed Since all the length information attached to the identifier will be wrong it is necessary to remove that information In order to be able to do this we need to recognize the properties which stem from the printing algorithm sketched above The simplest way to achieve this is to declare a global variable hash properties which is a list of all the 60 6 PRINTING AND READING WELL FORMED FORMULAS properties that have been used for printing so far This must be updated whenever PWFF is called The hope is that due to the limited number of static and dynamic parameters this list remains manageable in size An alternative would be to write the hashing function in such a way that all names produced by it start with a unique pattern say One can then systematically look for properties whose n
80. would make things more difficult for the user to remember This can cause problems in TPS3 unless programmers are careful You see we currently use the symbol s property list extensively to store things When a matingsearch command such as LEAVE is defined the actions that are to be taken when the user inputs the command are stored on LEAVE s property list It is important therefore that each category use different property names so that there is never a clash For example if we used the property ACTION for both review commands and matingsearch commands then LEAVE s property list could not hold both simultaneously but merely one or the other Better property names would be REVIEW ACTION and MATE ACTION 7 Output Some general tips for keeping the output as neat as possible e Avoid using the lisp function y or n p and stick to the TPs3 func tion prompt read so that the responses will go into work files cor rectly 14 1 INTRODUCTION e msg and msgf which is like msg but adds a linefeed if necessary are TPS3 functions for producing output These functions take a sequence of arguments and evaluate and print out each argument in an appropriate format an argument t means go to a new line See defutil msg e msg gwff1 gwff will print out the correct representation of the gwff whereas princ gwff1 will just print its internal rep resentation e To insert a call to runcount in the code msgf runcount e stringd
81. 116 OPTIONS VERBOSE Flag 16 order components Flag 125 otl help lisp File 97 P EdOp 40 parent 112 pass socket local pass Function 6 PC Parameter 62 PCALL Function 51 pdepth Data 52 pgroup Data 52 plength Data 52 pline 101 pmabbrev Syntax 35 PMABBREV Q Function 72 pmabbsym Syntax 34 pmpropsym Syntax 34 PMPROPSYM Q Function 72 pmprsym Syntax 34 positive 111 PP Enter Kset Function 55 PP SYMBOL SCRIBE Function 51 PP SYMBOL XTERM Function 51 PPlist 52 53 pplist Data 52 PPrinc Function 55 PPrinc0 Function 55 PPTyo Function 55 PPTyo0 Function 55 PPTyos Function 55 pptyox Function 18 PPWFFLAG Flag 63 PPWfflag Flag 48 51 PPWfflength Parameter 54 PPWfflist Parameter 54 PROO 167 pre process nonleaf leibniz connections Function 141 143 predecessor 112 116 predicate var 161 prettify bound bound legal p Function 161 prettify bound legal p Function 160 161 prettify bound rename Function 160 prettify etree Function 136 159 prettify free bound legal p Function 161 prettify free legal p Function 160 161 prettify free rename Function 160 prettify identify bound bound Function 159 161 prettify identify free bound Function 159 161 prettify process vars in etree Function 159 160 pretty var p Function 161 PreWff Data 83 prfw lisp File 14 prim vars 115 primitive tacl defn Syntax 196 primi
82. 16 Leaf Data 43 leaf list 117 leaf p Function 136 141 LEFTMARGIN Flag 48 50 Leftmargin Parameter 48 LEIBNIZ 116 lemmas 117 129 LerList Data 83 LerList Parameter 84 LezScan Function 83 LINE NO DEFAULTS Function 101 LINE NO DEFAULTS FROM Function 102 LINE NO DEFAULTS TO Function 102 linelength Function 6 LIST RULES Command 97 list rules Function 97 LOCALLEFTFLAG Flag 63 LocalLeftFlag Flag 48 51 locate 12 logconst Syntax 34 LOGCONST Q Function 71 LOWERCASERAISE Flag 62 LSYMBOL Q Function 71 maint lisp File 85 make assert a hyp Function 208 make clos setvar ind negf Function 181 185 186 make clos setvar ind negf 1 Function 186 189 make clos setvar ind negf 2 Function 186 187 make clos setvar ind negf 3 Function 187 188 make clos setvar ind negf 4 Function 188 189 make clos setvar ind negf 5 Function 189 make clos setvar lemma negf Function 175 179 make clos setvar lemma negf 0 Flag 175 make clos setvar lemma negf 0 Function 179 make clos setvar lemma negf 1 Function 181 make clos setvar lemma negf 2 Function 181 183 184 make clos setvar lemma negf 8 Function 181 182 make clos setvar lemma negf 4 Function 181 183 make clos setvar lemma negf 5 Function 182 183 make clos setvar lemma negf 6 Function 183 make clos setvar lemma negf 7 Function 188 184 make clos setvar lemma negf 8 Function 184 185 ma
83. 5 Unification The relevant files are ms90 3 node lisp ms90 3 unif lisp node lisp unif lisp Tps3 has four unification algorithms two for first order logic and two for the full type theory Here we are mainly concerned with the two type theory ones which differ as follows e UN838 is called by those procedures which do not use path focused duplication and by the Tps3 UNIFY top level Each variable is a symbol We use lazy reduction Head normal form General imple mentation Can use different strategies for searching the unification tree Default breadth first Requires storing almost the entire tree When called from mating search we search for a success node or generate the tree to a pre determined maximum depth e UN90 is called by those procedures which do use path focused du plication No interactive interface exists now Each variable has the form symbol number Terms are reduced to normal form as in Huet s paper Depth first search Stores only non failure leaf node Does not store the entire unification tree When called from mating search we search for the first non failure node within the pre determined maximum depth Search for a success node only when the mating is complete Major drawback Needs modification to implement subsumption 1 Data Structures 2 Computing Head Normal Form 3 Control Structure 4 First Order Unification 5 Subsumption Checking There is a subsumption checker for UN88 which uses th
84. 6 46 47 47 48 50 l 5l 52 54 54 55 56 57 57 58 58 CONTENTS 4 4 Implementation 4 5 Other Issues 4 6 How to save more in special cases 5 Entering and printing formulas 5 1 Parsing of Wffs 6 Printing Vertical Paths 7 Global Parameters and Flags 8 Simple MetaWffs in TPS3 8 1 The Notation 9 More about Jforms 10 Printing Proofs Chapter 7 Well formed formulae operators 1 Operations on Wffs 1 1 Arguments to Wffops 1 2 Defining Wffops 1 3 Defining Recursive Wffops 1 4 Defining a Function Performing a Wffop 1 5 Quick Test versus Slow Test 2 The formula editor 3 Example of Playing with a Jform in the Editor 4 Defining an EDOP 5 Useful functions 6 Examples 6 1 Global Parameters and Flags 7 The formula parser 7 1 Data Structures 7 2 Processing Chapter 8 Help and Documentation 1 Providing Help 1 1 Mhelp and Scribe 1 2 Mhelp and ATEX 2 The Info Category 3 Printed Documentation 4 Indexing in the Manuals 5 Other commands in the manuals 6 Converting Scribe to ATEX documentation 6 1 The latexdoc lisp file 6 2 Special Characters 6 3 BTE X Macros Chapter 9 Flags 1 Symbols as Flag Arguments 2 Synonyms 3 Relevancy Relationships Between Flags 58 59 60 60 60 61 62 63 63 64 64 67 67 67 68 70 70 71 TT TT 79 80 80 82 83 83 83 85 85 85 85 86 86 86 87 87 87 87 88 89 90 90 90 vi CONTENTS 3 1 Automatical
85. 8 GENERIC Style 49 GENERIC STRING Style 49 get best alt bound name Function 159 161 get best alt free name Function 159 161 get best alt name Function 161 get shallow Function 111 115 GETWFF SUBTYPE Function 67 global categorylist 23 global contezxtlist 23 global definelist 12 GO Command 99 111 goal Syntax 195 goal list Syntax 195 GroupList Data 83 GroupScan Function 84 GVAR Argument Type 67 GWFF Argument Type 67 hatom Syntax 38 heap space 7 Height Data 56 HELP object Command 85 AISTORY SIZE Flag 89 ALINE JUSTIFICATION Flag 109 ha natree aux File 215 hz natree cleanup File 216 hz natree debug File 216 hz natree duplication File 215 hz natree rulep File 215 hz natree top File 215 ha natree top lisp File 205 imitation eta Function 165 implication 115 in package 5 INCLUDE INDUCTION PRINCIPLE Flag 171 175 177 179 incomp clists 117 incomp clists wrt etree 117 individual var 161 Infix Property 46 InfizScan Function 84 INFO 86 INSERT LIB DIR Flag 234 INSERT TPTP Command 233 INSERT TPTP Command 233 inst exp vars params 116 117 interface style lisp File 246 248 ISTYLE 246 istyle characters 248 JAVA COMM Flag 246 JAVAWIN Command 246 JForm 55 jform 116 junctive 111 115 116 justification 115 Kset Data 49 label Syntax 35 LABEL Q Function 71 LAMBDA 115 LAZY2 118 leaf 1
86. A gt DB Lo XN 4 node A REW1 B IMP1 4 Lo XN nodes node nodes NEG1 gt A REW2 A A REW1 A NEG1 l node node The rest of the transformation rules are for two rewrite nodes Note that these are destructive operations REW1 A a i node gt A REW1 A node 7 RAISE LAMBDA NODES 153 AB REW1 A gt AREW1 A 4 4 A REW2 B AB REW2 Ay 4 4 node node Equiv Implics REW1 A gt A REW1 A 4 4 A REW2 B Equiv Implics REW2 Ay 4 4 node node Equiv Disjs REW1 A gt A REW1 A i A REW2 B Equiv Implics REW2 Ay i i node node After applying the transformation if the result is a A rewrite node REW then we move any connection from REW to its child Actually in the code we only do this if the original tree is a rewrite since in all other cases the top node of the resulting etree could only be a new A rewrite This is true because only the rewrite transformation rules are destructive We need to make sure the invariant holds If we are given a Subst rewrite node REW1 to process we start by pushing connections to A rewrite nodes below REW 1 to the child of the A rewrite This forces the invariant to hold no translation applies to a Subst rewrite In all other cases the invariant holds for the children because of the recursive call So we have a situation in which there are no connections to A rewrite nodes which are subtrees of the node N of interest Consider the
87. AF1 and LEAF2 Then LINE NODE LIST is updated to be 1 LEAF 1 2 LEAF2 3 IMPO and NAT XLATE is called recursively on lines 1 and 2 Since line 1 is justified by Hyp NAT XLATE does nothing Since line 2 is justified by Same as 1 NAT XLATE updates the value of MATING LIST to LEAF1 LEAF2 a connection consisting of the nodes which represent lines 1 and 2 In an nproof that is not cut free there will exist lines which do not arise from deepening the expansion tree which represents the last line of the nproof Currently NAT XLATE will get very confused and probably blow up The justification RuleP causes other dif ficulties because it generally requires that several connections be made involving lines whose nodes haven t been deepened to the literal level yet The function XLATE RULEP attempts to do this but does not always succeed This is true because RULEP can also be used to justify a line whose node is actually a child of the justifying line e g 218 18 PROOF TRANSLATIONS 45 A and B 46 A RuleP 45 Though XLATE RULEP can handle this situation it cannot handle more complex ones such as 16 A 17 A implies B 18 B RuleP 16 17 Ideally SAME IFY would identify these situations before the trans lation process is begun but it does not 4 Cut Elimination A cut elimination algorithm is worked out in Frank s Thesis First he defines a notion of expansion development a sort of se
88. AG Flag 63 FilLineFlag Flag 48 52 FinalScan Function 84 find 12 FIND ALT CHEAPEST CLIST SPANNING PATH Function 137 FIND CHEAPEST CLIST SPANNING PATH Function 137 finish up option search Function 116 First Order Mode Flag 60 FIRST ORDER MODE MS Flag 98 FIRST ORDER MODE PARSE Flag 62 FIRST ORDER PRINT MODE Flag 62 First Order Print Mode Flag 60 fiz shallow chain Function 140 flag deps lisp File 92 flagging lisp File 15 FlatSym Function 56 FlatWff Function 56 flavors 28 FLUSHLEFTFLAG Flag 68 FlushLeftFlag Flag 48 FlushleftFlag Flag 52 ForallIndent Parameter 62 free must avoid 160 free vars 111 free vars in etree 117 119 ftree seq File 210 ftree seq bring to front Function 227 ftree seq cut elim Function 205 224 ftree seq invert position list Function 226 ftree seq merge focus all pos Function 218 256 ftree seq miz elim 1 Function 223 ftree seq miz elim finish Function 225 ftree seq miz elim imitate rule Function 227 ftree seq miz elim principal Function 228 227 ftree seq replace equivwffs Function 225 ftree seq weaken early Function 205 ftree seq lisp File 209 ftree solve constraint set Function 174 ftree to etree Function 119 ftrees File 119 212 function var 161 GAR Function 71 garbage collection 8 GDR Function 71 genchar Data 52 gencharlist Data 52 generate java menus Command 24
89. ArgHelp lt help1 gt lt help2 gt Applicable Q lt fnspec gt Applicable P lt fnspec gt WffArgTypes lt type gt lt type gt Wffop Type lt type gt Wffop Typelist lt typesymbol gt lt typesymbol gt DefaultFns lt fnspeci gt lt fnspec2 gt MainFns lt fnspec1 gt lt fnspec2 gt Replaces lt wffop gt Print Op lt boolean gt Multiple Recursion lt boolean gt MHelp lt comment gt The keywords ArgTypes ArgNames ArgHelp DefaultFns MainFns and MHelp have the same meaning as for commands MExprs See Section 1 You have to mention ArgNames before Applicable P if you want to make use of the argnames without explicitly using lambda The other keywords are as follows RESULTTYPE is the only non optional part of the declaration and is used for printing the result of the wffop APPLICABLE Q is a quick predicate see Section 1 5 to decide whether the wffop is applicable to a given set of arguments If omitted or explicitly stated to be TRUEFN it means that the wffop can al ways be applied APPLICABLE P is a slow predicate which is supposed to check thor oughly whether the wffop is applicable Again if one wants to state explicitly that a wffop is always applicable use TRUEFN 1 OPERATIONS ON WFFS 69 WFFARGTYPES There must be exactly as many type entries as there are arguments to the wffop Each type entry may be either a type
90. At i CONJ1 va N REW Ct nodet LEAF1 Ct is converted to either only use one branch or a slightly modified tree with an explicit dual rewrite separate from expanding the definition The local variables in the loop are e junct DISJ1 or CON J1 e gsym A symbol standing in for the uninstantiated formula A First we check to make sure the rewrite node REW1 is a rewrite still in the etree Assuming it is in the tree we consider several cases 1 If the left child of junct is not needed in the mating then the uninstantiated definition is not needed We replace the junct node with its second child and call fix shallow chain to change the shallow formulas of the rewrites between REW 1 and junct to be the second conjunct If the right child of junct realrew is not needed in the mating then the instantiated definition is not needed We replace the junct node with its first child and call fix shallow chain to change the shallow formulas of the rewrites between REW1 and junct to be the first conjunct replacing gsym by A when necessary 3 Otherwise both are needed In this case we change the tree to have a form like N x Dual REWO A7 Equivwf fs REW1 AV A i REW AV A DISJ1 Va xX REW C7 node LEAF1 Cy This makes the dual rewrites easier to recognize and handle in the cleanup code 6 REMOVE LEIBNIZ 141 4 Prune Unmated Branches The purpose of the function prune unmated branches in the fi
91. Case Deduct This case is easy as are most of the introduction rules Suppose we have D T AFBT THADB By induction we have a derivation of a sequent Ty B where Set T1 C Set T U A Using the structural rules see the function ftree seq merge focus all pos we obtain a derivation with A at the front A T2 B where Set T2 C Set T Applying the D rule we have a derivation of fy gt ADB as desired Case MP This case is interesting because a naive algorithm would be forced to treat this case like a cut in the sequent calculus Suppose we have D TADB THAT TRB Since this is an extraction we must be given a derivation of D a sequent T gt C where Set T1 C Set T U B Applying structural rules to D1 we have a derivation D2 of a sequent B T2 gt C with Set T2 C Set T The first algorithm applied to gives a derivation of some 3 A where Set T3 C Set T If we apply D as follows T3 gt A B T2 gt C AD B T3 Gammaz gt C Deduct MP Pig then we can call the second algorithm on this derivation and D to obtain a derivation of some 4 gt C with Set T4 C Set L Case Backwards Coercion DD Bt beoercion 214 18 PROOF TRANSLATIONS with hypotheses I Suppose we are given a derivation of some 1 C where Set T1 C Set T U B Using structureal rules we obtain a derivation of a sequent B T C where Set T2 C Set T We want to remove B from the positive si
92. ELIST label LABEL P FLAVOR binder BINDER P VAR TYPE SCOPE TYPE WFF TYPE 3 Explanation of Properties The various properties mentioned above are as follows TYPE The type of the object Common are 000 for binary connec tives and I for individual constants PRINTNOTYPE In first order mode this is insignificant but if specified and T TPs3 will never print types following the object It is almost always appropriate to specify this INFIX The binding priority of an infix operator This will declare the connective to be infix The absolute value of INFIX is irrele vant only the relative precedence of the infix and prefix operators matters If two binders have identical precedence association will be to the left For example if R1 and R2 are operators with INFIX equal to 1 and 2 respectively p R1 q R2 r R2 s will parse as p R1 q R2 r R2 s PREFIX The binding priority of a prefix operator Binders are con sidered prefix operators see about binders below and thus have a binding priority The main purpose of a prefix binding priority is to allow formulas like a b to be parsed correctly as a b by giving precedence over PRT ASSOCIATIVE indicates whether to assume that the operator is left associative during printing You may want to switch this off specify NIL for an infix operator like equivalence say lt gt since 3 EXPLANATION OF PROPERTIES 37 p lt gt q lt gt r is often consider
93. File 150 MATING VERBOSE Flag 16 maz cgraph counter 117 maz incomp clists wrt etree 117 maz mates Flag 125 memory 7 menu 247 menuitem 247 258 merge all Function 136 137 merge debug 185 MERGE MINIMIZE MATING Flag 135 187 merge tree real Function 154 159 161 merged 117 MIN QUANT ETREE Flag 129 216 min quant scope Function 156 MIN QUANTIFIER SCOPE Flag 116 120 ML RULES 2 PROP 97 mode Function 15 modify dual rewrites Function 135 137 189 mon fn negf Function 178 180 monitor 95 monitor Command 96 monitor check 95 ms Function 127 ms director Function 127 ms propositional Function 127 MS03 7 231 MS04 2 231 ms04 search lisp File 183 MS91 6 116 MS91 7 116 ms91 basic lisp File 116 ms91 search lisp File 116 MS98 1 167 MS98 INIT Flag 167 MS98 TRACE Flag 205 msearch Function 125 msg 13 msg Function 14 msg Syntax 195 msgf Function 14 name 111 117 NAT ETREE Command 116 204 205 215 216 NAT ETREE VERSION Flag 204 205 215 216 NATREE DEBUG Flag 204 214 natree to ftree main Function 205 natree to ftree seq extraction Function 205 212 natree to ftree seq normal Function 205 212 negation 115 nomonitor Command 96 NONE 121 NORMALIZE PROOF Command 214 not alpha image 160 NUM OF DUPS Flag 177 178 one step mge bd Function 129 one step mge infix Function 129 option set
94. However note that CUT2 appears in both sides of the merge in MERGE a c Q CUT3 Q CUT4 Let us call these two occurrences CUT2 1 and CUT2 2 Eventually we will want to reduce one of these cuts in some reduced tree Q CUT2 1 CUT2 2 Suppose we reduce CUT2 1 Similar to the reduction of CUT1 above this will copy CUT2 2 so that there are CUT2 2 1 and CUT 2 2 2 The loop is evident It is conceivable that we could get around this loop by developing a notion of expansion DAG directed acyclic graph so that the CUT would not actually be duplicated But it isn t clear that this would eliminate all such loops 222 18 PROOF TRANSLATIONS Also there are some technical problems with the reductions above For instance a selection node may get copied which means the result will be a tree with two selection nodes that use the same selected variable This doesn t seem to be a serious problem because we could probably allow such a situation whenever the least common ancestor of two such selection nodes is a MERGE node But these details would have to be worked out to make this approach work 4 2 Cut and Mix Elimination in this Sequent Calculus There is a cut elimination algorithm implemented for the sequent calculus described in section 3 1 5 Suppose we have an instance of the cut rule D Do r gt A Ay A T2 gt Ag T T2 gt Ai As A cut elimination algorithm should take cut free derivations D and D2 and return a
95. ME IFY tries to add lines to the proof in such a way that each line corresponds to exactly one expansion tree node 3 NPROOFS TO EPROOFS 217 4 After the proof has been massaged by SAME IFY the initial root YS NS NS node of the expansion tree is constructed This node is merely a leaf whose shallow formula is the assertion of the last line of the nproof LINE NODE LIST is initialized to contain just the association of this leaf node with the last line of the proof and MATE LIST is set to nil Next the function NAT XLATE is called on the last line of the proof NAT XLATE depending on the line s justification calls auxiliary functions which carry out the translation and which usually call NAT XLATE recursively to translate lines by which the current line is justified When the justification Same as is found this indicates that the node associated with this line and the node which is associated with the line it is the same as should be mated in the expansion proof Example Suppose we have the following nproof 1 1 A Hyp 2 A implies A Deduct 1 SAME IFY will construct the new proof 4 1 A Hyp 2 1 A Same as 1 3 A implies A Deduct 2 Then a leaf node LEAFO is constructed with shallow formula A implies A and LINE NODE LIST is set to 3 LEAF0 NAT XLATE is called and because line 3 is justified using the deduction rule LEAFO is deepened to an implication node say IMPO with children LE
96. NIL starts a new line for each of the arguments of an infix operator even if only one of several arguments would be too long to fit on the remainder of the line FlushleftFlag T switches off indentation completely i e every line will be aligned with the left margin NIL indents the arguments of infix operators 3 2 Creating the PPlist The pretty printing is achieved in two steps During the first phase printing will be done without any formatting and the characters are not actually printed but appended to a list called PPlist In the second phase this list will then be printed The decisions when to start a new line how to indent etc are only made in this second stage The PPlist is of the following syntactical structure pplist aplicnlist pdepth pgroup plength gencharlist pdepth pgroup plength aplicnlist aplicn aplicnlist NIL aplicn MARKATOM aplicn pplist pplist plength 0 1 2 pgroup BRACKETS DOT NIL pdepth 0 1 2 gencharlist genchar gencharlist NIL genchar char ascnumber gencharlist char lt any non control character gt ascnumber 0 1 127 The PPlist contains a list of all the top level applications along with the grouping pgroup its print depth pdepth and its print length plength If the grouping is BRACKETS brackets will be printed around the formula A grouping DOT means that a d
97. Qi is an expansion tree proving B with lemmas corresponding to the value LEMMASi Note that if LEMMASi is NIL then the shallow formula of Q will be B the same as that of P Otherwise the shallow formula of Q will be of the form C A D D Bil We may use lemmas to handle some extensionality reasoning Consider the example Pololo A Tor Aste zV 1 D PA The jform for this example is of the form L1 P LAMBDA x A x OR FALSEHOOD L2 PA Number of vpaths 1 We would like to mate L1 with L2 but we cannot since they are not unifiable We need to use the fact that A and Az Ax V L are extensionally equivalent The mate command ADD EXT LEMMAS finds pairs of such propositional or set or relation terms embedded inside literals and includes an extensional ity lemma for any two such terms occurring in literals of opposite polarity In this example there are two extensionality lemmas added to the expansion tree e V 1 a T DAtL A Vzala TV1 Az DAs Arvl A This second lemma can be used to prove the theorem Both lemmas have an easy proof by expanding equality using extensionality For example the part of the expansion tree corresponding to the proof of the second lemma 11 LEMMAS IN EXPANSION PROOFS 131 is of the form IMP1 Via N EXPO Va Axor V L Ax EXT REW2 Axz Azv 1 A Ja 4 LEAF4 Aso VL Ax A REW1 Yza A Aoo V L z Ax SELO Y zo Aoi V L Ax x
98. REMOVE LEIBNIZ Flag 136 remove leibniz Function 141 143 144 remove leibniz nodes Function 141 remove spurious connections Function 141 149 remove unnecessary ab rews Function 160 rename all vars in etree Function 159 161 rename all vars in wff Function 161 replace non leaf leaves Function 136 141 REPORT 239 RESPONSE 246 RESTORE ETREE Command 120 reverse 115 rewrite 115 REWRITE DEFNS 121 REWRITE DEFNS Flag 118 156 REWRITE EQUALITIES Flag 118 121 REWRITE EQUIVS Flag 120 REWRITES Flag 126 RIGHTMARGIN Flag 48 50 93 246 Rightmargin Parameter 48 Rows Data 56 RULEP Command 97 RULEP Function 215 RULEQ 116 ruleq shallow 115 runcount Function 14 SAIL Style 49 SailCharacters Parameter 68 SAM 8 SAVE ETREE Command 120 260 SAVE FLAG RELEVANCY INFO Command 93 SAVE WORK Command 14 SAVEPROOF Command 100 scope problem p Function 161 SCRIBE Style 49 SCRIBE PREAMBLE Flag 64 SCRIBEPROOF Command 64 SCRIPT Command 14 sel var bound 160 selection 115 setup big aterm window Function 6 SETUP SLIDE STYLE Command 48 setup xterm window Function 6 shallow 115 116 SHOW RELEVANCE PATHS Command 93 shownotypes Command 63 skolem 115 116 skolem constants 117 SKOLEM DEFAULT Flag 115 117 121 136 skolem method 117 skolem node list 117 skolem term Data 115 skolemization 121 SLIDEPROOF Command 49
99. T v is a collection of literals are not negative literals with v at the head and W is a list of selection variables which occur in the sequent and are banned from occurring in the instantiation for v see section 4 In general v can occur inside the body of the literals in T and this case will be discussed below We do not allow v to occur in the argument terms f Maximal constraints can be defined and handled in a dual way We concentrate on minimal constraints for the present It is very easy to see that any collection of minimal constraints can be simultaneously solved since v AZT is a solution In interesting cases this instantiation will fail other conditions we might need v to satisfy What we would prefer to have is an optimal solution to the constraints In the case of minimal constraints optimal means a minimal solution First consider the case of a single minimal constraint of the form WIT gt vz where the arguments Z are distinct variables that occur in V Also assume that there are no other variables in Y In this case we can directly define the minimal solution as an intersection dz AT The notation A T means A A A An where each A is either a positive literal in IT or A is B where B is a negative literal in I The case where VW Z W for some extra selected variables W is only slightly more complicated We can directly define the minimal solution in this case as zIw I T This is a
100. TABLE uses the general function MAKE TABLE The latter takes three arguments as input a list of column indices a list of indexed entries row index column index entry and the maximum printing size of row indices With these it produces a ta ble of the entries EXER TABLE merely calls this on data it extracts from the record file for the DONE EXC event The definition for EXER TABLE follows defreport exer table source event done exc eventargs userid dproof numberoflines date argtypes date argnames since defaultfns lambda since cond eq since setq since since default list since default passed args since1 bin exerlis maxnam begin fn exertable beg do fn exertable do end fn exertable end mhelp Constructs table of student performance defun exertable beg since declare special since1 maxnam the only non Nil passed args setq sincel since setq maxnam 1 2 THE REPORT PACKAGE 241 defun exertable do userid dproof numberoflines date declare special sincel bin exerlis maxnam if greatdate date sincel progn setq bin cons list userid dproof numberoflines bin setq exerlis if member dproof exerlis exerlis cons dproof exerlis setq maxnam max flatc userid maxnam defun exertable end since declare special bin exerlis maxnam if bin progn make table exerlis bin maxnam msg t On exercises completed since write date since msg t pr
101. TPS3 Programmer s Guide Peter B Andrews Dan Nesmith Frank Pfenning Sunil Issar Hongwei Xi Matthew Bishop Chad E Brown R my Chr tien copyright 2000 Carnegie Mellon University All rights reserved This manual is based upon work supported by NSF grants MCS81 02870 DCR 8402532 CCR 8702699 CCR 9002546 CCR 9201893 CCR 9502878 CCR 9624683 CCR 9732312 CCR 0097179 and a grant from the Center for Design of Educational Computing Carnegie Mellon University Any opinions findings and conclusions or recommendations are those of the author s and do not necessarily reflect the views of the National Science Foundation Contents Preface Chapter 1 Introduction 1 Guidelines 2 Tps3 Conventions 2 1 Filenames 2 2 Lisp packages and export files 2 3 Implementation specific differences 2 4 TPs3 modules 2 5 File format 3 Maintenance 3 1 Porting TPs3 to a new Lisp 3 2 Building TPs3 3 3 Memory Management 3 3 1 Heap Size and Stack Size 3 3 2 Swap Space 3 3 3 Internal Limits in Lisp 3 4 TPs3 distribution 3 4 1 Making a tar file for Distribution 3 4 2 Distribution of TPS3 via http 3 4 3 Obsolete Information about Making tar tapes of TPs3 4 How to locate something 5 Utilities 6 Overloading Commands 7 Output 8 Compiling as much as possible 9 Writing New Code Without Making A Nuisance of Yourself 10 Debugging Hints 11 Miscellaneous 11 1 Counting Flags 11 2 Dealing with X Fonts
102. The output of LINE NO DEFAULTS FROM should be a default exists figured the output of LINE NO DEFAULTS TO is a list default exists figured default new figured in which all of the arguments have been filled in These functions may also do a THROWFALL if one of the requirements R for logical correctness cannot be satisfied in the given proof structure Also note that all lines in default exists have already been determined when default new is called The specification which must be meet by the LINE NO DEFAULTS x func tions can be grouped into three classes requirements which ensure the log ical correctness of the rule application R requirements which make the defaults sensible for the usual application of the rule D and should never be deviated from and desired properties which need not be satisfied but approximate what the user would like to see most of the time Note that the scope in this function is restricted by the fact that it does not examine the logical structure assertions or hypotheses of the lines in the proof This is accomplished by a completely different mechanism and is not the responsibility of the function For instance it is perfectly sensible for LINE NO DEFAULTS to suggest the first pline in the current support structure for the backwards conjunction rule even though it may not be a conjunction at all This may cause mayhem in rule tactics The latter assumes that if there is a correct default the default f
103. UBSET Q If all the types were omitted and TYPE IOTA MODE were NIL x would have been typed with the next available typevariable Using the same name for two variables of distinct type is legal but not recommended Consider for example FORALL x P x I AND Q x ID a Here the type of the very first occurrence of x will be assumed as II Leaving out the type of the third occurrence of x would have led to an error message Rather than assume that x II was really meant TPS assumes instead that the scoping must have been incorrect which seems much more likely All remaining type variables after a parse are automatically assumed to be of base type unless the flag TYPE IOTA MODE is set in which case they are assumed to be of type v In first order mode identifiers have only single characters Thus not Pxy is parsed as NOT P x y When a wff is read in and parsed each input token where the number of characters in a token is dependent on whether you are reading in first order mode or not is made into a lisp symbol which incorporates the token s 45 46 6 PRINTING AND READING WELL FORMED FORMULAS printed representation and type For example entering x A will result in a symbol being created whose print name is x lt A gt When you try to print a symbol like this first the part without the type information is printed then the type if necessary is printed E g first we print x then print A But the inform
104. W where wff2 is of the form Jw D w we create new selected variables w Let 62 psi w assoc send each w to w We call make clos setvar ind negf 4 with C and D w to obtain two positive ftrees PF and PF with shallow formulas C and D w2 We then return PF and SEL dw D w Lau PF The constraint v 0 has an empty Y so we proceed directly to make clos setvar ind negf 4 with pt0 and z 0 to get the two positive ftrees The constraint w v w v S w has the variable w in Y and Dis Jwz SwA pu We create a new selected variable w and call make clos setvar ind negf 4 with Vwi p w D pt Sw and zi Sw A pw The second positive ftree PF i constructed has shallow zt Sw A p w We return the first positive ftree along with SEL Au z SwA ptu Jw PF 3 SET CONSTRAINTS 189 e make clos setvar ind negf 4 We are given a constraint U T gt A and two wff s C and D where C is of the form VyC D A or VyA if T is empty where Y y Define a substitution 0 with dom 0 J as follows Either y y is az Y which occurs as an argument in A and so is in the domain of 0 psi z assoc or y isa w W in the domain of 62 psi w assoc In the first case let 0 y 01 y In the second case let O y 2 y We use these association lists to determine the expansion term for y Then make clos setvar ind negf 5 is called with 0 C or NIL hyp and
105. X This obstacle occurs for instance when using the Scribe Tabset function which has no strict equivalent in ATX where it is usually replaced by the Description environment 6 2 Special Characters The mhelp properties of many TPS3 ob jects present special characters when Scribe prevents the use of a lot of characters have to be escaped using a prefixed when generating a IATRX document The most common are 7 and amp Note that the backslash character in TFX is textbackslash and the character is textasciicircum 88 8 HELP AND DOCUMENTATION In order to prevent the programmer from editing every mhelp property a function princ mhelp latex is used instead of the regular princ mhelp This function simply replaces every occurence of a protected character by the correspondant TFX sequence 6 3 AT X Macros To facilitate the conversion from Scribe to ATEX com mands a great number of new commands and macros for ATX are defined in tps tex and tpsdoc tex These commands enable us to use commands such as greeka instead of the regular alpha CHAPTER 9 Flags Here is an example of how to add the flag neg prim sub to TPS3 Insert into the file prim lisp the code defflag neg prim sub flagtype boolean default nil subjects primsubs mhelp When T one of the primitive substitutions will introduce negation Actually that code almost worked but changing the flag did not have the desired ef
106. a JForm like the one above as an argument and returns a Descr JForm where we have the information about the height and width of the subformulas which we need in order to format the output explicitly attached to the parts of the JForm Quantifiers are handled similarly Multiple identical quantifiers are com bined in a list whose first element is the quantifier and the rest is the list of variables which are quantified VaVydzduA is in internal representation x FORALL y FORALL z EXISTS u EXISTS A and as a JForm it looks like FORALL x y EXISTS z u A The following is a formal description of what a JForm and a Descr JForm are Note that a descr jform is entirely an internal concept used by the file vpforms lisp for working out how to format a vpform a jform is a concept which is accessible to users e g users have commands to translate from gwffs to jforms and back JForm Literal SignAtom OR JForm 3 AND JForm FORALL Var JForm EXISTS Var JForm Literal LIT1 LIT2 SignAtom NOT Atom Atom Var lt any logical variable gt Atom lt any logical atom gt 56 6 PRINTING AND READING WELL FORMED FORMULAS It should be noted here that some programs might expect the arguments of a JForm starting with OR not to start itself with an OR the argument of a JForm starting with FORALL not to start with another FORALL etc b
107. a between the comment lines Menu Code BEGIN w a i and Menu Code END and save TpsWin java 4 On each machine find the Java directories which contains links to the main java files e g home theorem tps java and home theorem tps java tps cd to this directory and call javac TpsWin java to compile the new version 2 Adding a New Symbol The Java information for the fonts is contained in the files TpsSmall Fonts java and TpsBigFonts java The lisp information containing the code for the symbol is in interface style lisp To add a new symbol for the Java interface one should add a new code for the symbol to the variable istyle characters in interface style lisp For example the epsilon character was added by including epsilon 2 1 to istyle characters This means that epsilon will be communicated to the interface by sending the bytes 2 and 1 in symbol font mode Then one needs to add information about how to draw the new symbol to the variables blank xstartData ystartData widthData and heightData in TpsSmallFonts java and TpsBigFonts java Each of these variables is set to a multi dimensional array The 0 th element of each array corresponds to the normal fonts The rest are for symbol fonts For example the information for epsilon should be put in the 2 1 position of each array This information for epsilon in TpsSmallFonts java is as follows blank false the epsilon character is
108. a matter of style The following are general guidelines which do not address the definition of flavors see Section 6 1 All arguments to a wffop may be assumed to be of the correct type when the function is invoked This does not mean that the function never should check for an error but at least the function does not have to check whether an argument is well formed or whether an argument is a logical variable and not an application 2 Most user level wffops get by without using any slow predicates for constituents of a gwff Use the quick predicate and assume that the argument is a gwff 1 OPERATIONS ON WFFS 71 3 Make the name of a wffop as descriptive as possible The user will rarely have to type this long name since he will normally invoke wffops in the editor where they can be given short aliases See section 2 4 When using auxiliary functions make sure their name can be easily related to the name of the main function 5 Check the wff operations in the TPS3 Facilities Guide for Program mers and Users before defining new functions In particular you should often use GAR and GDR instead of car and cdr to manipu late wffs since the wffs may have labels 6 Always make sure you are invoking the quick test in the correct order since later tests rely on the fact that earlier tests failed 1 5 Quick Test versus Slow Test Most predicates which test for certain kinds of subformulas come in two inc
109. a subject PRIMSUBS The command LIST PRIMSUBS in TPs3 will list the relevant flags Some of the main flags that determine if and how primitive substitutions are generated are DEFAULT MS DEFAULT EXPAND and PRIMSUB METHOD Some examples in principle might require applying primsubs beneath primsubs An example discussed in ABB00 is the injective version of Can tor s Theorem X5309 2 Using Unification to Compute Setsubs If DEFAULT MS is set to MS98 1 MS98 INIT is set to 2 or 3 and PRIMSUB METHOD is set to PROO then unification is used to compute instantiations during pre processing 167 168 16 SET VARIABLES 3 Set Constraints An alternative to instantiating set variables in a pre processing step is to intertwine instantiating set variables with the mating search Let v be a set variable occurring at the head of some literal We write constraints in sequent form A constraint is in practice a list of positive and negative literals or expansion tree nodes Usually positive literals are written on the left of the sequent and negative literals are written on the right In some cases we write a positive literal on the right or a negative literal on the left and interpret it as the negation of the literal A sequent corresponds to a subset of a vertical path on a jform Minimal constraints for v are a collection of sequents these correspond to subsets of vertical paths in the jform of the form WIT v gt vt where
110. acter set switches the integers represent charac ters and CRLF represents a carriage return line feed combination LexList This is a list of lexical objects i e it contains name for logical objects which will appear in the fully parsed formula It also contains the brackets and It also contains the type symbols from the initial input These are distinguishable from the other items in the list since they are stored as lists Hence LexList is a flat list of these three things TypeAssoc This is an association list which associates to those iden tifiers in the LexList which got a type that type This is necessary so that an identifier which is typed explicitly at one place in the formula can have that type attributed to it at other non typed oc currences GroupList This is essentially the same as LexList except that the bracket identifiers are removed and nested s expressions are used to denote groupings Type symbols are also attached to the iden tifier preceding it Hence a GroupList contains only logical identi fiers some with types and some without grouped in a hierarchical fashion PreWff This data structure is like that of the wff structure except that not all items are correctly typed yet The full prefix organiza tion is present in this formula The types for polymorphic defini tions however are not yet computed 7 2 Processing Input is first processed into ByteStreams and then into LexLists by t
111. actual code The following are two different substitution functions SUBSTITUTE TERM VAR currently in wffsub1 lisp substitutes a term for a variable but gives and error if the term is not free for the variable in the wff SUBSTITUTE L TERM VAR currently in wffsub2 lisp also substitutes a term for a variable but renames bound variables if a name conflict occurs There may be a global variable say SUBST FN whose value is the function used for substitution by default or there may be a function SUBSTITUTE which checks certain flags to determine which function to call defwffop substitute term var argtypes gwff gvar gwff wffargtypes A A B resulttype gwff wffop type B wffop typelist A B argnames term var inwff arghelp term var inwff applicable p lambda term var inwff free for term var inwff mhelp Substitute a term for the free occurrences of variable in a gwff lAs of July 7th 1994 1 OPERATIONS ON WFFS 73 defun substitute term var term var inwff This function should be used with extreme caution There s an underlying assumption that TERM is free for VAR in INWFF which is true if TERM is a new variable or subst term var rec intern subst term var var inwff inwff defun subst term var rec term var inwff cond label q inwff apply label inwff subst term var rec term var inwff lsymbol q inwff if eq var inwff term nil boundwff q inwff if e
112. ain of explicit steps ex panding defns lambda normalization etc from the new shal low to the old shallow of the child of the rewrite Ruleq Ruleq Univ We again copy the node replace the shallow formula with shallow as usual and recursively call cleanup all expansions on the kid And again the tricky part is calculating what the new shallow formula of the kid should be What it does is let A newshallow be the gwff in the RULEQ SHALLOW slot of the node If this slot is empty it lets A be the min quant scope of shallow if the justification is Ruleq or just the same shallow if the justification is Ruleq Univ The new shallow of the kid is Qxz Qz A where Q is V if the node is positive and 4 if the node is negative and the x s are the variables introduced in this node Possible Bug It is not clear why this case is handled the way it is Truthp We copy the node replacing the shallow formula as usual Then recursively call cleanup all expansions on the kid and a shallow formula A V T RUTH obtained from the old 9 PRETTIFY 157 shallow formula A by replacing occurrences of FALSEHOOD with aT RUTH to obtain A Possible Bug One would expect the new shallow formula of the kid to be something like B V ATRUTH where B is the new shallow formula of the rewrite It s not clear that this causes a problem though 9 Prettify The code for PRETTIFY is in the file mating merge eq lisp Merging used to prettify the variabl
113. aining the expansion vari able for this expansion c prim vars 8 rewrite is an etree that rewrites the wff in some way Rewrite nodes have the following four additional attributes shallow justification ruleq shallow and reverse The justification attribute is a symbol which can currently be one of the following values this list may not be exhaustive a EQUIVWFFS Usually means there have been some definition expansions In case dual instantiation is being used it may mean the wff has been rewritten to a conjunction or disjunction of the wff and the instantiated form b LAMBDA BETA ETA The wff is the result of the appropri ate normalization 2 3 4 WS WS N 116 13 MATING SEARCH c EQUIV IMPLICS EQUIV DISJS An equivalence was expanded d LEIBNIZ Rewrites an equational wff using the Leibniz def inition of equality e EXT Rewrites an equational wff between terms of functional type using extensionality REFL Rewrites an equational wff of the form a a to TRUTH g RULEQ The only time this appears to be used is when MIN QUANTIFIER SCOPE is set to T in which case the quanti fiers in the wff are pushed in as far as possible h ADD TRUTH TRUTHP May conjoin the wff with TRUTH or disjoin the wff with NOT TRUTH See the flags ADD TRUTH and TRUTHVALUES HACK 9 leaf is an etree with the additional attribute shallow as in skolem above The components junctive and predec
114. ake advantage of the modification 3 3 The Original Nat Etree Note What follows is a descrip tion of how NAT ETREE used to work To use this code set NAT ETREE VERSION to OLD Legend has it that the code was written by Dan Nesmith and influenced by the ideas of Frank Pfenning Frank s thesis contains ideas for translating from a cut free sequent calculus to expansion tree proofs 1 Important files nat etr defines functions which are independent of the particular rules of inference used ml nat etrl and ml nat etr2 which define translations for the rules in the standard TPS 2 There are three global variables which are used throughout the SS translation process DPROOF which is the nproof to be trans lated LINE NODE LIST which is an association list which asso ciates each line of the proof to the node which represents it in the expansion tree which is being constructed MATE LIST which is a list of connections in the expansion proof which is being constructed At the beginning of the translation process the current proof is copied because modifications will be made to it It is restored when the translation is complete The copy is stored in the variable DPROOF Next the function SAME IFY is called This attempts to undo the effects of the CLEANUP function and to make explicit the connections in the proof This is done because in an nproof a single line can represent more than one node in an expansion proof SA
115. al characters superscripts etc are available and vertical path diagrams are correctly formatted although often too wide to print XTERM produces the special characters used by X windows You should set the value of RIGHTMARGIN to reflect the width of the window containing TPs3 SAIL SAIL is a style now all but obsolete used for printing on a Dover printer The font used is 10 point with 120 characters per line in landscape format used for vertical path diagrams and 86 in portrait format used for all other applications When you dover the file you have to remember size and orientation and specify it in the switches of your call of DOVER A SAIL file does not have subscripts but has as variety of other special characters From the information about the style the low level printing functions determine which sequence of characters including control characters to send to the selected output device If a symbol expands to a list of known symbols with different names e g EQUIVS expands to an EQUIV symbol with a superscript S then it has a property FACE which contains this information Various other properties give the way that the character is to be printed in different styles The CFONT property is a pair KSet AsciiValue Kset can be 0 1 2 or 3 although currently only the character sets 0 1 and 3 are used this gives the appropriate character for a Concept terminal Similarly the DFONT property is a string whatever whi
116. all the parameters given as arguments are PROG variables In the body of the PROG all parameters are set to the value specified in the call then the function PWff is called just with Wff as its argument All the other parameters and flags are now global or in LISP terminology special variables The function PWff performs two main tasks First a few special vari ables are set to the correct value After this is done PWff checks whether pretty printing is desired i e whether PPWfflag is T For an explanation of what happens during pretty printing see section 3 and in particular 3 4 Otherwise the recursive function Print WffPlain is called with the appropriate arguments At this point the current style is available to the functions in the flag STYLE The calling function has to make sure that LEFTMARGIN and RIGHTMARGIN will be bound They are important for the printing func tions in order to determine where to break lines and where to start formulas on the line This holds whether pretty printing is switched on or off Below PWff two functions appear PrintWffPlain prints a formula with out any delimiting symbols around it For example with STYLE SCRIBE x lt I gt FORALL OR P lt OI gt x lt I gt q lt O gt appears as Vz Paz V qo if BRACKETS T and as V2x Pot V qo if BRACKETS NIL PrintWffScope delimits a composite formula with a preceding dot if the ar gument BRACKETS is T and with brackets around it
117. alue while you are in the mating search top level is the node of the expansion tree that you are currently looking at Commands like D down UP move to root use and change the value of this variable see etrees wffops mating top mating move Actually the unification top level uses this same variable name of course 1 DATA STRUCTURES 113 rebinding it during the duration of the top level so there s no real conflict Now STATUS and PREDECESSOR are actually slots in every expansion tree node defined in etrees labels Each etree node has a status which is a nonnegative integer 0 means that the etree node should be ignored as if it and its descendants were not in the tree while positive values indicate the node is active and potentially the higher the value the more important it is By the etree s status you could rank certain expansions as more interesting than others I don t think that is now being used anywhere Originally this status was kept in a slot in each etree node I didn t really like this because then you can t share etree nodes among different virtual expansion trees For example during the MS90 9 search procedures there is really just one expansion tree which contains all of the expansions There are however many virtual expansion trees that is expansion trees with the same root but with different subsets of the expansions turned on Each one of these virtual trees is ke
118. ame starts with 4 6 How to save more in special cases There is a straightfor ward generalization of this to case where we would like to save information about the appearance of arbitrary labels The most general solution fails as demonstrated above but if we restrict ourselves to cases where the number of dynamic parameters is limited we can get somewhere We could make a case distinction of the kind save and use printing info for labels only if PrintDepth is 0 PrintTypes is NIL AllScopeFlag is NIL The only remaining dynamic parameter that comes to mind is the bracketing information which can take two different values This is what makes this fragment of the grand solution feasible Notice that this is not just of academic interest ETPS in first order mode satisfies all the criteria above 5 Entering and printing formulas 5 1 Parsing of Wffs Wffs can be specified in TPS3 in a variety of ways e g as strings and with or without special characters Regardless how a wff is specified there are general rules of syntax which always apply Sometimes one has to distinguish between first order mode and higher order mode with slightly different syntactic rules If the global variable First Order Mode is T all parsing will be done in first order mode Similarly the global variable First Order Print Mode determines whether wffs are printed as first order or higher order formulas It is important to note that wffs printed in higher or
119. ant kids relevant kids If we want to record the irrelevancy relationship in the DEFAULT MS MS98 NUM OF DUPS example above there are two ways The first is to record this in the definition of DEFAULT MS using the irrelevant kids slot as shown below 3 RELEVANCY RELATIONSHIPS BETWEEN FLAGS 91 defflag default ms flagtype searchtype default ms90 3 subjects mating search change fn lambda flag value pvalue when neq value pvalue update otherdefs value irrelevant kids neq default ms ms98 1 ms98 num of dups mhelp The format for this slot is a list of elements of the form lt pred gt lt sexpr gt where lt pred gt is a condition on the value of the flag being defined and lt sexpr gt is an s expression which should evaluate to a list of flags Alternatively we can specify the relationship when we define MS98 NUM OF DUPS using the slot irrelevancy preconditions defflag ms98 num of dups default nil flagtype null or posinteger subjects ms98 1 irrelevancy preconditions default ms neq default ms ms98 1 mhelp If some positive integer n we reject any component using more than n of the duplications The format for this slot is a list of pairs lt flag gt lt pred gt where lt flag gt is a flag and lt pred gt is a condition on the value of the flag lt flag gt Relevancy relationships can be specified in an analogous way defflag default ms flagtype
120. ant to mate a wff with an uninstantiated defn with a subformula of the instanti ated form However it is not clear that we can really rule out such connections since higher order quantifiers might cause a wff to be the negation of a subformula of itself However we can legally ban these connections if only for the reason that we do not need to use dual instantiation at all The value of this flag is used by quick unification connection to rule out some connections 5 ho banned conns list Similar to banned conns list but with a slightly different representation Instead of pairs of leaves each leaf corresponding to an uninstantiated definition is associated with a list of the leaves that occur beneath the instantiated form For example L4 L9 L8 L11 L15 L14 1 DATA STRUCTURES 119 6 unsubst exp vars This is a list of the expansion variables in the etree which are not instantiated This should usually be a list of the car s from the current eproof eproof slot free vars in etree Note that this variable is only set if some node was rewritten using dual instantiation Otherwise its value will be whatever it was the last time an etree was deepened using dual instantiation rew unsubst exps This is a list of expansion variables which occur free in some leaf corresponding to an uninstantiated definition 7 NS These are some other global variables 1 leibniz var list An association list of variables and r
121. arget A flag Fi is irrelevant when there is a path from a flag Fo to Fy where Fo is never irrelevant and at least one of the conditions on the path evaluates to true Consider the following session lt 2 gt default ms DEFAULT MS MS90 3 gt lt 3 gt ms98 num of dups MS98 NUM OF DUPS NIL gt 3 WARNING The setting of the flag DEFAULT MS makes the value of the flag MS98 NUM OF DUPS irrelevant lt 4 gt default ms DEFAULT MS MS90 3 gt ms98 1 lt 5 gt ms98 num of dups MS98 NUM OF DUPS 3 gt 2 lt 6 gt 3 1 Automatically Generating Flag Relevancy In addition to the relevancy information directly specified in the defflag declarations in the code there is now code in flag deps lisp to read and analyze the code in the lisp files to determine flag relevancy This code first reads the lisp files and records all defun and defmacro definitions Then it computes easy flag conditions in which flags and calls to other functions occur At present 3 RELEVANCY RELATIONSHIPS BETWEEN FLAGS 93 an easy flag condition is built from atoms of the form easy flag term easy operator easy flag term using boolean operations and IF An easy operator is one of lt gt lt gt eq eql equal or neq An easy flag term is a flag NIL T a number or any quoted term The important property these conditions should satisfy is that their values are static i e they do not depend on the dynamic environment Note that some flags such as
122. arnations as a quick test and a slow test As a general convention that should never be violated both functions have the same name except for the last character which is Q for the quick test and P for the slow test As arule of thumb quick predicates may assume a very restricted kind of argument e g a literal atom but may not work recursively down into the formula Slow predicates however may assume nothing about the argument they should always work and often have to do a recursion to see whether the predicate is true of the argument Quick predicates are most useful when in recursive functions that im plement a wffop Slow predicates are chiefly called inside the editor to test that certain transformations or applications will be legal before they are performed Speed is usually not important when working in the editor but wffops in general should be optimized for speed since time does make a difference in automatic mode A list of the most useful quick predicates in the order in which they must be called is supplied here See the comments attached to the predicates in the source file if this list is unclear or ambiguous It is absolutely essential to understand the role of quick predi cates and the order of their invocation to write bug free code LABEL Q gwff tests for a label The standard action in this case is APPLY LABEL GWFF wffop arg argn where wffop is the wffop we are defining and arg through ar
123. ation necessary to print x is really on the property list of the symbol whose print name is x So all wffs of the form x lt gt will be printed the same way except for the type So if you enter x1 A you get the symbol x1 lt A gt but no informa tion about a superscript is put on the symbol x1 Thus when you print it you get no superscript just xl Where do superscripts come from then Well when TPS renames a variable in order to get a new one such as alpha normalizing a wff it puts the superscript information on the new symbol s property list I e if we rename xl lt A gt we may get the symbol x2 lt A gt and on the property list of x2 we get the superscript informa tion Thus the next time the user types in x2 A or even x2 I the symbols created will have the superscript information This can be a little confusing because the x1 A that you originally entered still isn t superscripted but the renamed variables x2 x3 etc will be 2 Printing of formulas 2 1 The Basics In this section we will talk about how a formula in internal representation is printed on different output devices There are two main points to take into consideration how will the parts of the formula appear and where will they appear For the latter refer to section 3 the former we will discuss now 2 2 Prefix and Infix Since we deal with formulas of type theory we can regard every formula as built by ap
124. be called for one use and in one mode then bar will be called with a different use and mode Remember setting the value of the flag tactic verbose to T will cause the tactics to send informative messages as they execute 4 1 Implementation of tactics and tacticals 200 17 TACTICS AND TACTICALS 1 Main files in order of importance tactics macros tacticals tacticals NS NS macros tactics aux These files contain tactic related functions of a general nature Most tactics are actually contained in other Lisp packages When a tactic is executed two global variables affect its execution tacuse the tactic s use and tacmode the current mode Tacuse determines for what reason the tactic is being called Current uses are etree nat translation of eproof to natural deduction and nat ded construction of a natural deduction proof without any mating information A single tactic may be defined for more than one use Tacmode can have the value of either auto or interactive Each tactic should take this value into account during operation In general this means that when the value is interactive the user should be advised that the tactic is about to be applied and should be allowed to abort it When the value is auto the tactic should just be carried out if applicable For each use a number of auxiliary functions needed by the tacticals must be defined a get tac goal if a goal has not been specified get
125. cations of Conj rules The other two commands use Sunil s fast propositional search to find an expansion proof and uses the tactic BASIC PROP TAC to translate back to natural deduction to fill in the gap using only primitive propositional rules Of course the Same rule just propagates the annotation So with re spect to annotations there are two versions of this rule Ass SameAs SameAs Al A When converting normal natural deduction to expansion tree proofs we only consider formulas up to a conversion so we can ignore the correspond ing ND rule But effectively we allow this rule to be annotated in either of two ways as with the Same rule VrA 5 VaA f Vyly a A 4 Vyly a A f Neg NNF and NNF Expand can be used to make small first order inferences from Vz A to Jx A etc Since we only care about formulas up to af and negation normal form we can treat these rules the same way as the Same and AB rules Applications of Assert other than AssertRefl are replaced by ex plicit hypotheses in a preprocessing step by the function make assert a hyp So when building the natree there should be no instances of Assert other than Refl AssertRefl is annotated as a normal deduction AssertRefl A A t The idea is that we work backwards to an instance of reflexivity Definitions can be eliminated or introduced and the annotations re flect this Also elimination and introduction of definitions in
126. ccurrence of defconstant is not specified in the ANSI standard and as some Lisps such as Steel Bank Com mon Lisp do not consider this use as correct it is better to use the macro defconstnt which has been implemented to fit defconstant and still be used with SBCL Note that defconstnt has no a at the end Obsolete In 2004 Also if you are using Kyoto Common Lisp you will find that the way it represents directories is a little unusual all paths are relative unless specified not to be So for example tps3 sys should be changed to read setq news dir root usr tps setq source path root usr tps bin root usr tps lisp setq compiled dir root usr tps bin setq patch file dir root usr tps assuming the main TPs3 directory is usr tps 3 MAINTENANCE 7 3 2 Building TPs3 See the user s manual for a description of how to set up and build a new version of TPs3 ETPS The global variable core name currently contains TPS3 it is defined in tps3 sys which is generated by the Makefile All files news note ini sys patch exe use core name as their name File names and extensions should be strings rather than quoted symbols to avoid any ambiguity with the package qualifiers Changes to the code are put in the patch file tps3 patch until TPS3 is rebuilt ETPS and Grader have separate patch files When you change the file nat etr lisp for example put
127. ch will be printed into Scribe files as whatever The TEXNAME property does the same for the TRXoutput style There are some special fonts that are declared in the file tps mss A list of the available special characters for the Concept and for the Dover in a SCRIBE file are explicitly stored in the files cfont lisp and dfont lisp and loaded into TPs3 at the time the system is being built 50 6 PRINTING AND READING WELL FORMED FORMULAS Consider the following example SIGMA1 is a binder It has a property FACE of value CAPSIGMA SUP1 CAPSIGMA is a tex special character a scribe special character and a concept special character It has a property CFONT of value 3 83 It has a property DFONT of value gS It has a property TEXNAME of value Sigma SUP1 is a tex special character a scribe special character and a concept special character It has a property CFONT of value 1 49 It has a prop erty DFONT of value 1 It has a property TEXNAME of value sup1 In a scribe or tex file or on a Concept with special characters SIGMA1 will appear as Xt elsewhere it will be written as SIGMA1 The actual Scribe output produced will be g S 7 1 the actual TRXoutput will be Sigma 1 2 6 More about Functions In this section some more details of the functions which are used to do the printing are given As mentioned earlier the main connection with the rest of TPS3 is the MACRO PrtWf It expands into a PROG in which
128. cludes some 8 reduction Suppose the abbreviation A is defined to be Az Ap 1 Ln 3 NPROOFS TO EPROOFS 209 in the following annotated rule schemas lA By Bn 4 plylBi Brall t glA B Bn t When annotating A rules the arrows point in the direction of normal ization Defn Bl At Lambda Lambda Al Bt where A is the G7y normal form of B There are also rules Beta and Eta which are treated similarly 3 1 4 Equality Rules We assume that the proof has been preprocessed to remove applications of substitution of equals and applications of symme try so there is no need to annotate these rules for now As noted above reflexivity is treated as a normal deduction Refl A At There are two ways to apply extensionality consistent with the idea of annotations both correspond to expanding an equation using extensionality in the corresponding expansion tree Also there are two kinds of extension ality functional and propositional fba Iba 4 Po Qo1 Ext _ Ert Vig fx gx P Qo Vig fx gx P AN Sd Ext Fo Roi Ext ee fea Iga T Po Qof Leibniz equality is handled just like definition expansion Aa B 2 at Equiv eq Vdoa GA D qB Vdoa GA D qB f Aa Ba f 3 1 5 A Sequent Calculus In the file ftree seq lisp a sequent calculus is implemented The file contains code to convert sequent calculus derivations into expansion proofs and a cut eliminatio
129. coded in a separate function for each rule For the rule rule the function is called rule HYP DEFAULTS and is called when appropriate form within rule DEFAULTS 4 2 When the Algorithm is not Sufficient We must of course consider the case when a restriction like x not free in H is imposed upon applications of the inference rule Since we fill in Hupper for the hypotheses of the dlines which do not exist we must check whether x not free in Hupper It may be the case however that all dlines actually already existed In this case it would be sufficient for the validity of the rule application to check whether x not free in Hjgwer To see this may think of the rule as first a legal application of the inference rule leaving out the extra hypotheses then enlarging the set of hypotheses of the inferred line possibly with lines which contain x free This situation can also come up when not all the dlines are specified Then we may have been able to make the inference rule application legal by leaving out the lines H from Hupper which violate the condition x not free in the assertion of H This leads to a simple modification of the algorithm above which would need much more information about the rule namely the restrictions where we modify the definition of Hupper in step 2 by 2 Hupper Lplp Perist H E HpALp satisfies any restriction on H It seems more reasonable however not to place that
130. computed and stored under the appropriate name on the property list of the symbol In this case too the symbol itself will appear in the pplist During the actual printing phase of the pplist the necessary information about symbols is retrieved from the property lists of the identifiers This presents one additional problem we have to preserve the informa tion about the dynamic parameters in the pplist itself so that the correct property can be accessed This could be done in a very general way but for specific problems maybe wasteful way namely by including the name of the relevant property in the pplist Alternatively we may use the special circumstance that there are usually more identifiers without type We would then only mark those identifiers with type while all others are assumed to be printed without types The solution above requires some auxiliary data structures There should be a global variable say static printing flags which contains a list of all flags affecting the printing of symbols Then there must be a func tion hash printing flags which takes one argument signifying whether types are to be printed and returns an identifier coding the value of the static printing flags and the argument 4 5 Other Issues In the solution proposed above it is left open whether the actual ASCII character representation of a symbol should be computed once and for all for each set of static and dynamic parameters and saved in a list
131. connect socket accept socket conn and pass socket local pass are defined for each implementation in the file tops20 lisp The definitions of these functions for other versions of lisp should indi cate what the definition should be for the new implementation of lisp If you really do not know how to define the function you can always define the function as the following example indicates newlisp defun call system throwfail call system undefined in lisp lt newlisp gt Such definitions will limit some of the capabilities of TPS Among these functions tps3 save is the most vital for getting started The function tps3 save should create a core image file which will be used when starting TPs3 5 Call make tps If all goes well a core image file will be created and you are ready to run Tps3 Check the documentation for the implementation for lisp to find out how to start lisp with a given image file If you are using Allegro Common Lisp version 4 1 or later is used to separate user options from lisp options and hence the standard way of starting up the Grader program in X windows be comes xterm geometry 80x54 14 2 723 0 fn vtsingle fb vtsymbold sb n CTPS Grader T CTPS Grader e usr theorem bin run tps grader amp Use of defconstant Since files may be loaded more than once during compilation and building of TPs3 two identical uses of defconstant may occur As the behaviour of the second o
132. constructing the sequent calculus derivation A description of the algorithm is below The function natree to ftree seq normal is called on natree nodes which are considered normal These would be annotated with a ft The function natree to ftree seq extraction 206 18 PROOF TRANSLATIONS is called on natree nodes which are considered extractions These would be annotated with a J Frank Pfenning s ATP class contained notes on annotating intuition istic first order normal natural deduction proofs and gave a constructive proof algorithm that every natural deduction proof translates into a se quent calculus proof Also normal natural deduction proofs translate to cut free sequent calculus proofs The idea of using annotations carries over to classical higher order logic 3 1 1 Normal Deductions The idea of a normal deduction is that the proof works down using elimination rules and up using introduction rules meeting in the middle We can formalize this idea by saying that a natural deduction proof is normal if its assertions can be annotated so that the assertions involved in the applications of rules of inference are as described below Technically we are defining normal natural deductions by mutually defining normal deductions ft and extraction deductions 3 1 2 Annotations of the Assertions in a Proof First the basic rules which allow one to infer normal deductions ft Att UGen VaA tp where x is not
133. ction 4 Note that the merge process does or should anyway merge the mating appropriately as the tree collapses On THM131 this takes the time spent on merging from 7 days down to 12 minutes This is not so surprising because it begins with 113 million paths and after the merging of duplicate expansions it s down to around 442 thousand The matingstree top level has its own approach to merging which is essentially step 2 above in which all unused expansions are simply thrown away followed by a regular merge as detailed above Putting step 2 first here is necessary because the master expansion tree has many nodes which are irrelevant to any particular proof 3 Modify Dual Rewrites The functions for MODIFY DUAL REWRITES are in the file mating merge lisp This is only called when dual instantiation is used The main function modify dual rewrites is described in this section First this func tion uses the global variable hacked rewrites list The value of the global is a list of elements of the form lt rewrite node gt lt instantiated wff or symbol gt lt leaf with uninstantiated form gt This list is sorted so that the names of the rewrites are increasing The main body is a dolist considering each element of hacked rewrites list For each subtree of the form Equivwf fs REW1 A7 REW AV Ay i DISJ1 Vie N REW C7 node LEAF1 Cy 140 14 MERGING or Equivwf fs REW1 At i REW A
134. ction list Let EQi be these positive ftrees for these equations So after n steps we have a negative leaf leaf with shal low formula p ai n We create a positive leaf LE AF2 leaf2 with this shallow and mate the two leaves If IT is empty we return this positive ftree along with the positive ftree CONJ L N EQ pala CONJ a N BO EQin If T is nonempty then the rest of D after removing the equa tions should be a equal to C hyp We create a negative leaf LEAF3 and a positive leaf LEAF 4 with these shallows and return the positive ftrees IMP x xN LEAF4 LEAF2 and the positive ftree CONJ x N EQ aia i CONJ Lo EQi LEAF3 3 SET CONSTRAINTS 191 In the constraint LF1 T is empty and D is zt 0 The shallow formula of LEAF1 is p z We construct EQ as z 0 REW Leibniz i Az Yy Vq qx D qy zt 0 REW A i EXP q qz D q0 Ax pt x Az pt a z gt Ax 7 p a 0 REW A IMP a N NEG NEG 4 LEAF3 LEAF where LEAF3 mates to LEAF1 We also create a posi tive leaf LEAF2 with shallow formula pt0 and mate this to LEAF We return LEAF2 and EQ In the constraint LF3 LF2 D is z Sw A ptw As above a positive ftree EQ for the equation z S w giving a mate for LEAF 17 and a negative leaf LE AF37 with shallow formula p S w We also create a positive leaf LEAF2 to mate with LEAF3 The leftover part of D pt w corresponds to IT We create pos
135. ction proof calls natree to ftree main to build the ftree representation of the expansion tree and a complete mating Finally this is converted into an ordinary expansion proof which may optionally be merged Merging is appropriate if the user plans to translate back to a natural deduction proof but inappro priate if the user is trying to gather information about a potential automatic proof ceb proof to natree This is a modification of Hongwei s proof to natree see hx natree top lisp This function builds the natree changing some justifications to RuleP or RuleQ and changing the variables in applications of UGen and RuleC so they are unique in the entire proof i e the natree rules satisfy a global eigenvariable condition since the etree selection variables must be distinct natree to ftree main This function calls natree to ftree seq normal to build a sequent calculus derivation see 3 1 5 from the natree Then ftree seq weaken early modifies the derivation so that the weaken rule is applied eagerly This may eliminate certain unnecessary cuts and simplify the derivation Then the cut elimination function ftree seq cut elim see section 4 2 is used to make the derivation cut free Finally cutfree ftree seq to ftrees is used to obtain an ftree and a complete mating from the cut free derivation natree to ftree seq normal natree to ftree seq extraction These functions are mutually recursive and provide the main algorithm for
136. current tactic uses since any changes in the constructed proofs are made immediately not saved Validations must be modified as tacticals are executed since during their exe cution the order of goals may be changed For example a tactical may repeatedly apply a tactic to a goal then to all the new goals created etc until it fails on all of them When it succeeds on a successor goal the validation returned must be integrated into the validation which was returned for the first application of the tactic on the original goal The function make validation is used for this purpose CHAPTER 18 Proof Translations 1 Data Structures 2 EProofs to Nproofs Here is a summary of what happens after matingsearch has terminated with a proof The functions involved are located in the files mating merge lisp mating merge2 lisp and mating merge eq lisp 1 Apply Pfenning s Merge algorithm etr merge put resulting etree and mating in variable current eproof a Get a list of the connections from mating b Get a list of substitutions required i Extract substitutions from unification tree ii Replace occurrences of PI and SIGMA by quantifiers iii Lambda normalize each substitution iv Alpha beta normalize each substitution c Prune any unused expansions from the tree d Make substitutions for variables e Carry out merging merge al1 2 Replace skolem terms by parameters if applicable subst skol terms If remove leibniz is
137. d be PASSED ARGS initialized if the default NIL is not acceptable in the BEGIN FN NOTE The names of PASSED ARGS should be different from other arguments ARGNAMES and EVENTARGS Also they should be different from other variables in those functions where you use them and from the variables which DEFREPORT always introduces into the function for the report FILE INP and DO STOP The definition of the category of REPORTCMD follows defcategory reportcmd define defreport1 properties source event single eventargs multiple selected variables in the var template of event 240 20 TEACHING RECORDS argnames multiple argtypes multiple arghelp multiple passed args multiple values needed by DO FN init in BEGIN FN defaultfns multiplefns begin fn singlefn args argnames do fn singlefn 33 args eventargs special passed args end fn singlefn args argnames mhelp single global list global reportlist mhelp line report mhelp fn princ mhelp cat help A task to be done by REPORT The creation of a new report consists of a DEFREPORT statement DE FREPORT is a macro that invokes DEFREPORT1 and the definition of the BEGIN FN DO FN and END FN Any PASSED ARGS used in these func tions should be declared special It is suggested that most of the computation be done by general functions which are more readily usable by other reports In keeping with this philosophy the report EXER
138. d du plication has been used the expansion proof will often have a great deal of redundancy in the sense that the same expansion term will be used for a given variable many times More precisely if one defines an expansion branch by looking at sequences of nested expansion nodes attaching one expansion term to each expansion node in the sequence there will be many identical expansion branches So one can start by merging the tree in the sense of eliminating this redundancy see section 1 and then apply to this much simpler tree the procedure for deleting unnecessary expansion terms which we think is using so much time It turned out to be easiest to do this by throwing away the mating and reconstructing it by propositional search after the tree has been cut down to size Of course one could also preserve the original mating by merging it appropriately as one collapsed the tree The precise way in which this was done in the file mating merge lisp was 1 Don t do pruning of unnecessary nodes at the beginning of the merge when the tree is its greatest size 2 Instead do prune all branches that couldn t possibly have been used They are those that have a zero status This is probably not necessary but certainly makes debugging easier and doesn t cost much See section 1 3 MODIFY DUAL REWRITES 139 3 After merging of identical expansions has been done call the origi nal pruning function prune unmated branches see Se
139. d nodes And of course we cannot translate an application of cut 3 1 6 Translating from Natural Deduction to Sequent Calculus In the sequent calculus described above the order and multiplicity of formulas is important However in describing the algorithm below we are more inter ested in sets of formulas So let us use the notation Set T to denote the set of fomulas on the list IT Normal natural deductions are converted into the sequent calculus via two mutually recursive algorithms 1 natree to ftree seq normal Suppose we are given a line T F C f Then we can compute a derivation of a sequent C where Set T1 C Set T 2 natree to ftree seq extraction Given a line B and a deriva tion of a sequent T C where Set T1 C Set T U B Then we can compute a derivation of a sequent 2 C where Set T2 C Set T That is we have eliminated occurrences of B on the posi tive side 3 NPROOFS TO EPROOFS 213 We can show a few cases to demonstrate how the algorithms work Case Coercion Cl coercion C with hypotheses I We need a derivation of some Tz gt C We can apply the second induction hypothesis to the initial sequent C C with T1 empty to obtain a derivation of such a sequent 2 gt C Case Hyp Suppose the line is TFB where B is in I and suppose we are given a derivation of a sequent T1 gt C with Set T1 C Set T U B Since B T we have Set T1 C Set T and we are done
140. d the arguments args of the main lit eral of the constraint This function constructs three positive ftrees The first is an ftree Kuu Je which will be merged with Kuuf The second is NEGF where L is the list of extra literals L of the constraint or NIL if there are no extra literals If NEGF is not nil it is a positive ftree corresponding to a conjunct of the extra literals L of the constraint The third is a leaf inside the first ftree which will be used in the mating Recall that Kuu is Vz dul z SwrAuwlv z 0 D utz and this will be the shallow formula of Kuu JE The third ftree returned will be the leaf corresponding to utz beneath Kuu JE In the first constraint LF1 the args are 0 We use the list LF1 of arguments as expansion terms to create Kuu fs Here Kuuf will be EXP Kuu 10 IMP rae N Kuu ff LEAF u 0 where LEAF ut 0 will be the third ftree returned Kuuf E is a negative ftree constructed by make clos setvar lemma negf 5 In this case there are no extra literals so the second return value is NIL 3 SET CONSTRAINTS 183 In the second constraint LF3 LF2 the one argument is S w LF3LF2 In this case Kuu fs is EXP Kuu J Sw IMP Kuuf LF yl Sw where Kau is constructed along with NEGFEY by make clos setvar lemma negf 5 make clos setvar lemma negf 4 This simply makes a neg ative leaf and mates it to a given positive leaf make clos setvar lemma neg
141. de Actually a whole formula is represented by a tree each node of which is an etree At first current topnode is the root of the tree Each node in the tree can be one of the following structures all of which are derived from the structure etree described above We note only the differences between these structures and etrees 1 econjunction is just an etree without any additional new attributes components is a list containing two elements and junctive should be dis or con edisjunction is like econjunction implication is like econjunction negation is just an etree components contains one element and junctive is neutral 5 skolem is an etree with two additional attributes a shallow contains the shallow formula that the attribute skolem represents Never forget to make the corresponding changes in it if you have changed some other parts of this node otherwise the proof cannot be transformed into natural deduction style by etree nat since the function get shallow would not work nor mally b terms is a skolem term structure containing a term replacing the original variable and something else 6 selection is also an etree with attributes shallow and terms just as skolem etree nodes Whether the etree contains selection or skolem nodes depends on the values of SKOLEM DEFAULT 7 expansion is an etree with three additional properties a shallow is the same as in skolem b terms is an exp var structure cont
142. de Applying the first algorithm to we obtain a derivation of a sequent 3 gt B with Set T3 C Set An application of cut gives us the sequent we desire r3 gt B B T2 gt C T3 T2 gt C cut Remark We check equality of wff s up to a conversion and negation normal form Because we check up to negation normal form applications of Neg and NNF rules can be treated the same way as the Same and AB rules Note If NATREE DEBUG is set to T then at each step the code double checks that the derivation is well formed After we have a sequent calculus derivation cut elimination can be used to try to remove applications of cut see section 4 2 If we obtain a cut free derivation this can be translated into an ftree with a complete mating 3 1 7 Normalization of Proofs There is now a TPS3 command NORMALIZE PROOF that converts a natural deduction proof or a natural deduction proof with asserted lemmas which have natural deduction proofs in memory into a sequent calculus proof with cuts then uses the cut elimination algo rithm to obtain a cut free proof assuming termination and finally trans lates back to natural deduction The resulting natural deduction proof is normal If we decided to normalize natural deduction proofs directly without passing through a sequent calculus we would need to identify possible re dexes pairs of rule applications which must use backward coercion to be annotated and show how to reduce
143. deals with implication is a much simplified version of this strategy and helps understand the algorithm e hx natree aux contains the code of merge functions and the ones handling rewrite nodes Presumably there are some bugs in han dling rewrites and this can be found in the comments mixed with 216 18 PROOF TRANSLATIONS the code Also a new version of ETREE TO JFORM REC is de fined here to cope with a modified date structure ETREE hx natree cleanup contains the functions which clean up the expan sion proofs before they can be used by ETREE NAT This is tem porary crutch and should be replaced by some systematic methods For instance one could construct brand new expansion proofs ac cording to a constructed one rather than modify it to fit the needs of ETREE NAT This yields a better chance to avoid some problems caused by rewrite nodes hx natree debug contains some simple debugging facilities such as some display function and some modified versions of the main func tions in the code A suggested way is to modify the code using these debugging functions and trace them More facilities are needed to eliminate sophisticated bugs Selection nodes not Skolem nodes are used in the constructed expan sion trees The prevents us from setting the MIN QUANT ETREE flag to simplify a proof It is a little daunting task to modify the code for MIN QUANT ETREE but the benefits are also clear both NAT ETREE and non pfd procedures can t
144. definfo command definfo foo mhelp Help text attaches the given text to the symbol foo 2 Synonyms It is possible to define two flags with different names which are synony mous to each other using the defsynonym macro The advantage of this is that it allows the name of a flag to be changed from the user s point of view without requiring either a change in the code or extensive editing of all the modes saved in the library For example defsynonym SUBNAME synonym TRUENAME replace old T mhelp SUBNAME is a synonym for the flag TRUENAME defines a new synonym for TRUENAME The replace old property deter mines whether or not the new synonym is to be regarded as the new name of the flag if replace old is T and so to be recorded in the library etc or merely as an alias if replace old is NIL 3 Relevancy Relationships Between Flags When defining a new flag one can specify relevancy relationships between the flag and the values of other flags For example if the flag DEFAULT MS is set to MS90 3 then the flag MS98 NUM OF DUPS is irrelevant On the other hand if DEFAULT MS is set to MS98 1 then the flag MS98 NUM OF DUPS is relevant Since we expect the relevancy infor mation to be incomplete at any point in time it makes sense to explicitly record relevancy and irrelevancy information separately The slots used to record these relationships are irrelevancy preconditions relevancy preconditions irrelev
145. defun lcontr reduct cond label q reduct apply label reduct lcontr reduct lsymbol q reduct throwfail Cannot Lambda contract reduct gwff a logical symbol boundwff q reduct throwfail Cannot Lambda contract reduct gwff a bound wff t if lambda bd p car reduct substitute l term var cdr reduct gar car reduct gdr car reduct throwfail Top level application reduct gwff is not of the form LAMBDA x A t FREE FOR is a simple example of a predicate on wffs Here the type of the result is declared to be BOOLEAN defwffop free for argtypes gwff gvar gwff resulttype boolean argnames term var inwff arghelp term var inwff applicable q lambda term var inwff declare ignore inwff type equal term var applicable p lambda term var inwff declare ignore inwff type equal term var mhelp Tests whether a term is free for a variable in a wff defun free for term var inwff cond label q inwff apply label inwff free for term var inwff lsymbol q inwff t 1 OPERATIONS ON WFFS 75 boundwff q inwff cond eq caar inwff var t free in caar inwff term not free in var cdr inwff t free for term var cdr inwff t and free for term var car inwff free for term var cdr inwff TYPE currently in wffprim lisp returns the type of the argument The name is a very troublesome one and we may e
146. der mode can only be parsed in higher order mode and formulas printed in first order mode can only be parsed in first order mode e Operator precedence The parser for wffs is a standard operator precedence parser The binding priority of an infix or prefix opera tor is a simple integer and conforms with the usual conventions on how to restore brackets in formulas and serve as brackets and a period is to be replaced by a left bracket and a matching right bracket as far right as consistent with the brackets already present when brackets are restored from left to right For opera tions of equal binding priority association to the left is assumed In order of ascending priority we have or EQUIV 2 gt or IMPLIES 3 V or OR 4 A or AND 5 or NOT or 100 6 PRINTING VERTICAL PATHS 61 applications like Pxy or A x z t binders A V 3 e Types Function types are built from single letter primitive types Grouping is indicated by parentheses and The basic types are subomicron or O for truth values and subiota or I for indi viduals Any letter except T i e may serve as a typevariable A pair ap or AB is the type of a function from elements of type B to type A E g O OI or o a is the type of a collection of sets of individuals Association to the left is assumed so QAAA or gaaa is the type of a three place predicate on variables of type A e Identifiers in higher
147. dices are preimages of preimages though we must be careful to account for the changes in lengths of the two sides as A in the final sequents and A in the premisses are on opposite sides The diagram that applies here is D D 1 f D Dai elim elim D3 Da elim Ds A There are three relevant premisses here D has the form Diy Dio Ty A Ai Ty9 gt B A12 Ti 12 gt AA B Au Ars and D has the form A Doi A B ToA ANB T AOA Recursive calls eliminate the nonprincipal occurrences of the mix formula D3 D mix D2 to eliminate A A B T3 gt Ag where T3 contains residuals of A and B D4 M1 mix D to eliminate A A B T4 gt Ag where Ay contains residuals of A D5 D 2 mix D to eliminate A A B T5 gt As where As contains residuals of B A 230 18 PROOF TRANSLATIONS De M5 mix D3 to eliminate residuals of B Te gt Ag where Ig contains residuals of A D7 D4 mix De to eliminate residuals of A 7 A7 where there are no residuals of A B or AA B Since there were more recursive calls in this case we must compute preimages of preimages of preimages in some cases and preimages of preimages in other cases as indicated by the following diagram D112 Do D Dai Diu Do D5 D3 De De D7 Each wff in the sequent proven by D7 can be traced back to a non mix formula in either D or D using this diagram e V This case is similar to the A case but D has on
148. dren of an etree are also etrees We could use this attribute to check whether an etree is a leaf true or false 3 positive tells us whether the formula which an etree represents which is the formula given by get shallow appears positively or negatively in the whole formula This will be used to compute the vpform of the whole formula The vpform of a subformula may be not the same as the corresponding part of it in the whole formula because the positive property of the subformula is dependent on the context 4 junctive can be used for printing the vpform This attribute is linked tightly with the positive attribute and has to do with whether the node acts as neutral conjunction or disjunction 5 free vars is a list containing the free variables in whose scope the node appears When you skolemize a formula you should use this attribute 111 112 13 MATING SEARCH 6 parent is the parent of this etree 7 predecessor this slot tells you the leaf name from which the current etree was deepened It is mainly used for handling skolem constants 8 status has little to do with the system as currently implemented but you should be careful when you are creating commands which will change the variable current topnode You have two choices a Change the value of ignore statuses to T Then you need not worry about this attribute Of course what you are doing may then not be compatible with the future versions of the
149. e change there is a system flag XTERM ANSI BOLD with default value 53 the ASCII code for 5 The value of this flag can be changed to 49 the ASCII code for 1 when needed Perhaps in the future the flag could be deleted and the change hard coded once someone is confident of what the number should always be It would be nice if xterm could render more special symbols Though xterm would probably allow using more than the printable characters this appears very hard to get lisp to do in general that is it s hard to figure out how to get lisp to send these characters to the terminal This will probably never be completely implementation independent because character sets are a very unstable part of the lisp specification In TPs3 you will only be able to get the symbols between 32 and 127 basically because the lisp allows only those without some kind of great hackery A few lines of attack for getting more characters suggest themselves 1 Hack xterm to allow more than 2 fonts In fact Dan has done it to allow one more font and thinks it would be possible to add up to two more for a total of one normal font and three symbol lIn 2005 and previously a value of 53 worked for XTERM ANSI BOLD at CMU while using xterm version XFree86 4 2 0 165 and a value of 49 worked at Saarbrucken while using xterm version X Org 6 7 0 192 20 1 INTRODUCTION fonts The disadvantage here would be having to distribute the new version of x
150. e Leibniz rewrites to Subst rewrites destructively chang ing a positive subtree of the form Leibniz REW1 A Bt REW Yq qA D qBt EXP1 q qAi D q B Q t N Qn AREW Q4 D QIB gt o J IMP1 Z N to a subtree of the form REW A B Subst REW1 A By EXP1 q qAi D q B Qu K N Qim IMP1 er F N AREW AREW 4 4 where 1 lt i lt lt im In 2 pre process nonleaf leibniz connections Since the remove leibniz function may replace negative subtrees of the form Leibniz REW1 A B i SEL1 Vq qA D qB IMP1 L N d Equivwf fsREW qo At A Equivuf fsREW qo B L LEAF1 qo AG LEAF2 qo By with a leaf of the form LEAF A B we must have some way of dealing with connections to nonleaf nodes such as SEL1 and IM P1 This pre processing function replaces a 144 14 MERGING connection such as MP1 IMP3 with connections to the leaves LEAF1 LEAF3 LEAF2 LEAF4 where IM P3 is a positive subtree of the form IMP3 eg N Equivuf fsREW qo A d Equivwf fsREW qo BY 1 LEAF qo Ap LEAF4 qo B Such connections between leaves are dealt with in the function remove leibniz Note that we do allow connections to the node REW 1 to remain in the mating This connection may also be modified in the function remove leibniz 3 remove leibniz This function basically corresponds to the proof of Theorem 138 in Frank Pfenning s thesis Pfe87 Also Remark 140 in F
151. e appro priate form apply thm 146 This function corresponds to Theorem 146 in Frank Pfenning s thesis Pfe87 If the subtree passed to the function has 150 14 MERGING the form Subst REWn A Bt EXP1 V q q A gt qB tQ IMP1 L N i LEAF1 A Bt we can replace this subtree with LEAF 1 If the subtree passed to the function has the form Subst REWn A Bt EXP1 Vq qA D qBt 1Q IMP1 K XN ip LEAF2 A Bt we can replace this subtree with LEAF 2 7 Raise Lambda Nodes The functions of RAISE LAMBDA NODES are in the file mating merge2 lisp The main function is raise lambda nodes which calls the following auxiliary functions e raise lambda nodes skol commutes a A rewrite with a selection or skolem node raise lambda nodes auxl commutes one or two A rewrites with a conjunction disjunction or implication node raise lambda nodes neg commutes a A rewrite with a negation node raise lambda nodes ab commutes a A rewrite with an AB rewrite by destructively changing the justifications of the two rewrites and the shallow formula of the lower rewrite e raise lambda nodes equiv commutes a A rewrite with an EQUIV IMPLICS rewrite or EQUIV DIS JS rewrite by de structively changing the two rewrites Since A rewrite nodes may be destroyed during this process we may need to change the mating In fact we maintain the following invariant Invariant Once a tree has been processed there are
152. e control infor mation Control information is sent by first sending a null character ASCII 0 followed by a byte giving information The current possible byte values 1 GENERATING THE JAVA MENUS 247 following a null character and their meanings are listed below There are lisp functions in interface style lisp which send these bytes but anyone coding a new interface will need to know these values 0 e 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Switch to normal font mode In normal font mode each character is communicated by a single byte Switch to symbol font mode In symbol font mode each symbol character is communicated by two bytes allowing for many more symbol characters than normal font characters Start a prompt message string Start a prompt name string Start a prompt argtyp string This allows the interface to recognize some special finite argtyp s such as boolean Start a list of prompt options Start a list giving the prompt s default value Start a prompt help string End a prompt A note that a command has finished executing Start and end a string specifying the current top level Open a window eg for proof windows or vpforms and start sending a string giving the port value for a socket to connect to End a string giving a prompt for a window and start sending a string given a title for the window End the title of a window and start sending a string giving the widt
153. e editor top level compare the above definition with that of mexpr for example So all the commands that can only be seen from the editor top level are defined with the defedop command as follows defedop o alias invert printedtflag mhelp Invert PRINTEDTFLAG that is switch automatic recording of wffs in a file either on or off When switching on the current wff will be written to the PRINTEDTFILE Notice that the resulting file will be in Scribe format if you want something you can reload into TPS then use the SAVE command The top command ctree is used for command completion and the mhelp property is obvious This leaves top cmd decode which is the name of the function that is called by the command interpreter to for example fill in the default arguments for an edop 2 Command Interpreters Each top level has its own command interpreter The actual command interpreters in much of the code are older versions the code has since been simplified considerably New command interpreters which may in time re place the older versions and which should certainly be used as the models for the command interpreters of any new top levels are in the two files command interpreters core lisp and command interpreters auto lisp CHAPTER 4 MExpr s TPs3 provides its own top level It allows for default arguments and provides a way of giving arguments e g wffs in some external represen tation which is converted before the real
154. e nodes L1 L2 and L3 and the expansion variable v constrs has the value L1 L3 L2 v has the value v vsel is a new variable v of the same type as v 3 is f with vt not an expansion variable substituted for the ex pansion variable v e banned occurs ends up having the value w NIL representing Wy and Wo misc occurs is NIL assuming S and 0 are constants paths LF3 LF2 LF1 where each LFi is the ftree node in f3 corresponding to Li Then make ftree setvar soln is called with vsel v kind MIN paths LF3 LF2 LF1 banned occurs w NIL misc occurs NIL rec flag T A special dynamic variable clist is used to collect connections created in the process of building the lemma The function make min inv princ constructs the inversion principle Va vla gt dulz SwAv ulV z 0 Next make min setvar lemma posf constructs the full lemma dv Vw v2 w D v Sw A v0 AYzt v zt D Jule SwAv ulv zt 0 AVp w pw D p Swt A pO D Yz fvt x7 D pz and the positive ftree that will correspond to how the lemma can be used In particular connections to the nodes LF1 LF2 and LF3 are created which solve the constraints i e block every verticle path through LF1 and every vertical path through LF2 and LF3 3 SET CONSTRAINTS 175 In simpler examples which do not require a recursive definition using the Knaster Tarski Fixed Point Theorem the function make min setvar lemma negf constructs an
155. e premiss and D has two premisses D1 is D T A B A V Tr gt AVB A and Dz is Do Da2 A Ti gt Ay B T2 gt Ag AV B T1 T2 gt Aj Ao The following diagram indicates the order of the recursive calls Di Do D Doo D4 Ds D Da De Ds D7 e gt This is similar to the V case The following diagram indicates the order of the recursive calls Du Do D Da2 D Da Da D5 D3 De D7 V 5 Cut free Extensional Sequent Derivations to Extensional Expansion Proofs There is an implementation of the extensional sequent calculus in Chad E Brown s thesis cf Bro04 in Tps3 The structure ext seq defined 6 EXTENSIONAL EXPANSION PROOFS TO NPROOFS 231 in ext exp dag macros lisp represents sequent derivations in this extensional sequent calculus Sequent derivations can be created manipulated saved and restored in the EXT SEQ top level If a sequent derivation is cut free then the command CUTFREE TO EDAG implemented in ext seq top lisp will translate the sequent derivation to an extensional expansion proof The proof that this translation works is in Chapter 7 Section 10 of Chad E Brown s thesis cf Bro04 6 Extensional Expansion Proofs to NProofs TPs3 can translate an extensional expansion proof to a natural deduc tion proof using the code in the file ext exp dags nd lisp This code is auto matically called when either of the extensional search procedures MS04 2 or MS03 7 successful find a proof T
156. e slot subsumed in each node of the unification tree this is implemented in the file unif subs lisp The subsumption checker is passed the new node and a list of other nodes which might subsume it If SUBSUMPTION CHECK is NIL it returns immediately Otherwise it first checks the flags SUBSUMPTION NODES and SUBSUMPTION DEPTH and eliminates all nodes from the list that do not fit the criteria established by these two flags so it might for example 163 164 15 UNIFICATION pick out just those nodes at a depth of less than ten which lie either on the path to the new node or at the leaves of the current unification tree Since it is possible to add new disagreement pairs to the leaves of the tree under some conditions it also rejects any nodes that do not represent the same original set of disagreement pairs as the new node Then it computes a hash function somewhat similar to Goedel numbering by considering each wff in the set of disagreement pairs at a node The hash function has to ignore variables because we want to catch nodes that are the same up to a change in the h variables that have been introduced These hash numbers are calculated once and then stored in the subsumed slot in the following format for a dpairset A1 B1 A2 B2 we first calculate the hash numbers for each wff and generate the following list C A1 B1 A1 B1 A2 B2 A2 B2 Then for each disagreement pair if Bi lt Ai
157. e status 0 deletes children of expansion 2 DETECTING UNNEEDED NODES 137 nodes which have status zero The function substitute in etree is used to apply the substitution to the etree this puts the appropriate terms into the SUBST slot of the expansion variables Then the functions strip exp vars for etree and strip exp vars are used to replace all expansion variables by their SUBST slot Finally merge all is called The function merge all takes an expansion tree and a mating and de scends into the tree At each expansion node if two expansion terms are identical their corresponding trees are merged The resulting tree replaces the two original ones and the substitution returned is applied to the terms and trees The resulting tree and mating are returned The actual merging of two children of expansion nodes is carried out by treemerge The algorithm is described as Algorithm 84 in Frank Pfen ning s thesis Pfe87 The algorithm also must build a substitution replacing some selected variables with other selected variables and apply this to the tree The function returns three values the new etree the substitution merge theta for selected variables and the new mating 2 Detecting Unneeded Nodes The function unneeded node p is used both by modify dual rewrites and prune unmated branches It is defined in the file mating merge lisp The purpose of unneeded node p is to determine if a node is needed to have a complete mating If the
158. eanup leibniz expansions Start with a positive subtree eq rew node of the form Leibniz REW1 A Bt REW Yq q A D q B J EXP1 q qAi D qBt Qi L N Qh AREW Q14 D QIB oe o IMP1 lt N 148 14 MERGING A do loop early in the function pushes the justifications for the initial rewrites up one step and changes the shallow formulas from Vq qA D qB to be the uninstantiated equation A Bt Also the last rewrite in the chain is changed to have justification Subst So the intermediate tree has the form REW A Bt i Subst REWn A B EXP1 Vq qAi D q B Q t N Qn AREW QA DQi Bf gt o e if IMP1 lt N After the do loop the local variable exp node has value EX P1 Next for each child of the expansion node of the form AREW Qi A D Qi B J IMPi aaa we remove the initial A rewrites possibly adding a A rewrite be neath IMPi The function check shallow formulas does part of this work This replaces the corresponding son of EX P1 with a subtree of the form IMPi L N AREW Qi Ay AREW Qi BE i fi Next we call the function apply thm 146 This may replace some Subst rewrite nodes with leaves There is a description of this function later in this section The function corresponds to Theorem 146 in Frank Pfenning s thesis Pfe87 SS NS NS 6 REMOVE LEIBNIZ 149 As a final step which occurs in the code in the return por tion of the outermost dolist
159. ed line There are cases however where this may still be necessary see discussion below 4 DEFAULTS FOR SETS OF HYPOTHESIS 107 4 1 The Algorithm Here unlike in other parts of the OUTLINE mod ules we do not need to refer to the support structure Instead let us view the rule as if we were to infer the plines from all the dlines specified in the rule and let us disregard hypothesis lines hlines for the moment For a given line l in the rule specification we now let S stand for the set of lines in the hypotheses which were explicitly specified in the rule description corresponds to s1 Sn above and let L the actual list of hypotheses for the line which must either be matched or constructed de pending on whether the line existed or not Furthermore let H stand for the unique name for an arbitrary set of lines which appears in zero or more of the lines in the rule description Let us first consider the case that the hypotheses specified in the rule description do not contain H For dlines d we must check La E Sa and for plines p we need to check Sp C Lp For dlines d which contain H among their hypotheses we must satisfy LaCHU Sa and if we are filling in hypotheses for a new line we would like to choose Lq as large as possible so it satisfies this equation From another point of view namely when we match existent lines we find out some constraint on H La Sa CH On the other hand for any given pline p
160. ed to ensure 9 PRETTIFY 159 y does not occur free at all in M Equivalently y should not be free in M But this is precisely what condition Finally we have ensures R6 M B a z R amp N B a z p N o M O Now the algorithm should build and a for Q so that they satisfy and We also want 0 to be an injective renaming so that no two se lected variables will be identified Given partial renamings 0 and a satisfying and we need to know if an extension will continue to satisfy the conditions These tests are implemented by prettify free legal p and prettify bound legal p and are described below The actual PRETTIFY functions are e prettify etree This is the main function called by merge tree real This calls prettify process vars in etree to collect the names of free variables and bound variables in the etree We dis tinguish the ones which occur in the topmost shallow formula since these were supplied by the user and should not be renamed We also do not rename frees which are introduced by a rewrite This is a bit unusual but can happen An example is a rewrite instantiat ing the abbreviation PLUS which introduces the free S Here we are really thinking of S as being part of the signature but there s noth ing to explicitly indicate that S is not a variable We start off by sending each such free to itself and each such bound to itself It is easy to see that conditions
161. ed to mean p lt gt q amp q lt gt T FO SINGLE SYMBOL this is meaningful only in first order mode and declares the object to be a keyword in the sense that it may be typed in all upper or lower case Moreover the printer will surround it by blanks if necessary to set off surrounding text Also the parser will expect that the symbol is delimited by spaces dots brackets unless the symbol just consists of one letter in which case it doesn t matter You MUST use this attribute in first order mode for an identifier with more than one character MHELP An optional help string Properties specific to binders are described in the section below about binders Here are some more examples These examples do not actually exist under these names in TPS3 def logconst amp type 000 printnotype t infix 5 prt associative t fo single symbol amp mhelp Conjunction Note that the fo single symbol amp will make sure that spaces are printed around amp in formulas In the next example the character is used to make sure that the dis junction is printed in lowercase that is as v instead of V def logconst v type 000 printnotype t infix 4 prt associative t fo single symbol v mhelp Disjunction def logconst gt type 000 printnotype t infix 3 fo single symbol gt mhelp Implication We do not like spaces after negation So we do not declare it to be
162. ee nat tactic mode auto interactive tactic defn primitive tactic compound tactic primitive tactic lambda goal form This lambda expression should return four values of the form goal list msg token validation compound tactic tactical tactic erp tactic exp tactic a tactic which is already defined tactic use tactic use mode tactic mode goal goal compound tactic call command where command is a command which could be given at the TPS3 top level goal a goal which depends on the tactic s use e g a planned line when the tactic use is nat ded goal list goal msg string token complete meaning that all goals have been exhausted succeed meaning that the tactic has succeeded nil meaning that the tactic was called only for side effect fail meaning that the tactic was not applicable abort meaning that something has gone wrong such as an undefined tactic Tacticals are kept in the TPS3 category tactical with defining function deftactical Their definition has the following form deftactical tactical defn lt tacl defn gt mhelp lt string gt with 17 TACTICS AND TACTICALS tacl defn primitive tacl defn compound tacl defn primitive tacl defn lambda goal tac list form This lambda expression where tac list stands for a possibly empty list of tactic exp s should be independent of the tactic s use and current mode It should return val
163. elected variables See section 4 4 dissolve An alist of symbols representing connections between nodes in the etree which we assume will be in the final solution The code 1 DATA STRUCTURES 117 for building jforms from etrees etree to jform etree to prop jform will use this as well as the flag DISSOLVE to dissolve vertical paths from the jform Dissolution is described in MR93 The current dissolution code dissolves one connection at a time iterating the procedure for each connection free vars in etree This is an association list between free expansion variables which occur in the eproof i e those which have not be instantiated and the corresponding expansion node in which the variable was introduced Note that if expansion variables are EXP VAR structures An expansion variable is uninstantiated with the VAR slot is the same as its SUBST slot When an expansion vari able p introduced in expansion node EX P is instantiated with a term which introduces new expansion variables q e g a PRIM SUB the pair p EX P is removed from this slot and the pairs qi EX P is included in the list Also in such a case the value of the slots substitution list inst exp vars params all banned may change to reflect this instantiation 6 skolem constants An association list of skolem constants and in tegers representing their arities Note If SKOLEM DEFAULT is set to NIL then all skolem constant
164. en g is free in t Also y S q implies y is free in 6 q From this we have y is free in 6 t and we are done For the induction step suppose we have q lt g 9 q1 lt a o q By induction we have some t T such that for any to Ty O t lt eo O to Also we have some t Ty such that for any t Ty 0 t1 lt gqy A t In particular we have O t lt g g A t1 lt o O for any t Ty O Lemma 2 Let an expansion tree Q and a substitution 0 for Vg be given Suppose we have t t Og with 0 t lt gq A t There is a qd Ng U Vg with t Ty such that for any q E Ug U Vo with t T we have q lt Q q Proof By induction on the number of transitivity steps 5 EXPANSION TREE TO JFORM CONVERSION 125 For the base case suppose 6 t lt 9 Q O t Let y be a selected variable dominated by 6 t and free in t Since y is free in 0 t we must either have y free in t or y free in 0 v for some v Vg free in t In the first case let g t In the second case let q be some v Vg where y is free in 6 v and v is free in t So we have y S q Suppose we have any q with t Ty Since y is dominated by t we have y B q and we are done For the induction step suppose 6 t lt 9 Q O t1 lt 9 Q e t By induc tion there is a q with t Ty such that for any go with t Ty we have qo lt Q q Also by induction there is a q with
165. ength keep track of the characters virtually printed and the length of the formula virtually printed respectively 3 PRETTY PRINTING OF FORMULAS 55 On the very lowest level PPrinc and PPTyo perform a PRINC or TYO virtu ally by appending the appropriate characters to the PPWfflist Characters that do not appear in the final document or on the screen are virtually printed by PPrincO and PPTyo0 They prevent the counter PPWfflength from being incremented Similar functions are PP Enter Kset and PPTyos which correspond to Enter Kset and TYOS In the second phase of pretty printing as described in the previous section PrintPPlist is the main function If the remainder of a PPlist fits on the rest of the current line SPrintPPlist is called which just prints the PPlist without any counting or formatting 3 5 JForms and Descr JForms A JForm is an alternative way of representing well formed formulas and is used by the matingsearch package and for printing vertical path diagrams In JForms multiple conjunction are represented as lists and not as trees Consider the following example ANBACA DVEVF As a wff in internal representation this will be AND AND AND A B C OR OR D E F Obviously this is not a very suitable form for vertical path diagrams As a JForm however the above wff would read as AND ABC ORD EF which is already close to what we would like to see The function Describe VPForm takes
166. er any propositional connectives and skolem selection nodes This lifting stops at expansion nodes and rewrite nodes other than equiv implics equiv disjs and lambda This also moves connections to lambda nodes with the result that no connection in the mating involves a A rewrite node after this step is performed See section 7 e The etree is converted to a propositional jform including any non leaf nodes in the mating and the current set of connections is used to set active mating e The etree is cleaned up by calling cleanup etree This A normalizes expansion terms may remove some A rewrite nodes and may mod ify Subst rewrite nodes See section 8 e The etree is prettified by calling prettify etree See section 9 This renames bound variables and free variables in the etree that do not occur in the original wff We must be careful to avoid variable capture when doing this renaming There were bugs with the old code because of variable capturing 1 Applying Substitutions and Merging Duplicate Expansions The functions etr merge and merge all are in the file mating merge lisp These functions are used to preprocess the expansion tree in order to make the rest of merging more efficient For a discussion of why this preprocessing is done first see section 2 The function etr merge calls make mating lists to create the alist of mated nodes and the substitution for expansion variables corresponding to the mating Then the function prun
167. ere wffop has all the necessary type informa tion The typecase of wffop is irrelevant e Among arg arg each argument is either a gwff and may con tain other backquoted expressions or a Lisp expression which is considered a constant This is necessary to supply arguments which are not gwffs to a wffop Notice that it must be the internal repre sentation of the argument 9 More about Jforms Much of the code for handling jforms is in jforms labels lisp see defflavor jform in this file for the definition In the same file we see eval when load compile eval defflavor disjunction mhelp A disjunction label stands for a disjunction of wffs inherit properties jform include jform type disjunction This tell us that a jform can be a disjunction 10 Printing Proofs Proofs printed in Scribe or T Xare preceded by preambles which are de fined by the variables SCRIBE PREAMBLE and VPFORM TEX PREAMBLE The values of these flags are set in the tps3 ini file Since these preambles source files in the directory doc lib things must be done carefully to make sure that SCRIBEPROOF and TEXPROOF will insert the appro priate pathname when tps is distributed to other locations Note that the Makefile creates the file tps3 sys which contains the variable sys dir which shows where the tps was built When the Scribe preamble was changed to add LibraryFile KSets LibraryFile Mathematics10 LAt some point one c
168. erpreter singlefn print singlefn top level category singlefn top level ctree singlefn top cmd interpret multiplefns top cmd decode singlefn mhelp single global list global toplevellist mhelp line top level mhelp fn princ mhelp This shows a category whose individual members are defined with the deftoplevel command and whose properties include the prompting func tion a command interpreter and so on There is a global list called global toplevellist which will contain a list of all of the top levels defined and an mhelp line top level so that when you type HELP MATE TPS3 knows to respond MATE is a top level The mhelp fn is the function that will be used to print the 4 FLAVORS 23 help messages for all the objects in this category See chapter 8 for more information The chapters of the facilities guide correspond to categories Within each chapter the sections correspond to contexts In TPs3 global categorylist contains a list of all the currently defined categories 3 Contexts Contexts are used to provide better help messages for the user Each context is used to partition the objects in a category into groups with similar tasks For example the objects in the category MEXPR are grouped into contexts such as PRINTING and EQUALITY RULES Contexts are themselves a category of course the definition is in boot0 lisp New contexts are defined with the defcontext command and are invoked with t
169. es Inu F and ukuf are mated on clist make clos setvar lemma negf 1 This function constructs NEGF with shallow formula Vw ulwd ut Sw A u70 Along with this we will construct Kuuft and put connec tions onto clist So we build NEGF as a conjunction NEGF D nN EGE where NEGF and NEGF corresponding to the two constraints is constructed by make clos setvar lemma negf 2 We also use an integer n to keep up with which constraint we are considering make clos setvar lemma negf 2 Given a constraint C paths this function builds a negative ftree N EGF and changes Kuuf and clist First consider the constraint LF1 The shallow formula of NEGFUS should be u 0 Here we let make clos setvar lemma negf 3 do the work of constructing an ftree which will be merged with Kuuf Then make clos setvar lemma negf 4 creates the negative leaf NEGEEFY mated with part of the ftree generated in make clos setvar lemma negf 4 LF1 182 16 SET VARIABLES The second constraint we need N EGRESS LE to have shallow formula Yw ulw D ul Sw First we create a new selected variable w and let N BGR DE2 be SEL Yw u w D ul Sw Lw IMP NEGF LEAF3 where N EGR is constructed by make clos setvar lemma negf 3 along with an ftree to be merged with Kuuf and LEAF3 is a negative leaf with shallow u Sw mated to part of the new part of Kuuf make clos setvar lemma negf 3 Suppose we are given a con straint C T A an
170. es in the etree during the CLEANUP ETREE phase but there were some examples where prettify would lead to illegal variable captures To fix this the two phases have been separated and all PRETTIFY does is rename all the free and bound variables in the etree We go to great lengths to ensure that the renaming does not lead to variable capture First we should work out the theory of such renamings We would like to find two substitutions 0 and a taking variables to variables The intention is to use 6 for free variables and a for bound variables Let R be the renaming function on etrees and wffs On an etree Q R Q is simply the result of applying R to all shallow formulas expansion terms and selected variables Next we define R inductively on wffs R x a x if x is bound R y O y if y is free R c c if c is a constant R Ba M Ba x R M where B is a binder RG M N R M R amp N We would like the renamed wff to be the same as the result of doing some a conversions and substituting for the free variables So with respect to all the wffs M in the etree Q we need to know there is an M with M a M such that 0 is a legal substitution for M avoiding any variable capture and so that R M 6 M identical wffs We can guarantee this if we have 0 and a satisfy two conditions with respect to the etree Q For distinct variables x and z and any subwff B x B2 z2 M of a
171. es of problems The first only ap plies for formulae with equality and can be checked by trying again with the flag REMOVE LEIBNIZ set to NIL For the other two you need to use merge debug type setq auto merge debug t before calling merging and you can step through the process inspecting the etree at each step and omit ting the optional steps This can be a great help in discovering which part of the merging process is causing the bug Note In November 2000 merging was changed to handle the case when a mating contains nonleaf nodes The changes were to REMOVE LEIBNIZ and RAISE LAMBDA NODES Also the final phase was separated into a cleanup phase and a prettify phase The prettify code was for the most part rewritten In the process of making these changes this section of the Programmer s Guide was extended to reflect the current state of merging e The expansion tree is processed by the function etr merge see sec tion 1 which applies the substitutions for expansion variables in the expansion tree and merges duplicate expansion nodes It re turns both the new etree and an alist of nodes corresponding to the mating This is the part that actually corresponds to the merging algorithm Algorithm 84 in Frank Pfenning s thesis Pfe87 e Duplicate connections and connections between nodes that do not occur in the tree are deleted from the mating actually connection list in the local variable new conn list e If dual instant
172. essor attributes of leaf are all nil 10 true 11 false 12 empty dup info is an etree used by the NAT ETREE translation code not by the mating search There is also an eproof stored in the global variable master eproof In my experience this has been set to the same value as current eproof The only place I can find in the code where it may have a different value is when using option sets search procedures MS91 6 and MS91 7 see the files ms91 basic lisp and ms91 search lisp In particular there are option set structures which have an eproof slot These are set to copies of the eproof structure as opposed to the identical structure in master eproof Then in finish up option search current eproof is set to the value of such an eproof slot In addition to the etree slot there are numerous other slots associated with an eproof 1 jform Contains the jform associated with the etree see section 5 2 all banned A list of expansion vars and a list of selected vars whose selection node occurs beneath the expansion term This is needed to check acyclicity condition The value is set by the function fill selected 3 inst exp vars params An association list of expansion variables that occur instantiated in the etree and the selected variables that occur in the term This is needed to check the acyclicity condition when there are substitutions e g set variable substitutions made in a preprocessing stage that have contain s
173. ewrite nodes The variables are expansion variables actually only the symbol for the variable is used introduced by the Leibniz definition of equality i e the q in Yq q A D q B The rewrite node is the rewrite node in which the equality was rewritten using the Leibniz definition Note that the q is only an expansion variable if the rewrite node is positive after primsub This is just a toggle that is temporarily set to T after a primsub has been done so that it will be T while the new etree is deepened N xm There may be other global variables Needless to say it is difficult to build an expansion tree in any way other than using the deepening code that is already written see the file etrees wffops lisp and section 2 1 of this chapter because all these global variables and eproof slots need to be main tained 1 4 Functional Representation of Expansion Trees The imple mentation of expansion trees described above is well designed for mating search on a single tree An expansion tree has a great deal of global in formation associated with it such as what expansion variables and selected variables occur in the tree Also expansion trees contain circular references since each node is associated with both its children and its parent In par ticular since a node is associated with its parent we cannot coherently share a node in two expansion trees Unfortunately this makes it very difficult to build and modify expansion
174. f 5 Given a constraint C T gt A this function returns two values The first is a negative ftree Kuuf and the second is a positive ftree NEGF or NIL if T is empty constructed later For the constraint LF1 K uur should have shallow for mula dulz SwA ut wiv 0 Ol Cal of the form DISJ Via UNUSEDLEAF Kuu LF1 where Kuug is constructed by make clos setvar lemma negf 6 For the constraint LF3 LF2 K uu LF should have shal low formula Aw Sw SwA uw V Sw 0 So is Kuu oe DER is of the form In this case Kuu DISJ L N Kuut F UNUSEDLEAF where K uu LF is constructed by make clos setvar lemma negf 6 along with NEGE HY make clos setvar lemma negf 6 This function constructs K uu using the selected variables created in make clos setvar lemma negf 2 as expansion terms then calls make clos setvar lemma negf 7 to construct the rest of Kuu and NEGF At this 184 16 SET VARIABLES step we distinguish between arguments args2 that corre spond to these selected variables and remove one of the ar guments corresponding to each such selected variable The arguments sent to make clos setvar lemma negf 7 correspond to equations in the formula wff originating with the inversion principle Since the constraint LF1 has not banned variables K unl is constructed as K une by make clos setvar lemma negf 7 The constraint LF3 LF2 has the single banned variable
175. f an hline will always be strong and equal to hline anything else will result in an error even if perhaps logically correct Also in this case if CLEANUP SAME is T then CLEANUP will eliminate unnecessary hypotheses CHAPTER 13 Mating Search The top level files for matingsearch are mating dir lisp for ms88 ms90 3 top lisp for ms90 3 option tree search lisp for ms89 and ms90 9 ms91 search lisp for ms91 6 and ms91 7 and ms92 9 top lisp for ms92 9 and ms93 1 lisp for ms93 1 The lisp files with prefix ms98 are those used by ms98 1 The code for GO in mating top lisp shows what the main functions are Mat ing search with extensional expansion dags is different in many respects than mating search with expansion trees We delay this discussion until section 12 There are a lot of comments about the workings of the code embedded in the lisp files in particular there is an outline of ms90 3 at the top of ms90 3 top lisp 1 Data Structures See the section on flavors and labels section 6 for a discussion of some relevant information about the data structures below Among other things that section has the definition of the flavor etree 1 1 Expansion Tree The data structure etree defined in etrees labels lisp has the following properties 1 name the name of the etree We can use this attribute to identify which kind of structure an etree is 2 components is a list which contains all children of the etree The chil
176. fect The primsubs were stored in a hashtable which once computed was never changed again so the code had to be replaced by defflag neg prim sub flagtype boolean default nil change fn lambda a b c declare ignore a b c ini prim hashtable subjects primsubs mhelp When T one of the primitive substitutions will introduce negation Also put into the file auto exp the line export neg prim sub There are two ways to update flags One is to do it manually This is supported by function update flag The other way is to set flags automat ically For example you may have to do this in your ini file If XXX is a flag and you want to set it to YYY then you can add a line set flag XXX YYY in your ini file Sometimes you may use setq XXX YYY but this is highly discouraged because XXX may have a change fn associated with it which should be called whenever you set XXX Flag HISTORY SIZE is such an example Note that if the variable being set is just a variable and not a TPS3 flag then the setq form is correct 89 90 9 FLAGS 1 Symbols as Flag Arguments If your new flag accepts symbols as arguments and only certain sym bols are acceptable as in for example PRIMSUB METHOD or APPLY MATCH the symbols which can be used should have help messages at tached somehow This can either be done by defining a new category for the arguments such as ORDERCOM or DEV STYLE or it can be done using the
177. finition of Kp to the set w to obtain u C w D z F w Since u C v C w we have z F v So Kp is monotone Similarly suppose u C v and z K u Apply the definition of KF u to obtain a set w C u with z w Now since w C u C v this w can be used to witness that z K v So KF is monotone Let u p A be given We need to show Kp u C F u and F u C KF u Suppose z Kp u Apply the definition of Kp to u Since u C u we have z F u Next suppose z F u Then u can witness that z KF u Now suppose L is a monotone function such that for every u A L u C F u We need to show for every u p A L u C Kp u Let u p A be given and suppose z L u Since L is monotone for every w D u z E L w So z F w This shows z Kp u and we are done Now suppose R is a monotone function such that for every u A F u C R u We need to show for every u p A KF u C R u Let u p A be given and suppose z KF u So there is a w C u with z F w So z R w Since R is monotone and w C u we have z R u and we are done O 174 16 SET VARIABLES 3 2 Tracing Through An Example Consider again the two mini mal constraints gt v0 and w v w gt w S w where v has type ov Let us name the literals Li vO L2 vw L3 v S wu In the function ftree solve constraint set consider some of the local variables e f is an ftree containing th
178. following cases 1 Suppose N is not a rewrite In this case there are no connections to A rewrite nodes in N Since all the transformations for non rewrites can only create a new A rewrite node there will be no connections to A rewrites in the result connections can only involve nodes in the tree before the transformation Suppose N is a rewrite node Again there are no connections to rewrite nodes in N However there may be connections to N itself Since the AB Equiv Implics and Equiv Disjs transformations are destructive the node N may become a A rewrite if it was not already In these cases we have pushed the connections from N to the child of N We can see the child of N is not A rewrite nodes by examining the transformation rules N x 154 14 MERGING Remark about Subst Subst rewrites are processed further dur ing the CLEANUP ETREE stage in the function cleanup rewrite node de scribed in section 8 8 Cleanup Etree The code for CLEANUP ETREE is in the file mating merge eq lisp This terribly complicated procedure comes after merging because we assume that all exp vars and skolem terms have been removed leaving just ordinary wfts First a general description of the procedure 1 At each expansion term normalize it and reduce superscripts on the bound variables and make a new expansion which is a copy a remove unnecessary lambda norm steps b make the leaves the same name so mat
179. fore are left without having a proper type We must however be able to figure out the type of any given bound wff Thus each binder carries the information about the type of the scope the bound variable and the resulting bound wff with it See Section 5 for more details 33 34 5 REPRESENTING WELL FORMED FORMULAE The list below introduces syntactic categories of objects known to the parser only which are not legal in wffs themselves pmprsym Polymorphic Primitive Symbol pmprsyms are the STANDS FOR property of pmpropsyms but cannot appear in gwffs themselves Examples would be PI or IOTA pmabbsym Polymorphic Abbreviating Symbol pmabbsym are the STANDS FOR property of pmabbrevs but can not appear in gwffs themselves Examples are SUBSET UNION or IMAGE The following categories are the terminal objects of proper wffs The parser may not produce a formula with any other atomic in the Lisp sense object then from the list below logconst Logical Constants For example AND OR IMPLIES NOT FALSEHOOD TRUTH def logconst and type 000 printnotype t infix 5 prt associative t fo single symbol and mhelp Denotes conjunction propsym Proper Symbols For example P lt 0A gt x lt A gt y lt A gt Q lt OB gt x lt B gt are proper sym bols after parsing VaVy Poaaty A Qoa x This example demon strates part of the parser Since in a given wff a proper symbol may appear with more
180. free in any hypotheses z y A 4 JyA Ct RuleC C tt where x is not free in any hypotheses t x A f EGen JzA f A tt IDisj L IDisj R AVBt A B t t Conj AABT J Absurd S l Al Al Lf Lf Bt Indirect NegIntro Deduct At AA f ADB 3 NPROOFS TO EPROOFS 207 zar NegElim C tt Al B AVB Ct Ct Cases Ct TPS also has rules Cases3 and Cases4 which may be used to eliminate disjunctions with three or four disjuncts resp Such rule applications are replaced by iterations of the binary Cases rule in a preprocessing step using expand cases Next the basic rules which allow one to infer extraction deductions AAB AAB Conj Conj Al Bl VrA UI t a A ADB At MP By Notice that hypothesis lines are always considered extraction deriva tions Such lines may be justified by any of the following Hyp Choose Assumenegation Casel Case2 Case3 Cased We need a coercion rule as every extraction is a normal derivation At coercion A Ina TPS natural deduction style proof this coercion step will not usually be explicit Instead a single line will be given the property of being a coercion in which case we know it has both annotations and f and that these annotations were assigned in a way consistent with the coercion rule above Often when interactively constructing a natural deduction proof in TPS one finds that a planned line i
181. ftree proof of the lemma In this case the lemma does require a recursive definition so the function make clos setvar lemma negf is called If INCLUDE INDUCTION PRINCIPLE is T we need a strong form of the Knaster Tarski Theorem in which we know the set u is the least pre fixed point The function make knaster tarski leastfp lemma is called to construct an ftree proof for this lemma Otherwise make knaster tarski lemma gives an ftree proof of the simpler version The function make clos setvar lemma negf 0 uses the Knaster Tarski lemma to prove the set exis tence lemma Now let us examine each of these steps in more detail 1 make min inv princ Given v the paths LF1 LF3 LF2 and the banned variable information w NIL use the type of v to create a formula Vg vtz D gt InvP where InvP is formed by make min inv princ 2 Inv P needs to be positive with respect to v so that we can apply the Knaster Tarski Theorem later e make min inv princ 2 Given the bound list z1 x the atomic formula vzl vx as well as the information above form a disjunction of inversion principles corresponding to each path LF3 LF2 InvP Vv Inve where each InvP formula positive with respect to v is con structed by a call to make min inv princ 3 e make min inv princ 3 Given one of the constraints con struct equations and a substitution sending some banned vari ables those occurring as an argument of the main literal of
182. g 108 treemerge Function 137 true 116 TRUEFN Function 68 TRUTHP 116 TRUTHVALUES HACK Flag 116 136 TYPE IOTA MODE Flag 45 TypeAssoc Data 83 TYPESUBST Command 97 TYPESYM Argument Type 67 uni term Data 120 unif match lisp File 165 unification tree Data 120 UNIFY VERBOSE Flag 16 164 UNNEEDED NODE P Function 187 INDEX unneeded node p Function 137 141 UPDATE PLAN Function 104 UPDATE RELEVANT Command 92 93 UPWARD UPDATE NATREE Function 215 USE EXT LEMMAS Flag 132 use tactic MEapr 199 utilities 13 VAR 117 Vertical Paths 61 VPD FILENAME Flag 82 VPDiag Function 48 VPFORM JFORM FILE STYLE PRINTTYPES BRIEF VPFPAGE 61 VPForm Function 48 VPFORM TEX PREAMBLE Flag 64 upforms lisp File 14 55 VPFPage Parameter 57 VpfPage Parameter 62 wffrec 70 Width Data 56 XTERM 246 XTERM Style 49 XTERM ANSI BOLD Flag 19 sterm bold font Function 18 zterm normal font Function 18 sterm lisp File 18 19 y or n p Function 13 261
183. g and resetting this flag resets the default tactic mode Ideally a tactic operating in auto mode should require no input from the user while a tactic in interactive mode may request that the user make some decisions e g that the tactic actually be carried out It may be desirable however that some tactics ignore the mode compound tactics those tactics created by the use of tacticals and other tactics among them One may wish to have tactics print informative messages as they operate the flag tactic verbose can be set to T to allow this to occur and tactics can be defined so that messages are printed when tactic verbose is so set Each tactic should call the function tactic output with two arguments The first argument should be a string containing the information to be printed and the second argument T if the tactic succeeds and NIL otherwise tactic output will depending on the second argument and the current value of tactic verbose either print or not print the first argument 2 SYNTAX FOR TACTICS AND TACTICALS 195 2 Syntax for Tactics and Tacticals The TPs3 category for tactics is called tactic The defining function for tactics is deftactic The variable auto global tacticlist contains a list of all tactics Each tactic definition has the following form deftactic tactic lt tactic use gt lt tactic defn gt lt help string gt with components defined below tactic use nat ded nat etree etr
184. g A gt A T I2 gt Aj As In all these rules it is important that the formulas appear in the posi tions as indicated in the diagrams above The focus rule gives us the only way to reorder the formulas of the sequent This forces us to do some tedious shuffling in some places but makes it easier to perform recursion on the se quent derivations since we have a very good idea of how the rule application looks The sequent calculus is similar to the ftree representation of expansion trees see section 1 4 and the file includes a function cutfree ftree seq to ftrees which translates a cut free sequent calculus derivation of gt A to two lists of ftrees and A and a list of connections M For each wif A T there is a correponding positive ftree F I with shallow formula A For each wff A A there is a correponding negative ftree F A with shallow formula A The list of connections M gives a complete mating for the ftree A T gt A In particular a cut free sequent calculus derivation of gt A will be translated into a negative ftree with shallow formula A and a complete mating M Regarding this translation to ftrees the names of the logical and rewrite rules correspond to the construction of the corresponding ftree The weaken and focus structural rules are relatively easy to handle Applications of merge require the use of a merge algorithm for ftrees in the file ftrees The init rule corresponds to two mate
185. gative leaf LEAF1 p z We call make clos setvar ind negf 2 with Vw ptw D pt Sw A pt0 dw zt SwA piu Vv zi 0 and LEAF 1 to get two positive ftrees Pi and PE We return the positive ftree PF and the negative ftree IMP L N PF LEAF1 make clos setvar ind negf 2 We start with n constraints given by paths and banned occurs and two wffs one wff1 of the form Cy A A Chn and the other wf 2 of the form Di Z V V Dr We consider each constraint WT gt i conjunct C and disjunct D We distinguish between variables z W which occur as arguments in A and those w W which do not When z WV occurs as an argument in A the corresponding argument in the shallow formula of LEAF1 is a variable a created in the previous step Let 6 psi z assoc be the substitution sending 0 z a for each such z and a Those w WV which do not occur as arguments are stored on W This information is passed to make clos setvar ind negf 3 which returns two positive ftrees PFp and PFo and returns CONJ L XN PFo CONJ X N PFo x PFc 188 16 SET VARIABLES and DISJ Va N PFp DISJ L N PFp om L PFp In our example the two constraints are of the form v0 and wut w gt v S w Neither 0 nor S w is a variable in the corresponding W s so 6 is empty Y is NIL in one case and w in the other e make clos setvar ind negf 3 Given two wff s C wff1 and D wff2 and Y
186. gn are its arguments Always call this first since any given argument may be a label LSYMBOL Q gu ff tests for a logical symbol This could either be a variable constant or abbreviation This must come after the test for label but does not assume anything else There are several subtypes of lsymbol which assume that their argument is a Isymbol and must be called in the following order LOGCONST Q gu ff a logical constant which must have been declared with DEF LOGCONST 72 7 WELL FORMED FORMULAE OPERATORS PROPSYM Q gwff a proper symbol that is something that has not been declared a constant or abbreviation PMPROPSYM Q gwff a polymorphic proper symbol higher order mode only PMABBREV Q gwff a polymorphic abbreviation higher order mode only ABBREV Q gwff an abbreviation BOUNDWFE Q Test whether the wff starts with a binder of any type and assumes that we already know that it is neither label nor a lsymbol in Lisp terms it must be a CONS cell Access the bound variable with CAAR the binder with CDAR the scope of the binder with CDR Construct a new bound formula with CONS CONS bdvar binder scope T This is the otherwise case i e we have an application Access the function part with CAR the argument part with CDR Construct a new application with CONS function argument Remember also that all functions and predicates are curried Examples of Wffops The following examples are taken from
187. gt LIT1 These look the same but they are quite different 4 Defining an EDOP An edop does not define an operation on wffs it simply refers to one Thus typically we have a wffop associated with every edop and the edop inherits almost all of its properties from the associated wffop in particular the help the argument types the applicable predicates etc A definition of an edop itself then looks as follows enclose optional arguments DefEdop lt name gt Alias lt wffop gt Result gt lt destination gt Edwff Argname lt name gt DefaultFns lt fnspeci gt lt fnspec2 gt Move Fn lt fnspec gt MHelp lt comment gt In the above definition the properties have the following meanings ALIAS This is the name of the wffop this edop refers to It must be properly declared using the DEFWFFOP declaration RESULT gt This provides part of the added power of edops destina tion indicates what to do with the result of applying the wffop in ALIAS to the arguments destination can be any of the following omitted If omitted the appropriate print function for the type of result returned by the ALIAS wffop will be applied to the result EDWFF This means that the result of the operation is made the new current wff edwff in the editor EXECUTE This means that the result of the operation is a list of editor commands which are to be executed This may seem strange but is actually very u
188. gtypes string argnames prefix arghelp Marker string mainfns monitor chk mhelp Prints out the given string every time the monitor is called followed by the place from which it was called defun monitor chk string setq current monitorfn monitor check setq current monitorfn params string setq monitorfn params print msg Note that this accepts a marker string as input from the user other mon itor functions may look for a list of connections or flags or the name of an option set it may be necessary to define a new data type to accommodate the desired input It then calls a secondary function which in this case needs to do very little further processing in order to establish the three parameters which are required for every such function current monitorfn contains a symbol corresponding to the name of the monitor function current monitorfn params contains the user supplied parameters in any form you like since your func tion will be the only place where they are used and monitorfn params print contains the name of a function that can print out current monitorfn params 95 96 10 THE MONITOR in a readable way for use by the commands monitor and nomonitor The latter should be set to nil if you can t be bothered to write such a function 2 The Breakpoints In the relevant parts of the mating search code you should insert break points of the form if monitorflag funcall symb
189. h argument The left hand side contains the infix operator if one has to be printed in front of the argument the right hand side contains the argument itself Quantifiers are regarded as single applications where the left hand side is the quantifier plus the quantified variable while the right hand side is its scope Consider the following ex amples A B will be translated to aplicnlist lt gt lt A gt lt EQUIV gt lt B gt ANBACAD will be translated to aplicnlist lt gt lt A gt lt AND gt lt B gt lt AND gt lt C gt lt AND gt lt D gt EVF will be translated to aplicnlist lt gt lt E gt lt OR gt lt F gt YVr G will be translated to aplicnlist lt FORALL X lt I lt G gt where lt x gt denotes the PPlist corresponding to the subformula x and lt gt stands for the empty PPlist NIL 0 NIL 0 A generalized character genchar is defined to be an arbitrary non control ASCII character the number of an ASCII character in parentheses or another generalized character list in double parentheses When an ASCII character is printed it is assumed that the cursor advances one position while everything in the sub gencharlist is assumed not to appear on the screen or in the document after being processed by SCRIBE An aplicnlist with the structure aplicn MARKATOM signals that the aplicn is the internal representation of a logical atom For exa
190. h of the window End sending the width of the window and start sending the height of the window End sending window information for a window with small fonts End sending window information for a window with big fonts Clear the contents of a window Close a window Change the color This should be followed by another byte to indi cate the color For now this third byte can be 0 black 1 red 2 blue or 3 green 1 Generating the Java Menus There are two categories in the TPs3 lisp code for menus menu and menuitem Everytime a programmer adds a command mexpr flag or top level command a corresponding menuitem should be defined This menuitem should have a parent menu to indicate where the item lives The lines Java menu code is in the file TpsWin java between the comment Menu Code BEGIN and Menu Code END 248 22 RUNNING TPS WITH AN INTERFACE When you have added or changed menus or menuitems in the lisp code and want the Java interface to reflect these changes perform the following steps 1 Within Tps3 call the command generate java menus This will prompt for an output file e g menus java This command will create an output file with Java code which should be inserted into TpsWin java Delete the code between Menu Code BEGIN N and Menu Code END in TpsWin java Insert the contents of the output file of generate java menus e g menus java into TpsWin jav
191. he function LexScan 84 7 WELL FORMED FORMULAE OPERATORS GroupScan now operates on LexList in order to construct the GroupList This function has no arguments and uses a special variable called LexList to communicate with recursive calls to itself GroupScan is also responsible for building the TypeAssoc list InfixScan converts a GroupList into a PreWff This requires using the standard infix parser MakeTerm is used to build the prefix subformulas of the input Now that all logical items appear in their final positions the actual types of polymorphic abbreviations can be determined This is the job of FinalScan This function takes a PreWff and returns with a WFF This is not a very efficient algorithm A few of the passes could be joined together and a few might be made more efficient by using destructive changes The parser however is rather easy to upgrade CHAPTER 8 Help and Documentation 1 Providing Help When the user types the command HELP object TPS3 will first try to determine which category object is in it may be in several in which case it will produce a list of categories and then print the help for each separately Recall from the entry about categories section 2 that each category has mhelp line and mhelp fn properties The mhelp line is a short phrase that describes each object in the category for example the category PMPROPSYM has the mhelp line polymorphic proper symbol The mhelp fn is a func tion
192. he single line context whatever in the code all this does is to set a variable current context to whatever Here is a sample use of defcontext defcontext tactics short id Tactics order 61 92 mhelp Tactics and related functions The only property which is not immediately self explanatory is order this is used to sort the contexts into order before displaying them on the screen or in manuals Contexts are used in the facilities guide for example to divide chap ters into sections For example the line context unification occurs prior to the definition defflag max utree depth of the flag MAX UTREE DEPTH in the file node lisp and so this flag occurs in the section on unification in the chapter on flags in the facilities guide To see the contexts into which the commands for a given top level are divided just use the command at that top level Look at global contextlist in TPS3 to see all the contexts 4 Flavors Some TPS structures in particular all expansion tree nodes expan sion variables skolem terms and jforms are defined as flavors see the file flavoring lisp for the details These structures have many attached proper ties which allow wffops to be used on them as though they were gwffs for example the flavor exp var in etrees exp vars lisp has the properties type lambda gwff type exp var var gwff gwff p lambda gwff declare ignore gwff T which state that the type of an exp
193. he translation code can also be explicitly called from the EXT MATE top level using the command ETREE NAT There is an algorithm described in Chapter 7 Section 9 of Chad E Brown s thesis cf Bro04 which translates extensional expansion proofs to extensional sequent derivations This algorithm is not implemented as part of TPs3 but the same ideas are used for the algorithm translating from extensional expansion proofs to natural deduction proofs CHAPTER 19 Library The library commands are documented in the user manual A library can currently only occupy one directory i e subdirectories may not be used although users are given the ability to refer additionally to a common directory of basic definitions by using the BACKUP LIB DIR flag Many library commands are essentially written as two copies of the same function the first of which checks the default library directory and the second of which checks the backup directory The second piece of code is surrounded by unwind protect commands in order to make sure that the DEFAULT LIB DIR and BACKUP LIB DIR flags always end up correct Users may not write to the backup directory The index for each library directory is stored in the libindez rec file in that directory this file that is read every time the directory is changed Objects are removed from the library by deleting them from the appropriate lib file and removing their entry from the libindez rec file This may result in a
194. hem in the future Examples are such functions as msg and prompt read see the facilities guide for a complete list There aren t really very many utilities at the minute although it would be useful if more were defined since then we could avoid duplicating code in different places So if you write a useful macro or function foo or discover one already written please add a utility definition next to it in the code This should look like defutil foo Form Type function Keywords jforms printing Mhelp Some useful words of wisdom about the function foo Form Type should be either function or macro Keywords can be any thing you want since it is currently ignored by TPS3 Mhelp is of course a help message Note if your useful function is actually an operation on wffs it should be defined as a wffop or wffrec recursive wffop rather than as a utility utilities are really intended to be functions that are useful to TPs3 programmers but which do not fall into any other TPS3 category 6 Overloading Commands There are certain symbols in TPs3 that been overloaded that is they have been defined to have more than one meaning they may be simultaneously a matingsearch command review command and unification command This is done so that same symbol can have similar effect in different top levels For example LEAVE should leave the current top level as opposed to having a different exiting command for each top level which
195. here are some restrictions on events that should be respected if you plan to use REPORT to extract statistics from the files recording events Most importantly No two events should be written to the same file If you would like to record different things into the same file make one event with one template and allow several kinds of occurrences of the event For an example see the event PROOF ACTION below 1 1 Defining an Event If you are using ETPS it is unlikely that you need to define an event yourself However a lot of general information about events is given in the following description Events are defined using the DEFEVENT macro Its format is defevent lt name gt event args lt argl gt lt argn gt template lt list gt template names lt list gt signal hook lt hook function gt write when lt write when gt write file lt file parameter gt write hook lt hook function gt mhelp lt help string gt event args list of arguments passed on by SIGNAL EVENT for any event of this kind 235 236 20 TEACHING RECORDS template constructs the list to be written It is not assumed that every event is time stamped or has the user id The template must only contain globally evaluable forms and the arguments of the par ticular event signalled It could be the source of subtle bugs if some variables are not declared special template names names for the individual entries in the template These names
196. iation BA98 is used modify dual rewrites is called see section 3 Then connections between nodes no longer in the tree are removed from the mating e prune unmated branches is called If MERGE MINIMIZE MATING is set to T this function removes children of expansion nodes which are not needed to have a complete mating The function also calls 135 136 14 MERGING replace non leaf leaves on the etree which replaces empty expan sion nodes with leaves Note leaf p returns T on any node that has no kids except true and false nodes See section 4 e If the skolem method determined by SKOLEM DEFAULT used is not NIL then subst skol terms is called This replaces terms such as X MN with skolem terms SK whenever there is a skolem term SK with TERM slot X MN See section 5 e If the top of the tree is a conjunction whose first child is an ADD TRUT H rewrite then delete the conjunction leaving only the sec ond child as the expansion tree See TRUTHVALUES HACK and ADD TRUTH e If REMOVE LEIBNIZ is set to T then Leibniz rewrite nodes are removed This is a somewhat complicated process based on an algorithm described in Frank Pfenning s thesis Pfe87 See section 6 e subst vars for params is called This replaces skolem terms with the variable bound by the corresponding quantifier if this is possible If this is not possible then we replace the skolem term with the value of its PARAMETER slot e rewrite nodes are raised ov
197. ich may be stored with the etree It is not clear where or whether that jform is used it might be part of the interactive system msearch does the search It returns dup record ms90 3 mating unif prob unif prob represents the solution to the unification problem perhaps as a substitution stack This triple is then handed to the processes that translate things back to an expansion proof call merging and then translate to a natural deduction proof msearch looks at the flag order components read the help message for this flag for more information Each literal is a jform One of its attributes is a counter which gets adjusted to count how many mates that literal has this is compared with max mates The current jform being worked on is essentially represented as a stack which is passed around as an argument Indices are associated with 126 13 MATING SEARCH outermost variables which are implicitly duplicated These indices are also associated with literals in the scope of these quantifiers to keep track of what copy of the literal is being mated It is only when unification is called that these indices are actually attached to the variables to construct the terms unification must work on The functions check conn and conn unif p in the file ms90 3 path enum and related functions in that file may be relevant here The original code only created literals for leaves of the etree However it is possible to mate arbitrary nodes of an etree
198. ics is the print depth pdepth which is equal to the furthest extension of the formula to the right if printed with the above algorithm Whenever the pdepth is greater than the remainder of the line the indentation will be minimized to two spaces This is most useful if special characters are not available for example if V is printed as FORALL 3 4 Pretty Printing Functions Most of the functions used for the first phase of pretty printing i e for building the PPlist are already de scribed in section 2 6 The internal flag PPVirtFlag controls whether func tions like PrintFnTTY will actually produce output or create a PPlist Here it is now of importance what the different printing functions return some thing that was completely irrelevant for direct printing The general schema can be described as follows PrintWffPlain and PrintWffScope return a PPlist If called from PrintWff these PPlists are assembled to an aplicnlist and returned In this case PrintWff returns an aplicnlist The lower level functions PrintfnDover and PrintFnTTY return the gencharlist which contains the characters that would be printed in direct mode Note that therefore PrintWff will sometimes return a gencharlist instead of an aplicnlist These two are interchangeable as far as the defi nition of the PPlist is concerned and can hence be treated identically by PrintWffPlain which constructs a PPlist from them The special parameters PPWfflist and PPWfil
199. icts between different defaults for the same argument type in different functions Hence GetFn never computes the default IMore properties may become useful so the Deftype macro allows arbitrary property names Possibilities here include EdFn for editing this argument type or OutputFn to be able to read back a data object of the specified type 2 ARGUMENT TYPES 31 2 1 List Types The macro deflisttype defines a list from an exist ing type deflisttype filespeclist filespec This takes an existing type filespec and produces a type of lists of filespecs It is also possible to specify other properties the same properties as for deftype in which case these properties override those of the original type This is typically used to give the list type a different help message from the original type 2 2 Consed Types The macro defconstype defines a type as a cons of two existing types defconstype subst pair gvar gwff mhelp Means substitute gwff for gvar This takes two existing types gvar and gwff and produces a type subst pair of consed pairs gvar gwff It is also possible to spec ify other properties the same properties as for deftype in which case these properties override those of the original type This is typically used to give the cons type a different help message from the original type CHAPTER 5 Representing Well formed formulae 1 Types typeconstant Type Constant An identifier with
200. ies of the dev swapspace so that you will use the fast swap space before the slow one New swap space takes effect right away If you already have fs swapspace on all disks you can highlight the one you want to change and then choose Modify Swap Space from the Actions menu Increase the size as you want Modifications only take place after the next reboot but it is not necessary to reboot right away as it is for the heap and stack space 3 3 3 Internal Limits in Lisp As if all that wasn t enough your version of Lisp may also have some constraints on how large it can grow 1 CMU Lisp has no such limits as far as I know 2 Lucid Lisp has them and they are user modifiable type room t into a Lucid TPs3 to see what the current settings are Look for Memory Growth Limit if it seems too small type for exam ple change memory management growth limit 2048 into the TPs3 to allocate 128Mb 2048 64kb segments You can also make this permanent by adding lucid change memory management growth limit 2048 to your tps3 ini file Other parameters besides the overall size limit can also be changed see the Lucid manual for details Allegro Lisp also has a limit but in this case it is set at the ini tial building of Lisp Here you ll have to retrieve the build direc tory for allegro which is afs cs misc allegro build followed by the name of your system We have a copy of this on tps 1 called allegro4 2hp_ huge but it require
201. ilities 1 If the string gt COMMAND is received the next string is a com mand TPps3 should run or the response to a prompt if popups are disabled All the input does with this command string is attach it to the COMMAND symbol as the property RESPONSE The main process will accept this string as input from linereadp since when running through the interface a function read line sym will wait for this RESPONSE property to be set An exceptional case is when the string after COMMAND is INTERRUPT In this case the main process is killed and a new process with a top level prompt is created 2 If the string RIGHTMARGIN is received the next string received should be an integer giving the new value for the flag RIGHTMAR GIN This allows the interface to change this flag without having to interrupt another command that may be running 3 When popups are enabled some other string starting with PROMPT is received In this case the next string is put on the RESPONSE property of this PROMPT symbol This should be a response to a particular popup prompt So the code in external interface lisp handles receiving input from the interface The other problem is that of sending output to the interface This is handled by setting standard output to the socket stream and chang ing the STYLE to ISTYLE interface style defined in interface style lisp This style is similar to the XTERM style except with mor
202. ill later have a b z prettify free legal p This checks if it is legal to extend 0 to include O y y First we check to make sure 6 y is not already defined and that y is not in the codomain of 6 used frees since we want 0 to be injective Next to ensure condition will hold we make sure there is no bound z with a z y and z in the list free must avoid for y prettify bound legal p This checks if it is legal to extend a to include a z z First we check to make sure a z is not already defined Of course we do not mind if many bound variables are mapped to the same variable because this is often how we make the proof pretty so we do not need to check the codomain of a We must make sure that z is not on the list in the property not alpha image for z If it is then there is some free or bound x which is sent to 2 and occurs free in some subwff where it would be captured 9 PRETTIFY 161 by a binder Bz if this binder were renamed to Bz Also we must make sure there is no bound 6 on the list in the property bound must avoid for z such that a b z In such a case there would be an occurrence of z which would be captured by a binder for b upon renaming These checks ensure condition will hold prettify free bound legal p Checks if we can send both y and z to a variable v This involves a bit more checking than just checking that both commitments are independently legal Comments in the code exp
203. implement it roughly speaking as a requirement that no substitution term for an expansion variable should contain any of the selections which occur below that variable Extra slots called exp var selected on expansion variables and universal selected on universal jforms are used to record all of the selections below each quanti fier This is used in the unification check and more crucially in path focused duplication since skolem terms are stripped out of the jform this is our only way to remember where they were See the section on selected variables for more information about checking acyclicity of the dependency relation In SK1 and SK3 duplicating a quantifier above a skolem term produces a new skolem term consisting of the same skolem constant applied to dif ferent variables In NIL we obviously can t use the same skolem constant everywhere consider EXISTS X FORALL Y P X IMPLIES P Y if we persistently select the same Y every time we duplicate X the proof will fail This has two major consequences e Path focused duplication has to be changed We can no longer du plicate implicitly by changing the name of the quantified variable we must now make a copy of the entire scope of the quantifier and descend into it renaming all the selections as we go These copies are stored in a chain using the universal dup slot of the jform so the universal dup of the top jform contains the first copy whose universal dup contains the second
204. ind of heuristics used 2 DEFAULTS FOR LINE NUMBERS A SPECIFICATION 101 Below we give a description of the tasks to be done by a function LINE NO DEFAULTS which is called during the application of every inference rule in interactive mode We will set the stage by giving some not necessarily exhaustive examples of what meaning to assign to the data structures and what output to expect from the function 2 1 The support data structure At each stage in a proof we have associated with it a support structure which for any given planned line pline tells us which deduced lines dlines we expect to use in the proof of the pline Thus the support structure is of the form p1 di1 Say dix arene Pp dpi vai dpxp One may assume the following 1 The p are pairwise distinct 2 The dj are pairwise distinct for every fixed i and 1 lt k lt zi 3 For each i dig lt pi for all 1 lt k lt zi 4 The planned lines p are ordered such that the ones the user is expected to work on first appear closer to the front In particular p is the planned line worked on most recently 5 Similarly for a given i the dig are ordered such that the one the user is expected to use first appear earlier 3 4 With each rule definition there will be a description of how the support structure changes This is given as two support structure templates using the name given to the lines in the rule specification 2 2 Examples The e
205. ing systems like Unix it is easier to use utilities such as fgrep and gnu emacs tags vide infra to search for occurrences of symbols 3 MAINTENANCE 5 Each TPs3 source file should contain certain common elements First is a copyright notice whose purpose is self explanatory just copy it from another source file Make the copyright date current for any new code The first line of the file however should be something like 333 Mode LISP Package CORE The gnu emacs editor will use this line to put the buffer in Lisp mode and if you are using one of the gnu emacs interfaces to the lisp it will use the package information appropriately See the documentation for such gnu emacs lisp interfaces The first non comment form of each source file should be an in package statement to tell the compiler what package the file should be loaded in Re cent implementations of lisp will object if there is an in package command anywhere else in the file Next the TPs3 module of the file should be indicated by a part of statement like part of module name This should match the entry given in defpck lisp Don t forget context statements Basically they just reset the variable current context which is used by other functions to organize the docu mentation and help messages 3 Maintenance 3 1 Porting TPs3 to a new Lisp As discussed above the lisp implementation dependent parts of TPs3 are confined to a few files See the disc
206. ing at JZ P1 now has the form IMP1 L A Equivwf fsREW qo At Equivuf fsREW qo B7 ap LEAF Ag Af LEAF Ao Bo If A B is the same as Ag Bo up to a conversion we replace REW1 with LEAF2 Otherwise replace REW1 with a subtree of the form A REW1 A B 4 LEAF2 Ap Bo or Equivwf fs REW1 A B7 4 LEAF Ap Bo Since LEAF1 is no longer in the tree we must delete any connection LEAF1 LEAF3 We replace each such LEAF3 with Refl REW3 Ao Ap 4 TRU E1 and delete any connection with LEAF Consider the following examples which arise in a proof of THM15B using mode MODE THM15B C Leibniz REW5 FX X SKOL5 Vq q F X D qX lQ IMP7 A N LEAF37 Q F X LEAF38 QX with connections LEAF 28 LEAF109 LEAF38 LEAF33 LEAF3T LEAF32 LEAF29 LEAF 22 LEAF21 LEAF108 LE AF105 LEAF101 LEAF104 LEAF100 LEAF94 LEAF93 6 REMOVE LEIBNIZ 147 becomes LEAF38 F X X removing the connection LEAF37 LEAF32 Leibniz REW15 Aw F w X F Au u F X F7 SKOLS Yq q Aw F wX F Dq Au u FX F FRS IMP13 L X A REW19 Q A w F wX F A REW20 Q Au u F X F i LEAF93 Q F F X LEAF94 Q8 F F X with connections LEAF94 LEAF93 LEAF 38 LEAF 33 LEAF29 LEAF 22 LEAF21 LEAF 108 LEAF 105 LEAF 101 becomes A REW38 Aw F w X F Au u F X FO L Refl REW39 F FX F F xX L TRU E37 removing the connection LEAF94 LEAF93 4 cl
207. ing still holds 2 Remove original expansion In reality we just create a whole new expansion tree not sharing with original tree at all The main functions described below are 1 cleanup etree 2 cleanup all expansions 3 cleanup expansion 4 cleanup rewrite node 1 cleanup etree This is the function called by merge tree real It calls cleanup all expansions to rebuild the expansion tree cleaning up along the way cleanup all expansions Despite the name this actually builds a completely new copy of the etree with special attention paid to expansion and rewrite nodes The arguments are e etree the old etree node e shallow the shallow formula for the new node which may be only A equal to the old shallow e parent the parent for the new node being created e lambda normal p a boolean indicating if shallow is normal For each expansion term and corresponding kid call cleanup expansion to obtain the new A normal term and new kid For rewrite nodes call cleanup rewrite node 3 cleanup expansion We normalize the expansion term t to ob tain t If the new shallow formula is Vu A or Ju A then the new shallow newshallow for the kid is t uJA If newshallow is not A normal then rewrite nodes may need to be included between the expansion node and this kid Then we recursively call cleanup all expansions on the kid with the new shallow formula B 4 cleanup rewrite node There are cases for the different kinds of
208. ing will be printed regardless of its depth PrintDepth has to be an integer It is initialized to 0 The most useful application of this parameter is in the formula editor where one usually does not like to see the whole formula AtomValF lag This flag should usually not be touched by the user If it is true under each atom its value will appear AllScopeFlag This flag should be NIL most of the time If it is T brackets and dots will always be inserted i e no convention of associativity to the left is followed The precedence values of infix operators are also ignored It can be forced to T by calling the function PWScope GWff 2 4 Functions available There are of course a variety of occasions to print wffs For example in plans as lines after the P or PP command in the editor etc Associated with these are different printing commands given by the user Some of these commands override globally set parameters 48 6 PRINTING AND READING WELL FORMED FORMULAS or flags Internally however there is only one function which prints wffs This function PrtWff is called whenever formulas have to be printed The various flags controlling the way printed formulas will appear will either be defaulted to the global value or be passed to this function as arguments The general form of a call of PrtWff is as follows PrtWff Wff Parameter Value Parameter Value Before the actual printing is done Parameter Parameter will be set
209. ingle symbol subset defn lambda P OA lambda R OA forall x P x implies R x binder Variable Binders For example V J A 41 See the section below label A Label referring to one or more other wfts For example AXIOM1 ATM15 LABEL6 See Section 6 In principle the implementation is completely free to choose the repre sentation of the different terminal objects of the syntax The functions with test whether a given terminal object is of a given kind is the only user visible functions Once defined the particular implementation of the object should not be needed or relied upon by other functions It is explained more precisely what is meant by quick and slow pred icates to decide whether a given object is in a certain syntactic category in section 1 5 Here is a table of the different syntactic categories with the slow test function for it and the properties that are required or must be absent Keep in mind that the list reflects the current implementation and may not be reliable 36 5 REPRESENTING WELL FORMED FORMULAE Category Predicate Required Properties Absent Properties pmprsym PMPRSYM P TYPE TYPELIST DEFN pmabbsym PMABBSYM P TYPE TYPELIST DEFN logconst LOGCONST P TYPE LOGCONST propsym PROPSYM P TYPE LOGCONST STANDS FOR pmpropsym PMPROPSYM P TYPE POLYTYPELIST STANDS FOR a pmprsym pmabb PMPROPSYM P TYPE POLYTYPELIST STANDS FOR a pmabbsym abbrev ABBREV P TYPE DEFN TYP
210. ions 1 N x What if ap exists but is not a pline This case is delicate and perhaps frequently occurs if the user does not bother deleting some lines before backtracking after some previous mistake Here execute a PLAN AGAIN which may become smart about support lines This will make p into a planned line and we can handle it the usual way What if ap does not exist Then very likely a rule meant to be used backwards was applied in a forward way We can t do much here just ignore the relevant part of default exist completely What if ap is a variable but dip don t match anything in the current support structure This is already a special case of something discussed in the previous section What if a dig does not exist Then we must enter it as a planned line collecting d l lt k as its support lines What if a dix does exist but does not support p pi not a variable Then somehow dj was improperly erased form the supports of p Just treat dik as if it were supporting pi What if aq is a variable thus exists as a pline and matched a line number identical to a ejg Then we are closing a gap with a forward rule Do not enter the jt plan support pair into the support structure What if aq already exists as a pline In this probably very rare case we are reducing the proof of one planned line to the proof of another planned line Add the ejk as additional support lines als
211. is con straint the constraint has no literals in this constraint other than the main literal LF1 make min inv princ 6 makes no con tributation For the constraint LF3 LF2 InP s x Sw I nop where T nop is constructed by make min inv princ 6 using the extra literals of the constraint in this case LF2 e make min inv princ 6 Constructs a conjunct of formulas for each extra literal of the constraint In this case there is the one literal LF2 giving InP as v w In the general case when a literal LF does not contain the set selected variable vsel vt ToP R will be of the form A V I mePe tt ST gt where A is the shallow formula of LF or its negation if LF is negative This will also be the form if LF is a positive literal only containing v at the head The more complicated case is when vt occurs inside the body of the literal Suppose A v is the shallow formula of such a literal LF or its negation if the literal is negative Since we will want a formula which is positive with respect to vt we let LF LEAF InvP LF3LF2 be of a form such as Iwi Va w x D va z A Alw Of course in general ot may be any set type and x may be a list of variables See section 3 1 for a semantic description of this as a least monotone upper approximation THM2 is an example where such an approximation is necessary In the end we have constructed the inversion principle as fol lows Y
212. is for simplicity suppose we have a backward coercion Bt bcoercion B a negative expansion tree C with shallow formula C and a positive ex pansion tree B with shallow formula B By induction hypothesis we can obtain an appropriate positive expansion tree B with shallow formula B Then the algorithm returns CUT xy SN There is one more modification required in the algorithm When trans lating a hypothesis line we used the merge algorithm on expansion trees However merge is not defined on expansion trees containing CUT nodes So instead we make an explicit MERGE node with the two trees as chil dren The actual merging would be done during cut merge elimination That is if we are translating a hypothesis line A1 An F Ay then we must start with positive expansion trees A a positive expansion tree A and a negative expansion tree C Instead of associating the hy pothesis line with the merge of Aj and A we now associate the line with the node MERGE x N At At The most interesting case is eliminating a CUT between a selection and an expansion node There may be multiple expansion nodes and the acyclic ity of the dependency relation may affect the order in which the expansion terms are processed We were trying to reduce such a CUT by doing a global merge of two modified expansion trees Suppose the expansion tree is Q and contains a CUT node of the form CUT x 4 N C SEL Yx A x EXP Yz A x la
213. is of type B wffop typelist A B where A and B may be any types applicable p lambda var term inwff occurs declare ignore inwff occurs type equal term var mhelp defwffop substitute types 70 7 WELL FORMED FORMULAE OPERATORS argtypes typealist gwff resulttype gwff argnames alist gwff arghelp alist of types gwff mhelp Substitute for types from list old new in gwff 1 3 Defining Recursive Wffops The category wffrec is for recur sive wff functions Such operations are defined with the defwffrec function they have only three properties ARGNAMES MHELP and MULTIPLE RECURSION The point of this is that we needed a way of saving the ARGNAME in formation for functions which use an APPLY LABEL but are not wffops themselves These are defined as wffrecs Some examples defwffrec gwff q argnames gwff defun gwff q gwff cond label p gwff apply label gwff gwff q gwff lsymbol p gwff t atom gwff nil Cand boundwff p gwff gvar p caar gwff gwff q cdr gwff Cand gwff q car gwff gwff q cdr gwff defwffrec wffeq def1 argnames wff1 wff2 varstack switch multiple recursion t the function wffeq def1 is pages long so it s not quoted here Look in file wffequ2 lisp for details 1 4 Defining a Function Performing a Wffop There are some necessary restrictions on how to define proper wffops other conventions are simply
214. isp mswindows Microsoft Windows 4 1 INTRODUCTION There is one feature andrew that is added when we compile TPs3 ETps for use on the Andrew workstations machines in the domain an drew cmu edu This is because on those machines lisp implementations have problems interfacing with the operating system and getting the proper home directory of a user Thus special measures are taken in this case This feature is added in the sys files for the Andrew editions There are another two features which are added in the relevant sys files for ETPS and TPs3 these are TPS and ETPS This allows programmers to specify slightly different behaviour for the two systems for example when using the editor you may have a window that shows the vpform in TPs3 but not in ETps The files which use and are principally special erp boot0 lisp boot1 lisp tops20 lisp tps3 save lisp and tps3 error lisp 2 4 TPs3 modules TPs3 source files are organized into TPS3 mod ules Basically a TPS3 module is just a list of source files in the sequence in which they are to be compiled loaded All TPs3 modules are defined in the file defpck lisp Each source file should be in some TPS3 module and that module should be indicated in the file Conceivably one might define two different TPS3 modules which had files in common but we have never done that Some files are designated as macro files in the definition of the module in defpck lisp When a
215. ists nly i1 ig and nlg j j indicating the positions of the mix formulas in A and To There are several cases to consider e Mix Formula Does Not Occur That is nly nil or nlo nil If nly is nil then we can return D Tri gt Ai t 0 t n 1 where n is the length of Ty t 0 t m 1 where m is the length of A The indices indicate that every formula is the residual of a formula from the first sequent If nlg is nil we can return D and indices which indicate all the formulas are residuals of the second sequent e INIT Suppose D is an initial sequent A gt A A must be the mix formula or nl must be nil Ideally we would like to return Do and replace the indices for the mix formula occurrences in Ig by the index t 0 indicating the positive A from D The only problem is that each occurrence of the mix formula in T is a B equal to A only in the sense that the two formulas have a common reduct with respect to A reduction and expansions of abbreviations and equalities The function ftree seq replace equivwffs replaces each such B by A in Dg We can handle D initial similarly e FOCUS WEAKEN MERGE If either 9 or Do is a focus weaken or merge step we can simply recursively call with the premiss and the positions shuffled appropriately 226 18 PROOF TRANSLATIONS e REWRITES If either D or D2 isar EQUIVW FFS Leibniz or Ext rewrite and the mix formu
216. ith a CutShort label means that none of the remaining MainFn s will be executed and the value of the THROW will be handed on to the CloseFn s 2 ARGUMENT TYPES 29 CloseFns fnspecl fnspec2 is a list of function specifications They are called in succession with the value returned by the last MainFn Even if the MainFn s were FExpr s each fnspec has to describe an Expr Dont Restore boolean determines whether or not this command will be restored if it is saved using SAVE WORK For example com mands like HELP and should not be restored Print Command boolean is used by RESTORE WORK and EXECUTE FILE which both ask Execute Print Commands this is how they know which commands are print commands MHelp This has to be a string and will be available through UserHelp and the if no QuickHelp is available 2 Argument Types At the top level of TPs3 explicitly declared argument types are available Many of the more important ones are all declared in the file argtyp lisp They can be recognized by their ArgType property value which is T Each of argument type has at least three properties GetFn TestFn and PrintFn GetFn is responsible for translating the user s value into internal represen tation TestFn tests if some object is of the given type and PrintFn makes the internal representation intelligible to the user The defining command for the category argtype is actually deftype but all definitions of argty
217. itive and negative leaves LEAF 4 and LEAF5 with this shallow formula and mate the two We return the two positive ftrees IMP 4 N LEAF5 LEAF2 and CONJ L N EQ LEAF4 4 make knaster tarski leastfp lemma CHAPTER 17 Tactics and Tacticals Modify tactics tex 1 Overview Ordinarily in TPS3 the user proceeds by performing a series of atomic actions each one specified directly For example in constructing a proof she may first apply the deduct rule then the rule of cases then the deduct rule again etc These actions are related temporally but not necessarily in any other way the goal which is attacked by one action may result in several new goals yet there is no distinction between goals produced by one action and those produced by another In addition this use of small steps prohibits the user from outlining a general procedure to be followed A complex strategy cannot be expressed in these terms and thus the user must resign herself to proceeding by plodding along using simple often trivial and tedious applications of rules Tactics offer a way to encode strategies into new commands using a goal oriented approach With the use of tacticals more complex tactics and hence strategies may be built Tactics and tacticals are in essence a programming language in which one may specify techniques for solving goals Tactics are called partial subgoaling methods by GMW79 What this means is that a tactic is a
218. iz connections remove leibniz cleanup leibniz expansions remove spurious connections check shallow formulas apply thm 146 remove leibniz nodes This is the main function It collects neg ative and positive Leibniz rewrite nodes The connection list is pre processed by pre process nonleaf leibniz connections so that any mates to nonleaf nodes strictly below a negative Leibniz rewrite is replaced by mating the leaves below the node Next the function remove leibniz is called on each negative Leibniz rewrite ee O H 142 14 MERGING node possibly changing the connection list This has the result of changing negative subtrees of the form Leibniz REW1 A B uy SEI1 Vq qA D qB qo IMP1 x N A Equivwf fsREW qo AT A Equivwf fsREW qo B7 J LEAF1 q Ag LEAF qo Bo by trees of the form AREW A B a Equivwf fsREW A By Refl REW1 C C7 TRU E1 or simply a leaf LEAF2 A B Remark The notation REW C indicates a chain of rewrites starting with shallow formula C This chain may be empty a single node or several nodes In such cases connections to LEAF1 are deleted from the connection list The function deepen negated leaves is called but this apparently has no effect unless make left side refl returns NIL and it currently always returns T 6 REMOVE LEIBNIZ 143 Finally cleanup leibniz expansions is called in order to change positiv
219. k it by entering tar tvf o dev rst8 The same list of file names should pass by These are the names of the files which are now on the tape 6 Go back to the operator and ask for the tape 12 1 INTRODUCTION 4 How to locate something Sometimes you will be looking at code and will come across a function or variable whose purpose is not familiar to you If it is not a standard Common Lisp function for which the Lisp functions documentation and apropos may be useful as well as reference books and user manuals there are three ways to find where it is defined The first method uses the gnu emacs tags mechanism Periodically we run the etags program on the lisp files in the source directory One does this by entering the TPs3 lisp directory and then running the etags program usually this is done by typing M x shell command etags lisp This generates a file called TAGS with entries for each line of code which begins with def Then you can use the gnu emacs find tag function ESC unless you ve rearranged the emacs keys to look for the first occurrence of the symbol and the tags loop continue function ESC to find the each subsequent occurrence This can be slow if there are many symbols which begin with the prefix for which you are searching or if the symbol is overloaded by defining it for different purposes e g LEAVE is a matingsearch command a review command a unification toplevel command See the gnu emacs d
220. ke each label a Lisp structure The type of the structure is the name of the flavor thus an object s type can be used to determine that it is a label of a certain flavor If one wishes to have labels be symbols nothing must be done this is the default A flavor s labels will be structures only if one of two things is declared in the defflavor The first is that the property structured appears The second is if another flavor whose instances are structures is specified to be included in the new flavor When a flavor s labels are to be structures one will usually wish to specify the printfn property so that the labels will be printed in a nice way This function must be one which is acceptable for use in a defstruct It is also required that one specify the slots or attributes the structures are to have by including a list of the form instance attributes sloti defaulti slotN defaultN in the flavor definition 42 5 REPRESENTING WELL FORMED FORMULAE 6 2 Using Labels The function define label is a generic way to create new labels of a specified flavor The function call define label sym flavor name will do one of two things If flavor name is a flavor whose labels are symbols then the property list of sym will be updated with property FLAVOR having value flavor name If on the other hand flavor name is a flavor having structures for labels then sym will be setq d to the value of the result of calling the constructor
221. ke clos setvar lemma negf 9 Function 185 make ftree setvar soln Function 174 make ftree subst Function 189 make knaster tarski gfp lemma Function 172 make knaster tarski gfp negf Function 172 make knaster tarski leastfp lemma Flag 175 257 make knaster tarski least p lemma Function 172 179 191 make knaster tarski least p negf Function 172 make knaster tarski lemma Flag 175 make knaster tarski lemma Function 172 make knaster tarski negf Function 172 make left side refl Function 142 144 145 make mating lists Function 136 make min inv princ Function 174 175 make min inv princ 2 Function 175 make min inv princ 3 Function 175 176 make min inv princ 4 Function 175 176 make min inv princ 5 Function 176 make min inv princ 6 Function 176 make min setvar lemma negf Function 175 make min setvar lemma posf Function 174 177 make min setvar lemma posf 1 Function 177 make min setvar lemma posf 2 Function 177 make min setvar lemma posf 38 Function 177 make min setvar lemma posf 4 Function 178 make min setvar lemma posf 5 Function 178 make min setvar lemma posf 6 Function 178 make passive socket Function 6 make passive socket port Function 6 Makefile File 5 MakeTerm Function 84 master eproof 116 mating Data 120 mating list 117 mating merge eq lisp File 141 154 157 162 mating merge lisp File 136 139 141 mating merge2 lisp
222. ks Because of the restrictions of IaTRX concerning special command characters such as or the help processing may need special care an alternative to the usual princ mhelp function is provided to handle such characters in laterdoc lisp as princ mhelp latex 85 86 8 HELP AND DOCUMENTATION Many specific ATfXcommands are defined in the tps tex and tpsdoc tez files in the doc lib area The User s guide describes how the processing of such manuals is done You may need to set things up properly so that the entries you are introducing are put into the index of such guides Look at the file laterdoc lisp in the lisp area to see how the indexing is done 2 The Info Category There is a category of objects called INFO which is used solely for pro viding help on symbols that would otherwise not have help messages for example the various settings of some of the flags such as PR97 or APPLY MATCH ALL FRDPAIRS You can attach a help message to any symbol foo with definfo foo mhelp Help text 3 Printed Documentation The directories with root home theorem mentioned below are on gtps home theorem project doc files dir contains information about TPs3 documentation files home theorem project doc etps tps cs mss describes how to access TPS3 in the cmu cs cell home theorem project doc etps etps andrew mss describes how to ac cess ETPS in the andrew cell home theorem project doc lt topic gt manual mss is
223. ks that tactic is a valid tactic 4 USING TACTICS 201 b If a goal has not been specified calls get tac goal c If the tactic is an atom i gets the tactic s definition for the use ii if the definition is primitive and the goal is nil return nil Goals exhausted complete nil If the goal is non nil apply the tactic to the goal iii if the definition is compound call apply tactical on the definition and the goal d If the tactic s definition begins with a tactic call apply tactic recursively using those optional arguments e If the tactic begins with a tactical call apply tactical Whenever a tactic begins with a tactical the function apply tactical is used It takes two arguments a goal and a tactic It is assumed that the tactic begins with a tactical apply tactical works as follows a Get the definition for the tactical b If the definition is primitive a lambda expression funcall the definition on the goal and the remainder cdr of the tactic c If the definition is compound i e is defined in terms of other tacticals and begins with tac lambda expand the definition substituting the arguments provided in the tactic s definition for the dummy arguments in the tactical s definition Then call apply tactical recursively d Otherwise abort returning abort as the token third value re turned 4 Validations Though the validation mechanism is in place no use is made of them in the
224. l form of B If these are contain no abbreviations they should be identical so we change the etree to be REW2 A B Refl REW3 Ay B TRU E1 Otherwise let A gt and B be the result of instantiating all def initions except equiv in A and B1 resp These should be identical so we can change the subtree to be REW2 A B 1 Equivuf fs REW3 A By Refl REW4 Ag By ep TRU FE1 Finally we remove all connections involving LEAF 1 LEAF 2 or REW1 We still have a complete mating without these connections First note that any path which would have passed through any of these nodes would have passed through all of them Now the corresponding path in the jform for the new tree must pass through TRU E17 Possible Bug The point of Theorem 138 in Frank Pfen ning s thesis Pfe87 is to remove all Leibniz selected variables qo However in this case we are not substituting for the qo so there may still be references to it in the tree It s unclear however if this causes a problem in this special case If it does turn out to be a bug probably the fix is to substitute the value of subwff for qo 146 14 MERGING b If LEAF 1 and LEAF are not mated substitute the value of subwff for qo in the etree Assume subwff has value Ax Ao x or Ax x Bo Note that this substitution automati cally A normalizes and puts A rewrites above the leaves if they are needed So the subtree start
225. la is the principal formula we can usually simply recursively call the algorithm with the premiss and the same positions This is possible because we are only main taining that the mix formulas are the same up to having a common reduct via such rewriting steps The actual rewriting is done in the INIT step Exception Nonanalytic Uses of Extensionality Suppose one occurrence of the mix formula is A and another occurrence of the mix formula is A Vq q Mo D q Nj If A is the principal formula of an Eat rewrite then A and Alvq 4 Mo D q Nol do not have a common reduct The recursion fails at such a step In general this problem occurs when one instance of equality is ex panded as Leibniz and another corresponding instance is expanded as functional or propositional extensionality The real problem is that the extensionality rules by themselves without cut do not give a complete calculus for extensional higher order logic To get a cut free extensional calculus one needs to be able to have initial se quents i e mate modulo equations and then use decomposition rules and extensionality rules to solve the introduced equations Benzmuller s thesis gives rules for handling extensionality in the context of resolution The case for sequent calculi should be de scribed in upcoming technical reports by Benzmuller Brown and Kohlhase There should also be information in Chad E Brown s thesis NONPRINCIPAL LOGICAL RULE If
226. laced in the package CORE which is used by each of the other packages The package MAINT contains functions useful to the maintainer The package AUTO contains the automatic portions of TPs3 and the package ML contains the inference rules used in the mathematical logic system The TEACHER package contains files relevant to GRADER Within each Lisp package are several TPS3 modules groups of related source files which are clumped together These are defined in the file def pck lisp The CORE package contains functions such as those dealing with wff pars ing and printing proof manipulation and operating system interfaces such as basic file operations It also contains functions for dealing with Scribe vertical paths editing windows review etc The other packages include TEACHER for functions relating to the Grader subsystem AUTO for functions relating to automatic proof procedures such as mating search ML for things specific to the Math Logic courses such as exercises and proof rules ETPS contains part of the CORE package part of the OUTLINE package and part of the RULES package The idea is that only those symbols that are needed by other packages are exported from their home package In order to specify which symbols should be exported the files core exp auto exp etc These files one for each Lisp package are loaded at the beginning of the compilation process before any code is loaded This way any package conflicts are
227. lain the check prettify bound bound legal p Checks if we can send both x and z toa variable v This involves a bit more checking than just check ing that both commitments are independently legal Comments in the code explain the check prettify identify free bound Given a free y and bound z if both are already committed do nothing If one is committed to a pretty variable and the other is not committed send the other to the same pretty variable if this is legal If neither are committed compute alternative names for each If either have a pretty alternative v which is legal for the other send both to this v prettify identify bound bound Similar to prettify identify free bound except with a bound x and another bound z e pretty var p Returns T if the var does not have a superscript e get best alt name Given a variable and a legality test finds a new legal variable to replace it If the old variable is w w h or h then the new variable will be given a name based on whether the type is of a proposition predicate relation function or individual This de pends on the values of the globals proposition var predicate var relation var function var individual var The values of some of these globals were being randomly changed by a call to random vars at the beginning of merge tree real This may make prettify bugs difficult to reproduce We have decided to comment out this call to randomvars If the old variable is anything else
228. le mating merge lisp is to delete some children of expansion nodes by checking if it is really needed in the proof Each node is checked using unneeded node p see section 2 Before returning this function calls replace non leaf leaves on the etree which replaces all empty expansion nodes with leaves Note leaf p returns T on any node that has no children except true and false nodes 5 Subst Skol Terms The function subst skol terms in the file mating merge lisp is used to replace terms in the etree which correspond to skolem terms by the skolem term itself For example the etree might contain a skolem term SK with slots e PARAMETER for example Xog e TERM for example X2 MN oBaa The shallow formulas and expansion terms in the etree might contain sub terms of the form X aaMN up to A conversion These are replaced by the skolem term SK The function subst skol terms main actually does the work It takes an argument skol terms which is an list of pairs lt term gt lt skolem term gt where each lt skolem term gt is the skolem term for a skolem node and the lt term gt is the gwff in the TERM slot of the lt skolem term gt The function tra verses each term doing the destructive replacement in the shallow formula and or expansion terms 6 Remove Leibniz The functions for REMOVE LEIBNIZ are in the mating merge eq lisp The functions described here are remove leibniz nodes pre process nonleaf leibn
229. legal instantiation for v since all the variables in V are bound Next consider the more general case in which the minimal constraint is of the form WIT gt vt 3 SET CONSTRAINTS 169 where the arguments f need not be distinct variables from WV Let n be the length of t In this case the easiest way to directly define the minimal solution is da Ar Fw 2t th A Aa SERAT for new variables z of the same type as tf and w W However in practice it is easier if we distinguish between arguments tf which are actually variables in Y and those which are not So let us write Y Z w where each z t For each j l n if t Z and t tf for k lt j then let x tf Otherwise let zf be a new variable of the same type as tf Let Eqns xf t t is not the variable zf In this case we can directly define the minimal solution as Azt Ax F0 N Eqns a NCO Clearly if actually is the list Z then this solution is azam NC as before Now consider the case in which there are several minimal constraints of the form for i 1 n Again we assume v does not occur in T so we can directly define the minimal solution First let M be the minimal solution for each of the constraints individually defined as above aTa Eqnsi a AC Then we take the union of these i to get the minimal solution for the combined constraints Rather than diving into the general case in which v may occur in T first
230. les Information on how to write new rules using the rules module is in the User Manual The chapter about ETps chapter 12 has some discussion of how default line numbers and default hypotheses are generated in the various OUTLINE modules This should correspond fairly closely to the way in which the automatically generated rules generate their defaults The same chapter also has a discussion of support transformations which illustrates the way in which rules are defined Do not edit the automatically generated files This means any file whatever lisp that has a companion file whatever rules If you edit the lisp files directly your changes will be lost if the rules are ever recompiled There are some rules which are defined directly as commands mexpr s Examples are RULEP TYPESUBST ASSERT ADD HYPS and DELETE HYPS In order for these commands to show up when LIST RULES is called the programmer should modify the definition of list rules in otl help lisp to explicitly include them For example RULEP is explicitly included by the following code when get ML rulep mainfns push RULEP gsr1 push ML RULES 2 PROP gsr1 This includes RULEP in the context ML RULES 2 PROP 97 CHAPTER 12 ETPS 1 The Outline Modules The OUTLINE modules in TPS have two main subparts the bookkeeping functions and the GO command which gives sophisticated help or constructs a proof automatically They are collected in the module
231. loop the function remove spurious connections is called This function cleans the mating and expan sion tree and returns the new mating remove spurious connections This function finds connections between nodes with shallow formula A A bad conn One of these nodes must be negative so we can deepen it if necessary to replace it with a new tree of the form Refl REW1 A A TRU E1 Then we remove this connection from the mating After simplifying the mating in this way we delete any children of expansion nodes immediately beneath Subst rewrite nodes which are not used in the mating Then we call apply thm 146 because we may have simplified some Subst rewrite node to be of the appropriate form Finally we return the connection list check shallow formulas This function takes a positive equational rewrite node REW1 with shallow formula A B an expansion node EX P1 which is a child of REW1 and an implication node IM Pi which is a child of EX P1 Suppose IM Pi has shallow for mula C D D This function checks if the D can be obtained from C by replacing occurrences of A by B If not the relevant expansion term Q must have the form Az P So we add A rewrites are added beneath the implication to make the implication node have the form IMPi iL N A REW2 A P A REW3 B z P node nodes Note that we can clearly obtain B z P formula from A z P by replacing some occurrences of A by B So the tree has th
232. lting tree has been deepened Definitions Let Q be a given expansion tree e Let Ng C Og be the set of occurrences of expansion terms in Q which are not expansion variables e Let Vg be the set of expansion variables that occur in Q Note that Vg and Ng are disjoint sets e For each expansion variable v let T C Og be the set of expansion terms in which v occurs free Note that T is always nonempty For each t Xg let T t C Og For each q Ng U Vg let B q be the set of selected variables y whose selection node is dominated by the arc corresponding to t for some t T For each t Na let S t be the set of selected variables which occur free in t The set B t is the set of banned selection variables for t Ng U Vo The following definitions depend upon a substitution 0 This corresponds to the substitution found by unification during mating search 124 13 MATING SEARCH Definitions Let 0 be a given substitution for the expansion variables in Vg Note that we allow v v for some v Vg in order to make dom Vg e For each v Vo let S v be the set of selected variables which occur free in v e Define a relation lt Do on Xo U Vg by q xo ar iff there exists a y Big N S r e Let lt Q be the transitive closure of lt D o e The substitution is acyclic with respect to Q if the relation lt Q 0 is acyclic irreflexive Remark 1 It is important to note that 0 is a substit
233. ly Generating Flag Relevancy Chapter 10 The Monitor 1 The Defmonitor Command 2 The Breakpoints 3 The Actual Function Chapter 11 Writing Inference Rules Chapter 12 ETPS 1 The Outline Modules 1 1 Proofs as a Data Structure 1 2 Proof Lines as a Data Structure 2 Defaults for Line Numbers a Specification 2 1 The support data structure 2 2 Examples 2 3 The LINE NO DEFAULTS functions 3 Updating the support structure 3 1 support Structure Transformation in the Default Case 3 2 What if 3 3 Entering Lines into the Proof Outline 4 Defaults for Sets of Hypothesis 4 1 The Algorithm 4 2 When the Algorithm is not Sufficient 4 3 Hypothesis Lines Chapter 13 Mating Search 1 Data Structures 1 1 Expansion Tree 1 2 The Expansion Proof 1 3 Relevant Global Variables 1 4 Functional Representation of Expansion Trees 1 5 Other Structures 2 Operations on Expansion Trees 2 1 Deepening 3 Skolemization 4 Checking Acyclicity of the Dependency Relation 4 1 The Dependency Relation 5 Expansion Tree to Jform Conversion 6 Path Enumerator 6 1 Duplication Order 6 2 Backtracking 7 Propositional Case 7 1 Sunil s Propositional Theorem Prover 8 Control Structure and Interface to Unification 8 1 Sunil s Disjunction Heuristic 92 95 95 96 96 97 99 99 99 100 100 101 101 102 104 104 105 106 106 107 108 108 111 111 111 115 118 119 120 120 120 121 122 122 125 126 126 126 127
234. m Yeon D D v T for some formula D In words this principle says that any element T of v must be of a form satisfying D v T An induction principle is a statement that the solution really is minimal This would be a higher order statement of the form vp Ci p Dv Cp 3 where C is a condition similar to C defined above The general case below will explain the difference between Cj and C4 Consider the minimal constraints yl Ay gt vy and 2 B z gt vz Here C u is Vy Ay gt vy and C2 u is vz Bz gt vz The inversion principle here would be Va ua gt Azx v Ba The induction principle here would be Vp Ci p A Co p D v Cp In such cases where none of the minimal constraints are of the form WIT v gt vt where v does occur in I the set is fully determined by the conjunction of the C constraints and the inversion principle since these amount to an exten sional definition of v In the general case the induction principle determines 3 SET CONSTRAINTS 171 the set Though the inversion principle follows from the induction princi ple it may be helpful to have the inversion principle when we are trying to use the lemma to prove the theorem The flag INCLUDE INDUCTION PRINCIPLE controls whether the induction principle is included in set ex istence lemmas Let us return to the familiar example of the constraints gt v0 and w v w gt v S w In this case C41 v is v0
235. me examples of binders def binder lambda typelist A B var type A scope type B wff type BA prefix 100 fo single symbol lambda mhelp Church s lambda binder def binder forall typelist var type I scope type 0 wff type 0 prefix 100 fo single symbol forall mhelp Universal quantifier it The above definition is for math logic 1 where forall can only bind individual variables In math logic 2 the definition is as follows def binder forall typelist A var type A scope type 0 wff type 0 prefix 100 40 5 REPRESENTING WELL FORMED FORMULAE fo single symbol forall mhelp Universal quantifier 5 1 An example How to See the Wff Representations You can see examples of how wffs are represented by comparing the output of the editor commands P and edwff lt 44 gt ed x2106 lt Ed45 gt p FORALL x I R OI x IMPLIES P OI x AND FORALL x Q OI x IMPLIES R x IMPLIES FORALL x P x OR Q x lt Ed46 gt edwff IMPLIES AND x lt I gt FORALL IMPLIES R lt OI gt x lt I gt P lt OI gt x lt I gt x lt I gt FORALL IMPLIES NOT Q lt OI gt x lt I gt R lt OI gt x lt I gt x lt I gt FORALL OR P lt OI gt x lt I gt Q lt OI gt x lt I gt Another way to do this is as follows lt 3 gt ed x2106 lt Ed4 gt cw LABEL SYMBOL No Default gt x2106a lt Ed5 gt plist x2106a REPRESENTS
236. me variables are e Compile the file in several common lisps especially in cmulisp or tps3cmu and see if the error messages are helpful e Try to reproduce the bug in a simpler form e See in how many different contexts such as different matingsearch procedures it arises so you can isolate its essential features e Use the debugging features of your version of lisp e g step and trace e Change the values of the flags QUERY USER MATING VERBOSE UNIFY VERBOSE TACTIC VERBOSE OPTIONS VERBOSE etc to get more output se the monitor See chapter 10 se the lisp function plist to inspect the property list of an object Use inspect to see the values of the slots in a structure e Errors in translation or errors during verification of a merged jform The formula is not provable as there is no connection on the following path are usually caused by merging See the chapter on merging and in particular the note about using merge debug e The code in the file etrees debug can be useful for tracking down bugs involving etrees See subsection 1 1 e Errors of the form Wff operation lt wffop gt cannot be applied to labels of flavor lt label gt are almost always caused by attempting e U e U 11 MISCELLANEOUS 17 to use a wffop on a flavor for which the corresponding property is undefined See the section on flavors for more details e Errors in structure slot accessor are often of the form Structure for accesso
237. mount of swap space you need will depend not only on how big your TPS3 grows but also on what else is running at the same time 3 MAINTENANCE 9 Again on anything but an HP it s time to go whining to gripe cs and get them to do it On an HP start SAM as in the last section and double click on Disks and File Systems Now double click on Swap There are two sorts of swap space device dev and file system fs The former is faster and should be given priority over the latter Here is where I don t quite understand what s going on so if this infor mation ever becomes crucial it would be a good idea to check it I believe that device swap space is simply a partition of the internal disk drive and that it might be possible to create more space simply by rearranging the partition I have no idea how to do this For the time being then we ll restrict ourselves to filesystem swap space You can mount one filesystem swap space on each disk you ve got so take a look at the list that SAM has given you If there are no fs swap space listed or there is a disk that doesn t have one then you can create one by selecting Add Filesystem Swap from the Actions menu Give it a reasonable number you can use du and df to find out how much space there is on the disk at the moment and then choose some large fraction of that and allocate a priority that is lower which is to say a larger number 0 is highest priority than the priorit
238. mple ATM15 In case AtomValF lag is T the program notes the cursor position whenever it encounters such an aplicnlist during printing and prints the name of the atom in the next line at this position 54 6 PRINTING AND READING WELL FORMED FORMULAS 3 3 Printing the PPlist After the PPlist is created by the function PWff the actual output is done by the function PrintPPlist This func tion takes a PPlist and INDENT as arguments and has the following basic structure 1 Does the formula fit on the remainder of the line from INDENT to RightMargin If yes just print it from the PPlist If not go to 2 2 Is the formula composed of subformulas If not go to the next line and print it at the very right If yes go to 3 3 Is the formula a single application If yes call PrintPPlist recursively first with the function then with the argument such that the function will appear at INDENT and the argument right after the function If not go to 4 4 Print each application in the application list in a new line the operators at the vertical position INDENT and the arguments at the position INDENT maximal length of the operators This algorithm will be slightly different if the flags described above do not have their default values See section 3 1 for a description Some heuristics are employed to avoid the pathological case where the formula appears mostly in the rightmost 10 of each line Used in these heurist
239. mula eu Juo Vz P u z D uz A Yz uz D P u z Wea Vz P v z D vz D Yz uz D vz The node MONNEGEF 7 is constructed by the function mon fn negf This function implements the proof that set functions defined by positive formulas are monotone Let u be a new selected variable to use in the selection node BUF SEL EU fut CONJ N Kuuf CONJ va N uKuft If ft where EU is the right side of the top implication of the Knaster Tarski Theorem The node Kuuf is has shallow formula Kuu Vz dul z SwAwvu Vv z 0 D utz 3 SET CONSTRAINTS 181 and is constructed during the process of constructing NEGF below We start with Kuuf as a leaf The node uKuf is a leaf that corresponds directly to the inversion principle in the set existence lemma Vzlulz D wlz SwAululvz 0 Finally we will construct a node If f has shallow formula VoVz ulz SwA uu Vv z 0 D ut 2 gt Vz ulz gt D pz obtained from the fact that ut is the least pre fixed point The selected variable u is used as the expansion term N EGFy we want to construct for the set existence lemma So NEGFo will have the form EXP Set Existence La CONJ K XN NEGF CONJ VA N InvF IndF7 where NEGF is constructed by make clos setvar lemma negf 1 InvF 1 is a leaf corresponding to the inversion prin ciple and IndF indf constructed by make clos setvar ind negf corresponds to the induction principle The leav
240. n algorithm This is a two sided sequent calculus with sequents IT A The code refers to formulas in T as positive as opposed to left and formulas in A as negative as opposed to right to correspond to the parity of expansion tree nodes T and A are lists as opposed to multisets or sets so order and multiplicity are important Equiv eq 210 18 PROOF TRANSLATIONS There are many variations of sequent calculi for classical logic For ex ample consider the two variants of the negative rule for A To A A r B A T AAB A Tri gt A A T2 B Ag r r2 gt AA B Ay Ao Furthermore there is the issue of the positions of the main formulas i e must A and B be the first formulas on the list Different kinds of rules de termine what structural rules the sequent calculus should have The sequent calculus implemented in ftree seq has the following logical rules r gt 4 A DRAR A B T oA ny ee A r r2 gt AA B Aj Ao AAB TAA r gt A B A A Ti gt Ay B T2 gt Ao 2 n o yy TrT gt AVB A AV B T1 T2 gt Aj Ae AT gt B A r gt A A gt T gt ADB A B T2 gt Ao ot ADB T T2 gt Ay Ag A T oA To A A _ lt lt il _ a gt 7AA A AA T oA A a r gt A T gt A a A SEL SEL YrA x T gt A T gt VYrA x A A t T gt A T gt A t A EXP Ex P Ve A x T gt A Tr gt JrA x A There are also rewrite rule
241. ncipal To un derstand mix elimination first consider a few generic examples Suppose we have two cut free derivations of BoA C and D A gt E where A is the mix formula in negative position 0 in the first sequent and positive position 1 in the second sequent Mix elimination might return a cut free derivation of B D gt C E along with two lists of indices 1 t 0 nil 0 indicating that B corresponds to the 0 th positive formula of the first sequent and D corresponds to the 0 th positive formula of the second sequent 2 t 1 nil 0 indicating that C corresponds to negative position 1 in the first sequent and E corresponds to negative position 0 in the second sequent We say might because the return value depends of course on the given derivations not just the sequents Other possible outputs include a cut free derivation of BoC C with two lists of indices 1 t 0 again B corresponds to positive position 0 in the first sequent and 2 t 1 t 1 both occurrences of C correspond to negative position 1 in the first sequent The point is that we have eliminated the two mix formulas two occur rences of A and retain only residues of the other formulas in the two given sequents The other formulas may occur several times or not at all The lists of indices indicate where the formulas originally occured An index is a pair lt bool gt lt nat gt where e If lt
242. nd 208 email from Dan Nesmith about status 112 empty dup info 116 eproof 12 116 200 eproof Data 115 EQUIV DISJS 116 EQUIV IMPLICS 116 EQUIVS Data 49 EQUIVWFFS 115 ETA 115 etags 12 ETR INFO 204 etr merge Function 135 136 ETREE 216 Etree Data 43 etree Data 111 115 ETREE AUTO SUGGEST Command 204 ETREE NAT Command 215 216 231 etree nat Command 115 etree to ftree Function 119 etree to jform Function 117 ETREE TO JFORM REC Function 216 etree to prop jform Function 117 etrees debug File 16 114 etrees labels lisp File 111 etrees wffops lisp File 119 120 EVENTS 285 exit from lisp Function 6 EXP VAR 117 exp var Data 115 expand cases Function 207 expansion 115 expansion proof 115 EXPERTFLAG Flag 100 ext exp dag debug 132 ext exp dag macros lisp File 182 231 ext exp dag prettify Function 162 ext exp dag verbose 132 ext exp dags nd lisp File 231 ext exp dags lisp File 162 ext exp open dags lisp File 162 EXT MATE 182 231 ext mate top lisp File 132 ext search lisp File 183 255 EXT SEQ 231 ext seq 230 ext seq top lisp File 231 EXT 116 external interface lisp Command 246 external interface lisp File 246 EXTRACT TEST INFO Command 85 FACE Property 49 facilities short lisp File 16 facilities lisp File 16 failure record Data 120 false 116 fill selected Function 116 FILLINEFL
243. ne can count the number of flags in TPs3 as follows btps afs andrew mcs math TPS lisp grep i defflag lisp gt flagcount btps afs andrew mcs math TPS lisp ls 1 flagcount it Edit the file flagcount to eliminate lines which do not define flags btps afs andrew mcs math TPS lisp wc flagcount 210 421 8327 flagcount it The number of lines 210 is the number of flags btps afs andrew mcs math TPS lisp rm flagcount The above counts the number of flags defined in the source code The number currently present in a particular version of TPS3 can be found as follows defun discard list if null list nil if or listp car list memq car list cdr list 33 if it s a list it s a subject name and we don t want to count them if it appears later on we don t want to count it twice 18 1 INTRODUCTION may need to use franz memq rather than memq discard cdr list cons car list discard cdr list msg TPS has length discard global flaglist flags 11 2 Dealing with X Fonts e To enable your computer to find the fonts put into an appropriate such as Xclients file xset fp tps fonts using the appropriate pathname in place of tps fonts The fp adds the new directory at the start of the path because we ve had trouble in the past with old fonts with the same name being earlier on in the path e xset q shows the fonts e xset fp takes them out of the fontpath e xl
244. ne where the number of a line is stored on the line s property list WARNING This property is frequently changed destructively As a consequence it may never be empty and should be used for other purposes only in a copy PLANS pline supportlist pline supportlist This stores the important plan support structure of the current proof pline is a still unjustified line in the current proof supportlist is a list of 99 100 12 ETPS deduced lines supporting a pline A pline may never have a jus tification other than PLANi a sline support line must always be completely justified i e may not ultimately depend on a planned line The association list is ordered such that the most recently af fected pline is closer to the front of the list The order can be changed explicitly with the SUBPROOF command WARNING This property is frequently changed destructively As a consequence it may never be empty and should be used for other purposes only in a copy GAPS gap gap This is a list of the gaps between lines in the proof Each gap has the properties MIN LABEL line and MAX LABEL line NEXTPLAN NO integer This is just the next number that will be used for a planned line ASSERTION gwff This is the assertion being proven When a proof is saved using the SAVEPROOF command a checksum may be generated This is used by ETPS to verify that the saved proof has not been manually edited by a student otherwise it w
245. nformation from the remote user name email etc Once this information is given the perl script up dates the log file home theorem tps dist logs tpsdist_log For this to work apache should be the owner of this log file Then the perl script outputs instructions and a link to the tar file 3 4 3 Obsolete Information about Making tar tapes of TPs3 To make a tar archive onto a big round mag tape the kind that is seen in science fiction movies of the sixties and seventies always spinning aimlessly supposedly to suggest the immense computing power of some behemoth machine 1 Go to the CS operator s room on the third floor at the end of the Wean Hall 3600 hallway 2 Tell the operator that you wish to write a tar tape from the machine K GP Give her the tape and go back to the terminal room 3 MAINTENANCE 11 3 Log in on the K 4 At the Unix prompt enter assign mt This gives you control of the mag tape units 5 Enter cd home theorem project dist This puts you in the proper directory 6 Clean up home theorem project dist and its subdirectories by deleting each backup postscript or dvi file 7 Determine which device devname you wish to use This depends on the density which you wish to write For 6250 bpi let deuname be dev rmt16 for 1600 bpi let deuname be dev rmt8 and for 800 bpi let deuname be dev rmt0O Generally you can go for 6250 bpi unless the intended recipient has indicated otherwise 8
246. ng completed exercises defevent done exc event args numberoflines template status userid dproof numberoflines computed code status date dt template names userid dproof numberoflines computed code date daytime signal hook done exc hook write when immediate write file score file mhelp The event of completing an exercise defun done exc hook numberoflines The done exc hook will compute the code written to the file 3 Freeze the time of day right now declare special numberoflines 33 because of the eval list below setq dt status daytime setq computed code 0 setq computed code code list eval list get done exc template defflag proof file flagtype filespec default etps3 proof subjects events mhelp The file recording started and completed proofs defevent proof action event args kind template status userid kind dproof status date status daytime template names userid kind dproof date daytime write when immediate write file proof file mhelp The event of completing any proof defflag advice file flagtype filespec default etps3 advice subjects events mhelp The file recording advice defevent advice asked event args hint p template status userid dproof hint p template names userid dproof hint p 2 THE REPORT PACKAGE 239 write when 1 write file advice file mhelp Event of u
247. nkov editors Handbook of Automated Reasoning volume 2 chapter 15 pages 965 1007 Elsevier Science 2001 Matthew Bishop and Peter B Andrews Selectively instantiating definitions In Claude Kirchner and H l ne Kirchner editors Proceedings of the 15th Inter national Conference on Automated Deduction volume 1421 of Lecture Notes in Artificial Intelligence pages 365 380 Lindau Germany 1998 Springer Verlag http dx doi org 10 1007 BFb0054272 Sidney C Bailin and Dave Barker Plummer Z match An inference rule for incrementally elaborating set instantiations Journal of Automated Reasoning 11 391 428 1993 Errata JAR 12 1994 411 412 Christoph Benzmiiller Chad E Brown J rg Siekmann and Richard Statman editors Reasoning in Simple Type Theory Festschrift in Honour of Peter B Andrews on his 70 Birthday College Publications King s College London 2008 Reviewed in Bulletin of Symbolic Logic 16 3 September 2010 409 411 http www collegepublications co uk logic mlf 00010 W W Bledsoe and Guohui Feng Set Var Journal of Automated Reasoning 11 293 314 1993 Matthew Bishop Mating Search Without Path Enumeration PhD thesis De partment of Mathematical Sciences Carnegie Mellon University April 1999 Department of Mathematical Sciences Research Report No 99 223 W W Bledsoe Set variables In Proceedings of the 5th International Joint Con ference on Artificial Intelligence IJCAI 77 pages 501 510 MIT
248. no connections to any A rewrite nodes in that tree Note that it is legal to mate two nodes even if the shallow formulas are only the same up to A normal form 7 RAISE LAMBDA NODES 151 The function raise lambda nodes processes each child of a subtree unless the node is a Subst rewrite which is handled differently then simply returns the resulting tree and connection list except in certain cases These cases correspond to the trees on the left of the following diagrams We return the tree on the right Here A is the A normal form of A SEL1 Va A gt A REW2 Vxr A a 4 A REW1 a x A SEL1 Yz A 4 a node node SKOL1 YVYxz A gt A REW2 Yx A a J A REW1 a x A SKOLI Vx A a node node CONJ1 gt A REW3 AAB Lo N A REW1 A A REW2 B CONJ1 ul 4 Lo N node nodes node nodes CONJ1 gt A REW2 AAB Lo N 4 A REW1 A nodes CONJ1 1 Va N node node nodes CONJ1 gt A REW2 AAB Lo XN 4 node A REW1 B CONJ1 iL nodes node nodes DISJ1 gt A REW3 AVB Lo XN A REW1 A A REW2 B DISJ1 i an node nodes node nodes 152 14 MERGING DISJ1 gt A REW2 AVB A N A REW1 A nodez DISJ1 if na N node node nodes DISJ1 gt A REW2 AVB Lo XN node A REW1 B DISJ1 4 Lo nodes node nodes IMP1 gt A REW3 ADB Lo XN A REW1 A A REW2 B IMP1 4 4 x N node nodes node nodes IMP1 gt A REW2 ADB K XN A REW1 A nodez IMP1 di a N node node nodes IMPI1 gt A REW2
249. node has zero status then it is not needed If the flag MERGE MINIMIZE MATING is set to NIL then we insist that it is needed Otherwise we temporarily set the status to zero essentially removing the node from the tree and use SPANS to check if the mating still spans all paths The function SPANS calls SPANNING CLIST PATH which calls FIND CHEAPEST CLIST SPANNING PATH which FIND ALT CHEAPEST CLIST SPANNING PATH Note that even in x5207 which is relatively small 12 calls to UNNEEDED NODE P result in 295 calls to FIND CHEAPEST CLIST SPANNING PATH these functions are the main reason why merging can be so slow especially in proofs created by MS90 3 or MS90 9 You could possibly as was done at one time not test this spanning condition and just check to see if every expansion actually has a connection below it The problem here is that in ms90 3 by the time we get to the merging process we have mated every possible pair in the tree whether the connection is necessary or not That is why unneeded node p was modified to be more rigorous because otherwise it was almost useless Additionally there may be embedded falsehood nodes below it which are required to close some paths even if there are no mated nodes below it A better spanning function should be used though actually the one used is already propositional but of an earlier generation than Sunil s propo sitional search function One should realize however that the procedure
250. nodes iteratively until the tree does not have this form There is also code to make sure MP1 is the son of EX P1 but this should already be true because of the function cleanup leibniz expansions described in section 6 Finally check shallow formula described in section 6 is called in case we need to adjust the children of MP1 to have the appropriate form Equiv Implics Equiv Disjs In these cases we simply copy the node rewrite the equivalence in the new shallow formula in the same way as the old and recursively call cleanup all expansions on the kid Refl If the new shallow formula is of the form A A we simply copy the node and recursively call cleanup all expansions with new shallow TRUTH on the kid If the new shallow for mula does not have this form we build a chain of A AB EQUIVWFFS rewrites until it does have the form A A and end the chain with a true node An example where the new shallow formula does not have the form A A is THM144A with mode MODE THM144A Dual The shallow changes from A to AV A or AAA depending on the polarity of the node and make the recursive call Equivwf fs In this case essentially just copy the node re place the shallow formula with shallow as usual and recur sively call cleanup all expansions on the kid The tricky part is computing the new shallow formula for the kid We used to mimic deepening in particular using the value of the flag REWRITE DEFNS Now we find a ch
251. nvenient to be able to use simple notations from the metalan guage and include them directly in the input format for Wffs In TPs3 this is achieved by providing a notation for certain kinds of WFFOPS inside an external specification of a wff This method is not perfect but has other advantages as well as we shall see 8 1 The Notation The motivation behind the notation is an analogy to Lisp we use the backquote to introduce some Lisp form which is to be evaluated and inserted into the Wff One restriction is that the wffop must return a guff or a subtype like a guar The other is that TPs3 must have certain pieces of knowledge about the wffop used in order to be able to determine the type of the result of applying the wffop Some examples of external format and what they are parsed to forall x lcontr lambda x P x x x f x to Vx P f x f a f x forall x exists y lexpd z f x lexpd z f x Q f x f x y 1 t to YrIy Az Az2 Q 2t z y z f zx 64 6 PRINTING AND READING WELL FORMED FORMULAS Csubstitute types A O I P 0A subset Q 0A to Po ot C Qolo The latter could have been more easily specified as substitute types A OI P 0A subset Q but that is no longer possible when the formula is to be embedded in an other Here are the general rules e In an ordinary wff a backquote may precede something of the form wffop arg arg wh
252. ny wff in Q where z is free in M we must have a x 4 a z For any subwff B2 z M of any wff N in Q where a free occurrence of y in N is in M we must have a z O y In order to prove the result we want for any collection of variables I define an ordinary substitution by e d x a 2 if ET eo y Oly ify gr Clearly depends on T 6 and a We will omit the I superscript when possible The idea of course is that I is the context of bound variables Note that if IT is empty then dt 0 158 14 MERGING Proposition Suppose Q is an etree and 0 and a are variable renamings satisfying conditions and with respect to Q Let M be any occur rences of a subwff of a wff in Q Let I be the collection of variables bound in the context M occurs Let be defined as above Then there is an M satifying e M a M e for each x T a x is free for x in M e for each y Z T y is free for y in M these conditions together give that is a legal substitution for M e R M M In particular if M is a shallow formula expansion term or selected variable so that T is empty we have R M 0 M as desired Proof We can prove this by induction on M e Suppose M is a variable or constant Let M be M e Suppose M is N P By induction we have N and P with N a N P P and satisfying the other conditions It is easy to see that letting M be N P works
253. o matic search The new subject should be added to this list The purpose of the list is to make sure important flags have their default value if they are not explicitly set by the mode 1 INTRODUCTION e When a new part of TPs3 is developed an appropriate module should be defined in afs andrew mcs math TPS lisp defpck lisp If a new file is being added to an existing module just add it to the list in defpck lisp make sure the correct heading is on the file and export the filename from afs andrew mcs math TPS lisp lt package gt exp Actually the exporting should be done automatically by TPs3 but it won t hurt to do it manually as well e If a new package or module has been added it must go into all the build and compile files for ETPs and Tps3 See for ex ample afs andrew mcs math TPS common tps compile lisp In general it should go into the ends of the list of modules so that definitions it depends upon will be loaded first If a new module is added be sure to add it to the facilities lisp and facilities short lisp files otherwise it won t show up in the facilities guide e After installing new code remember to change the patch files the tpsjobs file and the tpsjobs done file and to send a mail message to the other people working on the program 10 Debugging Hints e Insert print commands in a temporary version of a file to see either which parts of the code are being used or what the current values of so
254. o be changed to ED or ED PRINTEDTFLAG SLIDES As PRINTEDTFLAG but the output is in Scribe 18 point style PRINTEDTOPS contains the name of a function which tests whether or not to print a particular wff to the PRINTEDTFILE VPD FILENAME is the equivalent of PRINTEDTFILE for vertical path diagrams PRINTVPDFLAG is the equivalent of PRINTEDTFLAG for vertical path diagrams 7 THE FORMULA PARSER 83 The flags and parameters listed below are the counterparts of flags described in full detail on page 7 They have the identical meaning except that they are effective in the editor while their counterparts are effective on the top level of TPs3 EDPPWFELAG If T wffs in the editor will generally be pretty printed Default is NIL EDPRINTDEPTH The value used as PRINTDEPTH within the formula editor It is initialized to 0 7 The formula parser 7 1 Data Structures ByteStream This list stores essentially the printing characters which are in its input CR LF and TAB characters are replaced with a space The ending ESC does not appear in this list All elements are INTERN identifiers See the function bytestream tty in wffing lisp RdCList This data structure appears to be all but obsolete the last remnants of it are in the file wffing lisp RAC refers to the Concept terminal This list contains either integers between 0 and 127 lists containing precisely one of 0 1 or 3 or the identifier CRLF The lists represent char
255. o of course pulling it to the front of the support structure What if aq exists as a dline Here we already proved what we need so leave this plan support pair out when constructing the new support structure What if aej exists as a dline Here we may be in a situation similar to 1 The justification of ejg will be changed according to the This is a small project in itself 106 12 ETPS current rule applied As far as the support structure is concerned we don t treat it specially 10 What if a ejk exists as a pline Here we are justify ing a planned line Delete the plan support pair for ejg from the current support structure The justification of ejg will be changed appropriately 11 What if a ejg exists as a hline If TREAT HLINES AS DLINES is T do what you would do to a dlines see 9 Otherwise nothing special is done 3 3 Entering Lines into the Proof Outline The descriptions in the previous section can when read carefully also serve as a guide to what should happen when entering a line into the proof outline Of course what should be done is clear if we are in the all default case Otherwise we may have to change some justifications as indicated in the previous section but otherwise existing lines are left alone Entering lines into the proof could be taken over by the same function if we handed it linelabels instead of line numbers in default exist and default new 4 Defaults for Sets of Hy
256. ocumentation Certain functions such as eproof statuses are defined implicitly and you won t find their definitions using the tags mechanism If you look at the definition of the structure eproof in the file etrees flags lisp however you will find defstruct eproof print function print eproof statuses make hash table test eq This defines the function eproof statuses The second method is to use the TPS3 export files Try examining the files with a exp extension Generally comments tell which file each symbol comes from This method will fail however if the symbol is not exported or if the symbol has been moved from the file in which it was originally defined without the exp having been updated TPps3 has many global lists the master list is called global definelist and in general each sort of TPS3 object will have an associated global list The last method is to use operating system utilities like grep and fgrep to find all occurrences of the symbol 5 Utilities Utilities are commonly used Lisp functions macros The functions or macros themselves are defined in the normal way and then a defutil com mand is added into the code beside the function definition The point of adding the defutil command is that utilities have their own TPS3 cate gory you can get online help on them and their help messages are printed 7 OUTPUT 13 into the Facilities Guide this will help other TPs3 programmers to find t
257. ogn msg t No exercises completed since write date since msg t CHAPTER 21 The Grader Program Programmers should be aware that the GRADER program has its own manual 1 The Startup Switch In theory adding the switch grader to the command line which starts up Tps3 should start up the Grader program directly The code which implements this is in tps3 save lisp In practice some modifications may be needed depending on the partic ular Lisp being used For example e When starting up in CMUlisp on an IBM RT the error Switch does not exist will be given This is just Lisp complaining that it doesn t recognize the switch it passes the switch on to TPs3 anyway so this is no cause for concern e When using Allegro Lisp version 4 1 or later a symbol is used to separate Lisp options from user options So on early versions of Allegro Lisp the line to start up grader is xterm lt many xterm switches gt e usr theorem bin run tps grader amp whereas for later versions it is xterm lt many xterm switches gt e usr theorem bin run tps grader amp 243 CHAPTER 22 Running TPS With An Interface There is an interface for TPS3 written in Java Running TPs3 through such an interface is similar to running TPS3 within an xterm window ex cept the Java interface supports menus and popup prompts The TPs3 lisp code now includes general facilities for communicating with such an interface when running unde
258. ol function current monitorfn params lt place gt lt alist gt The value of place should reflect what part of the code the breakpoint is at So for example it might be new mating added conn or duplicating The value of alist should be an association list of local variables and things that your monitor function will need For example alist might be mating active mating pfd nil it might equally well be just nil All breakpoints should have exactly this pattern By typing grep if monitorflag funcall lisp in the tpslisp directory you can get a listing of all the currently defined breakpoints 3 The Actual Function This is the function which will actually be called during mating search By convention it has the same name as the defmonitor function Normally it will first check the value of place to see if it has been called from the correct place it can then use the assoc command to retrieve the relevant entries from alist Theoretically it should be completely non destructive so as to ensure that the mating search continues properly of course you may be as destructive as you like provided you understand what you re doing The function for monitor check is as follows notice that this does not check place since it is intended to act at every single breakpoint defun monitor check place alist declare ignore alist msg current monitorfn params place t CHAPTER 11 Writing Inference Ru
259. on pp D1 ss gt pp D2 D3 ss 2 3 The LINE NO DEFAULTS functions There are two func tions whose job it is to determine defaults for line numbers The reason we need two functions is that some of the lines which appear on the left hand side of the support transformation may reappear on the right The way we handle these connections is that we first determine the defaults for lines which are supposed to exist the left hand side of the support transformation then substitute those values into the right hand side and call the second default function The function LINE NO DEFAULTS FROM is called with one argument line no defaults from default exist and LINE NO DEFAULTS TO is called with two arguments line no defaults to default exist default new where default exist is the left hand side of the support transformation spec ified for the rule with lines that we need the default replaced by a while the other lines are numbers which means they either have been figured out by an earlier default function or specified by the user Something which is neither nor a number is one of the variables d or p standing for other dlines or plines in the current support structure They must simply be returned in the proper place of course 2 DEFAULTS FOR LINE NUMBERS A SPECIFICATION 103 default new is the right hand side of the support transformation specified for the rule with the same interpretation as for default exist
260. onjunction introduction ATTACH DUP INFO TO NATREE called on N generates two recursive calls on N1 and N2 and get the expansion proofs corresponding to N1 and N2 respectively with which it constructs an expansion proof corresponding to N An important feature of ATTACH DUP INFO TO NATREE is that it can deal with all natural deductions with or without cuts in them This is mainly achieved by substitution and merge This essentially corresponds to the idea in Frank Pfenning s thesis though his setting is sequent calculus On the other hand the implementation differs significantly since natural deductions grow in both ways when compared with sequent calculus This is reflected in the code of ATTACH DUP INFO TO NATREE which travels through a natural deduction twice from bottom to top and from top to bottom to catch all the information needed to duplicate quantifiers correctly Overview of the files e hx natree top contains the definition of the data structure some print functions and top commands e hx natree duplication contains the code of ATTACH DUP INFO TO NATREE and some auxiliary functions such as UPWARD UPDATE NATREE Also many functions for constructing expan sion trees are defined here e hx natree rulep contains the code for handling RULEP This is done by using hash tables to store positive and negative duplication in formation Then cuts are eliminated by substitution and merge The case in ATTACH DUP INFO TO NATREE which
261. opositional proof checker to search for a mating This mating will be used to construct a proof in natural deduction style 2 NN es x There are still potential bugs in the procedure since various rewrite nodes in an expansion proof can interfere with flag MIN QUANT ETREE This has to be dealt with case by case 11 Lemmas in Expansion Proofs There are facilities to allow an expansion proof of a theorem A to depend on lemmas For the simplest case if a theorem A depends on a lemma B then the expansion tree has shallow formula B A B D A So a complete mating for this expansion tree gives a proof of B and a proof of B D A If an expansion proof does contain lemmas merging and translation must take this into account In general lemmas can themselves depend on lemmas The value of lemmas slot is a list structure containing symbols We can recursively describe these values as lt LEMMAS gt lt SYMBOL gt lt LEMMAS gt lt SYMBOL gt lt LEMMAS gt We can describe how these values correspond to lemmas in the expansion tree inductively An expansion tree proving A with lemmas corresponding to the value lt SYM1 gt lt LEMMAS1 gt lt SYMn gt lt LEMMASn gt 130 13 MATING SEARCH where n gt 0 is of the form CON J A N CONJ IMP VA N ah N Qi a CONJ Q 4 va N CONJ P a LN 4 Qn 1 Qn CONJ x N Ph 1 Pr where Q has shallow formula A each P has shallow formula B and each
262. or SEL EX P Suppose 9 is Dit T gt A t Ad EXP r gt JrA x Ai and Do is D21 A a To gt Ao SEL 4dr A x T2 Ao As described in the REW case we must first eliminate all the nonprincipal occurrences of the mix formulas by recursive calls using D and Dz to obtain D3 a A a T3 gt Ag and with 91 and D to obtain D4 T4 gt A t Ag Then we substitute t for a in D3 a to obtain D3 t and recursively call to eliminate the two occurrences of A t using D4 and D3 t Remark About Termination Usually A t will be smaller than JxrA x But of course in higher order logic there are cases where the formula is most certainly not smaller The most obvious example is when A is 3 o x and t is also 3 o x so that A t is CUT ELIMINATION 229 actually the same as 4x A x We don t actually know whether the algorithm always terminates As in the REW case computing the indices involves com puting preimages of preimages using the following diagram Diy Do D D21 elim elim D4 D3 Ds The SEL EX P case is similar a This case is relatively simple we first make two recursive calls to eliminate the nonprincipal occurrences of the mix formula A giving D3 elim T3 T A A3 and D4 A Ty gt A4 Finally we eliminate the two occurrences of A smaller than A in Dz and D the order is the opposite of the previous cases Again the in
263. order mode In higher order mode iden tifiers may consists of any string of ASCII and special characters Greek subscripts are reserved for type symbols and superscripts may only appear at the end of the identifier The following sym bols terminate identifiers lt Space gt lt Return gt lt Tab gt They may not appear inside an identifier Reserved for special pur poses package are lt gt and should therefore not be used Also with special characters V 4 and A are also single character identifiers In strings superscripted numbers are preceded by e Identifiers in first order mode In first order mode all identi fiers consist of a single letter Upper and lower case letters denote distinct identifiers In addition there is a set of keywords currently AND OR IMPLIES NOT FORALL EXISTS LAMBDA EQUIV which are multi letter identifiers and are always converted to all upper case They have to be delimited by one of the terminating charac ters listed above while all other identifiers may be typed without spaces in between e Type inference TPS3 implements a version of Milner s algorithm to infer the most general type of a wff with no or incomplete type information Internally every identifier in a wff is typed Only the first occurrence of an identifier will be typed in printing unless the same identifier occurs with different types in the same wff 6 Printing Vertical Paths The
264. ored in a list copies of which typically occur in many places in the pplist of a single wff Let us now look at some of the problems and possible solutions of the pretty printing problem 4 1 Static and Dynamic Parameters Crucial to finding a good solution is to understand which factors affect the appearance of wffs when printed These can be divided into two classes Static Parameters Static parameters are not changed during the printing of a given wff In particular their values are identical for a wff and their subformulas Of course they may be changed from one printing task to another but not within printing a particular wff Examples of such static parameters are Al1ScopeFlag Style KsetsAvailable PrintAtomnames etc One other characteristic of static parameters is that one frequently would like to and some times does expand the number of static parameters Dynamic Parameters Dynamic parameters are the ones which change from a wff to a subwff They are highly context dependent and are 58 6 PRINTING AND READING WELL FORMED FORMULAS often not explicitly available as flags but implicitly computed Ex amples of such parameters are should I print a type for this iden tifier PrintDepth should I print brackets or a dot An ex ample for the last question would be that we can sometimes write Qo fu LZ and sometimes Qo f x depending on the brackets in wff containing this as a subformula One can easil
265. ot will precede the formula otherwise the formula will just be printed without any delimiting symbols The plength is the total length of the formula if printed in one line including spaces brackets a s o but not control characters which are used to denote character sets or SCRIBE commands The pdepth is recursively defined as the maximum pdepth of the left hand sides plus the maximum pdepth of the right hand sides of the applica tions if the PPlist contains applications and the plength of the generalized character list gencharlist otherwise The plength of a gencharlist is its length after all members of the form gencharlist have been deleted This means that characters that have to be sent to the selected output device but do not occupy space in the final document will simply be enclosed in parentheses By this convention the function which then formats and 3 PRETTY PRINTING OF FORMULAS 53 actually prints the formula from the PPlist can keep track of the vertical position within a line The pdepth associated with each subformula is used to decide the amount of indentation as described below The list of applications aplicnlist typically contains contains only one pair with the left hand side a function and the right hand side the argument the function is applied to In case we have infix operators or multiple con junctions or disjunctions like A B AANBACAD or EV F aplicniist will contain a different pair for eac
266. other for all n Next for each one of these possible ways we take the two disagreement pair sets given and begin to rename the h variables in them We start at the left of both sets and build up a substitution as we move rightwards comparing each term to the other symbol by symbol Note that we are only replacing variables with other variables If we reach the end of the term without contradicting ourselves we output a and the new node is subsumed If we fail because the substitution is inconsistent or because 6 NOTES 165 we reach two different variables neither of which is an h variable we fail immediately and go on to the next arrangement if there is one Subsumption checking can be very slow set the flag SUBSUMPTION DEPTH with care Because of this it was necessary to add time checking to unification it was previously only done between considering connections The functions unify unify ho rec and subsumption check now check the time if they are called from within a procedure that uses time limits and in order to implement this many other unification functions have been given optional start time and time limit arguments that they do nothing with except passing them on to the next function 6 Notes The code that TPs3 uses to handle double negations is part of the uni fication code See imitation eta in the file unif match lisp CHAPTER 16 Set Variables Inductively a set type is either the type of propo
267. ou can tell roughly how much memory is being used in most versions of Lisp by turning on garbage collection messages and watching the numbers they report After a long search TPS3 may fail with an error message that mentions not having enough heap space or stack space or swap space Allegro Lisp is very good about indicating the real cause of the problem CMU lisp turns off errors while it garbage collects and unfortunately that s when most of 8 1 INTRODUCTION these errors occur so if your CMU based TPS3 seizes up in mid garbage collect and refuses to stop even for C then you ve probably run out of mem ory somewhere Lucid Lisp turns off garbage collection when it approaches the internal memory limits there is a good reason for this see the Lucid manual so if you get a message about garbage collection being off then the real problem is probably a lack of memory TPS3 never switches garbage collection off itself 3 3 1 Heap Size and Stack Size On a Unix system type limit into a C shell or whatever shell you re using to see a list of the upper limits on various things stored in memory The ones you re most interested in will be datasize and stacksize If you are superuser you can remove these restrictions temporarily by typing unlimit datasize stacksize or possibly unlimit h datasize stacksize To increase these limits permanently you need superuser privileges You will need to reconfigure the kernel and reboot your
268. ould be possible to edit out the planned lines and convince ETPS to issue the DONE command Since it takes time to generate the checksum it is only generated if the flag EXPERTFLAG is NIL This means that proofs written by TPs3 with EXPERTFLAG T cannot be read into ETPS with EXPERTFLAG NIL 1 2 Proof Lines as a Data Structure Proof lines in ETPS have a variety of properties REPRESENTS This is the wff asserted by the line In the original TPS this had to be an atomized wff of very particular structure which lead to numerous problems in higher order logic In ETPS this has been maintained for the present Our goal of course is to allow arbitrary gwffs as REPRESENTS of lines HYPOTHESES This is a list of lines assumed as hypotheses for the line The list of hypotheses is ordered lowest numbered line first but to my knowledge no function assumes this It simply looks better in the output No line should appear twice as an hypothesis this fact may actually be used here and there JUSTIFICATION RULE gwfflist linelist The line can be inferred by an inference rule RULEfrom linelist guwfflist has somehow been used to infer the line LINENUMBER The line number associated with the line 2 Defaults for Line Numbers a Specification There will never be an absolutely correct way of assigning default for line numbers we can merely make sure that the result will always be logically correct the rest is often a matter of style and the k
269. ould work at removing this restriction if types are handled properly 10 PRINTING PROOFS 65 some of the hacks in tps mss may have become obsolete but harmless Mathematics10 is a file from the standard Scribe library KSets is a file belonging to TPs3 CHAPTER 7 Well formed formulae operators 1 Operations on Wffs By definition operations on wffs differ from commands in that they return a meaningful value usually another wff or a truth value While com mands are usually given at the top level operations are usually used inside the editor In other respects operations on wffs are very similar to com mands in Tps3 The types of the arguments and the type of the result must be specified in the declaration of a wffop Moreover help for the arguments and help for the wffop itself is available Arguments for wffops may be typed exactly the way arguments for commands are one at a time after a short help message You may frequently have to refer to chapter 3 since it will be assumed below that you have a general idea of how the TPs3 top level interprets commands 1 1 Arguments to Wffops In principle arguments to or results of wffops can have any type defined inside TPs3 There are some argument types which are mainly used for wffops and rarely or not at all for commands They are the following GWEFE A generalized wf BOOLEAN NIL for false anything else for true Internally these are converted NIL and T firs
270. ounds to rename Each free has the prop erty free must avoid which is eventually set to all bound vars z such that y occurs free in the scope of a binder for z Each bound variable has a similar property bound must avoid Note that we need to use two different names for the properties since a variable may occur both free and bound in the etree If a free variable y is a selected variable in the etree it has the prop erty sel var bound which is set to the bound variable correspond ing to the outermost binder at the selection node for y Bound variables z also have properties bound try to equate bound and bound try to equate to free A variable x will be on one of these lists if there is a subwff of the form Azi Azn M At An where z is z and x is A for some i In such cases we will try to send these variables to the same pretty renamed variable prettify free rename This extends 0 to include 6 y y We also must propagate information about this commitment by includ ing y in the list used frees representing the codomain of 0 and by including y in the property not alpha image for any b in the property free must avoid for y We will use this to ensure no such b will later have a b y prettify bound rename This extends a to include a z 2 Again we propagate information by including 2 in the property not alpha image for any b in the property bound must avoid for z We will use this to ensure no such b w
271. ow formula of the rewrite in which the dual instantia tion was performed For example the value might be G162733 SUBSET lt 0 lt OI gt lt OI gt gt c lt OI gt d lt OI G162731 lt OII gt a lt I gt b lt I gt The symbols have no apparent meaning but are used internally as an identifier Note that this list contains both abbreviations and equations which have been instantiated using dual instantiation The value of this global is built during deepening see section 2 1 instantiated eqs list This global s value during deepening is an association list of symbols and the shallow formula of the equa tion being rewritten so its elements are a subset of the elements of instantiated defs list However the elements of this list are removed during deepening so that the final value after deepening an etree to literals seems to always be NIL 3 hacked rewrites list Its value is a list of elements of the form N x lt rewrite node gt lt instantiated wff or symbol gt lt leaf with uninstantiated form gt 4 banned conns list This is an association list of leaves which are not to be mated e g L4 L8 L4 L9 L11 L14 Lit L15 The value appears to be leaves which correspond to an uninstanti ated definition and the leaves which appear beneath the instanti ated form Since these leaves share a vertical path they could be mated Apparently the intuition is that we never w
272. p we created a positive ftree for the lemma along with connections to the nodes in the constraints We can follow the construction by the noting that the construction of the shallow formulas proceeded as w LEM v 3 SET CONSTRAINTS 179 Main v A InvP A IndP where Main v was constructed as Sh POSFEE 2 A Sh POSFE Vw sh POSF 342 A wt 0 Vw Sh POSF gt fu Sw A w0 Vw fut wt gt wt Swt A wt 0 where Sh N means the shallow formula of the node N make clos setvar lemma negf This produces a negative ftree with connections giving the proof of the set existence lemma Let us as sume INCLUDE INDUCTION PRINCIPLE is set to T The func tion make knaster tarski leastfp lemma constructs an ftree proof Knaster Tarski of the least fixed point version of the Knaster Tarski Theorem for the type of the set variable v In this case v has type ou so the Knaster Tarski Theorem generated is VK o o1r Voaz uz D v2 DJua Yz Kuz D uz A Yz uz D Kuz VWva Yz K vz D vz D Yz uz Dvz The function make clos setvar lemma negf 0 does the work of con structing the negative ftree node NEGF giving the proof of the set existence lemma The special variable expf is set to a positive ftree Knaster Tarski for the Knaster Tarski Lemma which is used to prove the set existence lemma Then we return the ftree CONJ N Knaster Tarski IMP vn Knaster Tarskit NEGF The new connections are added to
273. pearing in A are removed To make matters more confusing rewrite defns is dynamically set in this case to a simple list of abbreviations which may be rewritten before calling the function contains some defn This would be a value such as SUBSET REFLEXIVE TRANSITIVE The deepening code also sets many global variables as well as eproof slots in current eproof The code really assumes we are deepening the etree in the etree slot of current eproof 3 Skolemization There are three skolemization procedures in TPs3 SK1 SK3 and NIL Actually the latter is not skolemization at all but the selection nodes method from Miller s thesis However it still uses skolem constants inter nally The flag SKOLEM DEFAULT decides which one will be used in a proof and the help message for that flag explains the difference We assume familiarity with the way that SK1 is handled in TPS SK3 is broadly similar the only difference between the two is in the function create skolem node where the skolem variables are chosen differently 122 13 MATING SEARCH NIL the selection node method is very different Selections are rep resented as Skolem constants with no arguments and we now describe the additional machinery needed to make the search procedure work in this case During simplification in unification the requirement that a certain re lation should be acyclic is checked The exact statement of this relation is given in Miller s thesis we
274. pes should be made through the secondary macro DefType Its format is as follows enclose optional arguments DefType it name GetFn it fnspec TestFn it fnspec PrintFn it fnspec Short Prompt it boolean MHelp it comment it property1 it value1 it property2 it value2 In the above a fnspec is either the name of a one argument function or a list of forms which are to be evaluated as an implicit progn In the latter case name stands for the argument supplied name The name of the argument type It will get a property value of T for the property ArgType when the DefType has been executed GetFn Here fnspec defines the function used to process the argument as supplied by the user on the command line The value returned by it is then handed on to the main function executing the com mand No GetFn will ever receive a It is simply not called if the corresponding argument in the command line is defaulted A GetFn should signal an error if the argument is not of the correct type This will be implemented as it is right now as a THROW with the label FAIL A special case of fnspec for a GetFn is TestFn This sok 30 4 MEXPR S means the GetFn will test if the supplied argument is of the correct type If yes the argument will simply be returned otherwise an error will be signaled A GetFn may have side effects but this has to be declared under Side Effects PrintFn Here fnspec
275. plication and A abstraction from a few primitives In order to make formulas more legible and closer to the form usually used to represent formulas from first order logic we furthermore have quantification and definitions internally and quantification definitions and infix operators for the purpose of input and output The application of a function to an argument is printed by simply jux taposing the function and its argument As customary in type theory we do not have an explicit notation for functions of more than one argument Predicates are represented as functions with truth values as their codomain Infix operators have to be declared as such Only conjunction disjunc tion and implication are automatically declared to be infix operators In general infix operators will be associated to the left if explicit brackets are missing For example AA BAC will be AA B AC Internally every infix operator has a property Infix which is a number This number is the relative binding strength of this infix operator You will have 2 PRINTING OF FORMULAS 47 to specify it if you define a new connective to be infix The higher the pri ority the stronger the binding As usual A binds stronger than V which has precedence over D implication As an aside if you don t want conjunctions bound more tightly than disjunctions but want brackets to appear make the INFIX property of OR the same as AND Thus do GET AND
276. pothesis In Tps3 the user will rarely ever have to deal explicitly with sets of hypothesis However the detail can be controlled by a flag called AUTO GENERATE HYPS If this flag is T TPs3 will not only generate smart defaults for sets of hypothesis but make them strong defaults which means that the user will never be asked to specify hypotheses for a line There some restrictions on what the user of the RULES module may spec ify as hypothesis in a rule Ignoring for the moment the problem of fixed hypotheses like sets representing axioms of extensionality of an axiom of in finity the hypotheses for each line l may have the form H s1 5 where H is a meta notation for a set of lines and the s are labels for lines present elsewhere in the rule specification Let us use H for this set of specified hypotheses for line J Note the restriction that there may be only one variable standing for arbitrary sets of lines in any single rule description Defaults strong or not for the hypotheses of lines are only calculated after all line numbers have been specified This includes existent and non existent lines equally The algorithm below will always generate legal applications of the rule at the same time generating the correct set of hypotheses for each line The algorithm will almost always be adequate in the sense that the user will almost never need to explicitly add hypotheses to a deduced line or drop hypotheses from a plann
277. print top level category edop top level ctree ed command ctree top cmd decode opdecode mhelp The top level of the formula editor This says that the top level ed top identifies itself by the function ed top prompt which is one of the more complicated prompt functions in TPS3 its only purpose is to print the lt ed34 gt messages at the start of each line in the editor but the complications are necessary because the editor can be entered recursively The next line of the toplevel definition gives the name of the command interpreter function The print function is a function that gets called after every line in this case it s the ed print function which prints out the current wff if it has changed due to the last command The top level category is edop which is defined as follows defcategory edop define defedop properties alias single result gt singlefn edwff argname single defaultfns multiplefns move fn singlefn mhelp single global list global edoplist shadow t mhelp line editor command scribe one fn lambda item 25 26 3 TOP LEVELS maint scribe doc command format nil IndexEdop A symbol name item remove get item edwff argname get get item alias argnames or cdr assoc edop get item mhelp cdr assoc wffop get get item alias mhelp mhelp fn edop mhelp This category defines the sort of command found in th
278. properties of 44 5 REPRESENTING WELL FORMED FORMULAE etree including for example its print function unless they are explicitly redefined in the definition of leaf CHAPTER 6 Printing and Reading Well formed formulas 1 Parsing Frank has implemented a type inference mechanism based on an algo rithm by Milner as modified by Dan Leivant Type inference is very local The same variable say x will get different type variables assigned when used in different formulas Since multiple use of names with different types is rare the default could be changed so that after the first occurrence of an x during a session core image the type inferred the first time is remembered There are only a total of 26 type variables so you may run out during a session The function INITTYPES reset the way type variables are assigned and treats everything except O and I as type variables Normally a type variable once mentioned or assigned automatically becomes a type constant If TYPE IOTA MODE is NIL then TPS will assign type variables start ing with Z and going backwards as more are needed TYPE IOTA MODE defaults to T Polymorphic abbreviations like SUBSET now may be given a type so as to fix the type of other variables E g the following is legal FORALL x P x IMPLIES Q x IMPLIES P SUBSET O OC OC Q Note that x will be typed C Gamma The same typing could have been achieved by FORALL x C P x IMPLIES Q x IMPLIES P S
279. pt ina separate eproof structure For this reason the statuses are actually kept in a hashtable in the eproof structure as well so changing the status of a node in one virtual tree doesn t affect its status in other trees E g Assume we have a tree with root expansion node EXPO and children LEAF1 LEAF2 Then we have potentially 3 virtual trees one where LEAF1 has positive status and LEAF2 has 0 is not there one where LEAF2 positive status and LEAF1 has 0 and one where both LEAF1 and LEAF2 have positive status are thus both considered in the proof process Functions that do things like create the jform use the status to decide which nodes belong and which don t Because statuses are now kept separate from the nodes themselves the STATUS slot is an anachronism and actually can now be removed delete the form status 1 from the file etrees labels lisp PREDECESSOR is related This is a symbol the name of the etree node from which this node originated For example suppose we have a leaf node LEAFO If we deepen this node then we will get something like EXPO as a result Its PREDECESSOR slot will be LEAFO If we then change the status of all its expansions to 0 then this node is effectively a leaf node again and it will be printed out with the name LEAFO as before E g lt 34 gt mate exists y P y DEEPEN YESNO Deepen Yes gt no lt Mate35 gt etd 114 13 MATING SEARCH LEAFO EXISTS y I P OI y lt Mate36 g
280. q caar inwff var nil let mew wff subst term var rec term var cdr inwff if new wff cons car inwff new wff nil t let Cleft or subst term var rec term var car inwff car inwff right or subst term var rec term var cdr inwff cdr inwff unless and eq left car inwff eq right cdr inwff cons left right defwffop substitute 1l term var argtypes gwff gvar gwff wffargtypes A A B resulttype gwff wffop type B wffop typelist A B argnames term var inwff arghelp term var inwff mhelp Substitute a term for the free occurrences of variable in a gwff Bound variables may be renamed using the function in the global variable REN VAR FN defun substitute l term var term var inwff or subst l term rec intern subst term var var inwff inwff LCONTR currently in wfflmbd2 lisp does a Lambda contraction Notice the use of THROWFAIL and the use of general predicates like LAMBDA BD P rather than testing directly whether a given wff is bound by Lambda This way the function works even if the CAR fo the application is a label 74 7 WELL FORMED FORMULAE OPERATORS defwffop lcontr argtypes gwff wffargtypes A resulttype gwff wffop type A wffop typelist A argnames reduct arghelp gwff reduct applicable p reduct p mhelp Lambda contract a top level reduct Bound variables may be renamed using REN VAR FN
281. quent Calculus 3 1 6 Translating from Natural Deduction to Sequent Calculus 3 1 7 Normalization of Proofs 3 2 Hongwei s Nat Etree 3 3 The Original Nat Etree 4 Cut Elimination 4 1 An Example of a Loop in a Cut Elimination Algorithm 4 2 Cut and Mix Elimination in this Sequent Calculus 4 3 The Mix Elimination Algorithm 5 Cut free Extensional Sequent Derivations to Extensional Expansion Proofs 6 Extensional Expansion Proofs to NProofs Chapter 19 Library 1 Converting TPTP Problems to TPs3 library items Chapter 20 Teaching Records 1 Events in TPS3 1 1 Defining an Event 1 2 Signalling Events 1 3 Examples 2 The Report Package Chapter 21 The Grader Program 1 The Startup Switch Chapter 22 Running TPS With An Interface 1 Generating the Java Menus 2 Adding a New Symbol Bibliography Index 208 209 209 212 214 215 216 218 218 222 225 230 231 233 233 235 235 235 236 237 239 243 243 245 247 248 251 253 Preface The following is a T X actually ATEX version of the Tps3 Program mer s Guide The original version is in Scribe format CHAPTER 1 Introduction Tps3 has been developed over several decades by a number of people some of whom never actually met each other Attempts have been made to maintain documentation for the program but research progress was gener ally a higher priority and obsolete documentation was not always corrected or deleted Therefo
282. quent calculus for sets of expansion trees with rules for quantifiers merging and cuts Then he gives reductions on expansion developments with the hope that these reductions result in an expansion tree Frank s algorithm is not currently implemented as part of TPS Of course there are many ways of representing cuts and performing cut elimination 4 1 An Example of a Loop in a Cut Elimination Algorithm One approach we tried Chad Summer 2001 was to include explicit CUT and MERGE nodes in expansion trees defining redexes and doing cut elim ination by contracting redexes This section contains a brief outline of the approach and an example that shows how a loop can occur A CUT node is of the form CUT x N where C has shallow formula C and B and B have the same shallow formula but opposite polarity The polarity and shallow formula of the CUT node is the same as the polarity of the shallow formula of C The deep formula of the cut node is deep C A deep B V deep B A MERGE node is of the form MERGE xs A ATE where A and A have the same polarity and same shallow formula A The shallow formula of the MERGE node is also A The polarity is also inherited from the children The deep formula is given by deep A A deep A 4 CUT ELIMINATION 219 With these nodes the translation from a natural deduction proof see subsection 3 1 can now be extended to translate backward coercions Ig noring hypothes
283. r lt foo slot gt is not a lt foo gt For every structure lt foo gt there is a test lt foo p gt use it Of course you should also work out how something that wasn t a lt foo gt managed to turn up at that point in the program anyway often it s an exceptional case that you forgot to handle e In Allegro the function dumplisp can be used to save a Lisp image For example excl dumplisp name saved image dxl will create a large file named saved image dxl Then one can use lisp I saved image dx1 to start lisp specifying this as the image file This will start lisp in the same state e g the global variables will have the same values as when dumplisp was called This is especially useful if the bug shows up after running for a long time e If the bug is new for example if you know it wasn t there last month don t forget that the tpsjobs done file lists all of the files which have been changed along with the reasons for each change and the date of each change The tps tps old source directory should contain backup copies of the changed files Failing that snapshots of the entire lisp directory in the form of gzipped tar files made after each rebuild are stored in the tps tps tarfiles directory Use cload to restore the old copies of the most likely culprit files into a core image until the bug disappears then use ediff to compare the old and new files 11 Miscellaneous 11 1 Counting Flags O
284. r Allegro To start TPs3 with the java interface one can use the command line ar gument javainterface along with other relevant information as shown below lisp I tps3 dxl javainterface cd javafiles usr bin java TpsStart The command line arguments following javainterface should form a shell command which run the interface In this case the shell command would be cd javafiles usr bin java TpsStart Other command line arguments which have meaning for the java TpsStart command are listed below big Use the bigger sized fonts x2 Multiply the font size by 2 x4 Multiply the font size by 4 nopopups Do not use popup style prompts Instead the Java window should behave more like the x window interface The remaining command line arguments should be followed by a non negative integer screenx The initial horizontal size of the Java window screeny The initial vertical size of the Java window rightOffset The amount of extra room given to the right margin bottomOffset The amount of extra room given to the bottom mar gin maxChars The maximum number of characters to hold in the buffer This should be large enough that you can scroll back and see pre vious TPS3 output The default value of 20000 should usually be enough These other arguments should be preceeded by a command line other This tells TPS that the remaining command line information should be passed to the call to
285. r by a variable or a line number 3 1 support Structure Transformation in the Default Case If the rule is used completely in the default direction i e all plan support pairs in default exist exist in the current support structure and all pairs in default new consist of new lines then the effect of the rule application on the support structure is straightforward e Delete all pairs matching p di1 diz from the support structure and attach to the front the pairs qj e 1 ejy e A variable in place of a p matches any plan support pair in the current proof as long as the dig match the corresponding support lines 3 UPDATING THE SUPPORT STRUCTURE 105 e A variable in place of a dj matches the lines which are not matched by any of the line numbers If p is a variable every match for p produces a corresponding match of dik e A variable in place of q must occur as some p and as many copies of qj j1 jy are produced as there are matches of pj e A variable in place of ej must occur as some dig and the matched 3 2 list of lines in filled in What if We will go through all cases which differ from the default application of the rule and specify what should happen to the support structure Of course TPS3 can not always correctly predict what the user had in mind when applying a rule so the following must partly be considered heuristics but they will not always implement the user s devious intent
286. rank Pfenning s discusses the choice between using the formula subwff Av Ao z or Ax 7x Bo in the algorithm below One may lead to a more elegant proof In TPs3 the choice is made by a call to the function make left side refl which currently always returns T Suppose we are given a negative subtree of the form Leibniz REW1 A B J SELI1 Vq qA D q4 B7 qo IMP1 K N Equivwf fsREW qo At A Equivw f fsREW qo B a LEAF1 qo AG LEAF qo Bo Actually the selection node SEL1 might be a Skolem node but this is treated the same way The local variables in the function are given the following values param node SEL1 param qo imp node IM P1 new refl node LEAF1 or LEAF 2 if make left side refl were to return NIL subwff Ax Ao 2 or Axz x Bo if make left side refl were to return NIL e new non refl node LEAF or LEAF1 if make left side refl were to return NIL 6 REMOVE LEIBNIZ 145 e non refl branch the second son of IM P1 which is a rewrite node or LEAF 2 or the first son of IM P1 if make left side refl were to return NIL e mated to refl node list of nodes mated to LEAF 1 We consider two cases a If LEAF1 is connected to LEAF 2 then Ag and Bo must be identical Let lhs be A and rhs be B If these are identical wffs then simply change the etree to be Refl REW1 A B TRU E1 Otherwise let A 1hs be the A normal form of A and B rhs be the A norma
287. re this manual should be used with discretion The guid ance it provides may be very helpful at times but there is no claim that it is generally adequate and some of it may be misleading or incorrect The Guide is an indispensable companion to all those who are keen to make sense of life in an infinitely complex and confusing Universe for though it cannot hope to be useful or informative in all matters it does at least make the reassuring claim that where it is inaccurate it is at least definitively inaccurate In cases of major discrepancy it s always reality that s got it wrong Douglas Adams The Restaurant at the End of the Uni verse 1 Guidelines In addition to the information in this guide fragmentary documentation of the TPS3 code can be found in the tpsjobs done file This is included in the TPs3 distribution This guide assumes that the reader is familiar with Common Lisp and does not attempt to explain or summarize information that is available else where about the workings of that language in particular in Steele s Common Lisp the Language 2nd ed There are three major rules which should be followed whether maintain ing TPS3 code or just fooling around with it 1 Always keep a backup copy of the files you are changing so that when you realize how badly you goofed you can put things back the way they were 2 Don t get too tricky Clever hacks may be amusing and may indeed give some increase
288. re always recorded except for the T safety valve Events may be disabled which means that signalling them will always be successful but will not lead to a recordable entry This is done by setting or binding the flag event ENABLED to NIL initially set to T For example the line setq error enabled nil in your INI file will make sure that no MacLisp error will be recorded For a maintainer using expert mode this is probably a good idea 1 3 Examples Here are some examples take from the file ETPS EVENTS Interspersed is also the code from the places where the events are signalled defflag error file flagtype filespec default etps3 error subjects events mhelp The file recording the events of errors defevent error event args error args template status userid error args template names userid error args write when immediate write file error file a global variable eg C tpsrec etps error signal hook count errors count errors to avoid infinite loops mhelp The event of a Lisp Error DT is used to freeze the daytime upon invocation of DONE EXC so that the code is computed correctly The code is computed by CODE LIST implementing some trap door function defvar computed code 0 defvar dt 0 0 0 defvar score file defflag score file flagtype filespec default etps3 scores subjects events 238 20 TEACHING RECORDS mhelp The file recordi
289. re are a number of operations available in the editor and mate top levels for printing vertical path diagrams Also the following wff operation is available for printing vertical diagrams of jforms e VPFORM JFORM FILE STYLE PRINTTYPES BRIEF VPFPAGE The default values are e File defaults to TTY the terminal e Style defaults to the value of the flag STYLE e PrintTypes defaults to the value of the flag PRINTTYPES 62 6 PRINTING AND READING WELL FORMED FORMULAS Brief has three possible settings T means that only the names of logical atoms will be printed and not their values NIL means that under each atom its value will appear and e L means that just the atomnames will be printed in the diagram but a legend which contains every atom with its value will be appended to the first page of output VpfPage is the number of characters which fit on one line AndHeight is an optional global variable which is equal to the num ber of blank lines to be left for a conjunction It defaults to 1 ForallIndent is another optional global variable containing the num ber of columns the quantifier is set off its scope The default is 1 BRIEF can assume the values T for printing the diagram in brief format L for a brief diagram but with a legend atomnames with their associated values at the end of the first page LT for a legend with type symbols forced to print and NIL which gives the the full diagram Both of these function
290. restriction but rather give an error message Otherwise the user may only find out much later that some of the hypotheses he expected to be able to use have not been included in the dlines since they violated a restriction This makes it necessary however to give the user explicit rules which allow adding hypotheses to a deduced line or dropping hypotheses from a planned line 4 3 Hypothesis Lines There are two principal ways hypothesis lines hlines can be treated in TPs3 and since there is very little extra work re quired both are provided for The flag TREAT HLINES AS DLINES con trols how hypotheses lines are handled If TREAT HLINES AS DLINES is T an hline may have more hypotheses than simply hline Also hlines may have descriptive justifications like Case 4 DEFAULTS FOR SETS OF HYPOTHESIS 109 a or Ind Hyp for n The price you pay is that hypotheses lines become unique to a subproof and should not be used elsewhere In this case hlines are truly treated as dlines and in the above algorithm for determining default for lines we mean dline or hline whenever we say dline If TREAT HLINES AS DLINES is NIL every hline has exactly one hypoth esis itself Also the justification for any hline will be the same namely the value of the flag HLINE JUSTIFICATION by default Hyp What you gain in this case is that the same hypothesis line may used many different places in the given proof The default for the hypotheses o
291. rewrite nodes 2 NN 8 CLEANUP ETREE 155 e 8 7 We can skip this rewrite if the new shallow formula is already A normal Otherwise we copy the node normal ize the new shallow formula and recursively call cleanup all expansions on the kid e Subst Leibniz Ext and Both which is not cur rently fully supported We copy the node expand the equality in the new shallow formula and recursively call cleanup all expansions on the kid If the node is a positive Subst rewrite we process the new tree further We start with a tree of the form Subst REW1 A Bt EXP1 Vq qA D gt qBt Q J N Qn IMP1 oS we x N node nodes If node or nodeg is of the form A B n REW2 QiC node D where D is the A normal form or 6 or 7 normal form of QC then reduce rewrites modifies the node to be of the form 8 n REW2 E l node D and they both in fact should be where E is C z F and F is the A normal form or 8 or 7 normal form of Q z If in fact E and D are a equal we simply replace the node with node D removing the rewrite altogether That is we change the children of the implications so that the shallow formulas are normalized until all redexes are in A or B or are of the form A M or B M Next if the implication node is of the form IMP1 4 xN NEGI NEG3 4 4 NEG2 NEG4 4 4 node nodes 156 14 MERGING then remove double negations merge deletes the four negation
292. rguments where n is the number of mexpr arguments or else a lambda expression of n arguments Each fnspec must return a list of length n This list will then be handed on and processed by the next fnspec as if it were the list of arguments supplied by the user Any entry which is not a should be left unchanged The function is not allowed to have side effects As a general convention the arguments which are not used by a fnspec are not written out with their name but replaced by i This makes it easier to see at one glance which defaults are filled in by a certain defaultspec EnterFns fnspeci fnspec2 is an arbitrary list of function specifi cations They are called in succession with the value list returned by the last default fnspec before the MainFns are called MainFns fnspec1 fnspec2 is an arbitrary list of function specifica tions They are called in succession with the value list returned by the last default fnspec If none are specified it is assumed that there is a function named name which can be called Notice that at this stage no defaulted arguments may be left ComDeCode the com mand processing function will refuse to call any function unless all the defaults are determined This clearly divides the responsibilities between GetFn s DefaultFn s and MainFn s Any fnspec may abort with an error by doing a THROW with a FAIL label A THROW with a TryNext label will be handled like a normal return A THROW w
293. s gt ADB A BDA A _H _ REwW TS A B A AD B A BDA T gt A _ _ _ REW A B TA7A ee REW A 8 1 cea ae REW A 8 7 ne a Gy ape eee rj j rs an a Po a 7 IE T A A A T oA 3 NPROOFS TO EPROOFS 211 where A is either the G7 normal form 3 normal form or 7 normal form of A To A A A ToOA REW AB REW AB To B A BTA where A and B are a equivalent To A A REW EQUIVWFFS r B A A ATrT gt A REW EQUIVWFFS BTA where A is the result of expanding some abbreviations in B To A A A ToA REW Leibniz REW Leibniz T B A BoA where A is the result of expanding some equalities in B using the Leibniz definition of equality r gt A A AT gt A REW Ext REW Ext T B A BTA where A is the result of expanding some equalities in B using extensionality This does not provide a complete calculus for extensionality without a cut rule So sometimes cut elimination will fail if these extensionality rules are used The structural rules are r A ToA weaken weaken To A A A T oA 4 A A A A A T OHA merge merge To A A A T oA To Ai A Ao focus where A has length n r gt A Ai Ao 212 18 PROOF TRANSLATIONS Fis A To oA focus where T has length n A Ty Tg gt A Finally we have an initial rule and a cut rule r gt A A A T gt A
294. s array 117 contains some defn Function 121 core name 7 create skolem node Function 121 CRLF Data 83 current eproof 115 116 119 121 126 127 203 current topnode 115 cutfree ftree seq to ftrees Function 205 212 CUTFREE TO EDAG Command 231 datasize 8 Debugging 16 deepen leaf node real Function 120 121 deepen negated leaves Function 142 DEFAULT EXPAND Flag 167 DEFAULT LIB DIR Flag 233 DEFAULT MS Flag 167 defcategory 22 defconstant 6 defconstnt 6 defcontext 23 definfo 15 90 defmonitor Command 95 defpck lisp File 2 4 DEFPRTOP 69 DEFREPORT 240 DEFREPORTI1 240 defsynonym Command 90 defutil Command 12 DEFWFFOP Function 68 defuffrec Function 70 DELETE HYPS Command 97 Desc Jform Data 56 Descr JForm 55 Describe VPAtom Function 56 Describe VPForm Function 55 56 Describe VPLit Function 56 DFONT Property 49 dfont lisp File 49 dissolve 116 126 DISSOLVE Flag 117 126 DIY Command 132 dline 101 DONE Command 100 dproof 204 DUAL 118 dumplisp Function 17 econjunction 115 ED COMMAND Argument Type 79 edisjunction 115 edop Data 77 EDPPWFFLAG Flag 83 EDPRINTDEPTH Flag 62 83 EDSEARCH Function 80 edtop lisp File 14 edwff Data 77 edwff EdOp 40 eeod to eed node Function 162 ELIMINATE ALL RULEP APPS Command 208 ELIMINATE CONJ RULEP APPS Command 208 ELIMINATE RULEP LINE Comma
295. s concerned is merely a special case of gwff Labels thus stand for gwffs but may have additional properties and distinct representations Flavors are the classes into which labels are divided The definition of a flavor specifies some common properties of a class of labels usually the behavior of wffops and predicates Also a flavor s definition should specify what attributes each label of that flavor should have and how it should be printed 6 1 Representation Each flavor is represented in TPs3 by a Lisp structure of type flavor which has the following slots wffop hash table which stores the properties common to each instance of the flavor in par ticular how wffops are to behave constructor fun which is the name of the function to be called when a new label of the flavor is to be created name the flavor s name and mhelp a description of the flavor The values of these slots are automatically computed when TPs3 reads a defflavor declaration The flavor structures are stored in a central hash table called flavor hash table keyed on the flavor names This also is updated automatically whenever a flavor is defined or redefined There are two ways to represent labels instances of flavors and the choice is made during the definition of the flavor The first and more tradi tional way is to have each label be a Lisp symbol with the attributes of the label being kept on the symbol s property list The second way is to ma
296. s if it were an include specifier to a defstruct e g include existing flavor slot1 default1 6 4 Examples Here are some examples of flavor definitions defflavor etree mhelp Defines common properties of expansion tree nodes structured t instance attributes name type symbol components a node s children positive nil true if node is positive in the formula junctive nil type symbol whether node acts as neutral conjunction or disjunction free vars expansion variables in whose scope node occurs used for skolemizing parent parent of the node to keep track of nodes from which this node originated when copying a subtree predecessor nil status 1 printfn print etree printwff lambda wff bracket depth if print nodenames pp symbol space etree name wff printwff if print deep get deep wff get shallow wff bracket depth Many more properties Etree labels will be structures with several attributes The function used to print them will be print etree defflavor leaf mhelp A leaf label stands for a leaf node of an etree inherit properties etree instance attributes shallow include etree name intern str create namestring leaf name Leaf labels will also be structures with attributes including those of etree as well as a new one called shallow Note that the name attribute is given a default in the include form Leaf inherits all of the
297. s into separate problems but the path focused duplication search procedures do not When searching for a way to span a path TPs3 runs down the path from the top and considers each literal As a mate for that literal it considers each literal which precedes it on the path When TPs3 considers adding an essentially ffpair pair of literals which each start with a variable when one ignores any negations to the mating it simultaneously considers both orientations choices for which literal will be negative and which positive of the ffpair Roughly speaking it does this by putting a disagreement pair corresponding to the ffpair into the leaves of the unification tree and proceeding with the unification process If this process encounters a disagreement pair of the form lt A B gt where A starts with a constant but B does not it replaces this pair with lt A B gt 128 13 MATING SEARCH and continues In this way it finds whichever substitution works in a very economical fashion When a success node is found for a complete mating the associated substitution determines the orientation of the ffpair in the mating Here is some more detail about how this is actually implemented When TPS decides to mate a pair L K of literals which it considers as an unordered pair it seeks to unify L with K where L occurred before K on the path Whenever the unification process encounters a double negation it deletes it Thus in the case of a fir
298. s of the form OTL In ETPS only the bookkeeping functions are present The discussion below is aimed at understanding the OUTLINE modules independently of the system but we generally assume we are working in Etps If Tps3 differs this is noted We often talk about proofs even though they are properly only incom plete proofs or proof outlines It is assumed that the reader knows what planned lines plines and deduced lines dlines are This and general fa miliarity with ETPS are necessary to understand this discussion 1 1 Proofs as a Data Structure Proofs in ETPS are represented by a single atom with a variety of properties The global variable DPROOF has as value the name of the current proof In case you are working say on exercise X6200 DPROOF will have the value X6200 The current proof name then has a variety of properties LINES line line This is simply an ordered list of all lines in the current proof without repetition The order is such that lines with a lower number appear first in the list WARNING This property is frequently changed destructively As a consequence it may never be empty and should be used for other purposes only in a copy LINEALIASES line no line no This is an unordered association list correlating lines with their numbers No line should ever appear in more than one pair and neither should a number Try to think of arguments for and against this representation com pared to o
299. s slow for various reasons One of them is the tremendous amount of temporary list space used which takes time and more time through garbage collection Another is the forgetfulness of the printing routine which recomputes length and other information over and over again Below we will try to explore ways to improve the performance of the pretty printer without sacrificing any of the niceness of the output Let us recount which factors make pretty printing wffs more difficult than pretty printing Lisp S Expressions For once Lisp does not have infix operators and can therefore get by with a significantly smaller amount of lookahead Moreover the lookahead can be done during the printing where the extra time delay is hardly noticeable while TPS lookahead must all be done ahead of time before the first character is printed Secondly Lisp does not deal with a variety of output devices which makes counting symbol lengths as well as printing symbols much faster and more transparent The result of a first attempt at pretty printing is described earlier in this chapter The solution is nicely recursive and a lot of information is made available for deciding where to break and how to indent lines It is a sad fact that the algorithm does not reuse any information whatsoever For example the printed representation of identifiers is recomputed over and over again Even worse the characters comprising the printed representation of an identifier are st
300. s some hacking to make it build properly Follow the instructions in the README to build yourself a new Lisp core image with more than the standard 60Mb data es 10 1 INTRODUCTION limit If you aren t up to the hacking once again the solution is to whine at gripe cs who will forward your mail to the Allegro maintainer 3 4 TPs3 distribution 3 4 1 Making a tar file for Distribution You can execute afs andrew mcs math TPS admin tps from home theorem project dist The date is computed automatically The tar file is placed in home ftp pub KEY STEPS Login to gtps as root so that you can write a file in home ftp pub su You may need to do a klog e g klog pa01 c andrew cmu edu or klog cebrown c andrew cmu edu so you appropriate permissions cd home theorem project dist afs andrew mcs math TPS admin tps dist make tar exe This does tar cvhf home ftp pub tps3 date tar gzip home ftp pub tps3 date tar ln sf home ftp pub tps3 date tar gz home httpd html tps3 tar gz It s important that the tar file is not put into the same directory as is being tarred or it will try to work on itself cd home ftp pub move old tar file to the subdirectory old Delete the older one 3 4 2 Distribution of TPS3 via http There is a perl script home httpd cgi bin tpsdist pl which is used to distribute TPS3 via the gtps web site This perl script dis plays the distribution agreement and asks for i
301. s the same as a support line and finishes the subgoal using SAME This would correspond to the coercion rule above The backward coercion rule A tt bcoercion A is used to pass from normal deductions to extractions Backwards coercions correspond to instances of cut A separate interesting project in TPS would be to program a normalization procedure Such a procedure would find in stances of the backward coercion rule when annotating a proof identify to what kind of redex the backward coercion rule corresponds and perform the reduction For this to work we would need to define the notion of redex 208 18 PROOF TRANSLATIONS so that every proof which needs the backward coercion rule to be anno tated proofs that are not normal must have a redex Also we could not prove that reduction terminates a task equivalent to constructively prov ing cut elimination in classical higher order logic Instead the current code translates backwards coercion as an application of cut and then uses a cut elimination procedure which may not terminate to obtain a cut free proof See section 4 2 3 1 3 Some Nonstandard ND Rules There is code to replace fancy propositional rules like RuleP with subderivations using primitive rules See the commands ELIMINATE ALL RULEP APPS ELIMINATE RULEP LINE and ELIMINATE CONJ RULEP APPS The command ELIMINATE CONJ RULEP APPS only expands those RuleP applications which can be replaced by appli
302. s will have arity 0 7 substitution list A list of expansion variables which have been in stantiated 8 leaf list A list of the leaf nodes occurring in the etree 9 skolem method Corresponds to the value of SKOLEM DEFAULT 0 max cgraph counter 1 bktrack limit 2 connections array 3 incomp clists wrt etree 4 5 6 7 Ol a mating list incomp clists cgraph skolem node list A list of the skolem nodes which occur in the etree max incomp clists wrt etree 20 symmetry A hash table with etree nodes as keys and symmetry holder structures as values This information is built when the etree is deepened but does not appear to be used anywhere in the code 21 merged This is true if the etree has been merged 22 statuses A hashtable of nodes in the etree and their statuses See the discussion above and Dan s email in section 1 1 23 name This symbol is the name of the eproof 24 lemmas This slot contains information about the lemma structure in the expansion proof See section 11 118 13 MATING SEARCH 1 3 Relevant Global Variables In addition to the eproof slots there are some global variables which store information relevant to the current etree The following global variables are used by dual instantiation when REWRITE DEFNS or REWRITE EQUALITIES is set to LAZY2 or equivalently DUAL 1 instantiated defs list The value is an association list of symbols and the shall
303. s will prompt you for a comment after a few sta tistics about the diagram are given The comment will be spread across the top lines of the diagram with carriage returns placed where you type them 7 Global Parameters and Flags The following Lisp identifiers are either flags or values used by the func tions which read or write formulas CFontTable This is a two dimensional array which is used to trans late between special characters on the Concept screen and their internal name For example CFontTable 1 91 is AND FIRST ORDER PRINT MODE If T wffs will be printed in first order mode otherwise in higher order mode FIRST ORDER MODE PARSE If T wffs will be parsed in first order mode otherwise higher order parsing mode is in effect See the section on parsing for a more detailed explanation LOWERCASERAISE If this identifier is set to T then lower case letters will be converted to their upper case equivalents This con version is done when the formula is first parsed The default value is NIL PC A variable used by the formula printing functions It stores the previous character printed It is used to help determine spacing within the formula Set to NIL in prt lisp Not important to the user PRINTDEPTH When a formula is printed subformulas at a depth of more that PrintDepth are not printed but replaced by a amp In the formula editor it is set to EDPRINTDEPTH A PRINTDEPTH of 0 means that the formula will be prin
304. se as the bound variable in the shallow formula One child of this expansion node is POSF wee LEF with expan sion term w This child is constructed and is used to solve the constraint by calling make min setvar lemma posf 4 If NUM OF DUPS is greater than 0 we also have NUM OF DUPS many other children of POS por F2 expanded using expan sion variables These children could be used in the proof of the theorem e make min setvar lemma posf 4 This function creates more expansion nodes corresponding to the rest of the banned vari ables of the constraint Since LF3 LF2 only has one banned variable we skip directly to constructing an implication node POSF LF here the first child is a negative ftree POSF and the second child is a positive leaf with shallow formula vt S w the same as LF3 This is connected to the node LF3 The negative ftree POSF is constructed by make min setvar lemma posf 5 and used to block LF2 e make min setvar lemma posf 5 This function constructs a conjunction corresponding to the extra literals in a given con straint In this case we only have the one extra literal LF2 So POSF is POSH constructed by make min setvar lemma posf 6 e make min setvar lemma posf 6 Given a literal in a con straint this creates a corresponding leaf and mates it to the literal in the constraint Since LF2 is positive POSEL isa negative leaf with shallow formula vt w and this is mated to LF2 So to sum u
305. searchtype default ms90 3 subjects mating search change fn lambda flag value pvalue when neq value pvalue update otherdefs value irrelevant kids neq default ms ms98 1 ms98 num of dups relevant kids eq default ms ms98 1 ms98 num of dups mhelp or defflag ms98 num of dups default nil flagtype null or posinteger subjects ms98 1 irrelevancy preconditions default ms neq default ms ms98 1 relevancy preconditions default ms eq default ms ms98 1 mhelp The conditions given are compiled into a relevancy graph and an irrele vancy graph The graphs are labelled directed graphs where the nodes are 92 9 FLAGS flags and the arcs are labelled by the conditions These graphs are currently used in the following two ways 1 The relevancy graph is used by the UPDATE RELEVANT com NS mand The user specifies a flag to update Based on the value given for that flag the user is then asked to specify values for the tar get flags for which the condition on the arc is true For example consider the following session lt O gt update relevant default ms DEFAULT MS MS98 1 gt ms98 1 MS98 NUM OF DUPS NIL gt 2 lt 1l gt update relevant default ms DEFAULT MS MS98 1 gt ms90 3 lt 2 gt The irrelevancy graph is used to warn the user when an irrelevant flag is being set A flag Fo is never irrelevant if there are no arcs with Fo as the t
306. seful for commands like FI find the first infix operator or ED move to edit the first ill formed subpart The argument type ED COMMAND was introduced for this purpose only 80 7 WELL FORMED FORMULAE OPERATORS fnspec If the value is none of the above but is specified it is assumed to be an arbitrary function of one argument which is applied to the result returned by the edop EDWFF ARGNAME This is the name of the argument that will be filled with the edwff see the ARGNAME property of MExprs in section 1 for more information DEFAULTFNS See the arguments for MExprs in section 1 MOVE FN This means that the result of the operation will be the new current wff and moreover that the operation qualifies as a move namely that we should store what we currently have before execut ing the command and then use replace fn on the value returned after then next 0 or For example the editor command A moves to the function part of an application Moreover when we return via 0 or we need to replace this function part 5 Useful functions A useful function in defining edops is EDSEARCH EDSEARCH gwff pred icate will go through gwff from left to right and test at every subformula whether predicate is true of that subformula If such a subformula is found EDSEARCH will return a list of editor moving commands which will move down to this subformula If the predicate is true of the guff itself EDSEARCH will return P
307. ser asking for advice Here is how the DONE EXC and PROOF ACTION are used in the code of the DONE command We don t care if the PROOF ACTION was successful it will usually be but it s very important that the user knows when a DONE EXC was unsuccessful since it is used for automatic grading defun done if funcall get exercise testfn dproof do signal event done exc length get dproof lines msgf Score file updated msgf Could not write score file Trying again abort with G sleep 1 2 msgf You have completed the proof Since this is not an assigned exercise t the score file will not be updated signal event proof action done 2 The Report Package The REPORT package in TPs3 allows the processing of data from EVENTS Each report draws on a single event reading its data from the record file of that event The execution of a report begins with its BEGIN FN being run Then the DO FN is called repetitively on the value of the EVENTARGS in each record from the record file of the event until that file is exhausted or the special variable DO STOP is given a non NIL value Finally the END FN is called The arguments for the report command are given to the BEGIN FN and END FN The DO FN can only access these values if they are assigned to certain PASSED ARGS in the BEGIN FN Also all updated values which need to be used by later iterations of the DO FN or by the END FN shoul
308. sfonts lists the fonts available Because the font list is usually very long you may prefer to use xlsfonts grep lt fontname gt to check whether the font lt fontname gt is available e xfd fn lt fontname gt amp shows all the characters in lt fontname gt Dan Nesmith built the symbol fonts by starting with the vtsingle font because xterm requires a font that is exactly the same size as vtsingle How ever every character is now different from the original they were created by hand colouring the pixels It is however easier to edit an existing font than to create one from scratch The exact duplicates of the Concept ROM fonts are found in sym font1 bdf and symfont2 bdf for a total of 256 characters including a normal sized epsilon Unfortunately because of the limits below on xterm and lisp it was necessary to leave out some of these characters when making a single font now called vtsymbold The galsymbold font was created by splitting the font vtsymbold into 128 bitmaps then using an X10 program that would automatically blow each bitmap to the proper size then manually adjusting a few of the characters There is a real limitation of the xterm program in that you get only two fonts one for normal text and one for bold text We use the bold text font for symbols and switch back and forth between the fonts by sending the appropriate escape symbols The code for the appropriate escape symbols are in xterm lisp In this
309. sions in infix form return the list A B some message the token succeed and the validation lambda x y x y If now we solve the new goals A and B i e find their prefix forms and evaluate them and apply the validation as a function to their solutions we get a solution to the original goal A B When we use a tactic we must know for what purpose the tactic is being invoked We call this purpose the use of the tactic Some examples of uses are nat ded for carrying out natural deduction proofs nat etree for trans lating natural deduction proofs to expansion proofs not yet implemented and etree nat for translating expansion proofs to natural deductions A single tactic may have definitions for each of these uses In contrast to tac tics tacticals are defined independent of any specific tactic use some of the auxiliary functions they use however such as copying the current goal may depend upon the current tactic use For this purpose the current tactic use is determined by the flag tacuse Resetting this flag resets the default tactic use Though a tactic can be called with only a single use that tactic can call other tactics with different uses See the examples in the section Using Tactics Another important parameter used by a tactic is the mode There are two tactic modes auto and interactive The definition of a tactic may make a distinction between these two modes the current mode is determined by the fla
310. sitions o or a function type af where a is a set type and is any type A term of a set type represents either a proposition a set or a relation When we refer to set variables we generally mean an expansion variable of a set type To find an expansion proof with expansion variables we may need to instantiate these variables Propositional variables are relatively easy to instantiate since there are only two possible truth values T and L It is far more difficult to instantiate set variables which are not of propositional type Semantically these set types may correspond to infinite domains Syntactically we have logical constants and quantifiers which can be used to create terms of set types In some cases but certainly not in all cases the instantiations can be find using higher order unification For instantiations which require logical constants or quantifiers the original method used by TPs3 is that of PRIMSUBS see section 1 For references on PRIMSUBS see And89 ABI 96 and And01 Some work regarding instantiating set variables in other contexts include Ble77 Ble79 Ble83 BF 93 and BBP93 The SCAN algorithm see reduces some second order for mulas to equivalent first order formulas avoiding the need to instantiate the set variables 1 Primitive Substitutions Set variables can be instantiated in a pre processing stage using Prim itive Substitutions This depends on the value of several flags There is
311. st order problem TPS quickly starts to unify the atoms of the mated literals When the unification process encounters a flexible rigid pair which we designate by lt f H gt where the flexible term has head variable f and the rigid term has a head of the form H the following substitutions for f are generated 1 Projections 2 Aw Aw f where information is attached to ft which does not permit a substitution of this same type i e introducing a nega tion to be applied to ft 3 Aw Aw f where information is attached to f which does not permit the first two of these types of substitution to be applied to it The information is stored by putting the variables into the lists neg h var list and imitation h var list The restrictions on f assure that the dpair which is essentially lt f H gt can only be used to generate new substitutions for f if other substitutions reduce H to a form which does not start with a negation 8 1 Sunil s Disjunction Heuristic 1 If a matrix contains A V B and A has no mate then no mate for B will be sought 2 If a matrix contains A V B and A has a mate but no mate for B can be found then the search will backtrack throwing out the mate for A and all links which were subsequently added to the mating Remark This heuristic is also used by Matt Bishop s search procedure ms98 1 See his thesis Bis99 for more details
312. stitute l term var result gt edwff edwff argname inwff defedop db alias delete leftmost binder result gt execute edwff argname gwff defwffop delete leftmost binder argtypes gwff resulttype ed command argnames gwff arghelp gwff 82 7 WELL FORMED FORMULAE OPERATORS mhelp Delete the leftmost binder in a wff defun delete leftmost binder gwff let bdwff cmds find binder gwff append ldiff bdwff cmds member p bdwff cmds sub delete binder edwff defwffop delete binder argtypes gwff resulttype gwff argnames bdwff arghelp bound wff applicable q ae bd wff p applicable q ae bd wff p mhelp Delete a top level universal or existential binder defun delete binder bdwff cond label q bdwff apply label bdwff delete binder bdwff lsymbol q bdwff throwfail Cannot delete binder from bdwff gwff a logical symbol boundwff q bdwff cdr bdwff t throwfail Cannot delete binder from an application 6 1 Global Parameters and Flags The following are the flags and parameters controlling the output of the editing session Note that there are also editor windows which have separate flags type SEARCH EDWIN T to see a list of these PRINTEDTFILE The name of the file in which wffs are recorded PRINTEDTELAG If T a copy of the current editing in ED will be printed into the file given by PRINTEDTFILE The prompt will als
313. system On anything except an HP write to gripe cs and ask them to do it unless you re confi dent about being able to do such things On an HP you can use their SAM program when nobody else is logged in since you re going to do a reboot as follows 1 Log in as superuser and type sam 2 Double click on Kernel Configuration 3 Double click on Configurable Parameters 4 Highlight the parameter maxdsiz and select Modify from the Actions menu Increase the value as high as you want On our machines it was initially 0x04000000 and we increased it to 0x7B000000 If you choose too high a number it will be rejected and you can try again 5 Check that the value pending column shows your new value for maxdsiz If not pick Refresh Screen from the Options menu and do the last step again 6 Now do the same for maxssiz we increased it from 0x00800000 to 0x04F B0000 7 Choose Exit from the File menu You will get a barrage of questions say yes to all of them They will be something like create the kernel now replace the old kernel reboot the system When the reboot is done type limit to check that the values have increased 3 3 2 Swap Space Swap space is that part of the memory usually on disk where the operating system stores parts of the programs that are sup posed to be in memory This is how you can get away with running more programs than your RAM has space for Clearly the a
314. t T such that for any q with t Ty we have q lt g q Together we have q lt Q q lt Q q for any q with t T4 O Proposition Given an expansion tree Q and a substitution 0 for Vg the dependency relation for 6 Q is acyclic iff 0 is acyclic with respect to Q Proof Suppose we have q lt Q q for some q E Ug U Vg By Lemma 1 there is a t Ty such that for any t Ty O t lt o Q 0 t In particular a t lt a Q A t Suppose we have s lt gg s for some expansion term s in 6 Q Since 6 Q is obtained from Q by substitution and no deepening there is a unique expansion term occurrence t Og such that s 0 t By Lemma 2 there isa q EQ U Vg with t Ty such that for any q Ng U Vg with t T we have q lt q q In particular q lt Q q O After pre processing we can compute the set Q and Vg as well as the sets B t B v and S t for each t Q and v Vg So to check that the acyclicity condition is satisfied when unification generates a substitution 0 it suffices to compute S v for each v E Vo with respect to 0 and check for a lt Q cycle Efficiency Refinement Clearly if for some s 4 Q either B s or S s is empty then s cannot be part of a cycle with respect to any substitution 0 so we may disregard any such term 5 Expansion Tree to Jform Conversion In ms90 8 node the jform is computed directly from the etree without using the jform wh
315. t In particular if a wffop has been declared to return an object of type BOOLEAN this wffop may return anything but NIL is printed as NIL while everything else is printed as T TYPESYM A type symbol in string representation This is ex tremely useful for error messages inside THROWFAIL For exam ple the type inference program may contain a line throwfail Type ti typesym does not match t2 typesym For most settings of the STYLE flag this will print the types as true greek subscripts GVAR A general variable This is only one of a whole class of possi ble subtypes of wffs GWFF The GETFN for these special kinds of wffs can easily be described using the function GETWFF SUBTYPE which takes a predicate as the first argument an RWFF as the second 67 68 7 WELL FORMED FORMULAE OPERATORS As an example for the definition of a subtype of GWFF serves the definition of GVAR deftype gvar getfin getwff subtype gvar p gvar testfn gvar p printfn printwffhere side effects t no side effects edwff mhelp A gwff which must be a logical variable 1 2 Defining Wffops The format for defining a wffop is very similar to that for defining a MExpr The function that does the definition is called DEFWFFOP The general format is enclose optional arguments DefWffop lt name gt ArgTypes lt typel gt lt type2 gt ResultType lt type gt ArgNames lt namel gt lt name2 gt
316. t dp EXPO lt Mate37 gt etd EXPO LEAF1 y70 I LEAF1 P OI y 0 I lt Mate38 gt 1 LEAF 1 lt Mate39 gt mod status 0 lt Mate40 gt up LEAFO lt Mate41 gt etd LEAFO EXISTS y I P OI y PREDECESSOR is also used in case a node s name is not found in the statuses hashtable so effectively a node can inherit the status of the node from which it was created Dan The file etrees debug contains functions useful for debugging code deal ing with etrees The function check etree structure recursively checks struc tural properties of an etree and the function check etree structure break calls check etree structure and calls a break if the etree fails the structural test The idea is that one can temporarily insert check etree structure break lt etree gt in suspicious parts of the code to find out when an etree loses its integrity If the etree does not have structural integrity a break is called sending the user programmer to the debugger If one wants to insert this in several places in the code one may want to include a message as in check etree structure break lt etree gt unique identifying message to identify which caused the break 1 DATA STRUCTURES 115 1 2 The Expansion Proof In the mate toplevel we have an expan sion proof stored in the special variable current eproof which is an eproof structure current eproof has a attribute etree whose value is often used to update variable current topno
317. t gives the time and date stringdtl also inserts linefeeds e princ often puts messages into a buffer To get them to print out add the command finish output You may also have to do this when you use other output commands including msg e Windows proofwindows edwindows vpwindows all work by issu ing a Unix shell command which runs an xterm which in turn runs the Unix tail command recursively on an output file that TPs3 creates by temporarily redirecting standard output Compare such commands as SCRIPT and SAVE WORK which permanently redirect standard output See the files tops20 lisp prfw lisp edtop lisp and vpforms lisp for more information 8 Compiling as much as possible In defining new TPs3 objects we often define as a side effect new func tions For example when defining a new argument type we define a testfn and a getfn for that type based on the values for those properties that are given in the deftype form Currently all such functions are compiled by cleverly defining the def inition macros so that defun forms are created during the compilation of a file If you define new categories that will create such functions you will want to do something similar so that you aren t always running interpreted code See the files argtyp lisp and flavoring lisp for examples of how this can be done 9 Writing New Code Without Making A Nuisance of Yourself e Programmers should avoid referencing internal symbols of
318. tably SETUP SLIDE STYLE will change both the style and the default line length CONCEPT CONCEPT S this is the style used for a Concept terminal which might also occasionally also be useful to produce a file which can be displayed on the Concept terminal with CAT or MORE The difference between CONCEPT and CONCEPT S is that the latter assumes that your Concept is equipped with special charac ters and the former does not If special characters are available 2 PRINTING OF FORMULAS 49 you will then get types as greek subscripts the universal quantifier as V etc The default linelength is 80 GENERIC this style assumes no special features and defaults the linelength to 80 For example the existential quantifier shows up as EXISTS and types are enclosed in parentheses GENERIC STRING is much like GENERIC but prints in a format that can be re read by TPs3 SCRIBE corresponds to the style used by the Scribe text proces sor A file produced in this style has to be processed by SCRIBE before it can be printed All special characters superscripts and subscripts etc are available The main drawback of a SCRIBE file is that precise formatting as necessary for vertical path diagrams is impossible The font used is 10 point except when doing SLIDE PROOF when an 18 point font is used TEX is the output style used by the T gXtext processor A file produced in this style has to be processed by TFXbefore it can be printed All speci
319. tactic2 tactic3 tacticN on goal else re turn the result of applying tactic1 to goal then then tactic tactic2 Apply tactic1 to goal If this fails return failure else apply tactic2 to each of the subgoals generated by tactic1 If this fails on any subgoal return failure else return the list of new subgoals returned from the calls to tactic2 and the lambda expression representing the combination of applying tactic1 fol lowed by tactic2 Note that if tactic1 returns no subgoals tactic2 will not be called repeat repeat tactic Behaves like orelse then tactic repeat tactic idtac then then tactic1 tactic2 Defined by then tacticl then orelse tactic2 idtac idtac This tactical is taken from Fel86 then then tacticl tactic2 Acts like then except that no copying of the goal or related struc tures will be done ifthen ifthen test tacticl or ifthen test tactic1 tactic2 First evaluates test which may be either a tactic or if user is an expert an arbitrary LISP expression If test is a tactic and does 4 USING TACTICS 199 not fail or is an arbitrary LISP expression that does not evaluate to nil then tactic1 will be called on goal and its results returned Otherwise if tactic2 is present the results of calling tactic2 on goal will be returned else failure is returned test should be some kind of predicate any new subgoals it returns will be ignored by ifthen 10 seq
320. ted up to arbitrary depth PRINTTYPES If this is set to T type symbols will be printed at least once on all primitive symbols Otherwise no types are 8 SIMPLE METAWFFS IN TPS3 63 printed This defaults to T and can be toggled with the command shownotypes SailCharacters This is a list of pairs SYMBOL NUM Here NUM is the position in the SAIL character set for SYMBOL The following flags are used to control the way formulas are printed Usually the default setting of all these flags will be adequate For more information see the section on pretty printing in the TPs3 user manual PPWFFLAG if T formulas will be pretty printed This is the de fault setting except in the editor where you can achieve pretty printing with the Ited PP command FLUSHLEFTFLAG If T no line of a pretty printed formula will be indented The default is NIL FILLINEFLAG If NIL every argument of an associative infix op erator will have a separate line The default in NIL LOCALLEFTFLAG If T arguments of infix operators start in the same column as the operator The default is NIL ATOMVALFLAG If T the name of every atom will be printed below its value ALLSCOPEFLAG If T all punctuations will appear in the formulas to be printed No association to the left or precedence of logical connectives will be assumed 8 Simple MetaWffs in TPS3 Even though in TPps3 the principle metalanguage is of course Lisp it is often co
321. term and worrying about portability problems which actually should be minimal but with lots of different machines and versions of X out there not predictable But this approach is fairly easy to get to from the current state Give up on xterm There is a new version of gnu emacs version 19 which allows the use of more than one font in a buffer You can then run TPSs3 in a gnu emacs buffer and at the same time build in support for command completion hypertext documentation etc You can also get several windows from a single emacs now so you can still have the editor stuff pop up a separate window It might also be easier to bind in support for automatically running tex or scribe and displaying it This requires someone who can hack gnu emacs lisp but a lot of this stuff has already been done by somebody and it s just a matter of putting it together In this case you could probably roll the fonts together into a single large one CHAPTER 2 TPS Structures Notice that TPs3 has a command TLIST which outputs the same in formation as the Lisp command plist but formatted more readably So for example TLIST X2108 will show all of the slots in the structure X2108 which is a proof 1 TPS Modules See the introductory chapter for a discussion of what TPS3 modules are 1 1 The Tps3 Module Structure All modules are defined in one central file called DEFPCK You may want to look at this file to see examples of module definitions
322. that share a vertical path if we include literals corresponding to these nodes Currently the user may use the flag ALLOW NONLEAF CONNS to specify which nodes to include in the jform The flag ALLOW NONLEAF CONNS takes a list of symbols as its value If this list contains the symbol ALL then every node will have a literal in the jform If this list the symbol REWRITES is in the list then every rewrite node will have a literal in the jform giving a jform similar to the one dual instantiation produces though dual instantiation affects the structure of the etree If the name of any particular etree node is in the list then that etree node will have a literal in the jform The code also uses the slot allow nonleaf conns in the current eproof to decide which nonleaf etree nodes to include as literals in the jform After converting an etree into a jform TPS3 will perform dissolution see MR93 iteratively on each connection in the flag DISSOLVE and in the dissolve slot of the current eproof The resulting jform will not have any vertical paths that pass through these connections 6 Path Enumerator 6 1 Duplication Order Along a path the procedure stops at the first eligible universal jform A slot dup t in a universal jform tells whether it is eligible Then it starts from there to find the innermost universal jform on the path under the currently picked one and uses the innermost one as its candidate for next duplication This is fulfilled b
323. that will print out the help for a specific object For many simple categories e g context the function princ mhelp is sufficient this sim ply prints the mhelp string attached to the object in question Other cate gories need more complex help for example mexpr and so have their own specially defined mhelp functions An example of a mexpr which has some automatically constructed help information is the function EXTRACT TEST INFO in maint lisp When writing help messages or mhelp functions keep in mind that the information given should contain all the information that a user would want to know More detailed help for maintainers and programmers should be written down and incorporated into this manual not added into the online documentation 1 1 Mhelp and Scribe The online documentation can be used to generate a facilities guide so it is important that you be aware that the mhelp properties and mhelp functions you define for new objects or categories will be used to generate Scribe files Take a look at the files mhelp lisp and scrdoc lisp and see how this works You may need to set things up properly so that the entries you are introducing are put into the index of such guides Look at the file tpsdoc mss in the doc lib area to see how the indexing is done 1 2 Mhelp and ATEX The online documentation can also be used to generate a facilities guide using ATRX Take a look at the files mhelp lisp and laterdoc lisp and see how this wor
324. the constraint to the variables constructed by make min inv princ in this case x In this case we have two constraints First consider LF 1 i e v 0 In this case we only have one literal the main literal of the constraint Since 0 is a con stant and not a banned selected variable we make an equation x 0 between the argument of vt x and the argument of the main constraint literal v 0 In general there may be several arguments giving several equations Given these equations make min inv princ 4 constructs the formula I nvPP Second consider LF3 LF2 i e wut w vt Sw In this case the main literal is v S w If the argument S w were w we could replace w by x in the constraint for the purpose of computing the inversion principle Instead we make an 16 SET VARIABLES equation x S w and call make min inv princ 4 to construct the formula I nu P LER e make min inv princ 4 Constructs a formula existentially bind ing the remaining banned variables in the constraint Since LF1 LF1 LF1 has no banned variables Inv P is InvP con structed by make min inv princ 5 Since LF3 LF2 has the banned variable w I nuP LEF is of the form Jw Inve une where I nup LEF is constructed by make min inv princ 5 e make min inv princ 5 Constructs a conjunct of the equa tions generated in make min inv princ 3 corresponding to the constraint For the constraint LF1 InvPF is x 0 Since in th
325. the dynamic variable clist e make clos setvar lemma negf 0 Our goal is to construct a negative ftree NEGF for the set existence lemma qv Y wt v wt D v Sw A v20 AVal v2 a2 D 3wjrt SwA v w v zt 0 AYp Mwt pwt D p Sw A pO D Va2 v 27 gt p In general the set existence lemma universally binds the vari ables in misc occurs In this case there are no such variables We also need to construct the positive Knaster Tarskit node expf and add connections to clist between nodes in these two ftrees We proceed to the most important step instantiating the K in the Knaster Tarski Theorem The monotone function we want 180 16 SET VARIABLES can be extracted from the inversion principle The inversion principle inv princ is Yr vt z D Iw lr SwAvivi V z 0 and we construct the monotone function monfn AvtA z Jule SwA vtw V z 0 Let us write AvlAg P v x for this term Note that P vt is positive with respect to vt so the function will be monotone We substitute this for K so that Knaster Tarski is an expansion node EXP Knaster Tarksi Avt Ag P v A REW ate IMP N MONNEGF EUFt with one child using the monotone function as the expansion term The child of this node is a A rewrite passing to the normal form Below this is an implication node with two chil dren MONNEGEF providing a proof that v Ax P v is monotone and EUF a positive node with shallow for
326. the line qload nat etr into tps3 patch In general don t put qload nat etr lisp into the patch file or the uncompiled version of the file will be loaded However the export files exp do need their extension Entries such as qload auto exp which load exp files should come before those loading lisp files qload core exp should come before loading other export files Macro files should come before other files in the same module Putting the line setq core always compile source if newer T near the beginning of the tps3 patch file and setq core always compile source if newer NIL at the end of the same file will cause files to be compiled automatically whenever appro priate as one is starting up TPs3 but then restores the default value of always compile source if newer so that you will be able to decide whether or not to compile other files as you load them Example when ms91 6 and ms91 7 were introduced tps3 patch con tained qload core exp qload auto exp qload defpck qload contexts auto load module ms91 qload diy 3 3 Memory Management TPsS3 uses a huge amount of memory in the course of a long search and it may be necessary to rearrange either the internal memory available in your computer or the maximum space occupied by your version of Lisp Both of these things vary the former by system type sys to find out what system you are using the latter by the variety of Lisp Y
327. the next one e g the active planned line b copy tac goal copy the current goal into a new goal so that subsequent actions can be performed without destroying the current goal Allows later backtracking if necessary c save tac goal put the current goal into a form suitable for saving This is not actually used by current tactics d restore tac goal backtrack to the previous goal This is not actually used by current tactics e update tac goal given the old saved goal and the new goal on which some progress has been made update the old goal to reflect the progress made Tacticals must be independent of the value of tacuse They cannot make any assumptions about the structure of the goals etc The main function used in applying tactics is apply tactic This is a function that takes a tactic as argument and allows keyword arguments of goal use and mode If not specified the use and mode default to the global values of tacuse and tacmode If they are specified the values given then override the global values of tacuse and tacmode apply tactic and every tactic returns four values The first is a list of goals the second a string with some kind of message the third a token which indicates the result of the tactic and the fourth a validation which if non nil should be a function which specifies how solutions to the returned goals can be combined to solve the original goal apply tactic works as follows a Chec
328. these There are many such redexes The following is a typical example D A B Conj AAB D Conj gt A Al In first order logic one can show that some measure on the proof reduces when a redex is reduced so that the process will terminate with a normal proof In higher order logic showing termination is equivalent to showing termination of cut elimination Actually carrying this out is a possible future project Though this is much less important since we now have a cut elimination algorithm imple mented 3 NPROOFS TO EPROOFS 215 3 2 Hongwei s Nat Etree This is a brief description of Hongwei s code for NAT ETREE To use this code set NAT ETREE VERSION to HX ATTACH DUP INFO TO NATREE is the main function which is called recursively on the subproofs of a given natural deduction The goal of ATTACH DUP INFO TO NATREE is to construct an expansion tree with no mating attached corresponding to a given natural deduction The con structed expansion tree contains all the correct duplications done on quanti fiers and all substitutions done on variables A propositional search will be called on the generated expansion tree to recover the mating and generate an expansion proof Then ETREE NAT can produce a natural deduction corresponding to the constructed expansion proof The following is an oversimplified case Given natural deductions N1 and N2 with conclusions A and B re spectively and N derived from N1 and N2 by c
329. tive tactic Syntax 195 PRIMSUB METHOD Flag 90 167 princ mhelp 85 princ mhelp latex 85 Print Function 56 print symbol Function 51 PRINTDEPTH Flag 62 88 PrintDepth Parameter 47 48 PRINTEDTFILE Flag 82 PRINTEDTFLAG Flag 82 PRINTEDTFLAG SLIDES Flag 82 PRINTEDTOPS Flag 82 PrintFinDover Function 54 PrintFnTTY Function 54 PrintPPlist Function 54 PRINTTYPES Flag 61 62 PrintTypes Flag 47 48 PRINTVPDEFLAG Flag 82 PrintWff Function 51 54 PrintWffPlain Function 50 54 PrintWffScope Function 50 54 prompt read 13 prompt read Function 18 proof to natree Function 205 PROP FIND CHEAPEST PATH Function 138 PROP MSEARCH Function 188 prop msearch Function 127 proposition var 161 propsym Syntax 34 PROPSYM Q Function 72 prt lisp File 62 PrtWff Function 48 50 prune status 0 Function 136 prune unmated branches Function 135 187 139 141 PWff Function 50 54 PWScope GWff MExpr 47 QUERY USER Flag 16 259 quick unification connection Function 118 raise lambda nodes Function 150 151 raise lambda nodes ab Function 150 raise lambda nodes aurl Function 150 raise lambda nodes equiv Function 150 raise lambda nodes neg Function 150 raise lambda nodes skol Function 150 randomvars Function 161 RdCList Data 83 read line sym Function 246 reduce rewrites Function 155 REFL 116 relation var 161 remove double negations merge Function 156
330. trees using recursive algorithms One must constantly update global information There is an alternative representation called ftrees implemented in the file ftrees Ftrees are designed for functional programming the f is for functional Operations on ftrees are never destructive and the information carried at each node is minimal Functions etree to ftree and ftree to etree translate between the two representations Finally I was convinced that we needed a different representation de signed for functional programming I have implemented this alternative rep resentation ftrees and translations between the two representations 120 13 MATING SEARCH The new representation also allowed me to write code to save and re store expansion trees The commands are SAVE ETREE and RESTORE ETREE 1 5 Other Structures e A mating has certain attributes 1 A set of connections 2 A unification tree A unification tree is a tree of nodes e A uni term is an attribute of a node which is a structure it is a set of disagreement pairs e A failure record is a hashtable MS88 and the other non path focused procedures uses the failure record MS90 3 and the other path focused procedures does not use it this is one reason why when TPs3 abandons a mating and later returns to the partially completed eproof ms91 6 continues approximately where it left off and ms91 7 does not Links which all occur in the connection
331. truct by making an im plication whose antecedent is the conjunction of the line nodes of the supports and whose consequent is the line node of the planned line This will assure us that we have sufficient infor mation to carry out the translation It was observed that when path focused duplication had been used the expansion proof would often have a great deal of redundancy in the sense that the same expansion term would be used for a given variable many times More precisely if one defines an expansion branch by looking at sequences of nested expansion nodes attaching one expansion term to each expansion node in the sequence there would be many identical expansion branches In response to this mating merge lisp was modified in the following ways e Don t do pruning of unnecessary nodes at the beginning of the merge when the tree is its greatest size e Instead prune all branches that couldn t possibly have been used they are those that have a zero status This is probably not neces sary but certainly makes debugging easier and doesn t cost much e After merging of identical expansions has been done call the origi nal pruning function 3 NProofs to Eproofs There are three versions of NAT ETREE the command for translating natural deductions into expansion tree proofs The user can choose between the three by setting the flag NAT ETREE VERSION to one of the following values 1 OLD the original version 2 HX Hong
332. uence sequence tacticl tactic2 tacticN Applies tactic1 tacticN in succession regardless of their success or failure Their results are composed 11 compose compose tactici tacticN Applies tacticl tacticN in succession composing their re sults until one of them fails Defined by idtac if N 0 then tacticl compose tactic2 tacticN ifN gt 0 12 try try tactic Defined by then tactic failtac Succeeds only if tactic returns no new subgoals in which case it returns the results from applying tactic 13 no goal no goal Succeeds iff goal is nil 4 Using Tactics To use a tactic from the top level the command use tactic has been defined Use tactic takes three arguments a tactic erp a tactic use and a tactic mode The last two arguments default to the values of tacuse and tacmode respectively Remember that a tactic erp can be either the name of a tactic or a compound tactic Here are some examples lt 1 gt use tactic propositional nat ded auto lt 2 gt use tactic repeat orelse same tac deduct tac interactive lt 3 gt use tactic sequence call pall call cleanup call pall lt 4 gt use tactic sequence foo use nat etree mode auto bar use nat ded mode interactive Note that in the fourth example the default use and mode are overridden by the keyword specifications in the tactic exp itself Thus during the execution of this compound tactic foo will
333. ues like those returned by a primitive tac defn compound tacl defn tac lambda symbol tactic exp Here the tactic exp should use the symbols in the tac lambda list as dummy variables 2 SYNTAX FOR TACTICS AND TACTICALS 197 Here is an example of a definition of a primitive tactic deftactic finished p nat ded lambda goal if proof plans dproof progn when tactic verbose msgf Proof not complete t values nil Proof not complete fail progn when tactic verbose msgf Proof complete t values nil Proof complete succeed Returns success if all goals have been met otherwise returns failure This tactic is defined for just one use namely nat ded or natural de duction It merely checks to see whether there are any planned lines in the current proof returning failure if any remain otherwise returning success This tactic is used only as a predicate so the goal list it returns is nil as is the validation As an example of a compound tactic we have deftactic make nice nat ded sequence call cleanup call squeeze call pall Calls commands to clean up the proof squeeze the line numbers and then print the result Again this tactic is defined only for the use nat ded sequence is a tactical which calls the tactic expressions given it as arguments in succession Here is an example of a primitive tactical deftactical idtac defn lambda goal tac list values if goal
334. unction components aa FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 FORALL u lt I gt EXISTS v lt I gt OR LIT2 AND LIT3 LIT4 FORALL w lt I gt OR LIT5 LIT6 EXISTS u lt I gt FORALL v lt I gt OR LIT7 LITS8 lt Ed20 gt length bb 4 lt Ed21 gt car bb FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 lt Fd22 gt cadr bb FORALL u lt I gt EXISTS v lt I gt OR LIT2 AND LIT3 LIT4 lt Ed23 gt auto jform parent car bb AND FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 FORALL u lt I gt EXISTS v lt I gt OR LIT2 AND LIT3 LIT4 FORALL w lt I gt OR LIT5 LIT6 EXISTS u lt I gt FORALL v lt I gt OR LIT7 LIT8 lt Ed24 gt setq cc FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 FORALL X lt I gt OR FORALL Y lt I gt LITO FORALL Z lt I gt LIT1 4 DEFINING AN EDOP 79 lt Ed26 gt bb FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 FORALL u lt I gt EXISTS v lt I gt OR LIT2 AND LIT3 LIT4 FORALL w lt I gt OR LIT5 LIT6 EXISTS u lt I gt FORALL v lt I gt OR LIT7 LIT8 lt Ed27 gt car bb FORALL x lt I gt OR FORALL y lt I gt LITO FORALL z lt I gt LIT1 lt Ed28 gt cc FORALL X lt I gt OR FORALL Y lt I gt LITO FORALL Z lt I
335. unction will choose it Since a new line properly located is always a correct and possibly a useful default tactics may miss an opportunity to apply a rule Subsequently we will assume that default exist is p di1 diz Pp dpi dpzp default new is q1 11 1y dq q1 Cqyg Requirements for Logical Correctness Ry qj lt pi fr all 14 lt p 1 lt 7 lt gi Ro ejk lt qj forall 1 lt j lt q 1 lt k lt yj Rs dik lt pi for all 1 lt i lt p 1 lt k lt zri Sensible Defaults Requirements The requirements below only make sense if the lines specified by the user do not already violate them In that case they must be relaxed to apply only to the remaining unspecified lines D A plan support pair suggested for an element of default exist must always match a plan support pair in the current support structure of the incomplete proof D A plan support pair suggested for an element of default new must consist of entirely new lines and no two lines among all the sug gested defaults may have the same number 104 12 ETPS Wishful Thinking The following are constraints we would like to met but is of course not always possible Wy qj lt qj for alll lt j lt q We qj lt ej 1 k for all 1 lt j lt q and 1 lt j lt qj 1 W3 dik lt eg for all 1 lt i lt p 1 lt k lt zr1 lt j lt q 1 lt 1l lt yj Wa 3L max ejk lt L lt q for all 1 lt j lt q
336. using the WFF TYPE property of the binder An example may illustrate this process The binder LAMBDA has the following properties TYPELIST A B VAR TYPE B SCOPE TYPE A WFF TYPE A B When trying to determine the type of Ax Ro x TPS3 determines that A must be v and that B must be ov The type of the original formula is A B which then turns out to be ou Note that TYPELIST may be absent i e could be which amounts to stating that the binder has no variable types Currently we are not using 5 BINDERS IN TPS 39 such binders An example would be Foralin which can bind only variables of type a In addition to the properties mentioned above a binder except A would have a definition One can then instantiate a binder just as a definition can be instantiated The definition is to be written with two designated variables one for the bound variable and one for the scope For example THAT has definition Leon ADaSo Here the TypeList would be a designation for the bound variable would be ba designation for the scope would be So The internal representation for a binder inside a wff is always the same and simply bdvar binder scope but all of the above informa tion must be present to determine the type of a wff or to check whether formulas are well formed Fancy special effects such as Vx S A must be handled via special flavors of labels and are not treated as proper binders themselves Here are so
337. ussion above Section 2 3 and the portion of the user s manual on setting things up for more details The following is a list of steps for com piling TPs3 under a new lisp 1 Modify the line in the Makefile of the form lisp lt lisp executable gt so that lt lisp executable gt is the executable for the new lisp Try to perform make tps If you are very lucky this will work However probably the first problem you will encounter involves conflicts with respect to exporting symbols We explain how to resolve these conflicts in step 3 Once these conflicts are resolved go on to step 4 Suppose the implementation of lisp complains about a conflict with an exported symbol tps symbol We try to keep such conflicts localized to the file special exp If tps symbo1 is exported in the file special exp then add a compiler directive or to the export for tps symbol If tps symbol is exported in some other export file then move this export to special exp and add an appropriate compiler directive See Section 2 3 for more information about compiler directives and the features lisp variable The next thing you will likely need to do is add definitions for cer tain functions and macros For example the functions tps3 save N xm w iagt A N 6 1 INTRODUCTION linelength exit from lisp status userid call system setup xterm window setup big xterm window make passive socket make passive socket port
338. ut this is by no means essential for vertical path diagrams Desc Jform 5r lom Height Width Width Width Gen CharList PPlist OR Desc JForm Height Width Cols AND Desc JForm Height Width Rows EeRate Var Desc JForm Height Width Width GenCharList Height 0 1 2 Width 0 1 2 Cols 0 1 2 Rows 0 1 2 In a Descr JForm the second and third element Height and Width contain the height and width of the JForm that is described by the Descr JForm In case the JForm was a literal or a signed atom the next two elements are lists The left element of each of these sublists gives the width or print representation of the literal or atom the right element gives the width or print representation of the literal s or atom s value If the JForm was a conjunction or disjunction the last element of the corresponding Descr JForm is a list of the rows or columns in which the conjuncts or disjuncts begin If we deal with a top level quantifier in our JForm the last two elements contain the width and the print representation of the quantifier together with the quantified variables For a description of a GenCharList or PPlist see section 3 2 3 6 Some Functions The function which is called by VPForm and VPDiag is VPForm The handling of the comment and the different files that have to be opened is done here The main function which translates a JForm into a Descr JForm is
339. ution which does not create any new nodes in Q This is in contrast to substituting and then deepening the tree In particular the set of selected variables is the same i e Sg Sag Also for the same reason for each t Ne B t is the set of selected variables y whose selection node is dominated in 0 Q by the expansion term occurrence 6 t Similarly for v Vo y B v means there is some expansion term t Og such that v occurs free in t and y is dominated in 6 Q by 6 t Remark 2 Note that we do not consider A terms equivalent up to A conversion The expansion tree must have explicit rewrite nodes to normalize formulas For this reason if y is free in a term t in Q it will also be free in 0 t in 0 Q That is we can never project the y away because this would require not just substitution but also deepening 6 Q Lemma 1 Let an expansion tree Q and a substitution 0 for Vg be given Suppose q lt Q q Then there exists a t T such that for every t Ty O t lt 6 Q Ot Proof By induction on the number of transitivity steps For the base case suppose q lt Da q Then there is some y B q NS q Since y B q there is some t T such that y is dominated by t in Q So y is dominated by 0 t in 6 Q Now suppose t Ty We consider two cases If q Ng then t q and y S q So y is free in t By Remark 2 y is free in 6 t and we are done If q Vo th
340. v A InvP If INCLUDE INDUCTION PRINCIPLE is T we construct a positive ftree for Main v A InvP A IndP where IndP is an induction principle In this case the induction principle is Vp Main p D Vx v 27 D px The positive ftrees for the inversion principle and the induc tion principle are simply constructed by expanding the for mula as an ftree duplicating the outermost quantifier NUM OF DUPS times These are combined using conjunction nodes with POS F gt e make min setvar lemma posf 2 For each constraint P we make a conjunction of POSFP obtained by calling make min setvar lemma posf 3 In this case we have a conjunction of POSFETS FP and POSEY e make min setvar lemma posf 3 If there are banned vari ables in the constraint we will make an expansion node In this case we will have one child that corresponds expanding 16 SET VARIABLES using the banned variables so we can mate to the literals in the constraints In case we will later want to use this part of the lemma elsewhere in the proof we also duplicate NUM OF DUPS times The constraint LF1 does not contain any banned variables and has no extra literals So we simply let PosF be a positive leaf with shallow vt 0 the same shallow as LF 1 This leaf is connected to the negative node LF1 solving this constraint The constraint LF3 LF2 contains the banned variable w So POS po LF isan expansion node We create a fresh vari able w to u
341. v C w C A K v C K w Then there is a fixed point u of K Proof Let u v p A K v C v That is we define u to be the intersection of all the pre fixed points of K We need to show K u C u and u C K u First we show K u C u Suppose z K u To show z u we need to show z v for every pre fixed point v Let v be a pre fixed point By the definition of u we have u C v Since K is monotone K u C K v so 172 16 SET VARIABLES z K v But v is a pre fixed point so z K v C v Thus u is itself a pre fixed point In fact it is clearly the least pre fixed point Since u is a pre fixed point and K is monotone we have K K u C K u So K u is a pre fixed point Since u is the least pre fixed point we have u C K u This proof actually shows the following form of the theorem Knaster Tarski Fixed Point Theorem Least Suppose K A gt p A for a power set o A Further suppose K is monotone function in the sense that for every v C w C A K v C K w There is a least pre fixed point u of K which is also a fixed point of K A dual proof shows Knaster Tarski Fixed Point Theorem Greatest Suppose K 9 A gt p A for a power set o A Further suppose K is monotone function in the sense that for every v C w C A K v C K w There is a greatest post fixed point u of K which is also a fixed point of K These statements and proofs have a straightforward representation in type theory The same proof
342. vents TPs3 provides a function SIGNAL EVENT which takes a variable number of arguments The first argument is the kind of event to be signalled the rest of the arguments are the event args for this particular event SIGNAL EVENT will return T or NIL depending on whether the action to be taken in case of the event was successful or not Note that when an event is disabled see below signalling the event will always be successful There are basically three cases in which an event will be considered unsuc cessful if the SIGNAL HOOK is specified and does a THROWFALL if WRITE WHEN is IMMEDIATE and either the WRITE HOOK if specify does a THROWFALL or if for some reason the writing to the file fails if the file does not exists or is not accessible because it has the wrong protection for example It is the caller s responsibility to make use of the returned value of SIGNAL EVENT For example the signalling of DONE EXERCISE below 1 EVENTS IN TPS3 237 If WRITE WHEN is a number the evaluated templates will be collected into a list event LIST This list is periodically written out and cleared The interval is determined by EVENT CYCLE a global flag see description of WRITE WHEN above The list is also written out when the function EXIT is called but not if the user exits TPs3 with Note that if events have been signalled the writing is done without considering whether the event is disabled or not This ensures that events signalled a
343. ventually need to change it globally so as not to conflict with Common Lisp defwffop type argtypes gwff resulttype typesym argnames gwff arghelp gwff mhelp Return the type of a gwff defun type gwff cond label q gwff apply label gwff type gwff lsymbol q gwff get gwff type Cboundwff q gwff boundwfftype gwff t type car type car gwff The following are a sequence of functions which instantiate abbreviations One can either instantiate a certain abbreviation everywhere INSTANTIATE DEFN instantiate all abbreviations not recursively INSTANTIATE ALL or instantiate the first abbreviates counting from left to right INSTANTIATE 1 The functions are implemented by one master function one of whose arguments is a predicate to be applied to an abbreviation This predicate should return something non NIL if this occurrence is to be instantiated NIL otherwise Notice the subcases inside LSYMBOL Q and the order of the quick predicates in the OR clause defwffop instantiate defn argtypes symbol gwff resulttype gwff argnames gabbr inwff arghelp abbrev inwff applicable p lambda gabbr inwff declare ignore inwff 76 7 WELL FORMED FORMULAE OPERATORS or abbrev p gabbr pmabbsym p gabbr mhelp Instantiate all occurrences of an abbreviation The occurrences will be lambda contracted but not lambda normalized defun instantiate defn gabbr inwff
344. wei Xi s version written in the early to mid 1990 s 3 CEB Chad E Brown s version written in early 2000 Also note that setting the flag NATREE DEBUG to T is useful for debug ging the HX and CEB versions The subsections that follow describe each of these versions in greater detail After using NAT ETREE to translate to an expansion proof the user can use this expansion proof to suggest flag settings via the mate commands ETR INFO and ETREE AUTO SUGGEST or to trace MS98 1 using the 3 NPROOFS TO EPROOFS 205 flag MS98 TRACE See the User s Manual for a description of these facilities The User s Manual also has examples 3 1 Chad s Nat Etree To use this version of NAT ETREE set NAT ETREE VERSION to CEB The main functions for this version are in the files ceb nat etr lisp and ceb nat seq lisp The relevant functions are ceb nat etree This is the main function It preprocesses the proof to e remove applications of Subst and Sym e expand applications of RuleP and other propositional rules e g Assoc in terms of more primitive inference rules attempt to expand any applications of RuleQ replace instances of Assert by hypotheses which are discharged at the end of the proof e and replace applications of the Cases rule using more than two disjuncts by multiple applications of the Cases rule using two disjuncts The function then calls ceb proof to natree to build the natree ver sion of the natural dedu
345. xamples below are not complete in the sense that not the full description of the rule for the rules module is given we have merely extracted what is important in our context p and d are placeholders for a pline or any number of dlines respectively which are found in the support structure of the current proof but are merely copied in the application of the particular rule described Rule of Cases D1 H A 0 OR B 0 H2 H H2 A O Case 1 D1 P3 H H2 C O H4 H H4 B 0 Case 2 D1 P5 H H4 C O P6 H 1C 0 Cases D1 P3 P5 Support Transformation P6 D1 ss gt P3 H2 ss P5 H4 ss Note that the specified support transformation tells TPs3 what lines it expects to be there when the rule is applied and which lines should be new In this case P and Dab are expected to be new the others are to be 102 12 ETPS constructed Of course these are only defaults and the user can apply the rule with any combination of lines present or absent Induction Rule D1 H 1 PO H2 H H2 Pm Inductive Assumption on m D3 H H2 P Succ m P4 H FORALL n NAT n IMPLIES P n Induction D1 D3 Support Transformation P4 ss gt D1 ss D3 H2 ss Forward Conjunction Rule P1 H A 0 P2 H B 0 P3 H A 0 AND B 0 Conj P1 P2 Support Transformation P3 ss gt P1 ss P2 ss Backward Conjunction Rule D1 H A 0 AND B 0 D2 H A O Conj D1 D3 H B 0 Conj D1 Support Transformati
346. y calling function find nezt dup When testing please set flag max dup paths to an appropriate value so that you can suppress some unnecessary quantifier duplications It may save a lot of your searching time and make you aware if you are on the right track Always duplicating innermost quantifiers has the following advantages 1 producing shorter and clearer proofs and 2 lowering the values of flags max search depth max mates and num of dups sometimes 6 2 Backtracking When bactracking starts the search procedure re moves the last added connection A path attached to the connection tells the procedure where it should pick up the search This works efficiently since the following claim is almost always true With the help of disjunction heuristic the number of paths used to block jform is often a very small fraction of the whole paths in the jform This means that it is not a big burden to carry 8 CONTROL STRUCTURE AND INTERFACE TO UNIFICATION 127 the paths around all the time during searching The advantage is that the procedure knows exactly where it is without having to do heavy computa tion by using the information given by the current mating To make this work also carried with a path is an environment which stores the indices and partial substitutions for the variables in the path 7 Propositional Case In the file mating dir lisp you can see that the function ms director checks whether there are any free variables in
347. y of the leaf 3 SKOLEMIZATION 121 7 If A is a positive existential formula or negative universal formula create a skolem node or selection node depends on the value of SKOLEM DEFAULT with a leaf of the scope of the quantifier as its child 8 If A is a positive universal formula or negative existential formula create an expansion node with a leaf of the scope of the quantifier as its child 9 If A is an equation of the form t t and REWRITE EQUALITIES is not set to NONE then create a rewrite node with a leaf of TRUTH as its child 10 If A is an equation and REWRITE EQUALITIES is not set to NONE then rewrite the equation depending on REWRITE EQUALITIES and create an appropriate rewrite node 11 If A is a symbol introduced by dual instantiation of an equality replace it with the instantiated formula 12 Finally if there is a definition then there is a very complicated case which creates an appropriate rewrite node Anyone trying to figure out this part of the code needs to pay close attention to the value of REWRITE DEFNS Ordinarily REWRITE DEFNS is a flag whose value is a list of a form such as DUAL EAGER TRANSITIVE NONE INJECTIVE SURJECTIVE However at the beginning of deepen leaf node real the value of REWRITE DEFNS is dynamically set to a form DUAL SUBSET REFLEXIVE EAGER TRANSITIVE NONE INJECTIVE where all the abbreviations appearing the A are explicitly in the list and those not ap
348. y see that static parameters can be handled fairly easily while dynamic parameters can become a headache if we are trying to save information about the appearance of wffs and symbols 4 2 A grand solution and why it fails A first stab at a solution could be briefly described as follows During the printing of a wff we permanently attach relevant printing in formation like length depth or printing characters to each label and symbol in the wff When the label or symbol appears again somewhere else the information does not have to be recomputed We would then have to somehow code the information about the current static and dynamic parameters into the property of the label or symbol which stores this information With the aid of a hashing function this is straightforward for the static parameters since we can compute the name of the relevant property once and for all for the printing of a wff For dynamic parameters this is still in theory possible but in practice unfeasible We would have to recompute rehash the values of the dynamic and static parameters for each subformula To see that this is very difficult if not impossible consider the following example The simple wf Poaa La Ya May appear as Pry Poaa Y PXa Ya PLYa etc with almost endless possibilities for larger wffs All the information about which symbols should have types etc would have to be coded into the property name for say the printing length of a label This
349. z vtz gt InvP 3 SET CONSTRAINTS 177 Va vtz gt Inup 3E V InP Va v a gt 3w Inu p LF V InvPF Yg vtz D 3w z S w A InP v 0 Vg vtz gt 3wjz Su A fvt w v 2 0 2 make min setvar lemma posf Constructs the set existence lemma Jv Vwl v w D v Sw A v0 AVael v a gt Jwjr SwA v w V z 0 AYp Mwt pw D p Swt A p0 D V2 v2 x7 gt pz and a positive ftree used to solve the constraints In general this would start by universally quantifying any extra expansion variables and selected variables occurring in the constraints misc occurs In this case there are none We create a fresh variable v2 v to play the role of the set variable in the formula make min setvar lemma posf 1 is called to construct a positive ftree POSF The shallow formula of POSF is LEM v the body of the set exis tence lemma using the selected variable vt So we return a selected node POSF with shallow 3v LEM v and child POSF e make min setvar lemma posf 1 The main part of the lemma we need to construct is the part that solves the constraints The function make min setvar lemma posf 3 returns a postive ftree POSF with shallow formula Main v It also adds con nections between literals in POSF and the literals LF1 LF2 and LF3 of the constraints We have already constructed the inversion principle InvP If INCLUDE INDUCTION PRINCIPLE is NIL we construct a positive ftree for Main

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