Mini-Indexes for Literate Programs
Contents
1. 102 actually do fit This might happen if 100 and 101 have lots of references to variables declared in 102 Similarly we might be able to fit 100 with 101 if 99 had been held over from the previous spread But such situations are extremely unlikely and there is no reason to worry about them The one spread at a time strat egy adopted by ctwimac is optimum spacewise for all practical purposes On the other hand experience shows that unfortu nate page breaks between spreads do sometimes occur unless the user does a bit more fine tuning For example suppose the text of 7 in HAM had been one line longer Then 7 would not have fit with 6 and we would have been left with a spread containing just tiny little 6 and lots of wasted white space It would look awful And in fact that s the reason the three statements tark A v y goto advance now appear on a single line of the program instead of on three separate lines A bad break between spreads was avoided by manually grouping those statements using CWEB s command One further problem needs to be addressed the mini indexes must be sorted alphabetically TEX is es sential for determining the breaks between spreads and 12 consequently for determining the actual contents of the mini indexes but TEX is not a good vehicle for sort ing The solution to this problem is to run the output of CTWILL twice through TEX interposing a sorting program betwee2. No variable k is declared anywhere Indeed users who write comments involving expressions like f x might or might not be referring to identifiers f and or x in their programs they must tell CTWILL when they are making throwaway references that should not be indexed CWEAVE doesn t have this problem because it indexes only the definitions not the uses of single letter identifiers In the third place CTWILL will not recognize au tomatically that the vert parameter in the definition of not_taken 4 has no connection with the vert macro defined in 6 A fourth complication which does not arise in HAM but does occur in 3 and 7 is that sections of a WEB or CWEB program can be used more than once Therefore a single identifier might actually refer to several different variables simultaneously See for example 652 in 3 In general when an identifier is defined or de clared exactly once and used only in connection with its 5 unique definition CTWILL will have no problems with it But when an identifier has more than one implicit or explicit definition CTWILL can only guess which defi nition was meant Some identifiers especially single letter ones like x and y are too useful to be confined to a single significance throughout a large collection of programs Therefore CTWILL was designed to let users provide hints easily when choices need to be made The most important aspect of this design was to
3. Thus the following loop will print the names of all vertices adjacent to v for a v arcs a a a gt nezt printf s n a tip gt name An undirected edge between vertices u and v is repre sented by two arcs one from u to v and one from v to u Finally if g points to a Graph record then g n is the number of vertices in the associated graph and the Ver tex records representing those vertices are in locations grvertices k for O lt k lt gon 1 w name is actually typed v gt name in a C or CWEB program typographic sugar makes the program easier to read in print A Vertex record also contains utility fields that can be exploited in different ways by different algo rithms The actual C declarations of these fields quoted from 8 and 9 of the program GB_GRAPH 7 are as follows typedef union struct vertex_struct xV pointer to Vertex struct arc_struct A pointer to Are struct graph_struct G pointer to Graph char S pointer to string long J integer util typedef struct vertex_struct struct arc_struct arcs linked list of arcs out of this vertex char name string identifying this vertex symbolically util u v W Y Z multipurpose fields Vertex Program HAM uses the first four utility fields in or der to do its word efficiently Field u for example is treated as a long integer representing the degree of the
4. 2 t amp register amp Vertex 4 f mot _taken macro GB _GRAPH 10 tip amp Vertex 2 l v amp ftregister amp Vertex 2 l a amp fregister amp Arc 9 Here introduces an internal reference to another sec tion of HAM introduces an external reference to some other program typesets an identifier in text italics typesets an identifier in math italics amp typesets in boldface A special debugging mode is available in which TeX will simply typeset all the mini output at the end of each section instead of making actual mini indexes This makes it easy for users to check that CTWILL is in fact producing the information they really want No tice that mistakes in CTWILL s output need not neces sarily lead to mistakes in mini indexes for example a spurious reference in 6 to an identifier defined in 5 will not appear in a mini index for a spread that includes 85 Itis best to make sure that CTWILL s output is correct before looking at actual mini indexes Then unpleasant surprises won t occur when sections of the program are moved from one spread to another When TEX is finally asked to typeset the real mini indexes however it has plenty of work to do That s when the fun begins TEX s main task after format ting the commentary and C code of each section is to figure out whether the current section fits into the cur rent spread and if it does to
5. left hand or right hand page you are reading These footnote entries tell you whether the identifier is a pro cedure or a macro or a boolean etc 3 A similar idea is sometimes used in editions of liter ary texts for foreign language students where mini dictionaries of unusual words appear on each page 10 1 this saves the student from spending a lot of time search ing big dictionaries The idea of mini indexes was first suggested to the author by Joe Weening who prepared a brief mockup of what he thought might be possible 11 His proposal was immediately appealing so the author decided to implement it in a personal program called TWILL a name suggested by the fact that it was a two pass variant of the standard program called WEAVE TWILL was used in September 1985 to produce 3 and a companion book 4 The original WEB system was a combination of TeX and Pascal But the author s favorite program ming language nowadays is CWEB 6 which combines TeX with C In fact CWEB version 3 0 is fully com patible with C although the author usually restricts himself to a personal subset that might be called C One of the advantages of CWEB is that it supports collec tions of small program modules and libraries that can be combined in many ways A single CWEB source file foo wcan generate several output files in addition to the C program foo c for example foo w might generate a header file foo h for use by other mo
6. make CTWILL s default actions easily predictable The more intelligence we try to build into a system the harder it is for us to control it Therefore CTWILL has very simple rules for deciding what to put in mini indexes Each identifier has a unique current meaning which consists of three parts its type and the pro gram name and section number where it was defined At the beginning of arun CTWILL reads a number of files that define the initial current meanings Then whenever CTWILL sees a C construction that implies a change of meaning a macro definition a variable declaration a typedef a function declaration or the appearance of a label followed by a colon it assigns a new current meaning as specified by the semantics of C For ex ample when CTWILL sees Graph g in 2 of HAM it changes the current meaning of g to Graph x 2 These changes occur in the order of the CWEB source file not in the tangled order that is actually presented to the C compiler Therefore CTWILL makes no attempt to nest definitions according to block structure everything it does is purely sequential A variable declared in 5 and 10 will be assumed to have the meaning of 5 in 86 7 8 and 9 Whenever CTWILL changes the current meaning of a variable it outputs a record of that current meaning to an auxiliary file For the CWEB program ham w this auxiliary file is called ham aux The first few entries of ham
7. system bux ham aux and ham bux have been input CTWILL will know initial current mean ings of almost all identifiers that appear in HAM The 7 only exception is k found in 6 its current meaning is uninitialized and if the user does not take correc tive action its mini index entry will come out as k 227 0 Notice that d is declared in 4 of HAM and also in 8 Both of these declarations produce entries in ham aux Since CTWILL reads ham aux before looking at the source file ham w and since ham aux is read se quentially the current meaning of d will refer to 8 at the beginning of ham w This causes no problem because d is never used in HAM except in the sections where it is declared hence it never appears in a mini index When CTWILL processes each section of a program it makes a list of all identifiers used in that section ex cept for reserved words At the end of the section it mini outputs that is it outputs to the mini index the current meaning of each identifier on the list unless that current meaning refers to the current section of the program or unless the user intervenes The user has two ways to change the mini outputs either by suppressing the default entries or by inserting replacement entries First the explicit command ident gt tells CTWILL not to produce the standard mini output for ident in the current section Second the user can spec ify one or more temporary meanings for an i
8. update the mini index by merging together all entries for that spread Consider for example what happens when TEX typesets 10 of HAM This spread begins with 8 and TeX has already determined that 8 and 9 will fit to gether in a single column After typesetting the body of 10 TEX looks at the mini index entries If any of them refer to 8 or 9 TEX will tentatively ignore them because those sections are already part of the current spread In this case that situation doesn t arise but when TX processed 7 it did suppress entries for vert and ark since they referred to 86 TEX also tentatively discards mini index entries that match other entries al ready scheduled for the current spread In this case everything is discarded except the entries for advance and ark the others nezt t not_taken v and a are duplicates of entries in the mini output of 8 or 9 Finally TEX tentatively discards previously scheduled entries that refer to the current section In this case nothing happens because no entries from 8 or 9 refer to 10 After this calculation TEX knows the number n of mini index entries that would be needed if 10 were to join the spread with 8 and 9 TEX divides n by the number of columns in the mini index here 2 but 3 in 3 and 7 multiplies by the distance between mini baselines here 9 points and adds the result to the total height of the typeset text for the current spread her
9. vertex Notice the definition of deg as a macro in 2 this makes it possible to refer to the degree of v as v gt deg instead of the more cryptic v u J actually seen by the C compiler Similar macros for utility fields v w and x can be found in 84 and 86 The first mini index of HAM which can be seen be low 2 in the first column of the program gives cross references to all identifiers that appear in 1 or 2 but are not defined there For example restore_graph is men tioned in one of the comments of 1 the mini index tells us that it is a function that it returns a value of type Graph x and that it is defined in 4 of another CWEB pro gram called GB_SAVE The mini index also mentions that Vertex and arcs are defined in 9 of GB_GRAPH from which we quoted the relevant definitions above and that fields next and tip of Arc records are defined in GB_GRAPH 10 etc One subtlety of this first mini index is the entry for u which tells us that u is a utility field defined in GB_GRAPH 89 The identifier u actually appears twice in 2 once in the definition of deg and once as a variable of type Vertex The mini index refers only to the former because the latter usage is defined in 2 Mini indexes don t mention identifiers defined within their own spread The second mini index below 5 of HAM is similar to the first Notice that it contains two separate entries for v because the identifier v is used in two sen
10. 9 of the CWEB program We need not keep separate tables for more than 20 consecutive sections starting with the base r of the current spread because no spread can con tain more than 20 sections Token register 199 contains similarly entries that refer to sections preceding r and token register 220 contains entries that refer to sections r 20 and higher Token register 221 contains entries for identifiers defined in other programs Count register k contains the number of entries in token register k for 199 lt k lt 221 When count register k equals j the actual content of token register k is a sequence of 27 tokens 1mda cs lmda cs2 lmda cs where each cs is a control sequence defined via csname endcsname that uniquely characterizes a mini index entry TEX can tell if a new mini index en try agrees with another already in the current spread by simply testing if the corresponding control sequence is defined The replacement text for cs is the associated mini index entry while the definition of 1mda is def lmda 1 1 global let 1 relax Therefore when TEX executes the contents of a token register it typesets all the associated mini index entries and undefines all the associated control sequences Al ternatively we can say def lmda 1 global let 1 relax 11 if we merely want to erase all entries represented in a token register At the end of a spread containing p sec tions we generate the mini in
11. Mini Indexes for Literate Programs Donald E Knuth Computer Science Department Stanford University Stanford CA 94305 2140 USA Abstract This paper describes how to implement a documenta tion technique that helps readers to understand large programs or collections of programs by providing local indexes to all iden tifiers that are visible on every two page spread A detailed example is given for a program that finds all Hamiltonian circuits in an undirected graph Keywords Literate programming WEB CWEB C CTWILL TeX indexes hypertext Hamiltonian circuits 1 Introduction Users of systems like WEB 2 which provide support for structured documentation and literate programming 5 automatically get a printed index at the end of their programs showing where each identifier is defined and used Such indexes can be extremely helpful but they can also be cumbersome especially when the program is long An extreme example is provided by the list ing of TeX 3 where the index contains 32 pages of detailed entries in small print Readers of 3 can still find their way around the program quickly however because the right hand pages of this book contain mini indexes that will make it unnecessary for you to look at the big index very often Every identifier that is used somewhere on a pair of facing pages is listed in a footnote on the right hand page unless it is explicitly defined or declared somewhere on the
12. article and on the final left hand page Please take a quick look at HAM now before read ing further The program appears in five columns each of which will be called a spread because it is analogous to the two page spreads in 3 and 7 This arrange ment gives us five mini indexes to look at instead of just two so it makes HAM a decent example in spite of its relatively small size A shorter program wouldn t need much of an index at all a longer program would take too long to read HAM is intended for use with the library of routines that comes with the Stanford GraphBase so 1 of the program tells the C preprocessor to include header files gb_graph h and gb_save h These header files define the external functions and data types needed from the GraphBase library A brief introduction to GraphBase data structures will suffice for the interested reader to understand the full details of HAM A graph is represented by combin ing three kinds of struct records called Graph Vertex and Are If v points to a Vertex record v name is a string that names the vertex represented by v and v arcs points to the representation of the first arc em anating from that vertex If a points to an Are record that represents an arc from some vertex v to another vertex u then a tip points to the Vertex record that rep resents u also a nezt points to the representation of the next arc from v or a gt next A i e NULL if a is the last arc from v
13. aux are deg ham 2 macro gt arge ham 2 amp int o gt argv ham 2 amp char gt and the last entry is d ham 8 amp register amp int e gt In general these entries have the form e ident name nn type gt where ident is an identifier name and nn are the pro gram name and section number where ident is defined and type is a string of TEX commands to indicate its type In place of name nn the entry might have the 6 form string instead then the program name and sec tion number are replaced by the string This mechanism leads for example to the appearance of lt stdio h gt in HAM s mini index entries for printf Sometimes the type field says zip This situation doesn t arise in HAM nor does it arise very often in 7 but it occurs for example when a preprocessor macro name has been de fined externally as in a Makefile or when a type is very complicated like FILE in lt stdio h gt In such cases the mini index will simply say FILE lt stdio h gt with no colon or equals sign The user can explicitly change the current mean ing by specifying ident name nn type O gt anywhere in a CWEB program This means that CTWILL s default mechanism is easily overridden When CTWILL starts processing a program foo w it looks first for a file named foo aux that might have been produced on a previous run If foo aux is present it is read in and the gt co
14. dentifier all of which will be mini output at the end of the section Temporary meanings do not affect an identifier s current meaning Whenever at least one temporary meaning is mini output the current meaning will be suppressed just as if the gt command had been given Tempo rary meanings are specified by means of the operation which toggles a state switch affecting the gt command At the beginning of a section the switch is in permanent state and gt will change an identifier s current meaning as described earlier Each occurrence of changes the state from permanent to temporary or back again in temporary state the gt command specifies a temporary meaning that will be mini output with no effect on the identifier s permanent current meaning Examples of these conventions will be given mo mentarily but first we should note one further interac tion between CTWILL s and commands If CTWILL would normally assign a new current meaning to ident because of the semantics of C and if the command ident gt has already appeared in the current sec tion CTWILL will not override the present meaning but CTWILL will output the present meaning to the aux file In particular the user may have specified the present meaning with ident gt this allows user control over what gets into the aux file 8 For example here is a complete list of all com mands in
15. dex by executing token registers 199 and 200 p thru 221 using the former defi nition of lmda and we also execute token registers 200 thru 200 p 1 using the latter definition Everything works like magic A bug in the original TEX macros for TWILL led to an embarrassing error in the first 1986 printings of 3 and 4 Control sequences in token registers corresponding to sections of the current spread were not erased in other words the contents of those token registers were simply discarded not executed with the second definition of lmda The effect was to make TEX think that certain control sequences were still defined hence the macros would think that the mini index entries were still present such entries were therefore omitted by mistake Only about 3 of the entries were actually affected so this error was not outrageous enough to be noticed until after the books were printed and people started to read them The only bright spot in this part of the story was the fact that it proved how effective mini indexes are The missing entries were sorely missed because their presence would have been really helpful The longest fit method by which CTWILL s TEX macros allocate sections to pages tends to minimize the total number of pages but this is not guaranteed For example it s possible to imagine unusual scenarios in which sections 100 and 101 say do not fit on a sin gle spread while the three sections 100 101
16. dules that will be loaded with the object code foo o and it might generate a test program testfoo c that helps verify portability CWEB was used to create the Stanford GraphBase a collection of about three dozen public domain programs useful for the study of combinatorial algorithms 9 These programs have recently been published in book form again with mini indexes 7 The mini indexes in this case were prepared with CTWILL 8 a two pass variant of CWEAVE The purpose of this paper is to explain the op erations of TWILL and of its descendant CTWILL The concepts are easiest to understand when they are related to a detailed example so a complete CWEB program has been prepared for illustrative purposes Section 2 of this paper explains the example program Sections 3 and 4 explain how CTWILL and TeX process it and Section 5 contains concluding comments 2 2 An example The CWEB program for which sample mini indexes have been prepared especially for this paper is called HAM It enumerates all Hamiltonian circuits of a graph that is all undirected cycles that include each vertex exactly once For example the program can determine that there are exactly 9862 knight s tours on a 6 x 6 chess board ignoring symmetries of the board in about 2 3 seconds on a SPARCstation 2 Since HAM may be in teresting in its own right it is presented in its entirety as sort of a sideshow in the right hand columns of the pages of this
17. e the height of 8 9 810 With a few minor refinements for spacing between sections and for the ruled line that separates the mini index from the rest of the text TEX 10 is able to estimate the total space requirement In our example everything fits in a single column so T X ap pends 10 to the spread containing 8 and 9 Then after 11 has been processed in the same fashion TEX sees that there isn t room for 8 11 all together so it decides to begin a new spread with 11 The processing just described is not built in to TEX of course It is all under the control of a set of macros called ctwimac tex 8 The first thing CTWILL tells TEX is to input those macros TEX was designed for typesetting not for program ming so it is at best weird when considered as a programming language But the job of mini indexing does turn out to be programmable The full details of ctwimac are too complex to exhibit here but TEX hack ers will appreciate some of the less obvious ideas that are used Non TgXnicians please skip the rest of this long paragraph TEX reads the mini outputs of CTWILL twice with different definitions of and each time Suppose we are processing section s and suppose that the current spread begins with section r Then TpxX s token registers 200 201 219 contain all mini index entries from sections r r 1 s 1 for identifiers defined respectively in sections r r 1 7 1
18. l Stanford University Computer Science Department 1993 The Stanford GraphBase Available via anonymous ftp from labrea stanford edu in di rectory ftp pub sgb Pharr C 1930 Virgil s neid Books I VI D C Heath Boston Weening JS 1983 Personal communication Pre served in the archives of the T X project in Stan ford University Library s Department of Special Collections SC 97 series II box 18 folder 7 6 15
19. mmands of foo aux give current meanings to all identifiers defined in foo w Therefore CTWILL is able to know the meaning of an identifier before that identifier has been declared assuming that CTWILL has been run successfully on foo w at least once before and assuming that the fi nal definition of the identifier is the one intended at the beginning of the program CTWILL also looks for another auxiliary file called foo bux This one is not overwritten on each run so it can be modified by the user The purpose of foo bux is to give initial meanings to identifiers that are not defined in foo aux For example ham bux is a file containing the two lines i gb_graph hux i gb_save hux which tell CTWILL to input the files gb_graph hux and gb_save hux The latter files contain definitions of identifiers that appear in the header files gb_graph h and gb_save h which HAM includes in 1 For exam ple one of the lines of gb_graph hux is Vertex GB _GRAPH 9 amp struct gt This line appears also in gb_graph aux it was copied by hand using a text editor into gb_graph hux because Vertex is one of the identifiers defined in gb_graph h CTWILL also reads a file called system bux if it is present that file contains global information that is always assumed to be in the background as part of the current environment One of the lines in system bux is for example printf lt stdio h gt amp int gt After
20. n the two runs When TEX processes ham tex the macros of ctwimac tell it to look first for a file called ham sref If no such file is present a file called ham ref will be written containing all the un sorted mini index entries for each spread T X will also typeset the pages as usual with all mini indexes in their proper places but unsorted the user can there fore make adjustments to fix bad page breaks if neces sary Once the page breaks are satisfactory a separate program called REFSORT is invoked REFSORT converts ham ref into a sorted version ham sref Then when TEX sees ham sref it can use the sorted data to make the glorious final copy For example the file ham ref looks like this 11 GB _SAVE 4 restore _graph amp Graph GB _GRAPH 9 u amp util GB _GRAPH 8 I I amp long 12 lt stdio h gt printf amp int And the file ham sref looks like this J GB _GRAPH 10 amp Arc amp struct GB _GRAPH 9 u amp util GB _GRAPH 9 amp Vertex amp struct donewithpagel 2 l a amp tregister amp Arc JIGB _GRAPH 20 vertices amp Vertex donewithpage5 Each file contains one line for each mini index entry and one line to mark the beginning in ham ref or end in ham sref of each spread 5 Conclusions Although CTWILL is not fully automatic it dramati cally improves the readability of large collection
21. rtexts for others to read will want to give hints about the significance of identifiers whose roles are impossible or difficult to deduce mechanically Some of the lessons taught by CTWILL will therefore most likely be relevant to everyone who tries to design literate programming systems that replace books as we now know them References 1 Brown M Czejdo B 1990 A hypertext for liter ate programming In Lecture Notes in Computer Science Vol 468 250 259 2 Knuth DE 1984 Literate programming In The Computer Journal Vol 27 No 2 97 111 3 Knuth DE 1986 Computers amp Typesetting Vol B TEX The Program Addison Wesley Reading Massachusetts 4 Knuth DE 1986 Computers amp Typesetting Vol D METAFONT The Program Addison Wes ley Reading Massachusetts 5 Knuth DE 1992 Literate Programming CSLI Lecture Notes No 27 distributed by the Univer sity of Chicago Press Chicago 6 Knuth DE Levy S 1990 CWEB User Manual The CWEB System of Structured Documentation Computer Science Department Report STAN CS 90 1336 Stanford University Stanford Califor nia revised version available on the Internet via 14 10 11 anonymous ftp from labrea stanford edu in file ftp pub cweb cwebman tex Knuth DE 1993 The Stanford GraphBase A Plat form for Combinatorial Computing ACM Press New York Knuth DE 1993 CTWILL Available via anony mous ftp from labrea stanford edu in directory ftp pub ctwil
22. s of programs Therefore an author who has spent a year writing programs for publication won t mind spending an additional week improving the indexes Indeed a 13 little extra time spent on indexing generally leads to significant improvements in the text of any book that is being indexed by its author who has a chance to see the book in a new light Some manual intervention is unavoidable because a computer cannot know the proper reference for ev ery identifier that appears in program comments But experience with CTWILL s change file mechanism indi cates that correct mini indexes for large and complex programs can be obtained at the rate of about 100 book pages per day For example the construction of change files for the 460 pages of programs in 7 took 5 days during which time CTWILL was itself being debugged and refined Mini indexes are wonderful additions to printed books but we can expect hypertext like objects to re place books in the long run It s easy to imagine a system for viewing CWEB programs in which you can find the meaning of any identifier just by clicking on it Future systems will perhaps present fish eye views of pro grams allowing easy navigation through complicated webs of code See 1 for some steps in that direction Such future systems will however confront the same issues that are faced by CTWILL as it constructs mini indexes today An author who wants to create use ful program hype
23. serted by the author in order to correct or en hance CTWILL s default mini indexes for HAM e At the beginning of 2 deg gt deg f ham 2 u I gt 0 u GB _GRAPH 9 amp util gt to make the definition of deg read u l instead of just macro and to make the mini index refer to u as a utility field e At the beginning of 4 taken gt vert gt taken ham 4 v I gt 0 v GB _GRAPH 9 amp util gt v ham 2 amp register amp Vertex 0 gt for similar reasons and to suppress indexing of vert Here the mini index gets two temporary meanings for v one of which happens to coincide with its permanent meaning e At the beginning of 6 k gt t gt vert gt ark gt vert ham 6 w V gt ark ham 6 x AQ gt w GB _GRAPH 9 amp util gt x GB _GRAPH 9 amp util gt for similar reasons That s all These commands were not inserted into the program file ham w they were put into another file called ham ch and introduced via CWEB s change file feature 6 Change files make it easy to modify the effective contents of a master file without tampering with that file directly 4 Processing by TEX CTWILL writes a T X file that includes mini output at the end of each section For example the mini output after 10 of HAM is GB _GRAPH 10 next amp Arc x 7 fadvance label 6 fark x A
24. ses both as a utility field in the definition of taken and as a variable elsewhere The C compiler will under stand how to deal with constructions like v v I 0 which the C preprocessor expands from v taken 0 but human readers are spared such trouble Notice the entry for deg in this second mini index It uses an equals sign instead of a colon indicating that deg is a macro rather than a variable A simi lar notation was used in the first mini index for cross references to typedef d identifiers like Vertex See also the entry for not_taken in the fourth mini index Here not_taken macro indicates that not_taken is a macro with arguments 3 The operation of CTWILL It would be nice to report that the program CTWILL produces the mini indexes for HAM in a completely au tomatic fashion just as CWEAVE automatically produces ordinary indexes But that would be a lie The truth is that CTWILL only does about 99 of the work automat ically the user has to help it with the hard parts Why is this so Well in the first place CTWILL isn t smart enough to figure out that the w in the definition of deg in 2 is not the same as the u declared to be register Vertex x in that same section Indeed a high degree of artificial intelligence would be required before CTWILL could deduce that In the second place CTWILL has no idea what mini index entry to make for the identifier k that appears in 86
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