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For C Programmers - Trinity Computer Science Department

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1. Up till now we have applied unboxed inputs to verbs defined by defverbs and gotten useful results Why then must we box the atoms ofi 4 before giving them to plus The reason is that the result of a verb defined by defverbs is boxed Normally it is joined in an array with other boxed outputs Here the last result the one containing just the cell 3 is not produced by plus rather since plus is applied to a l element list the cell is the unmodified input cell If we had not boxed the atoms of i 4 i e if we had executed plus i 4 the unboxed 3 would be joined to the other boxed results and that would have given us a domain error Whenever you use u yoru y you must make sure that the result of u has the same type character numeric or boxed asy The dyadic form x u yhasrank 0O _ and applies u to infixes ofy An infix isa sequence of adjacent items x gives the length of the infixes The first item of the result comes from the infix of length x that is the absolute value of x starting with the first item of y and subsequent items of the result come from subsequent infixes If x is positive successive infixes start at successive items of y therefore they overlap and the last infix is the one that ends with the last item of y if x is negative infixes do not overlap each one starts with the item following the last item of the previous infix and the last infix may be shorte
2. The operation dyad is performed on each pair of cells Since each result is an atom and the frame is 02 2 2 the result is an array 46 with shape 2 2 Figure 7 Execution of i 2 2 i 2 2 Using fndisplay we have load system packages misc fndisplay ijs setfinform J defverbs plus 0 i 2 2 plus i 2 2 I0 plus 0 1 plus 1 2 plus 2 3 plus 3 As you can see we were correct when we asserted that the sum of two cells of the same shape is taken by adding their respective items For a second example we will introduce a new verb dyad the verb is the comma character x y creates an array that contains the items of x followed by the items of 37 y in other words it concatenates x and y Dyad has infinite rank which means that it applies to its operands in their entirety and so its detailed operation is defined not by the implicit looping we have been learning but instead by the definition of the verb in the Dictionary You are invited to read the entry for dyad to understand exactly what happens when x and y have different rank we are interested only in the cases where x and y are scalars or lists and a few examples will suffice 123 6 123 6 123 312 3 12312 3 4 6 46 123 0 6 12 3 As long as you have just scalars and lists the result of dyad is simply concatenation always resulting
3. lt 0 1 i 2 2 3 3 45 lt O 1 i 2 2 3 3 45 The results are identical As a final simplification if the selection is just a single item from axis 0 the left operand of dyad may be left unboxed This is the form in which we first met dyad Now that you have learned dyad completely take this quiz what is the difference between 0 1 yand lt lt 0 1 y 95 O1l i 6 01 lt lt 0 1 i 6 01 Answer the results are identical but because the left rank of dyad is0 0 1 y applies dyad twice once for each atom of 0 1 and collects the results into an array while lt lt 0 1 y applies dyad just once Split String Into J Words Monad Monad splits a string of characters into J words putting each word into a box of its own Each word is a list of characters so the result of monad is a list of boxed lists Words words and more words Words words and more words Monad is a handy way to get boxed character strings or to break an input stream into words if your language has word formation rules similar to J s Be aware that if the operand has an unmatched quote monad will fail Fetch From Structure Dyad Dyad this is a single primitive has left rank 1 right rank infinite It selects an atom from an array of boxes and opens it it is therefore analogous to the class
4. Fred 100 Joe 200 Fred 50 Sam 30 A small database of amounts we owe We can quickly get a total for each creditor Jb 1 a gt 1 a 150 200 30 1 a gives the list of names which we use as keys for the data given by 1 a For each subset we open the boxes and add up the results Note that gt would not do in place of gt and understand why It would be nice to have names associated with the totals 128 I a lt 0 b Fred 150 Joe 200 Sam 30 We box the totals using lt 0 before we join items of the totals to the items of the nub Recall thatx y concatenates each item of x with the corresponding item of y We take advantage of the fact that the results of u have the same order as the nub Apply On Partitions Monad u 1 and u 2 Unlike dyad u which applies u to scattered subsets of y u n applies u to sequential subsets of y In monad u n the subsets are computed from information in y itself in dyad u n x specifies the subsets u n is really 4 different cases distinguished by the number n we will start with the case wherenis1 2 1 or_2 Avoid the error of thinking that an operation on a subset of y somehow modifies y Never happens In J the result of a verb is always a new noun You may assign that noun to the same name as one of the opera
5. Joe 200 Here 0 0 creates a noun from the lines that follow Each line ends with an unseen LF character which provides a convenient fret for splitting the lines We use _2 to apply monad to the text of each line not including the fret which is dropped by _2 Monad splits each string into words and boxes the words The result is not the same as the array we created earlier because the second column is character rather than numeric We could have written noun define instead of 0 0 Monad u n hasrank 1 _whennis1 2 1 or 2 Apply On Specified Partitions Dyad u 1 and u 2 When n is 1 2 _1 or_2 x u n y has infinite rank and in the one dimensional case where x is a Boolean list of Os and 1s resembles u n y The difference is that the frets are given by the positions of 1s in x rather than by values of the items of y We can define u 1 yas y __1 y u 1 y andu n for the other values of n similarly 130 01010 1 20 30 40 50 60 70 110 As in this example some leading items of y may not be in any interval In the general case x is a list of boxed Boolean lists with j x supplying the frets for axis j The partition is then multidimensional O 1 1 1 001 lt 1 1 3 4 1456 7 18 9 10 11 In both the monadic and dyadic cases ofu 1 u _1 u 2 andu _2 the partitions to which u is applied have the same rank as y but the shapes along t
6. Our interest in u n is not in applying a verb several times usually we could just write the instances out if we needed to but rather in 4 special values ofn _1 0 1 and _ infinity u 0 meaning apply u 0 times is simple it does nothing with the result y in both dyadic and monadic forms u 1 means apply u once Thinking about that we see that if n is restricted to the values 0 or 1 n means If n u n yis y but modified by application of u ifn is 1 If we want to apply the verb u only on the items of y for which x is 1 we can writex u ly 86 1001 0 gt _1 1 2 3 4 5 223535 When n is _ u __ means keep applying u until the result doesn t change This is obviously a splendid way to perform a numerical calculation that converges on a result for example if you take cos cos cos cos y until the result stops changing you get the solution of the equation y cos y 2 o _ 0 0 739085 but that s not why we love __ Consider the verb u v _ either monad or dyad with the understanding that v always produces a Boolean result of O or 1 Itis parenthesized u v _ i e u v repeated until the result stops changing Now if v evaluates to 0 the result of u v will certainly be the same as its input because u will not be executed if v is 1 u v causes u to be executed once So this construct is like C s while v y y u y except that the J version also stops if the result ofu y is th
7. The characters and are not paired in J each is an independent verb This is jarring at first but you ll get used to it and are identity verbs they have infinite rank and y and y both result in y As dyads they pick one operand x yisx andx yisy They can be useful when you have a monadic verb that for one reason or another must be used as a dyad with an unwanted operand then x v y applies v tox andx v y applies vtoy Example 123 45 6 1 Here are some other uses of and We have already met the first one which is to display the result of an assignment a 12 3 produces no typeout but Ja 12 3 12 3 Second can be used to put multiple assignments on the same line since each application of ignores what is to its right 69 a 5 b abc c 0 Finally can be used instead of parentheses to separate numbers that would otherwise be treated as a list 5 0 1 2 syntax error 5 0 12 5 0 1 2 5 1 5 2 5 0 12 5 1 5 2 Making a Dyad Into a Monad u n and m amp v A dyadic verb takes two operands but if you know you are going to hold one fixed you can create a monadic verb out of the combination of the dyad and the fixed operand the monad s operand will be applied to whichever operand of the dyad was not held fixed The conjunction amp when one of its operands is a noun fixes an operand to a dyad m amp v y has infinite rank and is equivalent tom v y
8. monad 54 dyad 52 dyad 52 monad 57 Apply On All Subarrays conjunction 133 Apply On Partitions conjunction 129 Apply On Selected Partitions conjunction 130 Apply On Subarray conjunction 132 dyad 50 monad 48 u 3 and u 133 u 132 andu 130 u 1 andu 129 163 dyad 53 monad 56 dyad 53 dyad 54 monad 56 monad 83 dyad 90 monad 91 3 monad 96 _1 monad 172 167 69 101 69 N dyad 47 monad 83 dyad 47 monad 83 dyad 49 details 93 dyad 51 monad 56 monad 56 dyad 96 dyad 47 monad 48 163 dyad 53 monad 55 f monad 178 dyad 178 monad 55 dyad 51 monad 56 monad 56 Ney Nye eye e 60 monad 127 dyad 47 48 monad 127 2 l 2 dyad 47 monad 163 dyad 48 169 monad 163 163 monad 83 lt lt dyad 48 lt monad 72 lt dyad 48 lt monad 48 lt dyad 48 lt monad 48 dyad 47 48 gt gt dyad 47 gt dyad 48 gt monad 48 gt dyad 48 gt monad 48 1 128 2 dyad 137 9 91 18 178 91 19 178 A A dyad 171 A monad 171 a 73 addons 168 agreement of operands 39 aliasing 120 Amend adverb 143 Anagram Index verb A 171 anonymous enti
9. 142 26 Modifying an array m Modification of a portion of an array performed in C by the simple expression x m y fits uneasily into J To begin with 3 pieces of information are needed the array x the index m and the replacement data y Since J verbs take only 2 operands that means the primitive to modify the array will be an adverb so that its left operand can hold one of the 3 the verb derived from the adverb and its operand will perform the modification Moreover the adverb cannot be a modifier of the array s name as m is a modifier of x in x m y In J the array x may not have a name While in C every array is declared and named in J we summon up anonymous arrays in the blink of an i use them and let them disappear We expect modification of a subarray to be like other primitives in this regard which means that it must not be a form of assignment to a name using a copula or but instead a derived verb operating on an array to produce a result What will the result of the assignment be This is not much of an issue in C There the result of the assignment is y but it is seldom used in C after we have employed x m expression we are usually content to put down the pencil satisfied that we have described a statement s worth of computation In J we routinely use weapons of larger bore after we have modified the array we expect to be able to sort it or add it to another array or the like It is clear t
10. Figure 5 Execution of 2 i 23 4 3 i 234 12 14 16 18 20 22 24 26 28 30 32 34 The verb is applied to the single 3 cell Its items which are 2 cells are added leaving a single 2 cell as the result How abouti O 2 2 2 can you figure out what that will produce Notice I put parentheses around the numeric list 2 2 2 so that the rank 0 wouldn t be treated as part of the list 33 The verb rank is 0 and the noun rank is 1 so we will be applying the verb to 0 cells The frame fis 3 Think of the operand as a list of 3 0 cells i e atoms The verb monad i is applied to each cell 01 01101 Since each result is a list i e a 1 cell the result is a list of 3 1 cells each with shape 2 i e an array of shape 3 2 ooo PRP Figure 6 Execution of i 0 2 2 2 i 0 2 2 2 ooo PRP If you worked through that it might have occurred to you that the shape of each result cell depended on the value of the operand cell and that if those cells had not been identical there would be some rough edges showing when it came time at the end to join the dissimilar result cells together If so full marks to you That can indeed happen If it does then just before the cells are joined together to make the final result the interpreter will bulk up the smaller results to bring them up to the shape of the largest First if the ranks of the results are not identical each resu
11. and dyad i and the adverb For all of these you can specify comparison tolerance with 177 You may wonder whether an exact comparison using 0 is faster than a tolerant comparison The answer is Usually not but there is one important exception if the comparison is used for finding equal items whose rank is greater than 0 or are complex numbers exact comparison can be much faster So if x has rank 2 or higher it s worth the trouble to writex u 0 yorx i 0 y similarlyuse 0 y 0 y andx e 0 yify has rank greater than 0 The f in f can be no larger than about 2 _34 The reason for this is that there is much special code in J for handling integer operands and for speed it assumes that comparison tolerance cannot affect integer comparisons The foreign 9 19 y can be used to change the default comparison tolerance and 9 18 will return the current setting Right Shift Monad f One of my personal favorites is the infinite rank verb monad defined as _1 amp in other words it shifts y right one place discarding the last item and shifting an item of s into the first position Generalized Transpose Dyad Dyad hasrank1 _ x y rearranges y so that the axes given in x become the last axes of the result So if y has rank 3 0 y puts the axes of y into the order 1 2 3 OandO 2 yputs them into the order1 3 0 2 For example 178 2 6 0 m 12 13 14 15
12. 12 45 6 Evidently not A length error means that the operands to did not agree and you get an error if you try to add them We will shortly understand exactly what this means 422 3 012 345 A reminder of what monad i does 0100 i 2 3 0 1 2 103 104 105 Whoa The atoms of the left operand were applied to rows of the right operand Interesting This seems to be some kind of nested implicit loop Let s learn a couple of more verbs monad and monad Monad creates the binary representation of an integer i e a list of Os and 1s and monad is its inverse creating the integer from the binary representation For the longest time I couldn t remember which was which but at last I saw the mnemonic the verb with the single dot creates an atom from a list the verb with multiple dots creates a list from an atom 5 101 101 5 Yes they seem to perform as advertised They can be applied to arrays Ja 59 0101 1001 Look the result is not a rank 1 list but rather a rank 2 array where each item has the binary representation of one operand value and notice an extra leading zero was added to the representation of 5 The little trick with Ja will be explained later but for now just think of Ja as assign to a and display the result With a assigned we have a 5 9 This seems to be the desired result but on reflection we are puzzled how did the interpreter know to apply t
13. 12 3 6 As usual we can use ndisplay to explain what s happening 58 defverbs plus 0 plus 1 2 3 The great power of the adverb concept is that u can be any verb it s not restricted to or any other subset of verbs it can even be a user written verb What would monad gt mean Well gt 1 2 3 would be equivalenttol gt 2 gt 3 since each gt picks the larger operand the result is going to be the largest number so monad gt means maximum gt 314159 9 and of course minimum is similar lt 3 1 4 2 5 9 1 What about monad Convince yourself that it combines the first two axes of its operand pf Be 23 012345 defverbs comma comma i 2 3 12 13 14 15 16 17 18 19 20 21 22 23 j i 234 6 4 Ji 234 0 12 3 4 5 6 7 8 91011 12 13 14 15 16 17 18 19 20 21 22 23 How many atoms are in y Why y 59 4i 234 24 We can verify that the rows and columns of the following magic square sum to the same value 4 33 816 357 492 15 15 15 1 3 3 816 357 492 15 15 15 As this last example shows the items can be of any shape Applying to the rank 2 array added up 1 cells while applying 1 added up the 0 cells within each 1 cell Have you wondered what would happen if there is no cell or only 1 Good on you if you did The answer is if there is only 1 cell the result is just that cell if there is no cell the result is a cell of identity elements The identity elemen
14. E e dyad 51 E dyad 131 Element Of verb e 51 empty cells 78 empty operands 76 Enfile verb 54 entity 3 error handling 113 147 Extended integers 169 Extended Precision verb x 170 F f 201 Factorial verb 83 163 Fetch verb 96 FFT addon 168 fill 73 framing 34 verb 52 with empty operands 76 Find Sequence verb E 131 Fit conjunction 52 Flatten adverb 201 fndisplay 34 Foreign conjunction 112 form editor 181 format for printing 114 frame 18 28 36 From verb 49 G generalized hypergeometric function 166 gerund 88 Grade verbs 56 Greatest Common Divisor verb 169 H Head verb 56 hypergeometric function 167 I i dyad 50 i monad 24 i 179 identity element 60 Identity verbs 69 If conjunction 86 Index Of verb i 50 Infix adverb 80 input and output 112 integrated rank support 110 interrupting execution 9 inverse of verb 103 Invoke Gerund conjunction 136 item 19 Itemize verb 57 J J Forum 11 j dyad 163 j monad 163 L L monad 97 L 175 Labs 10 Laminate verb 52 LAPACK addon 168 Level verb L 97 Link verb 90 list 19 load verb 9 locale changing current 151 current 149 defined 149 locative 149 name lookup 150 search path 150 used for modular programming 154 z 154 Log verb 83 Lowest Common
15. amp lt 4 5 amp amp lt _1 The verb that is produced seems in order but notice one point the verb contains the value of init but the name of vb This is a rule the name of a verb argument is passed into a modifier but the value of a noun argument is passed Note that if these lines appear inside a verb the verb vb which is assigned by private assignment is not defined inside LoopWithInitial because LoopWithInitial is running ina different explicit definition from the one in which vb was assigned As we see above LoopWithInitial can pass vb into other modifiers but if LoopWithInitial tried to execute vb it would fail Before we move on I want to point out one tiny example of the beauty of J For amp amp lt 4 5 to work there must be some obverse of amp lt 4 5 that undoes its effect What would that be We can see what the interpreter uses amp lt 4 5 b 1 amp lt 4 5 It undoes the addition of a trailing item with which discards the last item Yes that makes sense the obverse has its own obverse which is the original verb Example A Conjunction that Analyzes u and v The conjunction u amp v expresses with great clarity the sequence of applying a transformation v then applying the operation u then inverting the transformation v The dyad x u amp v_ y applies the same transformation to both x and y but in many cases the transformation is meaningful only on one operand and what w
16. or at the end Since all primitives are system defined i e they are reserved words you may not put or in your names No space is required after a primitive The part of speech for each primitive is fixed Example primitives are i i for select case end The first step in processing a sentence is to split it into words The words correspond roughly to C tokens after making allowance for the special status of the and characters The space and TAB characters are treated as whitespace One big surprise will be that a sequence of numbers separated by spaces is treated as a single word which denotes the entire list of numbers We will be careful to distinguish periods used for English punctuation from the dot that may be at the end of a primitive When a J word comes at the end of an English sentence we will be sure to leave a space before the period For example the verb for Boolean Or is while the verb for addition is Numbers You do not need to trouble yourself with the distinction between integers floats and complex numbers If it s a number J will handle it properly There are a great many ways to specify numbers consult the Dictionary to learn details including among other things complex numbers extended precision integers and exponential forms Example numbers are 2 _2 underscore not is the negative sign 0 5 since is special it must not be the first character
17. we prepend gt to y then we use _1on 131 the string That will make the prepended gt the fret and so we will break the string into parts that start with gt throw away the first 8 characters of each part and box each trimmed down part We can even try it on a test bit extracthref abcdefghijk1 gt xxx Jijk1l Now we can look again at our original line rewritten lt a E html extracthref 1 html pagel html page2 html1 page3 html1 This makes some sense now The execution order is lt a E html extracthref 1 html WeuscE to find all the starting positions of lt a tags then for each one we split what follows the lt a into blocks terminated by gt and then we take the first one which will have the data before the gt that matched the lt a Apply On Subarray Dyad u 0 Dyadu Ohasrank2 _ x u 0 y uses x to specify a subarray of y the subarray is extracted and u is applied to it to produce the final result We will discuss the simple case where x is a rank 2 array The first item of x call that s gives the starting corner of the subarray The second item of x gives the length of each axis of the subarray If x is shorter than the rank of y unspecified axes are taken in full For example Ja a a i a i 44 abcd efgh ijkl mnop A cute little exp
18. 18 5 base This is how you query the name of the current locale Next we define two verbs vl_z_ verb define Locale at start of vl is j printf 18 5 gprintf n1 v2_result v2_locl_ nil Value returned by v2 is s printf lt v2_result Locale in vl after calling v2 is j printf 18 5 gprintf nl1 cocurrent lt loc2 v2 verb define Locale at start of v2 is j printf 18 5 gprintf nl y cocurrent lt loc2 gprintf n1 Locale at end of v2 is j printf 18 5 nl The verb v1 was defined in locale z because it was an assignment to a locative the verb v2 was defined in locale loc2 because it was an assignment to a simple name and the current locale at the time of its assignment was loc2 cocurrent lt loc3 v2 Now the verb v2 is defined in both locale loc2 and loc3 Next we define the noun n1 in each of our locales so we can see which locale a name was found in nl_locl_ nl in locl nl_loc2_ nl in loc2 nl_loc3_ n1 in loc3 Now run the verbs I will insert interpretation of the execution 152 v1 La J searches for the simple name v1 in the current locale loc3 not finding it there it looks in loc1 loc2 and z finally finding itin z J executes the definition of the verb found in z but without changing the current locale Locale at start of vl is loc3 Yes the current locale is still loc3 nl nl1 in loc3 and a n
19. At this point you may be impressed with the economy of the monadic fork but a bit confused about the details For example we said that VO V1 V2 Ny isnot the same as VO V1 V2 Ny and yet we said that mean is the sameas mean How can that be If we use the version without parentheses why doesn tmean 12 18 24 get evaluated like 12 18 24 0 333333 9 I could give you a simple rule of thumb namely that you can always imagine an extra set of parentheses around any value assigned to aname That would be true but misleading because it would encourage you to think that the values of defined names are substituted into a sentence before the sentence is executed That gets it backwards in reality the operands are supplied to the stored definitions In fact the execution of a J sentence is a subtle alternation between creating definitions and executing them We will take the next couple of chapters to give you a thorough understanding of execution after which we will return to see what magic we can work with forks and their brethren 186 35 Parsing and Execution I I hope your hunger for understanding will be enough to motivate you to read a couple of difficult chapters If you do you will learn something few J programmers know what really happens when J executes a sentence In this chapter we will analyze sentences from the top down to get an idea for the order of execution In the next chapter we will follow the
20. Verbs Have Rank iic 2i dices otsncsniatece tae dtiids ctaenienians 26 Examples of Implicit Loopssscccsccaciteniescotate hiv nlaniem et cane iesadeariete armada 26 Th Conceptof Verb Rank enisri redas nnt a r A a a a a nie 28 Verb Execution How Rank Is Used Monads c ccccsccccsssceessecessseeeesseeeesseeees 28 Controlling Verb Execution By Specifying a Rank cece ceeeeseceseeeeeeeeteeeneeees 30 iv Examples Of Verb Rank sic cintceckca cists sGuctin Ri ctaateeacss n A ecue 31 fndisplay A Utility for Understanding Evaluation cceceeeceeeseeteeeeeee 34 Negative Verb RANK iesadisincti teases ateeuc sine a bavi ee leas 35 Verb Execution How Rank Is Used Dyads s ssenseesesseseeseessessessrssressessssees 36 When Dyad Frames Differ Operand Agreement eceeccecsceceteceteceeeeeeeeeeseees 39 Order of Execution in Implied Loops eecceccceeseceeeceeseeeseeceseceeeeeeeeeaeecnaeenteenees 42 A Mistake To AVOID 35sec yicosshesitaacchsunagsae dues tated a tas sud tases sacs data a kisas aa 42 6 Starting To Write Im nenoras nen etal lye O eli al laa ana ta aa 44 diac More Verbs es tat cee ot oc nace indi dcnuasliaeacuamte E AEA AEN 47 Atithmetie TWA Seas se stak teks iae n i ntact sede sete E E E E E E naan 47 Boolean Dy ads ong ee a ai ied ws R ant Ais ag pias E AO 48 Mirand Max DyadS sei ce asian Actas ey e ge Seed ead ey E A dee E Cloaks 48 Afthme tic IVION AUS sa eeese ks eis eis tient Suna
21. s Short i int f float d double j complex or n placeholder a value of 0 is used and the result is ignored all but n can be preceded by to indicate an array of that type by itself indicates an array of unspecified type The right operand of cd is the actual arguments to be passed to the called function It must be a list of boxes in which the contents of each box holds one argument to the called function there is no box to correspond to the returned value An argument whose description does not include must correspond to a boxed scalar of the appropriate type An argument described with must correspond to a box whose contents are either an array or a boxed memory address see below The address of the array or memory area is passed to the DLL and the DLL may modify the array or memory area this is how you return an array from the function s and f are not native J types A scalar argument described by s or f is converted by the interpreter to the correct form while an argument described by s or f is assumed to point to a memory area with the correct format Note that if you get things wrong and the function scribbles outside the bounds of an array J may crash Note also that a vector of Os and 1s is held inside J as a vector of Booleans which are chars When the function calls for a vector of ints J will convert any Boolean to int In the example above _1 reserved space for an int 0 would have worked t
22. 0 defg 4 Jabc comma d abc comma e abc comma f abc comma g 4 You must thoroughly understand this example where one operand has only one cell because it occurs frequently The handling of the general case of dissimilar frames is uncommon enough that you do not need to understand it perfectly right now you ll 41 know when you need it and you can sweat out the solution the first few times Here are a few observations that may help when that time comes It is always entire cells of the operand with the shorter frame that are replicated A cell is never tampered with nothing inside a cell will be replicated And it is not the entire shorter frame operand that is replicated but cells singly to match the surplus frame of the other operand This fact that single operand cells are replicated is implied by the decision that the shorter frame must be a prefix of the longer frame the single cell is the only unit that can be replicated since the surplus frame is at the end of the frame rather than the beginning Take a moment to see that this was a good design decision Why should the following fail 123 i1 2 3 length error 123 i 2 3 The length error means that the operands do not agree because the frame prefix rule is not met Your first thought might be that adding a 3 item list to an array of 2 3 item lists should be som
23. 16 17 18 19 20 21 22 23 Oj i 234 12 13 14 15 wWwNBeEO 16 17 18 19 NOUA 8 20 9 21 10 22 11 23 Formally putting the axes into an order p means that lt p x p y isthe same as lt x y I wish I could give you an intuitive definition but I can t An item of x can be negative to count axes from the end The Dictionary shows how you can use boxed x to take elements along diagonals of y Monad i and Dyad i Monad i is like monad i but its interval is centered on 0 rather than starting at 0 i 5 _5 4 e ae 54321012 3 4 5 Monad i can also take a complex operand to specify a different spacing between items of the result 321012345 5 Dyad i is like dyad i but it gives the index of the ast occurrence or x if there is none Fast String Searching s Symbols If you find your program taking a lot of time matching strings you can create symbols representing the strings and then match the symbols rather than the strings themselves The interpreter uses special code to make symbol matching very fast 179 Symbol is an atomic data type like numeric literal and box In a noun of the symbol type each atom represents a boxed character string You create a symbol with monad s which has infinite rank s y takes an array of boxed strings y and creates an array of symbols of the same shape as y Jsym s 2 2 abc def ghi jk abc def ghi jk sym 22 The charact
24. 23 4 Referring To a Noun In a Tacit Verb Suppose you want a verb v that returns the current value of the noun n maybe it s a button handler Suppose you want v to be tacitly defined How would you do it You can t use v n because that would use the value of n at the time v is defined in fact v would be a noun and you want the value of n when v is executed Use this trick v n _ When this is executed the operand is ignored and replaced by the string n which is then executed by to produce the value of the noun n 200 38 Readable Tacit Definitions Heron s formula for the area of a triangle given the lengths of the sides a b and c is sqrt s s a s b s c where s is atb c 2 This can be written triarea 4 0 triarea 3 4 5 6 Regardless of your diligence in commenting your code and the level of J expertise in you and the sorry wretches who will have to read it long tacit definitions like this can become trackless wastelands Saharas of verb following verb unendingly with nothing to suggest an interpretation of the intermediate results I know of two ways to mark a trail through such a definition The first is to develop a regimen for using spaces and parentheses to help the reader group the parts I would write triarea 0 The second way is to split the line into multiple lines where each line can have a comment or a verb name indicating wha
25. 3 12 Since each result is an atom i e a 0 cell the result is a list of 2 0 cells i e an array of shape 2 Figure 3 Execution of 1 i 2 3 Examples Of Verb Rank Here are some more examples using a rank 3 array as data 31 2 6 0 m 12 13 14 15 16 17 18 19 20 21 22 23 1 i 234 6 22 38 54 70 86 The verb rank is 1 and the noun rank is 3 so we will be applying the verb to I cells The frame fis 2 3 0 cells i e an array of shape 2 3 Think of the operand 0123 FoG 891011 as a 2x3 array of l cells 12 13 1415 16 17 18 19 2021 22 23 The verb monad is 6 22 38 applied to each cell 54 70 86 Since each result is an atom i e a 0 cell the 6 22 38 result is a 2x3 array of 54 70 86 Figure 4 Execution of 1 i 23 4 32 2 i 234 12 15 18 21 48 51 54 57 The verb rank is 2 and the noun rank is 3 so we will be applying the verb to 2 cells The frame fis 2 0123 12131415 4567 16171819 891011 2021 22 23 Think of the operand as a list of 2 2 cells The verb monad is applied to each cell As we have learned this sums 12 15 1821 48 51 54 57 the items making each result a rank 1 list Since each result is a rank 1 list i e a 1 cell 12 15 18 21 the result is a list of 2 48 51 54 57 1 cells i e an array of shape 2 4
26. 4 3 10 2 4 Using to mean y divided by x we can have right to left execution without parentheses 2 4 2 When we know y contains exactly 2 items y is a convenient shorthand to subtract the second from the first without having to write y y To subtract the first from the second we simply invert the order of subtraction with The monadic form u y has infinite rank and is equivalent to y u y 1 e it applies dyad u with both the left and the right operands equal to the one operand of monad u As with dyad u most uses of monad u are esoteric but we know one already we can sort y into ascending order with either y y or our new equivalent y 3 14159 113459 61 9 Continuing to Write in J Now that we have a formidable battery of verbs at our command let s continue writing programs in J The data definitions are repeated here for convenience empno nemp in J just empno employee number for each member of the staff The number of employees is nemp payrate nemp number of dollars per hour the employee is paid billrate nemp number of dollars per hour the customer is billed for the services of this employee clientlist nclients every client has a number this is the list of all of them The number of clients is nclient emp_client nemp number of the client this employee is billed to hoursworked nemp 31 for each employee and for each day of the month the numbe
27. Dyad u _3 is useful in imaging applications The following example averages 2x2 pixel regions in an image using a simple box filter 133 31 14 40 19 47 41 13 21 47 34 44 image 88 58 14 31 38 13 78 67 2 75 26 75 75 65 84 56 62 20 88 45 90 2 29 15 59 25 12 2 9 34 25 75 2 68 95 48 65 90 16 45 86 2 2 55 38 88 38 55 46 62 42 7 80 47 38 14 96 80 14 36 93 50 100 49 98 12 24 94 25 65 60 amp 0 25 64 49 75 75 55 25 75 38 75 42 25 48 5 64 5 53 67 _3 image The rank of each operand presented to u is the same as the rank of y The shape of each operand presented to u is the shape of y with the leading items replaced by 1 x formally this is 1 x x y at all possible locations If x has rank less than 2 it is processed as if it were 1 x which will try the window x u 3 yissimilartox u 3 y but the window is positioned at every starting point that is within y even if the entire window will not fit within y For those positions at which the window will not fit the operand to u is truncated 134 24 When Programs Are Data One characteristic of maturity in programming is readiness to pass a program as an argument A well designed program does not exhibit a megalomanic urge to do everything the user may desire it is content to perform a limited function well
28. R r dyad 163 r monad 163 random numbers 167 rank 17 22 24 28 100 negative 35 Rank conjunction 31 Rational numbers 169 Ravel verb 54 Raze verb 91 recursion 173 reexecution of verb on cells 42 avoided by u n 42 Remove verb 50 result to produce no typeout 23 Reverse verb 55 Right Shift verb f 178 Roll verb 167 Root verb 82 Roots verb p 164 Rotate Left verb 53 S s monad and dyad 179 S 175 scalar 18 scripts supplied with J 168 sentence 4 setfnform 34 shape 17 Shift Left verb f 53 sockets 122 datagrams 125 IP addresses 123 sdaccept 125 sdasyne 122 sdbind 124 sdcleanup 126 sdclose 124 sdconnect 124 sdgethostbyaddr 123 sdgethostbyname 123 sdgethostname 123 sdgetpeername 123 sdgetsockets 126 sdgetsockopt 125 sdioctl 125 sdlisten 124 sdrecv 124 sdrecvfrom 125 sdselect 122 sdsend 124 sdsendto 125 sdsocket 122 setsockopt 125 socket options 125 socket_handler 123 Sort Using verb 53 Sort verb 54 sparse arrays 167 special code in the interpreter 109 Spread verb _ 1 172 Square Root verb 83 Stitch verb 52 string matching 179 Suicide verb 101 symbols 179 T Table adverb 173 tacit adverbs 199 tacit programming anonymous verb 187 explicit to tacit converter 204 fork 185 forks and hooks 196 improv
29. and the last one survives in the result Do not rely on this behavior As a general rule in J the order in which a parallel operation is performed is undefined and may change from release to release or from machine to machine The interpreter may incorrectly modify the first atom in the array if the atoms of m select portions of y that do not have the same shape You should avoid such m O 1 2 3 2 11 44655 ao uw WwW arrow U N UU aw uu Don t Modification In Place It is fundamental to the design of J that in general verbs produce output in a separate memory block from the inputs Thus if I execute gt y the result will be in a different memory area from y This takes more space than incrementing y in place but not much additional computation since every atom of y must be visited and incremented either way Usually the extra memory usage is immaterial to overall performance Sometimes we regret having to make a new copy for the output for example when we just remove the end of a long list with y or add a single character to the end of a long string withx try to figure out what happened avoid mixedm oO i ve 144 Dyad m s profligacy with memory bandwidth reaches an extreme even if y is huge and only one atom is modified the whole y must be copied to produce the result Sometimes this can be a noticeable drag on performance To help out in those cases the interpreter recognizes the specific forms nam
30. and to leave other functions to other programs A suite of such programs can be variously connected to perform a great variety of functions with each program doing its bit and passing control to the next one In C a program is passed to another program by pointer reference and invoked by pfi arguments J has no pointers but it has a great many ways to pass executable nuggets around the system We will learn them now Calling a Published Name The simplest way to get a verb f to pass control to another program g is to define f to call a verb with a public name say _subfn and then to define _subfntobeg In C this would be ridiculous because it would imply that _subfn is permanently bound to a single value of g with the result that all calls to from anywhere in your program would be stuck with the same value of g J overcomes this objection by allowing redefinition of f subfn Each invocation of looks like f subfn g f arguments When f is invoked elsewhere the correct value of f __subfn will be assigned similarly so each invocation of can pass control properly I find this technique hideous but I have to admit it is effective J uses it to get callbacks from DLLs It must not be used if f __subfn is going to be called in response to an event since its value may have been redefined by the time the event occurs Using the Argument To a Modifier If the verb isn t going to call a name of its own choosing
31. booltondx 0101001 13 6 Note that we are careful to give our verb a rank of 1 since it works only with lists To find the difference between the largest and smallest items in a list range gt lt 1 range 314159 8 To find the index of the largest item in a list indexmax i gt 1 indexmax 31415926535 5 To create a Boolean list with a 1 at each position that is different from the previous position changeflag 1 2 changeflag 1 1 2 2 7773 1010100101110 We could have done this without using forks with 1 amp 2 amp Which version you use is a matter of taste The number verbs _9 through 9 are very useful if you like forks rather than long conjunction chains Note that changeflag is executed as if parenthesized 1 2 Ifyou work with long trains of verbs like this you will soon notice that if you count the verbs from the right starting at 0 of 345 8 8 197 course the even numbered verbs have the original x and y applied and the odd numbered ones combine the results from the even numbered To replace multiple spaces by a single space delmb amp E 1 delmb abc nb abc nb To create an array in which each item is a list of value number of times that value appeared histogram histogram 3 1 4159265358979 AIMDHAHNWOU BE W PRPPRPWWEND To append the contents of the first item of x in front of y and the contents of the las
32. fit if only to save typing I have found it very useful in loading configuration parameters from a file the file contains both noun names and the values with the values being assigned to the names by multiple assignment Such a design is easily portable from release to release of a product since the file has no format it simply defines all the names it understands 92 A multiple assignment can produce verbs and modifiers as well as nouns You put a character before your list of names namel name2 list of atomic representations and each name is assigned the entity described by the atomic representation Each atomic representation is a noun but it may descibe any kind of entity Usually your entities will be verbs and then they can be converted to atomic representations by as in add subtract mult div Dyad Revisited Now that we know about boxes we can understand the full description of the selection verb dyad Inthe general form the left argument of x y is a box whose contents is a list of boxes Pictorially it is axis O selections axis 1 selections We will call the inner boxes 1 e the items of the contents of the box x the selectors The first selector gives the indexes to be selected along the first axis i e axis 0 the second selector gives the selections for axis 1 and so o
33. if a modifier does not contain any reference tou v m or n it is assumed to be an old style modifier and references to x and y are treated as if they wereu andv You may encounter old code that relies on this rule but you should not add any new examples of your own Modifiers That Refer Tox Ory Most of the modifiers you write will be refer tox andy The names x andy refer to the noun operands that are supplied when the modifier is invoked as x u adverb yor x u conjunction v y 160 Here is an example which is invoked as x u InLocales n y where uisa verb and n is a list of locale names it executes u y orx u y if the invocation is dyadic in each locale ofn InLocales 2 0 for 1 n do cocurrent 1 u y end for 1 n do cocurrent 1 x u y end This illustrates the important points The text of the definition is not interpreted until the x and y are available in other words until the verb defined by u InLocales nis invoked Since that invocation may be either monadic or dyadic two versions of the conjunction are given one for each valence The result of the execution must be a noun because the definition defines a verb and the result of a verb is always a noun That last point is important and I want to emphasize it It is true that InLocales is a conjunction and yet its text defines a verb How is this possible Because InLocales is executed as a conjunction at the time it gets its u a
34. in a few months you ll look in the mirror and be amazed at the programmer you ve turned into Good luck and I hope to see you on the J Forum 206 41 Glossary Adverb One of the primary parts of speech An adverb modifies the word or phrase to its left to produce a derived entity that can be any of the four primary parts of speech Anonymous Having no name Said of the result of an execution Atom Any single number single character or single box Also called a scalar An atom is a noun with rank 0 Array A noun comprising atoms arranged along one or more axes Each atom is identified by its index list J arrays are rectangular meaning that all 1 cells contain an identical number of 0 cells and all 2 cells contain an identical number of 1 cells and so on Axis One of the dimensions along which the atoms of an array are arranged The atoms of every noun are arranged along zero or more axes where each axis has a corresponding length and each unique list of nonnegative integers in which each integer is less than the length of the corresponding axis designates a unique atom of the noun Boolean Having a numeric value restricted to the values 0 and 1 Cell A subarray of a noun consisting of all the atoms of the noun whose index lists have a given prefix For positive k each k cell of a noun whose shape is s has the shape k lt s s and together they can be assembled to reconstruct the noun For negative k the k cells
35. indicate errors look in J_directory system main socket ijs for definitions The valid address families and socket types are defined in the same place sdselect You check the status of sockets with sdselect read list write list error_list maxtime The three lists are lists of sockets and maxtime is a time in milliseconds sdselect will delay for a maximum of maxtime milliseconds until it sees a socket in read_list that is ready for reading or one in write_list that is ready for writing or one in error_list that has an error When it finds a qualifying socket or when the time limit is reached it returns result_code read_ ready list write ready list error list which lists all the qualifying sockets A maxtime of 0 always returns immediately giving the current status of the specified sockets If sdselect is invoked with an empty operand sdselect it checks all sockets with a maxtime of 0 Asynchronous Sockets and socket_handler By default sockets created by sdsocket are blocking sockets an operation on such a socket waits until data is available This can tie up your J session while the remote computer is responding so if you are serious about your socket programming you will want to make your sockets nonblocking You do so with sdasync socket_number which marks that socket as nonblocking and requests notification of changes in its status All operations to a nonblocking socket return immediately with the error code EWOULDBLOCK
36. is applied to the successive prefixes Here are some commonly used forms i 6 013 6 10 15 means total the items so i 61s 0 O 1 O 1 2 0 14 2 3 i e the running total of the items ofy gt 95 3 10 3 2 20 9 9 9 10 10 10 20 For each item of y the result is the largest item occurring in the list up to that item of y 79 lt 0000110010 0000100000 lt y ona Boolean list i e one containing only 0 or 1 is a tricky way to turn off all 1s following the first See how it works lt y will produce a result of 1 only in the case where the last item of y is 1 and the rest are 0 1 e 0 lt 0 0 lt 1 so lt y produces a 1 for that prefix and 0 for all the others 111100111 111100000 Keep the leading 1s of y but set the rest toO u yis similar to u y except that it applies u to suffixes of y It applies u to allofy i e u 0 y to produce the first item of the result it then applies u to all but the first item of y i e u 1 y to produce the second item of the result and so on with the last item of the result beingu y 1 y that is u applied to the last item of y Example i 6 15 15141295 The running total now starts at the end and runs backward fndisplay shows the details and points out a subtlety of u defverbs plus 0 plus lt 0 i 4 I0 plus 1 plus 2 plus 3 1 plus 2 plus 3 2 plus 3 3
37. it supplies the text of the entity which again is given the part of speech indicated by m if n is 0 the interpreter interrupts what it s doing at the moment it executes the conjunction and reads lines from its input source until it finds one that contains only the word execution of the interrupted sentence then continues with the text of those lines becoming the right argument to and thence the text of the defined noun adverb conjunction verb If the verb is defined in a script file the input source for lines to satisfy m 0 is the script file otherwise the input source is the keyboard If you forget the in a definition at the end of a script file the interpreter will switch over to keyboard input and you will find the system unresponsive until you satisfy it by typing 99 So what s with this verb define lingo Simple when you start J you get a few names defined automatically and verb and define are two of them as you can see by asking the interpreter what their values are verb 3 define 70 so when you use verb define you are really executing 3 0 to produce a verb similarly dyad one line definition is the same as 4 one line definition which executes the conjunction to produce a dyadic verb result The point is that a verb definition sequence is just an instance of a compound verb produced by a conjunction and the resulting verb can appear anywhere a verb is allowed You may assign the verb to
38. matrix multiplication maybe or adding a list of numbers and that even if a wide range of programs were possible the set of operations supported must be too vast to be practical wouldn t we need an array operation for every possible program The first half of this book is devoted to showing you that it is indeed possible to write meaningful programs with array operations We take the approach of looking at the different ways loops are used and seeing what facilities J has for producing the same result using array operations We will find that J contains a couple of dozen array processing primitives and a dozen or so very cleverly chosen pipe fittings that allow those primitives to be connected together to provide the limitless supply of array processing functions needed for practical programming Interspersed with the elaboration of more and more intricate array operations are treatments of other matters of use in practical programming structure definition input and output performance measurement calling DLLs modular programming Eventually we will see how to use if then else and do while in J though you will have learned more elegant ways to get the same results The last portion of the book is devoted to the optional topic of tacit programming J s language for functional programming Tacit programming is an extremely terse way of expressing algorithms one that allows programs that have been compressed from a page of C into a few li
39. member operator in C 1 abc 1 2 3 i 22 123 Item 1 was selected and opened x y can go through multiple levels of structure referencing at once If x is a list of boxes the first box must be a valid left argument to dyad it is used to select an item of y which is then opened the next box of x selects an item from that opened box of y which item is then opened and so on till x is exhausted structl abc 1 2 3 Jstruct2 def struct1 4 5 6 def 14 5 6 abc 1 2 3 WR cer gees Here we have a structure in which item 1 is another structure 1 struct2 Jabc 1 2 3 We select and open item 1 resulting in the enclosed structure 96 1 1 struct2 12 3 We select and open item 1 of struct2 and then open item 1 of the enclosed structure 1 lt lt 1 0 struct2 1 2 3l abc If the last selection specifies a list of items as in this example the selected boxes are not opened Note that the Dictionary s description of dyad incorrectly indicates that the boxes are opened 1 lt lt 1 struct2 Even if the list contains only one item it is not opened Only the last box of x may specify selection of a list of boxes ja lt 0 i 3 3 1011121 1314151 1617181 a is a 3x3 array of boxes 1 1 a rank error 1 1
40. operands for general operands we can select using a selection vector sv with sv 1x y To replace lowercase a through f with uppercase A through F in a string that contains only a through f abcdef i y ABCDEF Extending the previous example to replace lowercase a through f with uppercase A through F leaving other characters unchanged abcdef a i y ABCDEF a To understand this you need to know the special noun a which is the character string containing all the ASCII characters in order Work through a simple example until you understand how this works it s a good example of how J thinking differs from C thinking A similar problem given a list of keys y and a list of data z with each item of y corresponding to an item of z and another list of search keys x and a default element d return the item in z corresponding to the item of y that matches the item of x or d if the item of x didn t match anything iny 84 yi x z 4d To evaluate the polynomial defined by x so that if for example x is 2 1 5 the result is 5y y 1 t x y i x and now you can see why 0 0 is 1 To evaluate the polynomial defined by x going the other direction so that if for example x is 2 1 5 the result is 2y y 5 y x The last example due to Roger Hui has a power and economy that amount to sorcery Suppose you had a list and you wanted to know for each item in the list how many identical items
41. p J supports 3 different ways of specifying a polynomial 1 asa list of coefficients of increasing powers starting with power 0 i e a constant term where each coefficient multiplied by the corresponding power of the variable produces a term the polynomial is the sum of the terms This form is an unboxed numeric list 2 asa multiplier m and a list of roots r representing the polynomial m x ro x r1 x rn This form is a 2 item list of boxes m r if mis 1 it may be omitted leaving just lt r 3 for multinomials a multinomial of n variables is represented as a list of terms each term consisting of a coefficient and a list of n exponents one exponent per variable The multinomial is the sum of the terms The form is a boxed rank 2 array in which each item is a coefficient followed by the list of n exponents This form is distinguished from the multiplier root form by the rank of the boxed array It is possible to have multiple multinomials that share a common exponent array by having more than one coefficient preceding each list of exponents but we will not pursue that here For example three ways of expressing the polynomial 3x 12x which can be written 3x x 2 x 2 areO _12 0 3 3 2 0 2 and lt 2 212133 p y hasrank and converts between coefficient and multiplier root form of the polynomial y Note that converting from coefficient form to multiplier root form solves for the roots of the polynomial p 0 12 0
42. 2g i 2 93 120 012 345 Here the second box contains a rank 2 array while the first box contains a rank 1 array The items of highest rank have rank 1 which means that rank 1 items are going to be lined up as the items of a rank 2 result Any contents of lower rank are brought up to the rank of the highest rank contents by adding single length leading axes that will mean of course that the items of all the modified contents have the same rank but not necessarily the same shape Ifthe shapes of any of those items differ verb fills are added to bring each axis up to the length of the longest axis then the items are assembled into a list whose rank is 1 higher than the rank of an item In this example the concatenated items have rank 1 and verb fill was added to bring the single item of the first box up to a length of 3 91 ed se oe 2 3 a 111 23 4 There is one amendment to the processing as described above if any of the contents is an atom it is replicated to bring it up to the shape of an item of the result before items are concatenated Here the atom 1 was replicated to become 1 11 FOGLE Ho 722 3 A 100 23 4 Here the first box was a item list rather than an atom so it was padded with fills rather than replicated When we discussed dyad we glossed over the treatment of operands with different shapes Now we can reveal that the padding and replication for dyad is just like what monad does on the contents of boxes In fa
43. 3 1312 2 0 p 3 2 0 2 0 1203 If the multinomial form has only one variable i e each item has length 2 monad p will convert it to coefficient form p lt 2 2 _ 12 1 3 3 0 120 3 Dyad p has rank 1 0 and is used to evaluate a polynomial in any of these forms If x is a polynomial in coefficient or multiplier root form x p y evaluates it with y giving the value of the variable 0 1203p 1 _9 3 _2 0 2 p 2 1012 090 90 the second evaluation applied the polynomial to 5 different values of the variable If x is a multinomial x p lt y evaluates it with the list y giving the values of the variables y must be boxed because the right rank of dyad p is 0 and in this case there is 164 more than one variable So to evaluate the binomial x 3x y 3xy y with x 2 and y 3 we have lt 4 38130 321 312 10 3 p lt 2 3 125 as expected Calculus d D D andp J provides support for differential and integral calculus u d n produces the verb that gives the n derivative ofu es ds 2 y is y squared the derivative is y which is y doubled amp 3 d 1 3 amp amp 2 amp 3 y is y cubed the derivativeis3 y 2 amp 3 d 2 3 0 Second derivativeis3 2 y Ro de al 00 0 0 33333 amp p The _1 derivative is the indefinite integral which is y 3 3 The form the interpreter uses is the polynomial form FS K fd 1 6 72 g gd 1
44. LOKAT ROK oero vausasseccadans les age EE iced dans EE O A R REEE 185 35 Parsing and Execution Itinera EA E EE E A R one 187 36 Parsing and Execution M eriiereaceisce nnii ne a a 190 Wears Tablesrrensa enoe E a tit gel E E Gs 190 Examples Of Parsing And Execution ccccccccesscesseceseceseceeeeeeacecsaeceseeseneenseees 191 Undefined WEIS 1 5 a cess eecses a ceases vctece ah a a aa a 195 37 Forks Hooks and Compound Advervs 43 2 1 26 tts wine eA Renae des 196 Tacit and Compound Adver DS 0sc5 sseissd atectescte nad hercdvsnivin daneedaeaaecateetaaviiens 199 Referring To a Noun In a Tacit Verb iaceicn sci aistaairinatiadacnuuiinnaianddainnen 200 38 Readable Tacit Definitions s ss ssesessseesseseeesressessrssessessteseessessrsseesseeseesresseeseesressre 201 Flatte a V rb Adverb Ersan onie i E E A OE E O EER 201 Using f to improve performance sesessessessseesresesseessesresstesresrrssressrssresseest 202 39 Explicit To Tacit Converter s ssesessseesseseeesressessessressesstssessessrssresseeseesresseeseesressre 204 viii Special Verb Forms Used in Tacit Definitions 40 Valedictory 41 Glossary 42 Index ix 1 Introduction This book will tell you enough about J for you to use it as a language for developing serious applications but it is about more than learning the J language it is also about thinking big in programming and how programming in J is fundamentally different from programming in
45. Note Names defined by private assignment using when a script is loaded are not available outside the script If you want to define names for use elsewhere make sure you use for your assignments within a script If you are used to debugging with Visual C or the like you will find the environment less glitzy and more friendly If you want to change a verb function you simply edit the script using the editor of your choice I use the built in editor provided with J and rerun it The verb will be updated but all defined nouns objects will be unchanged Even if you are running a large application yea even if the application is in the middle of reading from an asynchronous socket you can change the program without recompiling relinking or reinitializing If you d like to add some debugging code while the system is running go right ahead This easy interaction with an executing program is one of the great benefits of programming in J Interrupting Execution If a J verb is taking too long to run press the BREAK key CtrI BREAK in Windows to return control to the keyboard Names Defined at Startup When J starts a number of useful names are defined Rather than discuss them all I will show you how they come to be defined so you can study them when you need to When J starts it executes the script J directory system extras config profile ijs which then executes the script J directory system extras util boot ijs
46. Read initial balances set day to start of month sum of balances to 0 fid fopen acctfn for nacct 0 2 fscanf fid f f acctno openbal acct nacct ano acctno acct nacct openbal openbal acct nacct prevday 1 acct nacct currbal openbal acct nacct weightbal 0 nacct fclose acctfn Il Process the journal for each record look up the account Il structure add closing balance values for any days that Il ended before this journal record update the balance fid fopen jourfn while 3 fscanf fid f f f acctno xactnday xactnamt for acctx 0 acct acctx ano acctno acctx acct nacct weightbal acct nacct currbal xactnday acct nacct prevday acct nacct currbal xactnamt acct nacct prevday xactnday II Go through the accounts Close the month by adding Il closing balance values applicable to the final balance Il produce output record for acctx 0 acctx lt nacct acctx acct nacct weightbal acct nacct currbal daysinmo acct nacct prevday printf Account d Opening d closing d avg d n acct acctx ano acct acctx openbal acct acctx currbal acct acctx weightbal daysinmo fclose fid The corresponding J program would look like this NB Verb to convert TAB delimited file into numeric array rdtabfile 0 amp 2 TAB amp _2 ReadFile lt NB Verb to process journal and account files NB y is days
47. _1 if an odd number are needed 0 if y is not a permutation 171 32 Odds And Ends To keep my discussion from wandering too far afield I left out a number of useful features of J I will discuss some of them briefly here Dyad Revisited x y does not require that x be a Boolean list The items of x actually tell how many copies of the corresponding item of y to include in the result 1202 567 8 5 6 6 8 8 Boolean x used for simple selection is a special case If an item of x is complex the imaginary part tells how many cells of fill to insert after making the copies of the item of y The fill atom is the usual 0 ora depending on the type of y but the fit conjunction may be used to specify f as the fill 132 1 0j1 2 5678 5006088 1j2 1 0j1 2 99 5678 5 99 99 6 99 8 8 Finally a scalar x is replicated to the length of y This is a good way to take all items of y if x is 1 or no items ifxis 0 Boxed words to string Monad 1 _1 y converts y from a list of boxed strings to a single character string with spaces between the boxed strings _1 a list of words a list of words Spread _1 x _1 y creates an array with the items of y in the positions corresponding to nonzero items of the Boolean vector x and fills in the other items x must equal y 11001 1 abe ab c You can specify a fill atom but if you do you must bond x to rather than giving it as a left ope
48. a a a 146 EOL yids JEN Ge andor xa Or SIN ic esresstn Saeed lan nists was apart aaa 146 whi be de vend and wha LetiWdo wend ai cai n aaia 146 if do else end 1 do elseif do Ond veccccccccccccsssssssscesens 146 try Ca bch etd and Capel 7 LALOW anc aii a a een 147 SETECty Case POSS ee SNA A eae cnectaunstate eds T EE E soar an 147 POS TTA T E O E E A E AE E 148 ASSE TE E E Aa Sats E Dd Aad Re as a NE S 148 28 Mod l r Code Aass eiinesinak aa N A sear eae 149 Locales And Locative Simai iienaa e aE a asi aa aa 149 ASSENTE haiio A a T hae otal nici A E Aa TR 149 Name Look p asiaton aa a E n E N A once 150 Changing The Current Local nesinnes naan aiaa 151 TheShared Locale t ZA pelee taena ae a ae eE EE a 154 Usne Locales e a a tan a A e A a deans a a as 154 29 Writing Your Own Modifiers s sssseesseesseseesseesresessseeseesresssesrsstessessessressessessresseese 156 Modifiers That Do Not Refer To x Of Y eessssesrrssesrrssssssesssereessereesseserresssrreessee 156 Example Creating an Operating System Dependent Verb sssssesseeseeeeeeeeee 157 Example The LoopWithInitial Conjunction oe eee ceeeeseeneeereeeeees 158 Example A Conjunction that Analyzes u and V cc ceesseeceeeeceteceeeeneeeneeerees 158 An Exception Modifiers that Do Not Refer to u OF V vee eeeeeceeseeteeeeeeteeeeee 160 Modifiers That Refer To sry av secttits ertaess teh caiceaeedhcu tae atta a a 160 30 Applied Mathematics ii iss
49. a name but that s not required Here are some examples showing how a definition is just an ordinary part of a sentence addrow monad t y 1 We define the verb using the conjunction and then we give the resulting verb a rank using the conjunction This is a principle to live by Always make sure any verb you define has the proper rank Following this rule will save you untold trouble by guaranteeing that the verb is applied to the rank of cell that you intended The verb produced by the conjunction has infinite rank here we expect to apply our verb on lists so we assign a rank of 1 before we assign the verb to the name addrow 3 t y i 6 15 See we define a verb then we execute it It doesn t have to be assigned to aname The distinction between code and data is not sharp as it is in C a 4 x b O a y 3 b 12345 6 21 bis y so we can use it as the text of a verb which we then execute b l1 a y 3 b 123456 720 Here b is y Ina later chapter we will explore the delights that result from being able to change the text of a program while you are executing it Remember make sure the verbs you define have the proper rank Compound Verbs Can Be Assigned Since we now understand that verb define is just a conjunction producing a verb result and we know that we can assign its result to a name we wonder whether we are allowed to assign any compound verb to a name Yes indeed
50. advantage of your skill in grasping the essence of a problem indeed it will develop that skill considerably by making it easier for you to express what you grasp but you will go through a period during which it will seem like it takes forever to get things done During that period please remember that to justify your choice of J you don t have to be as expert in J as you were in C you only have to be more productive in J than you il were in C That might well happen within a month After you have fully learned J it will usually be your first choice for describing a program Becoming a J programmer doesn t mean you ll have to give up C completely every language has its place In the cases where you want to write code in C either to use a library you have in C or to write a DLL for a function that is inefficiently computed in J you will find interfacing J to DLLs to be simple and effective This book s goal is to explain rudimentary J using language familiar to a C programmer After you finish reading it you should do yourself the honor of carefully reading the J Dictionary in which you can learn the full language one of the great creations in computer science and mathematics Acknowledgements I am obliged to the reviewers who commented on earlier versions Michel Dumontier Ken Iverson Fraser Jackson David Ness Richard Payne and Keith Smillie Brian Schott Nicholas Spies and Norman Thomson exchanged emails with me at len
51. and I need it all lumped into a single verb so that I can have the adverb dyad modify the whole thing One solution would be to create the verb v dyad 1 04 x y after which v x y works but it s a shame to have to interrupt a J sentence just to define a verb with such a puny function I want magic words to let me say 1 04 abracadabra combine x y Jhas such magic words and we will learn a few now The magic words will join verbs and nouns together so they must be modifiers adverbs and conjunctions Before we start we need a little notation to help with the different cases we will encounter Given a conjunction c or adverb a we call its left operand m if it is a noun or u if it is a verb Similarly we call a conjunction s right operand n if it is noun v if a verb There are four possible ways to invoke a conjunction u c v m c v u c n andm c n and two for an adverb u aandm a and they are defined independently Moreover the derived verb produced by the invocation the derived entity may be a noun adverb or conjunction too but that is unusual can be used as a dyad e g x u c n y orasamonad e g m a y and those cases are defined independently as well You won t get the cases mixed up because verbs and nouns are so different that it will seem natural foru c nto be different from u c v just be aware that the variants are many and that we will be learning a tiny subset of J s toolkit The adverb is an exampl
52. and apply the verb to it the verb returns shape 2 3 and the overall result is a list of 0 items with that shape i 0 2 3 002 3 The cells of y have shape 2 3 so we apply the verb to a list of 0 of them i e with shape 0 2 3 The result of has the same shape as the input and the final result is a list of 0 items each with shape O 2 3 1 e with overall shape 0 0 2 3 81 14 Verbs for Arithmetic It s time to build your J vocabulary with a few more verbs Dyads x y rank 0 0 number of ways to choose x things from a population of y things More generally 1y x y x x y rank 0 0 The xth root of y x y rank 1 1 y evaluated in base x i e the atoms of y are digits in the base x representation x and y must have the same length unless x is a scalar in which case it is replicated to the length of y A place value list p is calculated in which each item is the product of all the subsequent corresponding items of x formally this is p x 1 the last item of p is always 1 and the first item of x is immaterial Then each item in y is multiplied by its place value and the results are summed to give the result formally this is y p In the simplest case x is a scalar 10 234 234 The digits 2 3 4 interpreted as base ten digits 16 23 4 564 The same interpreted as base sixteen digits 16b234 564 as expected 16b234 is 234 in base 16 equivalent to 0x234 Here is an example wher
53. and if v is a noun it is assigned to n User written modifiers are of two types those that refer to the variables x and y and those that do not Modifiers That Do Not Refer Tox Ory If the modifier does not refer tox or y its text is interpreted when its operands u and for conjunctions v are supplied and its result is an entity which may be any of the four principal parts of speech The result replaces the modifier and its operands in the sentence and execution of the sentence continues Usually you will want to create a verb but nothing keeps you from writing a conjunction whose result is another conjunction Here we will confine ourselves to verb results If a modifier does not refer to x or y it can be invoked without any x or y only the u and v for conjunctions are used The text of the modifier is executed and the resulting verb replaces the modifier and its operands in the execution of the sentence Let s write some of the utility modifiers referred to in earlier chapters Ifany was an adverb that executed u if y had a nonzero number of items 9 3 5 NB Do this once to select simplified display Ifany 1 u lt Ifany lt Ifany does not need to look at y it creates a verb that executes u only if y has items Here we have executed the adverb Ifany with the left operand lt and the result is a verb the compound verb lt We can execute that verb on a noun operan
54. and then parsed and code is generated from the parsed result Not so in J a sentence is broken into words but the sentence is not fully parsed rather parsing and execution proceed simultaneously scanning the text from right to left Parsing finds patterns in the sentence that are then executed Execution includes the usual supplying of noun operands to verbs to obtain a result from the verb but also other actions supplying verb and noun operands to conjunctions and adverbs to produce derived entities and recognition of other sequences that we will learn soon Execution of a bit of a sentence which we will call a fragment consists of replacing the fragment with an appropriate single word namely the result of executing the fragment In the simple case where the fragment is the invocation of a verb i e the fragment looks like verb noun or noun verb noun the word that replaces it is the noun that is the result of the verb If the fragment is the invocation of a modifier the result of executing it will be a noun or a derived verb adverb conjunction A noun is nothing but its value but the derived verb adverb conjunction will itself eventually be executed it is called an anonymous verb adverb conjunction and is saved by the interpreter in a private form and the single word used to replace the fragment is a reference to this anonymous verb adverb conjunction for the case of an anonymous verb you may think of the single word as a pointer to
55. appeared earlier in the list You could find out this way yz 2221214642 t i y t Jt y 0120310014 Take a little ttme maybe a long time to see how this works The yisan idiom we discussed earlier did you figure it out It gives the ordinal of each item of y in other words the rank of the item among the items of y If there are equal items they will occupy a block of successive ordinals In this example you can see that t does indeed hold the ordinals t 234051798 6 i y takes the index of each item of y within y itself in other words for each item the index of the first item with the same value i y 0003036760 Since the identical items of y are a block of successive ordinals and i y comprises indexes of first items in blocks we can find relative positions in blocks by subtracting the ordinal of the first item with a value from the ordinals of all the other items with the same value That is what this expression does Lovely In addition to the foregoing ad hoc means of varying the operation cell by cell J has some language features expressly designed for that purpose Power If DoWhile Conjunction u n and u v u n y has infinite rank It applies the verb u to y then applies u to that result and so on for a total of n applications of u in other words u u u n times y as we see when it is used with the gt increment primitive gt 5 6 gt 2 5 7 gt 3 5 8 fndispla
56. are ready to see forks used in some practical applications This will be a relief after the last two chapters of theory You have learned the rule for the trident called the monadic fork VO V1 v2 Ny is VO Ny V1 V2 Ny Now learn the other 3 bidents tridents involving only verbs The dyadic fork Nx VO V1 V2 Ny is Nx VO Ny V1 Nx V2 Ny The monadic hook VO V1 Ny is Ny VO V1 Ny The dyadic hook Nx VO V1 Ny is Nx VO V1 Ny The purpose of taking you through the 2 preceding chapters was for you to understand that is in these definitions is shorthand for replaces the parenthesized part with an anonymous entity that when executed on an x and y produces the same result as you don t have to be a politician to hesitate over the meaning of is It may be helpful to think of these bidents and tridents as ghostly conjunctions with no actual symbol that create an entity the bident trident out of the sequence of verbs The entity so created is quite real it is executed just like any anonymous verb created by a modifier You can see that for both the hooks and the forks the monadic case is derived from the dyadic for forks by omitting Nx and for hooks by replacing Nx with Ny The verbs produced by hooks and forks have infinite rank Using hooks and forks assisted by all the modifiers we have learned we can produce any function of 2 operands If we have more than 2 operands we can link them together into a boxed list usin
57. calculate withholding tax on each employee s earnings The tax rate within tax brackets will be assumed to be the same for all employees C code for this would look like 64 int renderuntocaesar shekels float shekels result withholding for each employee Il tax bracket table start of each bracket ending with high value float bktmin 4 0 10000 20000 1e9 Il tax bracket table rate in each bracket float bktrate 3 0 05 0 10 0 20 int earns nemp int i j empearnings earns get earnings for each employee for i 0 i lt nemp t ti shekels i 0 0 for j 0 j lt sizeof bktrate sizeof bktrate 0 j if earns i gt bktmin j float bktval bktmin j 1 if earns i lt bktval bktval earns i shekels i bktval bktrate j In J we will sum over the tax brackets and we must create a suitable array for the summation The items in this array will be the amounts earned in each tax bracket Corresponding to the two if statements we will use conditional operators to discard amounts earned outside the tax bracket The conditionals will operate on each row so they will have rank 1 and they will look something like 0 gt bracket_top lt earned bracket _bottom which will give the amount earned in the bracket set to 0 if the earnings are below the bracket and to the size of the bracket if above We will create bracket_top by shifting the brackets left and filling with infinity this correspond
58. can we do it Ja lt 0 i 6 1011121314151 The easy way is 104 lt gt a 115 but after you get used to amp you will find that amp gt a 115 seems clearer because it expresses the temporary nature of the unboxing reboxing As an exercise take the time to see why amp gt a gives the same answer but amp gt a does not Defined obverses u v u v has the same ranks as u and it produces the same result as u except that the obverse ofu visv By defining verbs with appropriate obverses you make it possible to use amp and _1 with them For example in a finance application it is necessary to deal with dates in both year month day form and market day form for example if Friday is market day number 1200 the following Monday will be market day number 1201 If you have written routines to convert between them ymdtomd dyad definition mdtoymd dyad definition you would be well advised to make them obverses of each other ymdtomd dyad definition mdtoymd mdtoymd dyad definition ymdtomd so that if you want to find the y m d date of the next market day after the date ymd you simply code 1 amp amp ymdtomd ymd and J will convert ymd to a market day add one and convert back to y m d form u amp v and u amp v x u amp v y has infinite rank and is the same as v x u v y Itresembles u amp y but
59. cells using another level of implicit looping one that provides considerable additional programming power The formal description that follows is not easy to follow you might want to skim over it and read it in detail after you have studied the examples that follow J requires that one of the frames be a prefix of the other if the frames are identical each is a prefix of the other and all the following reduces to the simple case we have studied The common frame cf is the part of the frames that is identical namely the shorter of the two frames its length is designated rcf If we look at the cells of the operands relative to this common frame i e the rcf cells we see that for the operand with the shorter frame these cells are exactly the rank that will be operated on while for the operand with the longer frame each rcf cell contains multiple operand cells First the rcf cells of the two operands are paired one to one because they have the same frame leaving each shorter frame operand cell paired with a longer frame rcf cell Then the longer frame rcf cells are broken up into operand cells with each operand cell being paired with a copy of the shorter frame operand cell that was paired with the rcf cell an equivalent statement is that the cells of the shorter frame operand are replicated to match the surplus frame of the longer frame operand This completes the pairing of operand cells and the operation is then
60. coefficients II Code taken from Numerical Recipes in C 1 e include lt math h gt define PI 3 141592653589793 void chebft a b c n func float a b c float func int n int k j float fac bpa bma f 300 bma 0 5 b a bpa 0 5 b a for k 0 k lt n k float y cos PI k 0 5 n f k func y bmatbpa fac 2 0 n for j 0 j lt n j double sum 0 0 for k 0 k lt n k sum f k cos PI j k 0 5 n c j fac sum J version chebft adverb define f u 0 5 y y 2o0o o 0 5 i 2 x 20 0 0 5 i x 0 1 i oe x x x amp X 4 Declarations J has no declarations Good riddance No more will you have to warn the computer of all the names you intend to use and their types and sizes No more will your program crash if you step outside an array bound You specify the calculations you want to perform if along the way you want to assign a result to a name J will allocate enough space for the data It will free the space when the name is no longer needed Seasoned C programmers have learned to use declarations to create a web of type checking making sure that objects pointed to are of the expected type This is an example of making a virtue of necessity Since J solves the problem much more directly by not having pointers at all you will soon lose your uneasiness with weak typing Arrays But you ask with
61. different One difference is obvious the rank of _1 is infinite while the rank of is 1 but that is immaterial in Undery The other difference is subtle but it could be significant the two forms may have different performance in compounds The interpreter recognizes certain compounds for special handling the list grows from release to release but it s a pretty safe bet that will be selected for special treatment before _1 and lt before gt _1 and so on So we would like to replace the v _1 with the actual inverse of v We can get the inverse of v by looking atv b _1 which produces a character string representation of the inverse of v We can then convert this string to a verb by making it the result of an adverb we can t make it the result of a verb because the result of a verb must be a noun So we are led to Undery 2 a 1 v b _1 u v 1 u b 0 2 v b 0 where we defined the adverb 1 v b _1 and then immediately executed it with an ignored left operand a to create the desired verb form Now we have 0 1 Undery 0 1 0 0 which we can be content with An Exception Modifiers that Do Not Refer tou orv In very early versions of J modifiers could not refer to their x and y operands In those days a modifier used the names x and y to mean what we now mean by u and v Modern versions of J continue to execute the old fashioned modifiers correctly by applying the following rule
62. dynamically It is possible to clear the entire symbol table using y 0 s 10and10 s y but doing so invalidates any symbols previously created bys y If you would like to do high speed matching but what you want to match is not a string consider converting to strings using 5 5 lt y which converts the variable named y to string form 180 Unicode Characters u 2 byte unicode characters can be represented by variables that have the unicode atomic data type Such variables are created by the verb u Its use is described in the Dictionary Window Driver And Form Editor Designing user interfaces is quick and painless with J s Form Editor The Lab named Form Editor will show you how 181 Tacit Programming 33 Tacit Programs There is another language within J a microcode for J as it were Like Moliere s M Jourdain who was astonished to find he had been speaking prose for forty years without knowing it you have been using a small subset of this language unwittingly throughout our investigations The hidden language describes a way of coding called tacit programming and it is time for you to learn it in full J s tacit language is the irreducible essence of a programming language It describes your algorithm using only the ordering of the J primitives you have already learned It has a grammar without words which you use to write programs without visible operands yet its renouncing these seemingly essential components is
63. ee Eg ee eo 115 Format binary datar3 bs tis saad sti eesan de nin desea ay a ness 116 ORs sprintf and Go ee ferjeien oe scan ch Tonle a siens 117 Convert Character To Numeric Dyad oo eeecccccesceesceeseeeeeceeeceeeeeeeeeeseeeaeens 117 21 Calling a DLL Under Windows secistiains Gsiehesictasccs Mts desaastnscaseigstadsioidacese ceases 118 Memory Viet ae CMe UE 5s dates natie aie cg Saccacvencenee ns aac Ea iE a AEE ceases 119 Ahasing OF Variables assess cancns yarns oneview hake 120 Aliasing of Mapped Nouns s isiccccciseccesshssccteadeserisssccsasbavcses shsactsscteuatentietsaceaees 120 Aliasing of DEL Operands scicceta ssacaiseanctliascaaaedaweaioxniina a aai 121 225 Socket PROD PAINS ea e A ig a A R aa 122 SOS SCE E E E E E 122 Asynchronous Sockets and socket handler s sssssssssrsrseseseseseseseserrsseses 122 Names and IP Addresses csscsccssiesscts sssxsndagoatecinasteunedeantedysieestvsevducge teteceetoavececntess 123 ROTI COND ps2 cai ac tees esietecie tints Na tet call Dt al a a te ence pt ai 124 BIS fet in n ee rst ear Seat ehe ne ethane a Sota oa adh aoe eats cian se 124 Oiaoi SOOKE EV CU sage T EE EA T 125 Datagraf S spsa d5ccuunsianhlsteute spaeietesyaadastacabieaspacidsneitaaatiaqageranshsteamessanedtaatannedessenliees 125 Socket OPOS eaii aoai E AEREE a E A ER Ea 125 Housekeeping eneng e a a dees Peeenacuanccnen 126 23 Loopless Code V Partitions seseesseeseesseeseessesseseesseesessresseestesresseeseesresseesrese
64. form Similarly farther along we see that x i amp 1 lt y has special coding so we know to prefer that form over x lt y i 1 This list changes from release to release so you should review it occasionally J s performance is very good even if you pay no attention whatsoever to which compounds have special coding but if you re going to code a lot of J you might as well learn the interpreter s preferred idioms 109 Use Large Verb Ranks and Integrated Rank Support Think big is a watchword not just for program design but for coding as well Starting a primitive has a small cost but if you start a primitive for each atom of a large array the cost will add up To reduce the time spent starting primitives apply them to the biggest operands possible This means Use as large a verb rank as you can See what a difference a tiny change can make a i 100000 10 Ts a a 3 96384 4 19552e6 Ts a a 0 12801 8 3895e6 These two verbs produce identical results but is 30 fold slower than on this large operand The reason is that has rank 0 taken from the rank of while has infinite rank Rank 0 means that each pair of atoms is fed individually through the verb So when is executed two primitives are started for each pair of atoms one to add and the other to change the sign Execution of requires only two primitive starts for the entire array You do not need to worry much about the ra
65. i int i j for i 0 i lt nemp tti for j 0 earns i 0 j lt 31 j earns i hoursworked i j earns i payrate i When we write the pseudocode we will change the algorithm just a bit rather than multiplying each total by the billing rate just after the total is calculated we will make one loop to calculate the totals and then a second pass to multiply by the billing rate This is a case where good J practice differs from good C practice Because of the implicit looping that is performed on all verbs you get better performance if you let each verb operate on as much data as possible You may at first worry that you re using too much memory or that you might misuse the processor s caches get over it Apply verbs to large operands The pseudocode is 45 for each employee for each day take the sum of hoursworked for each pair of wage_rate and sum multiply the pair The first two loops are just 1 hoursworked as before The last loop clearly multiplies scalars so itis 0 We note that dyad has rank 0 so we don t need to specify the rank and we get the final program empearns monad payrate 1 hoursworked Problem 3 How much profit did each employee bring in C code void empprofit profit int profit result profit i is profit from employee i int i j temp for i 0 i lt nemp tti for j 0 temp 0 j lt 31 j temp hoursworked i j profit i temp billrate i pay
66. i Mow tees 101 TS S icide Verb Penerima ai e a E E e i 101 Multi Line Comments Using 0 sO neessesessssessessssssessessrssressessrsseessesstesresseeseeseessee 102 IED WS CENT i oee eae aes an cath Pa hate eat anes E 102 18 a WR AN SN eee a amin OAA ATO E ea eae 103 Pes OB Verse 5 Ms sce teamencnaitee it ea A E E A ER EARR 103 Apply Under Transformation u amp v and U amp iV coeceeeeeceeseceteeeteeeeeeeeseeesseeneenees 103 Defined obyers si t Say Vons aea emit ee eevee 105 DEV And E e a a RACETAB a e ee rae e eC EAT E 105 An observation about dyadic Verbs ccccccessccesecesecseseeeseecsaeceeeeeeeeeseecaeeneensees 106 19 Performance Measurement amp Tips ssesesssesssssesseeseesseesresresseesresersseessessrssressessessres 107 Compounds Recognized by the Interpreter c ce ecccesceeeeeeeeeeeeseceteeeeeeeeseees 109 Use Large Verb Ranks and Integrated Rank Support 0 cceeeeceesseeeteeeteeeees 110 20 Input And Output senesi ti E cos mete nie haa cok E RE te ata eure 112 POLCTONS cc areca ss usar curate E seco tu cect e S E 112 File Operations 1 n Error Handling cece cccccesecsseceseceeeceeeecsseceseesseeeenees 112 Error Handling u v 13 11 and ON EB hess liseeedectbdtnadeeddvihadentabess 113 Treating a File as a Noun Mapped Files ccecceesceesceesceceeceeeeeseeeeeeeeeseeenaeens 113 Format Data For Printing Monad And Dyad oo eee ceecceesceenseeeeeeteeeeeeenaees 114 Monad Tirer a ED
67. i a The first selection took item 1 of a This was a 3 item list of boxes and it is inadmissible to open the list and perform further selections Note that if x is unboxed dyad first boxes it and then uses it for selection The Dictionary s description does not mention the boxing step Report Boxing Level Monad L Monad L has infinite rank and tells you the boxing level of its operand Boxing level is defined recursively if y is unboxed or empty its boxing level is 0 otherwise its boxing level is one greater than the maximum of the boxing levels of its opened items 97 1 12 4 1 i LISIA II at U o Sss L 1 lt 2 lt 3 lt 4 3 L 1 lt 2 lt 3 lt 4 1 You can use monad L to decide how many levels of boxing to remove gt L lt lt lt 6 6 Note that an empty boxed list shows boxing level of 0 but the type revealed by 3 0 is boxed Also fill elements for an empty boxed list are the boxed fill element a L O a 0 3 0 0 a 32 3 O a IILI 98 17 Verb Definition Revisited Before we discuss verbs in more detail let me point out that the colon character which you have seen used mostly as a suffix to create new primitives has its own meaning when it is used alone or as the first character of a primitive is a conjunction andsoare and The dot also has a meaning but we won t be discuss
68. in a list Now see if you can figure out what i 3 3 1 0 i 3 will do before reading the explanation that follows The verb dyad 1 0 has left rank 1 and the left operand has rank 2 so the operation will be applied to 1 cells of the left operand The verb has right rank 0 and the right operand has rank 1 so the operation will be applied to 0 cells of the right operand The left frame is 3 the right frame is 3 Think of the left operand as a list of 3 1 cells and the right operand as a list OEE A ee SEE ARE of 3 0 cells The corresponding left and right operand cells are 01210 345 1 678 2 paired The operation dyad is performed on each pair of 0120 3451 6782 cells Since each result is a 1 cell of shape 4 and the frame is 3 the result is an array with shape 3 4 oO WO a BR UI N NBEO Figure 8 Execution of i 3 3 10i 3 38 defverbs comma i 3 3 comma l1 0 i 3 I0 1 2 comma 0 3 4 5 comma 1 6 7 8 comma 2 When Dyad Frames Differ Operand Agreement The processing of dyads has an extra step not present for monads namely the pairing of corresponding cells of the left and right operands As long as the frames fand rf are the same as in the examples so far this is straightforward If the frames are different J may still be able to pair left and right
69. matter how many there are A pleasant side effect of this way of coding is that the verbs you write can be applied to operands of any shape write a verb to calculate the current value of a loan and you can use that very verb to calculate the current value of all loans at a branch or at all branches in the city or all over the state We will write a number of J verbs starting from their C counterparts so you can see how you need to change your thinking For these examples we will imagine we are in the payroll department of a small consulting business and we will answer certain questions concerning some arrays defined as follows empno nemp in J just empno employee number for each member of the staff The number of employees is nemp payrate nemp number of dollars per hour the employee is paid billrate nemp number of dollars per hour the customer is billed for the services of this employee clientlist nclients every client has a number this is the list of all of them The number of clients is nclient emp_client nemp number of the client this employee is billed to hoursworked nemp 31 for each employee and for each day of the month the number of hours worked 44 To get you started thinking about cells rather than loops I am going to suggest that you use C style pseudocode written in a way that is easily translatable into J Your progress in J will be measured by how little you have to use this crutch Prob
70. of a number le2 16b1f equivalent to 0x1f _ infinity ___ negative infinity represented by two underscores A noun whose value is one of the numbers 0 and 1 is said to be Boolean Many verbs in J are designed to use or produce Boolean values with 0 meaning false and 1 meaning true but there is no Boolean data type any noun can be used as a Boolean if its values are 0 or 1 A word is in order in defense of the underscore as the negative sign x means take the negative of the number x likewise 5 means take the negative of the number 5 In J the number negative 5 is no cloistered companion accessible only by reference to the number 5 it is a number in its own right and it deserves its own symbol _5 Characters An ASCII string enclosed in single quotes is a constant of character type examples a abc There is no notation to make the distinction between C s single quoted character constants and double quoted character strings There are no special escape sequences such as n If you need a quote character inside a string double the quote cannot can be shortened to can t Character constants do not include a trailing NUL 0 character and NUL is a legal character within a string Valence of Verbs Binary and Unary Operators C operators can be unary or binary depending on whether they have one or two operands for example the unary operator means pointer dereferencing p while the binary operato
71. of the text of C and with y as the operand Such confusion is almost always harmless but let s go through a few examples so you can see the layers of execution A a 2 cus We execute 2 u and the result is an anonymous conjunction that we ll call acl The display of ac1 shows where it came from When ac1 is executed its result will be its left operand ti 2 te Tust s Here2 u is executed first to produce ac1 then acl is executed with left operand of and right operand of The result is an anonymous verb that we ll call av1 its value is the verb which was the left operand to ac1 2 u 5 10 Remember 2 u is a conjunction the conjunction we have called ac1 and conjunctions are executed before verbs so this is executed as 2 u 5 which becomes av1 5 We execute av1 with the operand 5 Monad doubles its operand and the resultis 10 We know that a conjunction can produce a conjunction result That s how explicit conjunctions are formed executing the conjunction with left operand 2 asin 2 n produces a conjunction There is nothing special about 2 n any conjunction is allowed to produce a conjunction result 2 amp 2 amp We execute 2 amp and the result is an anonymous conjunction that we ll call ac2 The display of ac2 shows where it came from the in the display of ac2 is harmless a reminder that internally the anonymous entity resulting fr
72. of the verb v1 is complete We should be back in locale loc3 from which we executed v1 153 18 5 Loc3 The Shared Locale z It is a J convention universally adhered to that every locale s search path must end with the locale z Any name in the z locale can then be referred to by a simple name from any locale making names in the z locale truly global names Using Locales You have my sympathy for having to read through that detailed description of name processing I refuse to apologize though because you really can t write programs if you don t know what names mean But you wonder How do I use locales You won t go far wrong to think of a locale as akin to a class in C When you have a set of functions that go together as a module define all their verbs and nouns in a single locale The easiest way to do this is to put a line coclass lt localename at the beginning of each source file for the module coclass is like cocurrent but it supports inheritance Then every public assignment in the file will automatically be made in the locale localename The names defined in the locale are the equivalent of the private portion of class To provide the public interface to the class you need to put those names in a place where they can be found by other modules The traditional way to do this is to define them in the locale z by ending each file with lines like epname z_ epname_localename_ H
73. point to that new block Aliasing of DLL Operands The J interpreter uses aliasing for boxed cells of an array so that if you execute b i 10000 10000 a b 5 item 0 of a simply contains a pointer to the data block of b rather than a fresh copy of the 400MB array In addition when a list of boxes is used as the right operand of ed as in dll spec cd root key 0 0 sam 0 _1 _1 any array operands to the DLL function are passed via a pointer to the data in the boxed list with no separate copy of the data being made This means that if the DLL modifies one of its arguments any nouns aliased to that argument will also be modified if the DLL function called above modifies its argument 1 the noun key and any noun aliased to key possibly including private nouns in suspended verbs will be changed To protect yourself from such side effects you can use 1 key in place of key in the invocation of cd 121 22 Socket Programming J provides the standard set of socket functions This chapter assumes you already understand socket programming and provides a description of J s facilities First you must load the script that defines the socket interface require socket require is like load but if the file has already been loaded it does nothing You are then ready to create a socket sdsocket AF _ INET jsocket_ SOCK_STREAM jsocket_ 0 0 96 sdsocket returns return_code socket_number Nonzero return codes
74. rank was then inherited by m v 5 uname 0 _5 05 Remember that monad is the signum function returning _1 for negative 0 for zero and 1 for positive operands For each atom of y execute 5 uname y ify is zero 5 y ify is positive and pass y through unchanged 5 y ify is negative uname must be defined elsewhere This expression makes use of negative indexing if y is negative verb number _ 1 the last item is taken from the gerund m v obviously can be used with a small rank to afford great control over what operation is performed cell by cell but if you do that it will have to apply J verbs on small operands which is inefficient After all we ve been through I feel confident that I can trust you not to use m v with small rank unless it s absolutely necessary 89 16 More Verbs For Boxes Dyad Link And Monad Raze Dyad has infinite rank It boxes its operands and concatenates the boxes abc 12 3 Jabc 1 2 3 Dyad is the easiest way to create a list of boxes abc 123 i 2 2 Jabc 1 2 3 0 1 12 3 Did you notice that I gave you an inaccurate definition If dyad just boxed the operands and concatenated its result would be like dyadsemicolon dyad lt x lt y abc dyadsemicolon 1 2 3 dyadsemicolon i 2 2 11 2 3 0 1 II 12 3 That s not the list of boxes we wanted Actu
75. sequence of words that cannot immediately be executed to produce a noun result Type An attribute of a noun numeric literal also called string or boxed Valence Of a verb definition an indication of the number of noun operands that the definition can accept monadic if 1 dyadic if 2 dual valence if either 1 or 2 209 Vector A sequence of cells arranged with a leading axis An array can be construed as a vector of its items Verb One of the primary parts of speech A verb operates on the noun to its right and its left if a noun is to its left to produce a noun result Verb Fill Fill added during processing of a verb The fit conjunction can often be used to specify the fill atom to be used for verb fill Word A sequence of characters in a sentence recognized as a lexical unit A word is either a name a primitive a constant which may be a number or a character or a list of either or a synthetic word used to refer to the result of an execution 210 42 Index dyad 47 dyad 82 169 monad 83 163 l lt 52 177 l 112 mg dyad 115 117 monad 138 dyad 114 dyad 50 172 monad 24 dyad 82 monad 27 dyad 82 monad 27 _1 dyad 172 dyad 20 monad 21 dyad 167 monad 167 173 dyad 47 dyad 82 211 monad 83 dyad 47 monad 48 dyad 48 169 monad 163 monad 83 3
76. shift only when x and y are both negative i e right shift of a negative number in which case the vacated bit positions are filled with 1 If you use shift and rotate you may need to know the word size of your machine One way to do that is PZ sl a 32b 1 32 Operations Inside Boxes u L n u S n u amp gt is the recommended way to perform an operation on the contents of a box leaving the result boxed It is the idiom used most often by J coders and the first one to be supported by special code when performance improvements are made in the interpreter Sometimes your operations inside boxes require greater control than u amp gt can provide For example you may need to operate on the innermost boxes where the boxing level varies from box to box In these cases consider using u L n which has infinite rank It goes inside the operands and applies u to contents at boxing leveln The monadic caseu L n y isthe simpler one It is defined recursively If the boxing level of y is no more than n the resultisu y Otherwise u L nis applied to each opened atom of y and the result of that is boxed The effect is that u is applied on each level n subbox and the result replaces that subbox with outer levels of boxing intact For example 175 Ja 0 1 2 3 4 5 lt lt 6 7 8 9 4 0 111 213 4 5 4 E ie a SAS 1617 8 9 I I1 it tt t
77. so here the scalar x is replicated to the shape of y before the unboxing occurs fndisplay shows the details where open means the inverse of open defverbs plus 0 open 0 lt 1 plus amp open 4 5 open open 1 plus open 4 open open 1 plus open 5 lt amp 10 amp 4 4 43 4 89 4 4 4 4 8 lt y finds the largest integer not greater than y by temporarily multiplying by 10 we truncate y to the next lower tenth We can easily define a verb to take the arithmetic mean i e the average of a list mean monad y y mean 1 2 4 2 33333 If we want to take the geometric mean we could define a new verb to multiply and take the root or we could just take the arithmetic mean of the logarithms and then undo the logarithm mean amp _ 124 2 Note that we had to use a verb of infinite rank as v so that u would be applied to the entire list This is a common enough pattern that the conjunction amp is provided which is just like amp but with infinite rank We could have used mean amp here To add 10 minutes to a time represented as hours minutes seconds we can transform to seconds after midnight do the addition and transform back O 10 O amp 24 60 60 amp 13 55 0 1450 Suppose we had a list of boxed scalar numbers and we wanted to add them and leave the result boxed How
78. sum is 80000 times slower than What happened Recall what monad sum is really doing It applies monad sum to the first item of the list then to the list made of the first 2 items then the list made of the first 3 items and so on At each evaluation of monad sum the dyad sum verb is interleaved between the items and the result is evaluated right to left The problem is the interpreter doesn t analyze sum to know that it is associative thatx sum y sum z is the same as x sum y sum z so it doesn t know that it can use the result from one subset as an 108 input to the operation for the next subset and it winds up performing every single addition for the 400 item it adds all 400 numbers together That s why its time increases as the square of the length of the list Monad is fast because the interpreter knows that dyad is associative and therefore it reuses the result from one subset as input to the next producing each item of the result with a single addition Well then can we give a hint to the interpreter that sum is associative Alas no but we have another trick up our sleeve Consider monad sum which applies monad sum to successive suffixes If the interpreter is clever it will notice that if it starts with the smallest suffix the one made up of just the last item and processes the suffixes in order of increasing size it will always be evaluating x sum previous suffix result and right to left e
79. this program by giving it operands 1 2 3 add2x3y 4 5 6 14 19 24 Instead of simply displaying the result we can assign it to a noun a 1 2 3 add2x3y 4 5 6 We can inspect the value assigned to the noun by typing the name of the noun a 14 19 24 We can use the noun in an expression 2 a 28 38 48 We can create a new verb that operates on the noun twicea monad 2 a twicea 28 38 48 Notice the after the invocation of twicea Remember to invoke a verb you must give it an operand even if the verb doesn t use an operand is just an empty string 0 or any other value would work too If you leave out the operand J will show you the value of the name since twicea is a verb its value is the definition of the verb twicea 3 2 a Of course in any practical application you will need to have most of your programs in a library so you can quickly make them all available to J J calls these libraries scripts filename extension ijs and runs them with the Load verb for example load system packages misc jforc ijs load reads lines from the script and executes them These lines will normally be all the verb and noun definitions your application needs possibly including Load commands for other scripts A script may end with a line executing one of the verbs it defined thereby launching the application or it may end after defining names leaving you in control at the keyboard to type sentences for J to execute
80. trains Examples are CO C1 A2 V0 V1 andthe VO V1 V2 we are considering now Understanding tacit programming will largely be a matter of understanding how trains are parsed and executed You will learn that VO V1 V2 is a new verb that can be applied to noun operands and you will learn how it applies to nouns To begin with observe that there is no reason that VO V1 V2 N should be the same as vO V1 V2 N which as we know is VO V1 V2 N The meaning J assigns to VO V1 V2 Ny is VO V1 V2 Ny is VO Ny V1 V2 Ny This substitution goes by the name monadic fork I think finding this definition was a stroke of brilliance in the design of J An example of the use of the fork is 4 6 8 6 which calculates the mean of the operand It is processed using the substitution rule above as 4 6 8 amp 4 6 8 6 which divides the sum of the items in the list by the number of items in the list You can use fndisplay to help yourself see how the substitutions are made 185 defverbs plus 0 div 0 tally plus div tally 4 6 8 The sequence isaverb It can therefore be assigned to a name mean or mean and then used by that name mean 12 18 24 18 Neat eh With just 4 symbols we described a program to take the mean of a list of numbers or a list of lists The beauty and the power are in the way the operands and verbs are connected that s what we ll be learning in the next few chapters
81. usual that the verb will apply to r cells if possible but with r negative the rank of those r cells will depend on the rank of the operand After the first step of processing the verb which decides what rank of cell the verb will be applied to verbs with negative rank are processed just like verbs of positive rank 35 t 1 i 3 012 Here the _1 cells are atoms so applying monad SumItems on each one has no effect 82 ae 23 3 12 The operand has rank 2 so this expression totals the items in each 1 cell t 24 223 3 12 21 30 The operand has rank 3 so this totals the items in each 1 cell leaving a 2x2 array of totals Verb Execution How Rank Is Used Dyads We are at last ready to understand the implicit looping that is performed when J processes a dyadic verb Because a dyadic verb has two ranks one for each operand and these two ranks interact with each other as well as the ranks of the operands you should not read further until you thoroughly understand what we have covered already We have learned that the rank conjunction u n is used to specify the rank of a verb Since each verb has the potential of being invoked monadically or dyadically the rank conjunction must specify the ranks for both valences This requires 3 ranks since the monad has a single rank and the dyad a left and a right rank The ranks n may comprise from 1 to 3 items if 3 ranks are given they are in order the monad s rank the dyad s left rank a
82. we can as we saw in the assignment to addrow in the previous section Any verb can be assigned to a name 100 dotprod 1 Dot product of two lists is the sum of the products of respective pairs of elements in the lists 1 2 3 dotprod 1 2 3 14 dotprod takes the dot product veclength dotprod 1 The length of a vector is the square root of its dot product with itself veclength 1 2 3 3 74166 Dual Valence verbs u v u v also defines a verb of infinite rank but it is completely different fromm n The defined verb is u if it is used monadically but v if used dyadically bv monad y dyad y x bv i 7 21 2 bv i 7 2101234 You can see that the appropriate valence was chosen based on whether the verb was executed as a dyad or as a monad u v is often used to assign a default left operand to a dyadic verb pwr 2 amp pwr dyad x y If you execute pwr as a dyad you get x y If you execute it as a monad you get 2 pwr y which is then executed using the dyadic valence of pwr to become 28 ye 3 pwr 4 81 pwr 4 16 The Suicide Verb The verb fails if it is executed Use it as one side ofu v to produce a verb that fails if it is used with the wrong valence Example i 100 110 120 130 Jout of memory 1 100 110 120 130 Oops we thought we were looking those values up in a list We re lucky that we just ran out of memory sometimes using the wrong valence of a ve
83. will be applying the verb to 0 cells The frame fis 2 2 Think of the operand as a 2x2 array of OU 0 cells 21 3 0 2 The verb is applied to each cell 4 6 Since each result is an atom i e a 0 cell the 02 result is a 22 array of 0 cells i e an array of 46 shape 2 2 Figure 1 Execution of i 2 2 Another example Ja 224 50011 0001 0100 0010 0011 0001 0100 0010 This is a rank 3 array 29 3 1 42 The verb rank is 1 and the noun rank is 3 so we will be applying the verb to l cells The frame fis 2 2 Think of the operand as a 2x2 array of CP e200 Keel 0100 0010 311 The verb is applied to each cell 4 2 Since each result is an atom i e a 0 cell the 31 result is a 2x2 array of 0 cells i e an array of 42 shape 2 2 Figure 2 Execution of 224 0011 00010100 0010 Controlling Verb Execution By Specifying a Rank The implicit loops we have used so far are interesting but they are not powerful enough for our mission of replacing all explicit loops To understand the deficiency and its remedy consider the new verb monad which creates the total of the items of its operand just think of it as monad SumI tems 12 3 6 The result was1 2 3 as expected i 203 01 3 4 NON 35 7 The result wasO 1 2 3 4 5 as expected remember that the items are added and the items of i 2 3 are l cells Adding together a pair of 1 cells
84. will need to finish this book before trying to use J Phrases The J Dictionary the official definition of the language Release Notes for all releases of J A description of foreign conjunctions A description of the operands to the wd verb Windows interface 10 Getting Help Your first step in learning J should be to sign up for the J Forum at www jsoftware com A great many experienced J users monitor messages sent to the Forum and are willing to answer your questions on J from the trivial to the profound 11 3 A First Look At J Programs Before we get into learning the details of J let s look at a couple of realistic if simple problems comparing solutions in C to solutions in J The J code will be utterly incomprehensible to you but we will nevertheless be able to see some of the differences between J programs and C programs If you stick with me through this book you will be able to come back at the end and understand the J code presented here Average Daily Balance Here is a program a bank might use It calculates some information on accounts given the transactions that were performed during a month We are given two files each one containing numbers in lines ended by CR LF and numeric fields separated by TAB characters they could come from spreadsheets Each line in the Accounts file contains an account number followed by the balance in the account at the beginning of the month Each line in the Journal fi
85. 172 Spread a gan ae em ir as aa nO i a A necro een One 172 Choose From Lists Item By Item monad M csccssccesscessccssccsscsesecesaceeensees 172 Recursion SF ea eiccsaacas ase galy cata dsngdcc vate saya send des Sage asad Ala E ETE EEE EEEE 173 Make a Table AGver dyad U sxacssseekt tcscuscaesyaal natch niall dcaecaisvcocednaeicaipnowtels 173 Boolean Functions Dyadm D ssaiissuccikteatissisceduseeons oisiahios Gate ontesiaed 174 Functions on Boolean operands cccccescceeseeeseeesseceeceeeeeeseeeseeceaecneeseeeenseees 174 Bitwise Boolean Operations on Integers ccccecsseceseceeeceeeeeeeecsseeeeeeeeeeensees 175 Operations Inside Boxes u L N U S Tai hal ans beans avtecrueaed 175 Comparison Tolerance Ao ij ssazasseestedstssdatlaszele ARER RENEE TEESE Ms ARE 177 Right Shitt Monad Jels fat cmessniecis adsze Macatee tea salva as Ste aaa tena eaters 178 Generalized Transpose Dyad lt cc cscccetivectesesuvateta lon ceusncsreswaieeamncestienaaon 178 Monad i and Dyad 20405 canst ys apdiscteuus vanes tan cy Saacawaee EE EE 179 Fast String Searching s Sym bOls 5 erese sescasccsaeasees dsscusstotusta sevens vaatataserseteneed 179 Umicode Characters ai pecenie a SaaS ashes tastes cts ce te eit gated Sl tse 181 Window Driver And Form Editor cccecccecccsscesscecesecesecseeeeseeecescecaeceeeeeeeeeaees 181 Tacit Programming 182 33 Tacit Programs recreere a a a a So Sioa twee ti cade deen Wakes asain 183 34 First
86. 20 3 20 withresult23 a 23 withresult23 The names a and b are assigned when the assignment fragments are executed What a verb function looks like As we saw a J verb function is defined by lines that look like name verb define J sentences here The result of the verb define is a verb and normally you will assign the result to a name so you can execute the verb by name when you need it Subsequent lines starting with the one after verb define and ending before the next line containing only the word are read and saved as the text of the verb heaven help you if you leave out the The verb is not compiled only the most rudimentary syntax checking is performed the text is saved and will be interpreted when the verb is executed Each line of the verb is a sentence statement The result of the last sentence executed becomes the result of the whole verb this is not precisely true but it s close enough for now details will be revealed in Control Structures Since a J verb has only one or two operands there is no need for you to provide a list of parameter names as you do in a function definition in C instead J names them for you At the start of a verb s execution the private name y is initialized with the value of the right operand of the verb Ifthe verb is dyadic the private name x is initialized with the value of the left operand Many programmers like to start their verbs by assigning these valu
87. 22S3S25 55 4 A boxed noun L a 3 Its boxing level is 3 L 0 a 1 121311 1 4 1 1 Jt t F1 1121212101 deal ar The contents of each innermost box where boxing level is 0 is replaced by the number of items there L 1 a 1112 I IESII Each level 1 boxed entity is replaced by the number of items L 2 a 1112111 Similarly for level 2 entities L 2a 1112 I IESII Negative level n means level of y minus n Note that this does not mean n levels down into y That would result in u s being applied at different levels in the different items of y instead the level at which u is to be applied is calculated using the level of the entire y The dyadic casex u L n y is similar but you need to know how the items of x and y correspond During the recursion as long as both x and y have a higher boxing 176 level than the one specified in n the atoms of x and y are matched as they would be matched in processing a verb with rank 0 0 with replication of cells if necessary If either operand is at the specified level it is not changed as the items of the other operand only are opened When both operands are at or below the specified boxing level u is applied between them The results of each recursion are boxe
88. 30 12 remember that monad converts an integer y to its binary representation producing a Boolean list we are taking x bits of that and then converting back to integer The binary code for 30 is 11110 the 3 low order bits are 110 and the result is 6 But when we have list arguments we get an incorrect result 3 4 3 0 1 Undery 32 31 30 01512 The result for 30 is wrong because was applied to the entire y 110 was extended with framing fills to become 1100 and the result is 12 instead of the expected 6 To get the right result we need to apply the verb to cells of the correct size _3 4 3 Undery 0 32 31 30 015 6 and naturally we would like to make Undery automatically produce a verb with the correct rank The way to find the rank of the verb u is to executeu b O In our conjunction u and v are verbs and we can use their ranks to produce an Undery that gives the correct rank 159 Undery 2 v _1 u v 1 u b 0 2 v b 0 We have selected the left rank of u and the right rank of v and put them as the ranks of the verb produced by Undery This produces the desired result 3 4 3 0 1 Undery 32 31 30 0156 and we can see the verb produced by Undery with its ranks 0 1 Undery 1 0 1 0 0 This version of Undery produces correct results but we should add one small improvement the inverse of monad should be monad rather than monad _ 1 because the two forms are
89. 6 288 u d n produces ordinary derivatives and evaluates its u with rank 0 For partial derivatives useu D n where u has rank greater than 0 Each cell of y produces an array of results one result for each atom of the cell giving the partial derivative with respect to that atom For example the length of a vector is given by veclength amp 1 which squares the atoms adds them and takes the square root veclength 3 4 5 7 07107 The derivative of the vector length with respect to the individual components is given by veclength D 1 3 4 5 0 424264 0 565685 0 707107 The result of u need not be a scalar Here we define the cross product 165 xp dyad 1 x _1l y _11 x 1 y 1 2 amp xp D 1 4 2 0 N O 00 2 0 00 00 Oo Each row is the vector valued partial derivative of the cross product 0 0 2 xp 4 2 0 with respect to one component ofy The interpreter will bend every effort to find a derivative for your function but it may fail or you may not like the derivative it chooses m D n when m is a gerund uv produces a new verb which executes like u but whose n derivative is v a D 1 Here a is defined to be except that its derivative is a D 1 a D 1 5 5 Sure enough the derivative is If you don t want to express the derivative you can have the interpreter approximate it for you x u D n y approximates the derivative at y by evaluating u at y and
90. C C programs deal intimately with scalars single numbers and characters and even when they combine those scalars into arrays and structures the operations on the arrays and structures are defined by operations on the scalars To ensure that each item of an array is operated on loops are created that visit each element of the array and perform a scalar operation on the element Writing code in a scalar language makes you rather like a general who gives orders to his troops by visiting each one and whispering in his ear That touch of Harry kind of generalling can achieve victorious results and it has the advantage that the orders can be tailored to the man but its disadvantages are significant the general spends much mental energy in formulating individual orders and much breath in communicating them individually more significant his limited attention is drawn toward individuals and away from the army as a whole Even the great Rommel was overtaxed at times The J programmer is in contrast a general who stands before his army and snaps out orders to the army as a whole Every man receives the same order but the order itself contains enough detail for the individual men to act appropriately Such a general can command a corps as easily as a platoon and always has the big picture in mind OK maybe you re not Rommel but you are a working programmer and you suspect that very few practical programs can be represented as array operations
91. J For C Programmers Henry Rich 2002 10 02 Copyright 2002 Henry H Rich All rights reserved Send comments to glasss bellsouth net Foreword You are an experienced C programmer who has heard about J and you think you d like to see what it s all about Congratulations You have made a decision that will change your programming life if only you see it through The purpose of this book is to help you do that It won t be easy and it certainly won t be what you re expecting You ve learned languages before and you know the drill find out how variables are declared learn the syntax for conditionals and loops learn how to call a function get a couple of examples to edit and you re a coder Fuggeddaboutit In J there are no declarations seldom will you see a loop and conditionals often go incognito As for coding from examples well most of our examples are only a couple of lines of code you won t get much momentum from that You re just going to have to grit your teeth and learn a completely new way to write programs Why should you bother To begin with for the productivity J programs are usually a fifth to a tenth as long as corresponding C programs and along with that economy of expression comes coding speed Next for the programming environment J is an interpreted language so your programs will never crash you can modify code while it s running you don t have to deal with makefiles and linking and you can test you
92. Multiple verb 169 M m n 99 mb 174 mD n 166 m H n 167 m n 57 m amp v dyad 205 m amp v monad 70 m v 88 m 6 136 m dyad 143 m monad 172 mapped files 113 Match verb 54 matrix verbs 163 memory management 119 mema 119 memf 119 memr 119 memw 119 modification in place 144 172 modifier 4 user defined 156 Modifier order of execution 58 Modify adverb 143 modular code 149 multinomials 164 N name 4 Nub Sieve verb 127 Nub verb 127 Number Of Combinations verb 82 169 Numbers verb 117 O o trigonometric function 48 obverse of verb 103 defined by u v 105 discussed under 103 operand passing by name 158 by value 158 order of execution in implied loops 42 P p dyad 164 p monad 164 p dyad 166 p monad 166 p monad 170 parts of speech 3 passing a verb as an argument 135 performance measurement 107 permutation parity 171 permutations direct representation 170 standard cycle representation 170 Permute verb C 171 plot package 167 Polynomial Derivative verb p 166 Polynomial Integral verb p 166 Polynomial verb p 164 polynomials 164 calculus 166 Power conjunction 85 Prime Exponents verb q 170 Prime Factors verb q 170 Primes verb p 170 primitive 4 printf 117 priority of number representation 169 punctuation 4 Q q dyad 170 q monad 170
93. NS ace E a a aa 48 Boolean Monad orei aeai a AA E REER EE AARE EA 49 Operations OM AITAYS eosar iea A E E A medicine R D Grates 49 Dyads cca tat are ate Reh E R E A E tk is 49 Monad Sn e scecind cides Sede A E A Cetin 54 8 Loopless Code II Adverbs and sssssssssssesssssessseessesesssesseserssesseserssressessessressesse 58 IMIG ITI CS access sacs nea aae eaea aa T A ERE E SEER E E A 58 TheAdv rb Monad ig orea e E A aa me a 58 TES AUVs oe at aun EEE E A O E EE 60 De Continuing to Write inasre iuen ni ae a a agaia a iins 62 10 Compound Verbs senenin a EA ER R AAE 68 Verb Sequences u v and UQV seseesessesesserseeersseseesesseseesessererssestesessestesesseseeseeses 68 Making a Monad Into a Dyad The Verbs and ossonnssesnssseessessesseesseesseseesseseess 69 Making a Dyad Into a Monad u amp n and M amp V sessssssessessssssesseesreseesseessesersseessesess 70 Tel Boxing S ct res j eenaa a E E E A A AE aT 72 Terminolo SY sinr a ARE A ARE EE a EA EEA aE E a AA 74 Boxing As an Equivalent For Structures In C uo cecccceeseeseeseeeeeseeeseeseeeeeeesees 75 V22Empty Operands staen e e e A cane ance E A R tase a a A i 76 Execution Ona C l Or Fills prann nna r a A 76 Empty cels 5 sc ieenaede cece ona a a a E a A a a A a aA 78 If Fill Cells At Not Enot ghisee areen aaa iaaiiai 78 13 Loopless Code II Adverbs and 5 sarccsccanseasteniu tinecignecarcadesasasesatecunbeeraetsiiensses 79 LAW TS ESASTAN EAE A ENE E E E
94. The following table may help verb function or operator noun object copula assignment punctuation separator adverb untranslatable conjunction untranslatable In keeping with the grammatical flavor of the vocabulary we say that every word token in a J program has a part of speech name type which is one of the following noun verb adverb adjective copula or punctuation The primary parts of speech are noun verb adverb and conjunction Every name we can create and every word defined by J except for the copulas and and punctuation will be a definite one of the primary parts of speech In this book the term entity is used to mean something that can be any of the primary parts of speech An entity can be assigned to a name but most entities are anonymous appearing and disappearing during the execution of a single sentence just like intermediate results in the evaluation of C expressions A noun holds data a verb operates on one or two nouns to produce a result which is a noun an adverb operates on one noun or verb to produce a derived entity a conjunction operates on two nouns or verbs to produce a derived entity Adverbs and conjunctions are called modifiers A word on punctuation under J s definition it consists of the characters and end of line written LF but representing either a single LF character or the CRLF combination along with the comment delimiter NB and a few oth
95. The integer was made into a rational number so it keeps its precision 3 0 4r5 3 0 8 The floating point constant forces the rational number to floating point 2 3r4 3r2 The operation was performed on rational operands with a rational result Results of verbs are given a higher priority representation if necessary 169 4r9 2r3 5r9 0 745356 Explicit conversions between extended rational and floating point can be performed by the infinite rank verb x x y converts floating point y to rational or integer y to extended integer The inverse x _1 y converts in the other direction A rational number can be split into numerator and denominator by 2 x y rank 0 2x 1r3 5 13 5 1 Factors and Primes Monad p Monad and Dyad q p y rank 0 gives the y prime prime number 0 is 2 q y rank 0 gives the prime factors of y x q y rank 0 with positive x is the first x items or all items if x is _ in the list of exponents in the prime factorization of y _ q 700 2021 p 0123 2021 700 x q y with negative x returns a 2 row table The second row is the nonzero items of x q y i e the nonzero exponents in the prime factorization the first row is the corresponding prime numbers q 700 25 22 e N Permutations A and C In the direct representation of a permutation p each item i p of the permutation vector indicates the item number that moves to position i when the permutation is appl
96. a function that performs according to the definition of the anonymous verb In all cases the word replacing the fragment has a definite part of speech and if it is a verb a definite rank The Parsing Table Execution of a sentence begins by breaking the sentence into words The words with a beginning of line marker shown here as prepended become the initial contents of the unprocessed word list A push down stack will also be used during execution it is initially empty Execution of the sentence is performed by repetition of the parsing step which comprises 1 examining the top 4 elements of the stack to see if they match one of the 10 executable patterns 2 if a match was found executing the executable portion of the stack what we called the executable fragment in the last chapter resulting in a single word which replaces the fragment on the stack 3 if no match was found moving the rightmost word of the unprocessed word list into the leftmost position of the stack pushing the rest of the stack to the right Execution finishes when there are no unprocessed words and the stack does not contain an executable pattern Note that execution of a fragment may leave the stack matching one of the executable patterns so several sequential parsing steps may perform an execution without moving anything onto the stack After all words have been processed the stack should contain a beginning of line marker followed by a single word which beco
97. ach integer in the range 0 to y 1 is similar but works in descending order Example 1 3 T4 T 50 9 130245 Read this result as follows the smallest item of y is item 1 with value 1 the next smallest is item 3 1 then item 0 3 then item 2 4 then item 4 5 then item 5 9 Monad defined this way turns out to be surprisingly useful As a limbering up exercise in the use of permutation vectors and an example of how compactly J can express ideas see if you can describe in words what two applications of monad will give 56 3144159 203145 Add An Axis Itemize re Y The result of y has rank one higher than the rank of y with one item which isy The shape of y is the shape of y with 1 prepended in plain J y is Constant Verb m n ee 57 We have met the rank conjunction applied to verb left arguments applied to noun left arguments it produces a verb whose result is always the value of the noun The created verb which can be used as either a monad or a dyad has ranks given as the right operand of Examples 5 _ i 44 5 The simplest and most common case Since the verb has infinite rank it operates on the entire operand and produces the scalar value 5 0 i 3 555 Here the verb is applied to each 0 cell of the list giving the scalar result for each cell 1 2 3 02 4 3 12 3 12 3 12 3 Here the result at each cell is the left argument of the list 1 2 3 Ifyo
98. adds the respective atoms as we will soon learn is 2 3 This application of monad to a rank 2 array corresponds to the C code fragment for j 0 j lt 3 j sum j 0 for i 0 i lt 2 i for j 0 j lt 3 j sum j array i j Suppose we wanted to add up the items of each row as in the C code fragment 30 for i 0 i lt 2 i sum i 0 for j 0 j lt 3 j sum i array i j to produce the result 3 12 How can we do it in J What we have learned so far is not enough but if we had a way to make monad apply to 1 cells if we could make monad have rank 1 our problem would be solved the implicit looping would cause each row to be summed and the results collected You will not be surprised to learn that J does indeed provide a way to apply monad on 1 cells That way is the rank conjunction We will learn all about conjunctions later on the syntax is a little different than for verbs but for now we ll try to understand this It s used like this u n to produce a new verb that is u applied to n cells individually This is a simple idea but its ramifications spread wide As a first example 1 i 2 3 3 12 This is what we were looking for It happened this way The verb rank is 1 and the noun rank is 2 so we will be applying the verb to 1l cells The frame fis 2 Think of the operand as a list of 2 1 cells 012 345 The verb monad is applied to each cell
99. adians or degrees between the real axis and a line in the complex plane from the origin to the point representing the number larl 0 54030250 841471 1ad90 051 A number of verbs are available for operations on components of complex numbers All have rank 0 y conjugate of y y creates a 2 atom list of the real and imaginary components of y y creates a 2 atom list of the length and angle in radians of the polar form of y y magnitude of y j Y Ojl y r y ojl y x j y x j y i e xis the real part y is the imaginary part xr y x r y i e polar form where x is the magnitude and y the angle in radians y factorial of y more generally the gamma function T 1 y Matrix Operations J has primitive verbs for operations on matrices some using the conjunction y gives the determinant of y x y is the matrix product of x and y y is the matrix inverse of y provided y is nonsingular x y is the projection of xontoy Inx y arank 1 x is treated as a matrix with one row and a rank 1 y is treated as a matrix with one column but the result rank which is 2 when rank 2 matrices are multiplied is 1 when one operand has rank 1 and 0 when both do x yisarough and ready way to get a least squares approximation of x as a linear combination of the columns of y If your y is singular or close to it avoid and use methods based on singular value decomposition 163 Polynomials
100. al sum seems to be about 50 times slower than Let s just try adding a list to itself remember that u y is equivalent to y u y 107 1000 Ts al0 2 68191le 5 896 100 Ts sum al0 0 00033021 2560 Yes sum is definitely slower than though only by a factor of 10 or so this time Why should it be slower The answer is Because it deals with atoms Since J verb definitions are not compiled but interpreted line by line on each execution every single time we add two numbers with sum the interpreter has to parse x y and perform the addition Why it s a miracle that it only slows down by a factor of 10 The lesson is that if you define verbs with small rank the interpretive overhead will be significant Still that doesn t fully explain why sum is so much slower than Let s investigate further by increasing the size of the operand a20 i 20 1000 Ts a20 2 68191e 5 1280 Ts sum a20 0 00728641 3648 is unchanged when we move to a list of 20 items the operation is so fast that time is being spent starting the verb rather than running it but sum slows down noticeably Interesting let s try bigger and bigger operands a40 i 40 1000 Ts a40 2 76572e 5 1408 Ts sum a40 0 0299561 4160 a100 i 100 1000 Ts al100 2 76572e 5 1664 Ts sum al100 0 185741 5184 a400 i 400 1000 Ts a400 3 77143e 5 3200 Ts sum a400 3 00367 11328 Holy cow On a 400 item list
101. al treatment in J The definition of an empty operand is not as obvious as you might think An array is empty if it has no atoms i e has a 0 in its shape but whether an operand is empty depends on the rank of its cells it is empty if there is a O in the frame with respect to the cells operated on by the verb Consider an array with shape 3 0 4 This is an empty array because it has no atoms As an operand of a verb with rank 0 it has frame 3 0 4 so there are no cells and the array is an empty operand As an operand of a verb with rank 1 it has frame 3 0O and again is empty As an operand of a verb with rank 2 though it has frame 3 and is not empty there are 3 cells each of which has shape O 4 In this case each cell is an empty array but the operand is not empty As an operand of a verb with rank 3 or higher the frame is empty and each cell has shape 3 0 4 so there is one cell and the operand is not empty though each cell is an empty array You can see that an operand may have items and yet be an empty operand if the verb operates on cells smaller than items as is the case in this example for a verb of rank 0 or 1 We are left with the question What is executed when an operand is empty For something must be executed It is fundamental in J that if a verb produces a result with shape s when applied to a single cell executing that verb over an array of that cell even an array of none of them produces an array of results each wit
102. ally dyad is more subtle x y always boxes x but it boxes y only if y is unboxed That produces the behavior we want most of the time the exception is when the last item in the list is boxed already lt abc lt def lt ghi ghi Jabc def If we expected all the items to be boxed the same we are disappointed We must develop the habit that when we use a sequence of dyad s we box the last operand unless we are sure it is unboxed and even then we might do it to reinforce our habit 90 lt abc lt def lt lt ghi Jabe def ghil Monad infinite rank removes one level of boxing from an array of boxes concatenating the contents of all the boxes into one long list The shape of the operand of 1s immaterial If the items do not have a common shape the items of the contents of the operand boxes mind you not the contents themselves they are brought up to a common shape as described below Examples abc x d H ef labcidlef A list of boxes gt abc d ef abcdef The items which are scalars are joined into one long list 1 234 1234 It works for numbers too i 2 3 i 2 3 012 345 012 345 The items are lists so the lists are concatenated into a rank 2 array PAL
103. ame lookup uses the current locale as the starting point Locale at start of v2 is locl v1 has executed v2_locl1_ J starts searching for the name v2 in locale loc1 and its path eventually finding itin z v2 is executed using the definition found in z but with the current locale set to the explicit locale of the locative namely loc1 Note that v2 was also defined in loc3 as which would give an error but loc3 was never searched The operand of v2 was n1 note that the lookup for n1 is completely independent of the lookup for v2 n1 is sought and found in loc3 and it is that value that becomes y at the start of execution of v2 nl nl1 in locl y nl1 in loc3 Simple name lookups start in the current locale loc1 The private name y has the value it was given on entry nl n1 in loc2 Here we have switched the current locale to loc2 using cocurrent and the name is found in the new current locale Locale at end of v2 is loc2 Value returned by v2 is nl in loc2 Here v2 has finished and control has returned to v1 Note that the value returned by v2 is simply the result of the last sentence executed it is a noun Here it is the value of n1 at the end of v2 at which time the current locale was loc2 Locale in vl after calling v2 is loc3 Note that when v2 finished the current locale was restored to its value before execution ofv2 The cocurrent in v2 has no effect once v2 finishes nl nl1 in loc3 Execution
104. ard mathematical notation give multiplication precedence over addition We will discuss the parsing rules in detail later for now know that modifiers are bound to their operands before verbs are executed and that if the left operand of a modifier has a conjunction to its left e g x c y a the conjunction is bound to its arguments first and the result of that becomes the left argument to the modifier x c y ais x c y a not x c y a Inother words modifiers associate left to right So 1 in which is a conjunction and is an adverb is the same as 1 not 1 The phrase 1 4 i 3 3isexecutedas 1 4 i 3 3 in accordance with the rule right to left among verbs but applying modifiers first Note that I had to put parentheses around the 4 because 1 4 would have been interpreted as rank 1 4 collecting adjacent numbers into a list is done before anything is executed J includes a rich set of modifiers and even allows you to write your own though many J programmers will never write a modifier We will begin our study of modifiers with the adverb monad u which goes by the mnemonic Insert The Adverb Monad u Monad u by which we mean with a verb left operand used as u y rather than asx u y which is dyad u and is completely different note thatm y where m is a noun is different yet inserts u between items of y Monad u has infinite rank As a simple example 1 2 3 is equivalenttol 2 3
105. are defined to be the 0 gt k 8 cells Conjunction One of the primary parts of speech A conjunction modifies the words or phrases to its left and right to produce a derived entity that can be any of the four primary parts of speech Copula One of the parts of speech signifying an assignment of a value to aname The copulas are and Derived Entity The result of executing an adverb or conjunction If the part of speech of a derived entity is known it may be called for example a derived verb Dyad dyadic A verb with left and right operands which must be nouns Any verb may be used as a monad or dyad depending on whether it has a left noun operand when it is executed Entity A noun verb conjunction or adverb Execution The process of replacing a verb and its operands on the execution stack with the result from applying the verb to those operands or of converting a fragment into a derived entity in accordance with the definition of the fragment and replacing the fragment on the execution stack by a single word referring to the derived entity 207 Execution Stack The set of words of a sentence that have been examined by parsing in its search for an executable fragment as modified by the replacement performed by execution Fill An atom appended to a noun to extend the length of the noun to a required length especially when the noun must be extended because it is being made a cell in an array whose cells are longe
106. asy in J but it can often be avoided by ordering the axes properly in the first place Negative Cell Rank Items In some cases you may know the length of the frame and want to define the cells to have whatever rank is left over for example you may have a noun with one cell per employee but you may not know the rank of the cells We say that you are selecting the cells of the noun relative to the given frame Negative cell ranks indicate this A _1 cell has the shape corresponding to a frame of length 1 5 6 in our example a _2 cell has the shape corresponding to a frame of length 2 6 in our example and so on If the specified frame is longer than the rank of the noun the entire shape of the noun is used for the frame and the cells are 0 cells i e atoms In our example a _3 cell a 4 cell a _5 cell etc all refer to atoms As an important case of this the _1 cell of an atom is the atom itself _1 cells are so important in J that they are given the name items Our example has 4 items each of shape 5 6 Anatom has one item itself Remember an atom has one item itself The index of an item in an array is the sequence number of the item from the beginning of the array The first item in an array has the index 0 just as in C Lists When we choose to view an array as a collection of its items we say that the array is a list of its items In our example above the items of the 4x5x6 array are 5x6 arrays and we say that t
107. ate In other words Jinity k p 25 10 100 25 100 50 5 10 Here we used the last item of k as a placeholder The values don t matter but we want the right shape so the program will still work if we change the number of coins ink Applying this value to the first knave we get 50 50 knave inity 100 100 50 25 10 5 50 50 Yes that s right the first two items of the result are the two dollar coins the knave took and he threw his coins in at the end Before we go on we can t help noticing that taking the first two items of splate and then dropping those same items that s needless work We can simplify knave to knave dyad x y x Now we need to apply knave sequentially on all items ofk We have learned enough J to write a sentence to do that but because this is a recurring problem I have written a conjunction LoopWithInitial to hide the complexity we ll look at its details in a moment This conjunction takes the verb knave the initial value inity and the array k and applies knave repeatedly with each item of k taking a turn as x with the y set to the result from the previous invocation of knave Jres knave LoopWithInitial inity k 100 100 50 25 10 5 50 50 50 50 50 25 10 5 100 25 100 50 25 25 10 5 5 10 25 251010 55 25 50 50 251010 55 25 10 We see the result after each application of knave if you want to see the input alongside the result typek amp lt _1 res The contents of the plate are included i
108. be executed can be quite complex and can have a mixture of named and anonymous components as in the next example Sentence mean amp 1 4 5 6 find the difference between arithmetic and geometric mean unprocessed word list mean amp 1 456 mean amp 4 1 456 result av defined as 4 1 mean amp Cavl 456 result av 8 mean amp av1 456 result av2 3 defined as mean av2 amp av 456 result av3 5 defined as av2 mean av3 amp av 456 result av3 8 mean av3 amp av1 456 result av4 4 defined as av3 amp av1 mean av4 456 result av5 5 defined as mean av4 Cav5 456 result av5 8 av5 456 result 0 0675759 0 0 0675759 Again there was only one execution of a verb It happened at the very end after av5 was created it was executed and its execution included the execution of everything else Sentence inc a amp twhereais4 5 6 unprocessed word list inc a amp inc 456 amp result 4 inc 4 amp result 4 inc 4 amp result av defined as 4 amp NI o o inc ay result av inc is assigned av av 193 This illustrates an important point Even in the middle of a complex definition verbs are applied to nouns wherever
109. been selected for the nub ofy Green grow the lilacs 1 11011101101100110111 a 110110 a a RPNOFr Oo WWNHEe yisequivalentto y y Apply On Subsets Dyad u Dyad u has infinite rank x u y applies u to subsets of y for which the corresponding items of x called the keys are identical I will skip the formal description in favor of a verbal one First find the nub of x call it nx Then for each item of nx 127 find all matching items of x make an array of the corresponding items of y this array will always have rank one more than the rank of an item of y and apply u to that array the result becomes one item of the result ofx u y Theitemsofx u y thus correspond to items of the nub of x Note that the subsets of y may have different shapes they will all have the same rank being made of items of y but each subset may have a different number of items For this reason we usually have u produce a boxed result to avoid framing fills Ja 2010 2 lt Hello Ho el 1 The subsets were created and each one was boxed Note that each subset is a list even the one with only one element amp gt a 1212111 30303 100 1 200 2 300 600 3 The summing was performed for each subset Note that the result is sorted not on the value of the keys but rather on the smallest index of each key in x Ja _2 Fred 100 Joe 200 Fred 50 Sam 30
110. boot ijsin turn executes a series of scripts in J directory system main which define the starting environment Look at these scripts to see what they define If you want to add your own initial definitions do so either by adding commands at the end of profile ijs or by creating your own startup script in J directory system extras config startup ijs Step By Step Learning Labs The Labs are interactive demos describing various topics in J To run the lab for printf start a J session on the menu bar select Studio Labs then select the lab you are interested in then press Run The lab provides explanatory text interspersed with examples executed in your J session which you are free to experiment with as you step through the lab I recommend that every now and again you tarry a while among the labs running whichever ones seem interesting Much of the description of the J system can be found only there J Documentation The J documentation is available online Pressing F1 brings up the Vocabulary page from which you can quickly go to the Dictionary s description of each J primitive At the top of each page of documentation are links to the manuals distributed with J these are The Jndex to all documentation The User Manual which describes components of J that are not in the language itself including system libraries and external interfaces The J Primer an introduction to J J Phrases a collection of useful fragments of J you
111. boxes both explicitly using dyad and dyad or implicitly using dyad i and dyad e but there is an arithmetic flavor to the operation if the contents of corresponding components of the boxes are both numbers tolerant comparison is used 72 Most primitives that accept boxes as operands do not examine the contents of the boxes but instead perform their operation on the box atoms themselves Any deviation from this behavior will be noted in the definition of the verb we have not encountered any yet Examples 3 lt abc labc abc abc The dyad was applied to the box creating a list of identical boxes lt 1 2 lt 5 lt abc 1 2 5 abc The boxes were concatenated resulting in a list of boxes Note that the contents of the boxes do not have to have the same shape or type 101 lt 1 2 lt 5 lt abc 1 2 abc The selection performed by dyad is performed on the boxes not on their contents Since applying a verb may result in the addition of verb fills and framing fills we need to meet the fill element used for boxed nouns It is the noun a a is defined to be lt 0 0 i e an atom which is a box containing an empty numeric list note that this is not the same thing as a box containing an empty string or an empty boxed list a II 3 lt 5 I5I1I a was used for the fills added by overtaking
112. bs m and then executes that verb m v requires that m be a valid gerund It produces a verb which can be used monadically or dyadically and whose ranks are the ranks of v The operation of this verb is as follows v y if monadic orx v y if dyadic is evaluated it must produce a scalar result r that is a valid index into m i e m lt r andr lt m Then item r m is selected it is the atomic representation of one of the verbs that went into m and that atomic representation is converted toa verb u Finally u y if monadic orx u y if dyadic is executed and its result is the result of the execution ofm v 88 So verb0 verb1 verb2 v y evaluates v y resulting in r and then executes verbr y The dyadic case x verbO verbl verb2 v y evaluates x v y resulting in r and then executes x verbr y The verbs may be any valid verb a primitive a compound verb or a named verb Examples 1 amp amp 2 2 amp 1 O i 6 1 13153 This added 1 to each even number and subtracted 2 from each odd number Note that we had to assign rank 0 to the overall combined verb because otherwise the rank of 1 amp amp 2 2 amp would have been the rank of 2 amp which is infinite because m amp v has infinite rank 5 3 113 5 0 amp gt 0 2 T _ 5 3357 Subtract 2 from elements of x that are negative add 2 to elements that are nonnegative Here we assigned the rank to the selector verb in m v that
113. c2_ d a_loc3_ a The name was not defined in loc3 so the path was used and the name was found in loc1 a_loc2_ b The value in loc2 can be retrieved if we start the search there c_loc3_ c If the value is found in the starting locale no search is performed c_locl_ Jvalue error c_locl_ We have not defined a path for loc1 so loc2 is not searched Changing The Current Locale The current locale can be changed in two ways explicitly by executing the cocurrent verb whose purpose is to change the current locale and implicitly by executing a verb named by a locative cocurrent y sets the current locale to y Simple as that cocurrent uses the foreign 18 4 Do not use 18 4 directly It is intended to be used under an alias and it has side effects 151 Executing an entity named by a locative almost always a verb but it could be a modifier as well saves the current locale changes the current locale to the explicit locale of the locative before starting the entity and resets the current locale to the saved value when the entity finishes Note that the entity always runs in the explicit locale of the locative even if the search for the name found the entity in some other locale in the search path Whenever a named entity finishes execution the locale is restored to its original value even if the entity changed the current locale Here are examples of actions affecting the current locale load printf
114. cent numbers that you don t want to have made into a list Unlike in C an array in J may have one or more axes of length 0 Such an array has no atoms but its rank is still the number of its axes 17 A single number or character is called an atom object of basic type which is said to have the type numeric or character as appropriate Actually there are types other than number and character including a type that resembles a structure but we won t get to them for a while An atom is also called a scalar An atom is defined to have rank 0 therefore its shape is an array with 0 items i e an empty array of rank 1 Just as in C every atom of an array must have the same type Cells Because the execution of every J verb involves breaking the argument into pieces presenting the pieces to the verb and assembling results J has a vocabulary for describing these operations A rank 3 array of shape 4 5 6 such as the one defined in C by the declaration int q 4 5 6 can be thought of as an array of 4 elements each with rank 2 and shape 5 6 or as a 4x5 array of elements each with rank 1 and shape 6 or as a 4x5x6 array of rank 0 atoms The term cell is used to indicate the rank of the elements that will be operated on A 0 cell is an atom a I cell is an element of rank 1 a 2 cell is an element of rank 2 and so on a 3s B cell a Ro item i cell Anis 2 Once you have picked a cell size you can think of your noun a
115. ck do block end The T block is evaluated and its result A that is the result of the last sentence in the T block is saved block is executed once for each item item not atom of A If you use the handy for_x form where x represents any valid name of your choice the private variables x and x_index are created and every time block is executed x_index is assigned the index of an item of A and x is assigned x_index A The break and continue control words do what you would expect while do end and whilst do end The allowed forms are while T block do block end whilst T block do block end while corresponds to while and whilst corresponds to do while The st in whilst stands for skip test the first time through so you get one free pass through the loop just as with do while The break and continue control words do what you would expect if do else end if do elseif do end The allowed forms with optional components underlined are 146 if T block do block else block end if T block do block elseif T block do block end The flow of control is as you would expect T blocks and blocks are both sequences of zero or more J sentences The result of the last sentence in a T block provides the result of the T block The result of a T block tests true if its first atom is nonzero or if it is empty or if the T block was empty The flow of control is as you would expect based on the test
116. ct x y is equivalent to lt x lt y When you have an array of boxes the difference between opening it with monad gt and with monad is that monad gt keeps the frame of the array of boxes and brings every opened box up to the same shape while monad just runs the items of the contents together into one long list with no regard for the shape of the array of boxes Verbs With More Than 2 Operands Multiple Assignment Dyad is part of the standard J method of passing many operands to a verb The invocation of the verb normally looks like this verbname opl op2 lt opn the lt is needed only if opn is boxed and the verb that is so invoked looks like verbname monad define opl op2 opn y remainder of verb The line op1 op2 opn y is J s handy multiple assignment When the target of the assignment is a string the string is broken into words and the words are matched with items of the value being assigned they must match one for one or a length error results Then each word from the string is used as a name which is assigned the corresponding item of the value If the value being assigned is boxed each item is unboxed before it is assigned When defined and invoked as shown above the variables op1 op2 etc during execution of the called verb will hold the values of op1 op2 etc in the calling verb Multiple assignment is not restricted to parameter passing you may use it as you see
117. d 156 lt Ifany 1 2 3 Remember that Ifany is an adverb so it has precedence and the line is executed as if lt Ifany 1 2 3 Theverb lt Ifany which has the value lt is applied to 1 2 3 and produces the boxed result lt Ifany An empty y is left unboxed u Butifnull n was a conjunction that applied u if y had items otherwise it produced a result ofn It could be written Butifnull 2 n _ u Again will check whether y has items and this time the result will be used to select the appropriate verb to execute lt Butifnull 5 5 _ lt When Butifnu11 is executed with operands it produces a verb lt Butifnull 5 abc Jabc lt Butifnull 5 5 The verb it produces can be applied to its own noun operands Example Creating an Operating System Dependent Verb The great thing about modifiers that do not refer tox or y is that they are fully interpreted before the x and y operands are supplied so there is no interpretive overhead during the processing of the data Here is a more complex example taken from the J system The goal is to define a verb playsound that can be used to play a wav file under Windows NB y is the file data to be played playsound adverb define select 9 12 NIL case 2 do winmm dll sndplaysound i c i amp 15 0 amp 1 case 6 do NB 2 nodefault 4 memory 16b20000 file winmm dll PlaySou
118. d stats subdirectories have scripts for mathematics and statistics other subdirectories cover topics such as finance printing graphics and interfacing to Windows 168 31 Elementary Mathematics in J Verbs for Mathematics All the verbs have rank O 0 x y Lowest common multiple of x and y x y Greatest common divisor of x and y x y number of ways to choose x things from a population of y things More generally y x y x The verbs dyad e Member of and dyad set difference are useful in working with sets Extended Integers Rational Numbers and x In J numbers are not limited to 32 bit integers or 64 bit floating point Extended integer and rational are atomic data types like numeric literal and boxed that allow representation of numbers with arbitrary accuracy An extended integer constant is defined by a sequence of digits with the letter x appended a rational constant is two strings of digits numerator and denominator separated by the letter r examples are 123x and 4r5 The various representations of numbers in J can be given a priority order boolean low integer extended integer rational floating point complex high When a dyadic arithmetic operation is performed on operands of different priorities the lower priority operand is converted to the higher priority representation The simplest example arises in a list constant 12345678901234567890 4r5 12345678901234567890 4r5
119. d this will give each the deeper boxing level of the two operands at each application ofu An example O 1 lt 2 3 L 0 10 20 10 21 112 22 13 23 t Poss y was passed through and applied to each level 0 entity O 1 lt 2 3 L 0 lt lt 10 20 Once again y was applied to each entity but because it has boxing level 2 all the results have boxing level 2 The conjunction S is like L but instead of preserving the boxing of the operands it accumulates all results into a list 0 1 lt 2 3 8 0 lt lt 10 20 10 21 12 22 13 23 Comparison Tolerance f Like a diamond earring that adds a sparkle to any outfit the fit conjunction is a general purpose modifier whose interpretation is up to the verb it modifies We have seen f used to specify the fill atom for a verb and to alter the formatting of monad Its other important use is in specifying the comparison tolerance for comparisons A comparison like x y calls two operands equal if they are close where close is defined as differing by no more than the comparison tolerance times the magnitude of the larger number If you want exact comparison you can set the comparison tolerance to 0 using 0 1 0 1 000000000000001 0 1 1 000000000000001 1 Tolerant comparison is used in the obvious places verbs like dyad dyad gt and dyad and also in some unobvious ones like the verbs monad monad
120. d a dyad you will usually find 3 numbers the first is the rank of the monad the other two are the left and right rank of the dyad For example click up the page for and you will see 10 which means that monad has infinite rank while dyad has left rank 1 and right rank 0 For any verb including user written verbs you can ask the interpreter the rank by typing verbname b 0 b 0 1 0 Verb Execution How Rank Is Used Monads The implicit looping in J results from the interplay of verb rank and noun rank For monads it goes like this 1 Figure out the rank r of the cells that will be operated on this will be the smaller of rank of the verb and the rank of the operand This rule applies even if the verb has infinite rank r will be the rank of the operand which is another way of saying that the verb applies to the operand in its entirety 2 Find the frame fof the operand with respect to cells of rank r 3 Think of the operand as an array with shape f whose items are cells of rank r Apply the verb to each r cell replacing each cell with the result of the verb 28 Obviously this will yield an array of shape f whose items have the shape of the result of applying the verb to an 7 cell Let s look at some simple examples i 22 01 2 3 This will be the right operand i 2 2 0 2 46 The steps to get this result are The verb rank is 0 and the noun rank is 2 so we
121. d by that try block returns control to the catcht block of the try Just as an error in execution causes the catch block to be executed a throw causes the catcht to be executed select case fcase end The form is select T block0O case T blockl1 do blockl end T block0O is evaluated then the T blocks of the case control words are evaluated sequentially until one is found that matches the result of T block0O the 147 following block is then executed after which control passes to the sentence following the end fcase is like case except that after the block of an fcase is executed control passes to the next block rather than to the sentence following the end A T blockn matches the result of T block0 if the result of T block0O is an element of the result of the T blockn So a T blockn could be 2 3 5 and any of those three values would match it Before this check is made each side of the comparison is boxed if it is not boxed already An empty T blockn matches anything return return ends execution of the definition that is running The result of the last sentence not ina T block is the result Example if T block do return value return end assert assert sentence assert fails with an assertion failure error if the result of executing the sentence contains any atoms that are not equal to 1 Note that there is no end corresponding to the assert so there may be only a single sentence rather tha
122. d in the formm H n where mis the numerator list andn is the denominator list The resulting verbm H n has rank 0 The monadm H n y takes the limit of the sum of the generalized hypergeometric series the dyad x m H n y takes the sum of the first x terms Formally the generalized hypergeometric function is A Mo Mi k M cm Di Y i to Ao A r can K Ifm contains 2 items and n contains 1 item m H n defines a hypergeometric function where a a a 1 a k 1 Generalized hypergeometric functions can be used to calculate a great many functions of interest Legendre polynomials Laguerre polynomials Chebyshev polynomials and Bessel functions of the first kind are all special cases of hypergeometric functions Ewart Shaw in http www ewartshaw co uk data jhyper doc gives a number of examples of uses of H For example the error function and the cumulative distribution function are given by erf 3 2p_0 5 y y 1H 1 5 y nOlcdf 3 gt erf y 2 NB CDF of N 0 1 where I have rewritten Shaw s formulas to use elementary J 2p_0 5 is 2 sqrt n Sparse Arrays Monad and Dyad y converts the array y into a sparse matrix representation which can save a lot of space and time if most of the atoms of y have the same value _1 y converts sparse y back to normal dense form A large but incomplete subset of operations is supported on sparse arrays look at the description of if
123. e x m name name x m name and executes the modification in place i e without creating a new copy of name For the modification to be in place there must be nothing else in the sentence either before or after the form shown above and the two appearances of name must be identical x and m may be any expressions that evaluate to nouns 145 27 Control Structures Your reward for persevering through two dozen chapters on J is to be shown the direct equivalent of if then else for and while I have waited until I am sure you realize that you don t need them and will choose them not because you see no other way to solve a problem but because you think they are the best way if while and the rest are classified as control words in J Their part of speech is punctuation like a parenthesis or LF character They are allowed only inside definitions that is to say inside right operands of the conjunction If you want to use them from the keyboard or in a script you must define a verb adverb conjunction and then execute it A control word ends any sentence to its left as if the control word started with an end of line character We will cover only a few important control sequences here to find the rest bring up the J Vocabulary by pressing F1 and click on Controls which is hidden in plain view next to the heading Vocabulary for do end and for x do end The allowed forms are for T block do block end for _ x T blo
124. e creates an entity having the simple name in the locative residing in the explicit locale given by the locative A public assignment to a simple name creates the named entity in the current locale Entities in a locale are not affected by completion of an explicit definition they have a life of their own and can be referred to by any verb that knows how to reach the locale they are in The following examples illustrate assignments the interpretation given is correct if the lines are entered from the keyboard Simpl 5 NB public outside of explicit definition vbl verb define NB public isimp simpl NB private referring to public simpl simpl 8 NB private inside definition locl_z 10 NB public locative simp2 12 NB public isimp simp1 NB result vb1 NB execute vbl see result 5 8 Note that simp1 was set to 8 by the explicit definition Because this was a private assignment the public value was not changed simp1 5 The public value is still 5 as it was before the explicit definition was executed locl z_ 10 simp2 12 The other public assignments leave their results in the locale they were assigned to isimp value error isimp The entities assigned by private assignment were destroyed when vb1 finished Note that the load verb which runs scripts is an explicit definition Any assignment using executed during load will be lost Use to define names in scripts Name Lookup Names a
125. e same as y The great thing about having a verb to do this loop rather than a statement is that we can give the verb a rank and apply it to cells with independent loop control on each cell 2 100 amp gt _ 0 1 3 5 7 9 11 128 192 160 112 144 176 Read this as for each atom of y double it as long as the value is less than 100 u _ 1 is also of great interest but we will discuss it later One last point gt 1 2 4 5 679 As you can see n may be an array in which case u n1 y is repeatedly evaluated with n1 assuming the value of each atom of n and the results are assembled using the shape of n as the frame with framing fills added as needed Pop quiz express the preceding English sentence in J Solution u n y is equivalentton u 0 _ y andx u n yis equivalent ton x amp u 0 _ y Ifyou can make sense of the answer you should be content with your progress If you came up with either half on your own you are entitled to claim Apprentice Guru status Tie and Agenda switch The Tie Conjunction u v u n mv mn s The backquote character is the conjunction named Tie is one of the few conjunctions that produce a noun so it is neither monadic or dyadic If an operand of is a verb it is converted to its atomic representation which is a noun form from which the verb can be recovered then the two operands m and n both nouns now since any verb was converted to a noun are joined by executingm n S
126. e we have learned about monad u but dyad u is very different as is m Verb Sequences u v and u v uQ v creates a derived verb of infinite rank that applies v to its argument s and then applies u to the result In other words u v yisthesameasu v yandx u v y is the same asu x v y Examples 314159 1 Monad produced the permutation 1 3 0 2 4 5 of which we took the first item 68 123 12 3 14 Dyad produced 1 4 9 whose items we then summed fndisplay shows the details defverbs plus 0 times 0 1 2 3 plus times 1 2 3 u v is like u v except that the rank of the derived verb is the rank of v also expressible as u v v because u v is defined to have the function of u with the rank of v My advice is to stick to and avoid unless you re sure you need it 1 2 3 12 3 149 What happened We thought we were multiplying the vectors and then taking the sum but because we used rather than the derived verb had the rank of dyad namely 0 which means that the derived verb was applied to each cell at each cell we multiplied and then took the sum of the single cell In fndisplay form defverbs plus 0 times 0 1 2 3 plus times 1 2 3 1 times 1 2 times 2 3 times 3 plus never got executed because plus was applied to 1 element lists leaving in each case the single element Making a Monad Into a Dyad The Verbs and
127. e daily balance for an account tasks that clearly require loops and yet there is nothing resembling loop indexes This is one of the miracles of J loops are implied in C terminology they are expressions rather than statements and so they can be assembled easily into single lines of code that replace many nested loops We will be spending a lot of time learning how to do this We also note that there is nothing in the J code corresponding to the define MAXACCT 500 in the C This is one of the things that makes programming in J so pleasant you don t have to worry about allocating storage or freeing it or wondering how long is long enough for a character string variable or how big to make an array Here even though we don t know how many accounts there are until we have read the 14 entire Accounts file we simply read the file split it into lines and numbers and let the interpreter allocate as much storage as it needs to hold the resulting array The last thing to see and perhaps the most important is that the C version is just a toy program It searches through the Accounts information for every record in the Journal file We can test it with a small dataset and verify that it works but if we scale it up to 10 000 accounts and 1 000 000 journal entries we are going to be disappointed in the performance because its execution time will be proportional to A J where A is the number of accounts and J the number of journal entries It
128. e me a boxed list If I have a list whose atoms are boxes e g 74 lt 1 lt 2 is that also a boxed list Unfortunately writers on J have not agreed on terminology for these cases In this book a boxed list will be a list that has been put into a box it is therefore an atom and a list of boxes is a list each atom of which is a box Higher ranks are described similarly If we say that a noun is boxed that simply means that its atoms are boxes Boxing As an Equivalent For Structures In C A C struct corresponds to a list of boxes in J where each box in the list corresponds to one structure element Referencing a structure element in C corresponds to selecting and opening an item in J For example struct int f 1 2 3 char g abc float h 1 0 is equivalent to lt 1 2 3 lt abc lt 1 0 A C n dimensional array of structures is equivalent to a J array of boxes of rank n 1 The last axis of the J array corresponds to the structure and its length is the number of structure elements in the C structure C has no exact equivalent for a single box 75 12 Empty Operands Every programmer knows that errors lurk near boundaries for example at the beginning and end of arrays or at the point in execution where an array becomes empty Our discussion of verb rank omitted one important situation suppose there are no cells An operand with no cells is said to be empty and it gets speci
129. e where x is a scalar The result has the same shape as y with the items of y rotated x places to the left with wraparound If x is negative the items are rotated to the right Examples 2 Ven 3d Ao 5 9 931 le tw 33 A I WOO amp RP N ONnNor UU y Shift Left When the fit conjunction is used any item that is rotated across the beginning of y is replaced by the fill This turns the rotate into a shift where f gives the value to be shifted in 2 x abcde cdexx _2 x abcde xxabc Sort x y Sort Up Using x y Sort Down Using 53 The left rank is infinite x and y must have the same number of items The items of x are records and the items of y are keys x y is the records x sorted into ascending order of corresponding keys y x y sorts into descending order Examples 314159 012345 314159 y was already in order 314159 543210 95141 3 Sorting into reverse order 314159 0101 20 2 30 345119 x in order of ascending y i 4 3 10 20 1 2 6 7 8 9 10 11 O 1 2 3 4 5 Items of x in order of ascending y 135 78 12 45 351 The keys y do not have to be single numbers they don t even have to be numeric Here 1 2 is lowest then 4 5 then 7 8 The Dictionary gives complete rules for ordering y Because sorting is not so easy in C C programmers are not quick to recognize applications of and The J implementation of and runs in linear t
130. e would like is a 158 conjunction Undery such that x u Undery v y produces v _1 x uv y For example to encipher the characters of y by replacing each one by the letter x positions earlier we would use 5 1 3 2 Undery abcdefghijkl amp i 0 hijk to perform the function t abcdefghijkl amp i 0 hijk t 5 13 2 t t abcdefghijkl chgi With that x stuck in the middle of the desired result v _1 x u v y it appears that we will have to refer to x in our conjunction but actually we can use an advanced feature of J to make the x disappear The sequence u v produces a verb that when executed as the dyadx u v y gives the result ofx u v y you will learn about this and more if you persevere with the part of the book devoted to tacit programming So the verb we are looking for is v _1 u v and we can write Undery 2 v 1 u v 5 13 2 Undery abcdefghijkl amp i 0 hijk chgi Before we pat ourselves on the back for this achievement we should consider whether the verb produced by Undery has the proper rank We see that it does not Undery applies v to the entire y and u to the entire x and the result ofu y when really we should be performing the operation on cells of x and y where the cell size of x is given by the left rank of u and the cell size of y is given by the right rank of v For example if we wanted to take the x least significant bits of y we could use _3 0 1 Undery
131. e x is a list converting time in hours minutes seconds to seconds since midnight 24 60 60 12 30 0 45000 The list p was 3600 60 1 The first item in x has no effect on the result x y rank1 0 This is the inverse of x y except that if the first digit of the result is not between 0 and x it is changed modulo x to lie within that range For example 24 60 60 45000 12 30 0 The normal case converting the number of seconds since midnight back to hours minutes seconds 82 24 60 60 36 30 0 131400 24 60 60 131400 12 30 0 Here the result would have been 36 30 0 but 36 is outside the range 0 to 24 so it is replaced by 24 36 whichis 12 If you want the first item of the result to be unconstrained make the first item of x either O or_ O 60 60 131400 36 30 0 Note that monad and monad are similar to the dyadic forms with x set to 2 To keep and straight in your mind remember that the one with a single dot produces a single scalar result while the one with multiple dots produces a list result Monads all rank 0 y factorial of y more generally the gamma function I 1 y y same as the base of natural logarithms e y y same as the base of natural logarithms e y y sameasy 2 y sameasy 2 y sameasy 2 y sameas2 y You can see the common feature in the arithmetic primitives ending in 83 15 Loopless Code IV In our quest to write l
132. eaeeaceeeteareeeets sean ee 82 Dyads orrei n E E E as ae ci EARE OE RE a 82 Mon ads all rank O aneian a E E A a 83 15 Loopl ss Code IV isemi a a e a E E A A a A aati wakes 84 Power If DoWhile Conjunction u nand U2 V essssesssssessseesseseesseessesersseessesees 85 Tie and Agenda SWitCh neesesenseseeeesseseesesseseossesesssssestesesseseesssssenessesesessesresres 87 The Tie Conjunction ie a m v M N e s ssssssssseesseseessressessrsseessessrssressesse 87 The Agenda switch conjunction M V sssssessssessseesesessesesesresesessesestsresesesseses 88 16 More Verbs FOr BOXES rnaen a naana iea band AE E A ar aa as 90 Dyad Link And Monad Raze eeneesessseessesesessessrssresseesresresseeseeserssessesess 90 Verbs With More Than 2 Operands Multiple Assignment eeeeeeees 92 Dyad Revisited racte a aa a a 93 Split String Into J Words Monad s1 coscsch necracastaisnmaltuceesaneaiawekaemeaties 96 Fetch From Structure Dyad sesesssssssesessssessessrssressessrssresseestesresseesresrrsseessesres 96 Report Boxing Level Monad L ssssesssssssssssssessessrssressessrssresseesresresseeseserssessesess 97 I7 Verb Definition Revisited isinin nhen ieoi a a a oia 99 What really happens during m n and verb define n essssssssrseesessesersesseseesees 99 Compound Verbs Can Be Assigned ccccccsecessceeeeeesceeeeeeseeceaeeneeeeeeeeseeesaeens 100 Dual Valence verbs 2 0 aiicesen wed tostce etueeaee i
133. eeessecsseceseceeseeeseecsseceeeeees 7 R nnimg aJ protam ien e aa a i a a 8 Interrupting EXECU ON ie rronisrieorii e isidi ANA RAY LET EA TR AEEA 9 Names Defined at Startup a icucsicetidivnccticctsceapaseiedinahes Aas todaeasncedeeteaen 10 Step By Step Learning Labs s a siscasedsenndssveiasesacseansesdadeuse so cebesataesceasdseountessacen sense 10 J Documentation de eco Gd Sa Ooo sack cear e aie anata Gere EN as ls 10 Getinge Wel pics caste aru a Ga veo dese E E ew andes Maeva is 11 Se A First Look At J Progrannss jc ssstsqreessadsbaitusssigaeiashandud aN aaO SEa OREORE SERRAS EGES 12 Avetase Daily Balanc cenenenessin ai a EE el E E E 12 Calculating Chebyshev Coeth cient 55 c0cecsecsssesensads eaasneecasedacuseenenauecousssacteesnnees 15 4 Declarations oneen e ni e E A E E E A Eai 17 AAYE a ese sas EEEE E AE E E E EE TEE E E Aa 17 Cells a r cease an E A A E has Sauces S 18 Choosing Axis Orders ict asinn ennn a a a a tea a a 19 Negative CellsRatiks Items sisirin i E aa 19 EiStS apie pe ia spade ad uae A ace S ESE O A a A AA a 19 Constant ES EG sts tet cc atoue ease E E A ae R A a ala Re 20 Array creating Verbs dees ede ies a eraciedes e N ER a e ya a ayes rene 20 Dyad hape and monad Shape Of ccceccecssecsseeseeeseeseeeseceeeseeeeeeeees 20 Monad Fally oases tea arses hus dete e e oes A E A SOUR 24 Monad i Integers ido Aacse gash ol mare eats See c aaa shea roel Soe ug ne tad Seta a ane 24 5 Loopless Code I
134. eire etnio r n EA ach tetas KAREE Rina 163 Complex Numbers 2 5 arainn a a a aa aias 163 Matrix Operatii ase care teeta inns a eae a a E n 163 Polynomials peee eani reed i a vee a R E a eatin ds 164 Calcul s sdn Day DE aNd Dre n a aS Bt a NOH IE a Up ie 165 Taylot Series to ES ANG Te os tate ioca a a a aa 166 vii Hypergeometric Function with H scsiecstiseeasievageateedekaie ndoatdeerusaidvwonedanamsauneiens 166 Sparse Arrays Monad and Dyad sssssssessssessessssssessessrssresseesreseesseesseseessressesess 167 Random Numbers 2 resserre erasa E A EREEREER R A 167 Pl fran inerneta eae ea T a AEE A aa 167 Comp tational Addons scc aa a lati oe 168 Useful Scripts Supplied With J ss sesseseseesseessesseeseesseessseesseesresessseessesersseessesess 168 31 Elementary Mathematics in J s22 5 2sassccanetueadieccsbancaxessaeicassdsaadioabaacdsaelecanias ydadoawnces 169 Verbs for Mathematics icss ccsdhascctuck ten sandeatiarseaieiaina i E ewe 169 Extended Integers Rational Numbers and x o eeeceesceceseceeeeeeneeeeeeeeeeesseeneenees 169 Factors and Primes Monad p Monad and Dyad do eee eecceessecsteceteeeteeeenees 170 Permutations As andi C sy 1ccsssbabeveaereccseastanessenticashand EERE TA ERRER neds 170 DDS OCS And Ends hari E E EE E TE TEE A A a a 172 Dydd 4 REVISIVCG sriti irinin ae aeoe EATS AARE SAA R AA ETAS CE ES AESi 172 Boxed words to string Monad 1 u sessseeeeieisissisrsrsrsrststststststnrntnenenenerens
135. ened is that is handled by special code in the interpreter but tally is not The fact that tally is defined to be is immaterial the interpreter doesn t know that at the time it creates the anonymous verb for tally The penalty is an almost tenfold increase in execution time 6 2 a tally b 0 0167351 By flattening tally we cause it to be replaced by its definition and then the special case is recognized 202 b 0 i 100000 6 2 i b 0 0011616 6 2 tally index tally b 0 00385943 Another example i is handled by special code but the names prevent the interpreter from recognizing the situation 6 2 tally f index f tally f b 0 0018061 If we flatten every name we get good performance but what an effort It s much better to use the J primitives directly so the interpreter can do its job effectively 203 39 Explicit To Tacit Converter Q What s the most common cause of blindness among J programmers A Conjunctivitis In the early weeks complex tacit definitions are torturous to write and next to impossible to read blurred vision and sleepless nights are the occupational hazard of the programmer who dives into tacit J If a co worker is banging into doors as he stumbles to refill his tankard of coffee here s how to check him out hold up three fingers and ask him how many he sees If he says three he s merely fallen in love or is working to a deadline and he will
136. er special words like if andcase There area lot of other characters that you think of as punctuation namely that J uses to do work You will be especially surprised to find that and are independent rather than matched pairs but you ll get used to it Sentences Statements The executable unit in J is called the sentence corresponding to the C statement The sentence delimiters in J corresponding to the semicolon in C are the linefeed LF and the control words like if that we will learn about later A sentence comprises all the characters between sentence delimiters since LF is a sentence delimiter it follows that a J sentence must all fit on one line There is nothing corresponding to lt CR gt in C that allows you to split a sentence across lines All comments start with NB and run to the next LF The comment is ignored when the sentence is executed Word Formation tokenizing rules J s names identifiers are formed much as in C Names must begin with an alphabetic underscore is allowed and upper and lowercase letters are distinguished Names that end with an underscore or contain two consecutive underscores are special and you should avoid them until you know what a locale is The ASCII graphic characters for example are called primitives operators in J You will learn their meanings as we go on Any name or primitive identifier or operator can be made into a new primitive by adding
137. ere epname is the name of a verb and localename is the name of the module s locale Take a minute to see what happens when some other locale invokes epname The name search for epname will end in the locale z where this definition is found Execution of this definition immediately results in execution of epname_localename_ which switches the current locale to localename and runs the definition of epname found there The benefit is that the calling module doesn t need to know the locale that epname is going to be executed in If you tire of writing out the public definitions one by one I have included in jforc ijs a verb to do it for a list of entry points using a sentence like PublishEntryPoints publicl public2 public3 If you want to create multiple copies of objects derived from a class you should consult the Lab on Object Oriented Programming There you will learn about numbered locales and how to create and destroy objects We will not discuss these topics here By following the guidelines given above you will be able to emulate the class facilities of C Because J is interpreted you can do much more if you want you can change search paths dynamically you can use locales and paths to create a high performance network database you can pass locales as data and use them to direct 154 processing you can peek at a module s private data You can even modify a module s private data from outside the module but if you are struck by ligh
138. eric type lt 2 3 0 3 4 100 3 0 3 0 lt 2 3 0 3 4 100 32 Here the verb monad lt was applied to the same fill cell c 3 4 0 but it produced a boxed scalar result shape empty so the shape of the overall result is the frame 3 0 concatenated with an empty list i e shape 3 0 and type boxed as indicated by the 32 returned by 3 0 Note that the contents of each box in an empty array of boxes are all empty In the example lt 2 3 0 3 4 100 execution on the fill cell produced lt 3 4 0 and if the frame weren t empty the box would contain an array but when the frame is empty the value of the result is discarded all that remains is its type If executing the verb on a cell of fills results in an error execution continues as if the verb had returned the scalar O 541 domain error 5 Trying to add 5 to a space is nonsense 5 ss 0 3 0 5 4 but adding 5 to an empty list of characters produces an empty numeric list The addition is attempted with a cell of fills for the empty operand the values added are 5 on the left on the right the addition fails and the error fallback result of scalar 0 is used appending the longer frame 0 gives shape 0 numeric Note that y here is an empty operand but it nonetheless has the longer frame the scalar x has empty shape and perforce empty frame Note there is much special code in the interpreter to handle the cases of empty opera
139. ers 999 12 5 3 6 4 1 000 x25 12 0 5 3 6 4 1000 999 All legal numbers except for x25 117 21 Calling a DLL Under Windows Interfacing to a DLL is one thing you can do the old fashioned way by getting a copy of a working program and editing it You can find starter programs in the J distribution A good one to start with is system packages winapi registry ijs You can glance through this program to see what it does It starts with require dll which defines the verb cd Calls to the DLL appear in lines like re Advapi32 RegCreateKeyExA i i o AS i i i i i cd root key 0 0 sam 0 _1 _1 this is a single line in the program I have shown it here as 2 lines because it won t fit on the page as a single line This example exhibits the elements of a call to a DLL The left operand of cd is a character list describing the function to be called and what arguments it is expecting Here we are calling the entry point RegCreateKeyEXxA in the library Advapi32 The sequence of is and cs describes the interface to the function The first item in that sequence describes the type of value returned by the function the other items are the arguments to the function and are a one for one rendering of the argument list that would be passed in C So the line above is appropriate for calling a function defined with the prototype int RegCreateKeyExA int char int char int int int int int The descriptors can be c char
140. ers are a clue that sym is an array of symbols The value of the top left atom of sym is not abc or abc it is a value understood only by the interpreter The interpreter chooses to display the text associated with the symbol but that text is actually stored in the interpreter s private memory y ins y can be a character string which is chopped into pieces using the leading character as a separator each piece is then converted to a symbol This is a handy way of creating a short list of symbols s abc ghi abc ghi Symbols can be operands of any verb that does not perform arithmetic in addition comparison between symbols is allowed with less than defined to mean earlier in alphabetical order a s abc def ghi jk defines a list of 4 symbols ai s lt ghi 2 We create a symbol to represent ghi and find that in the list ai lt ghi 4 Note the boxed string lt ghi is not a symbol so it is not found in the list Dyad s has a number of forms for operating on symbols The only one of interest to us here is 5 s y which converts each symbol in y to its corresponding boxed string 5s 31l1f a jk def When a string is converted to a symbol the interpreter allocates internal resources to hold the string s value and other information There is no way to tell the interpreter to free the resources for a single string this can be a problem if your symbol table is large and changes
141. es to more descriptive names If your verb is going to define only a monadic or dyadic form you should use monad define or dyad define instead of verb define If you are going to define both valences the way to do so is name verb define monadic case here dyadic case here where a line with the single word separates the two cases If you use verb define and don t have the the verb will be monadic If your verb is only one line long not at all unusual in J you can define it all in one line by using the appropriate one of the forms name monad text of verb name dyad text of verb Running a J program No compiling No linking No makefiles No debugger You simply type J sentences and the interpreter executes them and displays any result At the very simplest you can use it as a desk calculator 22 55 77 J prints 3 spaces as a prompt so when you scroll through the log of a session your input will be indented 3 spaces while J s typeout will be unindented The result of a sentence typed on the keyboard is displayed except that to avoid excessive typeout nothing is displayed if the last fragment executed in the sentence is an assignment If you are at the keyboard while you are reading this book you can type the examples and see the responses or experiment on your own Here is a simple program to add twice the left argument to three times the right argument add2x3y dyad 2 x 3 y We can run
142. et_number 0 2 65 80 203 8 2982 where the result is return_code address_family local_IP_addr local_port 123 Connecting Reading from a Web site you use the sequence sdsocket sdconnect sdsend sdrecv sdclose sdconnect socket address family IP_addr port will connect your socket to the remote machine addressed by address _family IP_addr port Ifthe socket is blocking sdconnect completes when the connection has been made If the socket is nonblocking sdconnect will return immediately with the error code EWOULDBLOCK_jsocket__ and socket_handler will be called when the connection has been made An example is sdconnect 184 2 64 58 76 229 80 With the connection established you can send data with data sdsend socket flags where the flags are any of the MSG_ values from socket ijs usually 0 The result is return_code number_of bytes_sent Fewer bytes may be sent than were in your data you will have to resend the excess You receive data with sdrecv socket count flags where count is the maximum number of bytes you will accept and flags are any of the MSG_ values from socket ijs usually 0 The result is return_code data Ifthe socket is blocking sdrecv will wait until it is ready for reading if the socket is nonblocking sdrecv will immediately return presumably you issued the sdrecv only after using sdselect to verify that the socket was ready for reading In eithe
143. ething that a fancy language like J would do without complaining if so think more deeply J does give you a way to add lists together just tell J to apply the verb to lists IL 2 3 4L is 2 3 135 468 Operands in which one shape is a suffix of the other as in this example are handled by making the verb have the rank of the lower rank operand that single operand cell will then be paired with all the cells of the other operand By requiring dissimilar frames to match at the beginning J gives you more control over implicit looping because each different verb rank causes different operand cells to be paired If dissimilar frames matched at the end the pairing of operand cells would be the same regardless of verb rank Order of Execution in Implied Loops Whenever a verb is applied to an operand whose rank is higher than the verb s rank an implied loop is created as we have discussed above The order in which the verb is applied to the cells is undefined The order used on one machine may not be that used on another one and the ordering may not be predictable at all If your verb has side effects you must insure that they do not depend on the order of execution Current versions of the interpreter apply the verb to cells in order but that may change in future releases A Mistake To Avoid Do not fall into the error of thinking that v r is v with the rank changed to r It is not Nothing can ever change the rank of the verb v v r
144. exes are selected Such a selector is considered to be specifying the list of non excluded indexes so the corresponding axis remains in the result Example lt 0 1 lt lt 1 O 1 II I Vite II I I il lt 0 1 lt lt 1 i 2 2 3 35 We select a single 2 cell and from that a single 1 cell and within that we select all except item 1 The result is a 2 item list Note that we had to put an extra lt after the last to ensure that the contents of the last selector was boxed 94 lt 0 lt 0 0 2 1O 121 II IIE Ul II I Il lt 0 lt 0 0 2 i 2 2 3 25 Complementary indexing can be used to select all of an axis as in this example We request all of axis 1 except the named items and then we name an empty list we get all the items This trick is called for when we need to specify a selector for some axis after the axis we want to take completely trailing axes can be taken in full simply by omitting their selectors If our use of x y does not require any selector to specify a list i e each selector is a scalar we are allowed to omit the boxing of the selectors This leaves x as a boxed numeric list or scalar in which the successive items indicate the single index to be selected from each axis This form in which x is lt i j k corresponds to C s array i j k lt 0 1 I0 1 lt 0 1 1 O 111
145. fill using the fit conjunction 5 9 3 1 4 31499 _5 x abc xxabc The fit conjunction creates a new verb in this case f is a verb that looks just like but uses f for the verb fill The fit conjunction is by no means reserved for specifying verb fills it is available for use on any primitive to make a small change to the operation of the primitive If has a meaning for a primitive that meaning is given in the Dictionary entry for the primitive Joining Arrays x y Stitch The left rank is infinite x yisequivalenttox _1 y That means that dyad is applied to the corresponding items of x and y making each item of the overall result the concatenation of the corresponding items Example 345 789 U W O N x y Laminate The left rank is infinite x y is a list of 2 items item 0 is x and item 1 isy If x and y do not have the same shape they are brought to a common rank and padded with fills to a common shape Dyad concatenates x and y along an added axis in contrast to dyad which concatenates them along their leading axis 345 789 345 789 Contrast this with dyad above or dyad which would produce3 45789 12 34 56 12 34 5 6 Take a moment to understand why in this example the first verb is dyad and the second is dyad 52 Rotate Left and Shift Left x x y Rotate Left The left rank is 1 but we will discuss only the cas
146. fore the first NUL 0 character encountered Aliasing of Variables When a noun is assigned the value of another noun as in a b 5 a single memory area is used to hold the value common to both nouns and the two nouns are said to be aliases of each other Aliasing obviously reduces the time and space used by a computation The interpreter takes care to ensure that aliasing is invisible to the programmer if after the statement above we execute b 6 the interpreter will assign the new value to b only leaving a unchanged What actually happens is that the new value is created in a data block of its own and the descriptor for the noun b is changed to point to the new block Almost all verbs create their outputs in newly allocated data blocks As of release 5 01 the exceptions are and u when it is used in one of the forms that produces in place modification Increasing the number of cases recognized for in place execution is a continuing activity of the J developers If there were nothing more to say about aliasing I would not single it out for mention from among the dozens of performance improving tricks used by the interpreter What makes it worth considering is the effect aliasing has when elements outside the J language touch J s nouns This can occur in two ways when a noun is mapped to a file and when a noun is modified by a DLL Aliasing of Mapped Nouns When a noun is mapped to a file the descriptor for the noun points t
147. from this boxed noun The contents of a box can be any noun including a box or array of boxes 73 lt abc lt lt 1 2 abe 11 211 I The contents of a box can be recovered by opening the box with monad gt Monad gt has rank 0 since it operates only on boxes which are atoms and its result is the contents of the box gt lt abc abc The contents of the box is the character string a lt 1 2 lt lt 5 lt abc gt 0 a 1 2 gt 11 a 15 Here we recover the contents of the selected box which may be a box gt lt 1 2 3 lt 4 12 3 400 Remember that monad gt has rank 0 so it is applied to each box and the results are collected using the frame Here framing fills were added to the shorter result 0 was used as the framing fill because the contents of the boxes were numeric gt a domain error gt a Here the results on the different cells were of different types so it was impossible to collect them into an array If y is unboxed the resultof gt yisy Terminology Before we go further with boxes let s agree on some terminology Because every atom in an array must have the same type it is reasonable to speak of an array as being boxed if its atoms are boxed unboxed otherwise some writers use open as a synonym for unboxed Trouble arises with a phrase like boxed list If I box a list e g lt 1 2 3 does that giv
148. g dyad and then extract the pieces as needed using dyad For the rest of this chapter we will show examples of functions turned into tacit verbs using hooks and forks If I don t show the expansion using the bident trident rules you should produce it yourself To find how much x has changed from y as a percentage of y pctchg 100 amp 12 pcetchg 10 20 This becomes 100 x y y Note the use of to select the original y operand Similarly can be used to select the original x operand Tacit verbs make heavy use of and Another way to code petchg is pctchg 100 where we used the constant verb 100 __ which produces 100 no matter what its operands are Which of these forms you prefer is a matter of taste 196 fndisplay is very helpful in understanding tacit verbs The two versions of pctchg are displayed as defverbs times 0 minus 0 div 0 defnouns x y x 100 amp times minus div y I encourage you to use Endisplay to expand any tacit definitions that are troublesome To find the elements common to x and y keeping the same order as in x setintersect e 3 141 5 9 setintersect 4 6 9 49 This becomes x e y x You can see that the identity verbs and are useful for steering operands through hooks and forks As an exercise see how the alternative version produces the same result To list all the indexes of the 1s in a Boolean list booltondx i 1
149. g the number of digits following the decimal point If either w or dis less than zero the result is in exponential form and the absolute value of w or d gives the corresponding width otherwise the result is in standard form Examples 114 Oo i 44 0 1 2 3 4 5 6 7 8 910 11 12 13 14 15 Note that extra spaces were added to the one digit values to make them line up Note also that enough space includes a leading space to keep adjacent results from running together 1 i 44 0123 4567 89 kkk When you specify a width you are expected to mean it and no extra space is added Here two digit results would not fit and were replaced with 0 033 03 3 100 i 3 3 00 010 2 000e 2 00 040 5 000e 2 00 070 8 000e 2 A complex number has its parts separated by j Here the first field is an integer the second has 3 decimal places and the third is in exponential form When dyad is applied to an array the result has the same rank as y If you need to write that result to a file you will need to use monad to convert it to a list possibly after adding CR or LF characters at the ends of lines Monad Monad resembles dyad with a default x except that x depends on the value of the corresponding atom of y The simple description of monad is that it formats numbers the way you d like to see them formatted The full description is as follows The precision d is given by a system variable that can be set by the fo
150. gerund operations are not difficult so I am going to keep your interest with a couple of new tricks 136 callback dyad x y Howzat x Yeah this means that the value of x tells what variable will be assigned x can be any valid assignment target a name a multiple assignment or other exotic forms given in the Dictionary calledfn monad 0 y 6 1 y So calledfn is expecting an argument of at least 2 boxes The first one will be the gerund to execute and the second one will be the argument to pass to that verb We open the second box with but we leave the first as a box so that 6 can turn it into a verb calledfn vbl amp callback lt 25 The first challenge is figuring out what the argument to calledfn is sees a verb on its left it converts that to a gerund It sees a noun to its right so it appends it unchanged to the gerund from the left This produces vbl amp callback lt 25 1251 amp ana naps I callback O vb1 III os The first box is the atomic representation of the verb just as mysterious as it was billed to be and the second box has the 25 Now what do we get when we pass that in as the argument to calledfn try to work it out before you peek at the answer calledfn vbl amp callback lt 25 25 Did you get it We executed vb1 amp callback 25 which t
151. gth to smooth over rough spots David Steele conducted a painstaking review of several early drafts and suggested many changes great and small Bj rn Helgason translated the text into Icelandic finding a number of errors along the way Kip Murray s review became more of a dismantling cleaning and reassembly operation in which large sections of prose were rewritten as he pointed out to me their essential meaninglessness the reader should be as grateful to him as I am Without the patient explanations of my early teachers in J Raul Miller and Martin Neitzel I would have given up on J I hope that this book pays to others the debt I owe to them My current happy career as a J programmer would not have been possible without the work of the staff at Jsoftware Inc who created J For the patriarch Ken Iverson I am unworthy to express admiration I have only awe I hope his achievement eases the lives of programmers for generations to come To the rest both Iversons and non Iversons I give my thanks The implementation of the J interpreter has required diverse skills architectural vision careful selection of algorithms cold eyed project management to select features for implementation robust and efficient coding performance optimization and expertise in numerical analysis Most improbably all these talents have resided in one man Roger Hui il miglior fabbro J gives us all a way to have a little of Roger s code in our own We should a
152. h shape s The only way to find out what shape a verb is going to produce is to execute it and see and that is what J does Execution On a Cell Of Fills If an operand is empty i e its frame contains a 0 and it therefore has no cells the verb is executed on a cell c of fills The shape of c is the shape of a cell of the corresponding operand and the value of each atom is the appropriate fill for that operand O for numeric operands for characters a for boxes The shape s and type t of the result of executing the verb on c are noted Then the shape of the overall result is the frame of the operand the longer frame for dyads concatenated with s and the result is given type t The result will necessarily be empty because it will have a 0 in its shape from the frame which contained a 0 Example 2 3 0 3 4 100 30 4 Remember that a cell has a shape even if there are none of them Here the verb monad 2 is applied to 2 cells each with shape 3 4 The frame 3 0 contains a 0 so the verb is executed on the fill cell c which is 3 4 0 Monad adds the items of the cell producing a list with shape 4 The frame 3 0 is concatenated with 4 to give the shape of the result 3 0 4 76 3 0 2 3 0 3 4 100 4 The verb 3 0 one of dozens of special goodies provided by the conjunction and documented under Foreigns tells you the type of its operand Here 4 means numeric type the result has shape 3 0 4 and num
153. hat the result of the modification of the array must be the entire modified array This reasoning justifies J s design is an adverb Modification of an array is performed by dyad m which produces a derived verb of infinite rank x m y creates a copy of y and installs the atoms of x into the positions selected bym mspecifies portions of y in the same way as the left operand ofm y Even though dyad has left rank 0 m may have any shape the atoms of m are processed to accumulate a list of selected portions of y x must either have the same shape as the selected portion s m y or have the same shape as some cell of m y in which case x is replicated to come up to the shape ofm y Examples are a recap of the forms of the left operand of dyad with a couple of twists O lt 1 1 446865 5 0 5 5 e The simplest example We select an atom and modify it 0123 0 44 5 143 O 0 44855 0000 5555 5555 5 5 55 x is shorter than m y but x can be replicated to match the shape of m y O 1 0 446865 length error 01 0 4 485 Here the shape of x is 2 while the shape of m y is 4 so the operands do not agree 1 2 1 1 2 2 44655 aununu anunu aur wu U N UU Here m has 2 atoms each selecting a single atom of y x has two atoms and they are stored into the selected positions 1 2 1 1 11 44655 5555 5255 555 5 5555 The same element is modified twice As it happens the modifications are performed in order
154. he clone_socket Other Socket Verbs Datagrams The sequences given above apply to sockets of type SOCK_STREAM jsocket_ e g TCP sockets Connectionless sockets with type SOCK_DGRAM_ jsocket_are also supported and could be used for UDP transfers The verbs to use to transfer datagrams are sdrecvfrom socket count flags where count is the maximum number of bytes you will accept and flags are any of the MSG_ values from socket ijs usually 0 The result is return_code data sending address If the socket is blocking sdrecvfron will wait until it is ready for reading if the socket is nonblocking sdrecvfrom will immediately return presumably you issued the sdrecvfrom only after using sdselect to verify that the socket was ready for reading The returned data is the datagram which may have a length of 0 data sdsendto socket flags address_family IP_addr port where the flags are any of the MSG_ values from socket ijs usually 0 The datagram is sent to the remote address given by address_family IP_addr port and the result is return_code number_of_bytes_sent Socket Options Verbs to query and set socket options are sdgetsockopt socket option_level option_name with result return_code option_value Examples sdgetsockopt sk SOL_ SOCKET jsocket_ SO DEBUG jsocket_ sdgetsockopt sk SOL SOCKET jsocket_ SO LINGER_jsocket_ sdsetsockopt socket option_level option_name value_ list note that the operand is an unboxed list The option i
155. he J Dictionary uses the terms local and global instead of private and public I think private and public are more accurate terms because there is another dimension to name scope in J using the J notions locale and path that causes public variables to be visible only in certain entities It will be a long time before we learn about locales until then public names will be global Order of Evaluation Forget the table of operator precedence All J verbs functions and operators have the same priority and associate right to left For example a b c is equivalenttoa b c not a b c Use care when copying mathematical formulas Note that the negative sign _ is a part of the number not a verb _5 _4is_9 while 5 4is_1 The executable bits of a sentence statement are called fragments subexpressions A verb with its operand s is a fragment as is a copula with its name and value We will meet other types of fragment later Execution of a sentence consists of the right to left execution of its fragments with the result of each fragment s execution replacing the fragment and being passed as an operand into the next fragment The result of the last execution becomes the result of the sentence This result is usually a noun but it can be any of the primary parts of speech As an example execution of the sentence a 3 b 4 5 consists of execution of the following fragments 4 5 withresult20 b 20 with result
156. he column labeled line indicates which line of the parsing table was matched by the stack The fragment is marked in boldface and underlined Note that when the noun a moved onto the stack its value was moved when a named verb adverb or conjunction is moved onto the stack only the name is moved Note also that the noun s value 1 2 3 here is a single word From now on we will omit the lines that do not contain an executable fragment Sentence mean unprocessed word list mean mean result av1 defined as 3 mean avl result av2 defined as av1 5 mean av2 result av2 mean is assigned av2 7 av2 I want to emphasize that what is assigned to mean is the result of parsing It is not the sequence but rather a single verb which performs the function described by the fork Now you see why putting parentheses around the definition doesn t matter av2 would be parsed the same either way Sentence mean 4 5 6 unprocessed word list mean 4 5 6 mean 456 result 5 0 5 Since mean is the result from parsing it is executed without further ado As you can see a single execution step can trigger a cascade of processing as each verb 192 referred to by an executing verb is executed in turn Here execution of mean does the entire processing of the fork returning the result 5 The verb to
157. he leading axes are reduced by the partitioning Find Sequence Of Items Dyad E Dyad E has infinite rank and is used to slide a pattern across an array and look for positions at which the pattern matches the items of the array x and y should have the same rank and the sliding occurs along all axes of y giving a Boolean result with the same shape as y We will consider only the cases where y is a list is E Mississippi 01001000000 The 1s in the result tell where the pattern starts The result of E is often used as input to u r is E Mississippi lt 1 Mississippi Jiss issippi A more ambitious example html 0 0 lt th gt lt a href pagel html gt Press here to go back lt a gt lt th gt lt th gt lt a href page2 html gt Press here to go home lt a gt lt th gt lt th gt lt a href page3 html gt Press here to go away lt a gt lt th gt lt table gt lt center gt lt a E html lt 8 amp _1 gt amp 1 html page1 html page2 html page3 html That looks like a useful result but what a mess to produce it If you want you may try to understand that but a better approach is to break it up extracthref lt 8 amp _1 gt amp This seems understandable with effort the execution order is lt 8 amp _1 gt amp
158. he same length as the shape of a noun designating an atom of the noun by giving its position along each axis Interval A sequence of consecutive indexes or cells Item A_1 cell ofa noun An array is a vector of its items An atom has one item itself List An array ofrank1 List of An array whose items are as in list of 3 item lists Local See private Locale A namespace in J The locale in which a public name is defined is an attribute of the name A locale is identified by a locale name which is a character string not containing an underscore Locative A name including both a simple name and an explicit locale Modifier An adverb or conjunction which modifies its operand s to produce a derived entity 208 Monad monadic A verb with no left operand Any verb may be used as a monad or dyad depending on whether it has a left noun operand when it is executed Name Either a simple name or a locative to which an entity can be assigned Noun One of the primary parts of speech An atom or array of atoms Nouns hold the data that verbs operate on Parsing The right to left search for suitable patterns in a sentence When a suitable pattern is found that subset of it that constitutes an executable fragment is executed Part of Speech One of the six categories into which words are classified or the word or entity so classified Every word has a part of speech for primitives the part of speech is defined by the language fo
159. he whole array is a list of 4 items each with shape 5 6 equivalently we say that it is a list of four 5x6 arrays So many of J s primitives operate on items of their operands that we will find ourselves usually thinking of an array as a list of its items 19 When the word list is used without any indication of what the list contains the list is assumed to contain atoms So the list x refers to an array of rank 1 one dimensional array 0 3 5 isa numeric list Note that J s use of the term list has nothing to do with linked lists such as you are familiar with where an element in the list contains a pointer to other elements Since J has no pointers at all you will not need that meaning and you can get used to calling rank 1 arrays lists A list can also be called a vector Constant Lists A character or numeric list can be created simply by including the list in a sentence We have seen that a sequence of numbers separated by spaces is recognized as a single word representing the list Similarly a character or a character list can be represented directly by a quoted string C distinguishes between single character constants such as a and strings such as abce using single quotes for characters and double quotes for strings J uses only single quotes for defining character constants the character is a primitive in its own right If exactly one character is between the quotes the value is an atom if none or more tha
160. hen executed vbl 25 which has the result 25 which comes back as the result of calledfn The assignment was public vbl 25 Passing the Definition Of a Verb 128 2 Apply As an alternative to passing a gerund and invoking it with m 6 you could pass the string representation of a verb and make a verb out of it with 3 nor4 n Better yet if you can make do with a monadic verb you can use the foreign dyad 128 2 which has rank 1 _ and goes by the name Apply x 128 2 y takes the string x which must describe a verb and applies the verb so described to y as a monad It is therefore similar to 3 x y with the restriction that x must describe a one line verb without assignments The advantage of using 128 2 rather than the method in the next section is that y does not have to be converted to its string representation 137 Passing an Executable Sentence Monad and 5 5 As the ultimate in flexibility you can pass an entire J sentence as a character string and then execute it with monad It is executed exactly as if it had been a line of the executing verb or from the keyboard if that s where was entered For example My et SS ae A 012 3 The sentence was executed a 0123 The assignment was performed mo PPTA E a 6 The operand of monad must be a string so before we can take its total we must convert a to a sequence of characters that will have the value of a when executed For a large operand conver
161. ich yields O x y exponentiation x to the power y 0 0 yields1 y logarithm base x logarithm of y x y modulus remainder when y is divided by x For the longest time I had trouble remembering dyad it seemed that the divisor should be y by analogy to dyad 3 The inconsistency is that J defines x y as x divided by y to match accepted practice in mathematics that makes dyad anomalous in J because we have y operating on x The comparison verbs have rank 0 and produce Boolean results in which 1 means true 0 means false They use tolerant comparison which means that two values that are very close to equal are considered equal This saves you the trouble of adding small amounts to mask the effects of floating point rounding 1 3 1 3is1 unlike 1 0 3 0 1 0 3 0 whose value depends on the compiler If you need exact comparison look under Comparison Tolerance to see how to do it x y equal x y not equal if you squint the colon looks like an equal sign x gt y greater than 47 x lt y less than x gt y greater than or equal if you squint the colon looks like an equal sign x lt y less than or equal Boolean Dyads These verbs have rank 0 and are applied to Boolean arguments to produce Boolean results x y Boolean AND x y Boolean OR x y Boolean XNOR 1 if the operands are equal x y Boolean XOR 1 if operands differ Min and Max Dyads These verbs are useful for perfor
162. ied Applying the permutation in the direct form is as simple as writing p y The standard cycle representation of a permutation gives the permutation as a list of cycles sets of elements that are replaced by other elements of the set The standard cycle form is a list of boxes one for each cycle with each cycle starting with the largest element and the cycles in ascending order of largest element Monad C y rank 1 converts between direct and standard cycle representations of the permutation y 170 fe 3 4159 1302 5 C 314159 13 20 11415 C C 314159 130245 1 4 x C y rank1 _ permutes the items of y according to the permutation x which may be in either standard cycle or direct form other nonstandard forms are also supported as described in the Dictionary There are n possible permutations on n items so it is possible to give each one a number between 0 and lt n Imagine the table of all possible permutations in lexicographic order the anagram index of a permutation is its index in that table A y rank 0 gives the anagram index for the permutation y which may be in either direct or standard cycle form x A y rank 0 _ permutes the items of y according to the permutation whose anagram index is x a 314159 168 A 123456 24135 6 The monadC 2 y gives the parity of y 1 if an even number of pairwise exchanges are needed to convert y to the identity permutation i y
163. ight to left and the interpreter has not seen the reference to w at the time w is assigned A similar statement in C for example X w w 4 would be undefined Of course even though it s legal in J some would cavil we will eventually learn ways to do this without defining a variable at 66 all but I leave it to you to decide how far you will detour to avoid jaywalking Once sw has been calculated the rest of the program is trivial The final result is NB Version 2 Discard multiple whitespace NB and return chars words lines wc2 monad define f ReadFile y sw 1 w we fe TAB LF sw LF f 67 10 Compound Verbs On New Year s Day my rich uncle gives me x dollars which I add to the y dollars I already have earning 4 interest How much money will I have at the end of the year Simple in J I just write 1 04 x y andI have the answer whether x and y are scalars or arrays That s nice but here s my problem I expect his largesse to continue and in my anticipation I have estimated his gifts for the next few years as the list x I want to know what I ll be left with at the end I need to pass each year s starting balance into the calculation for the next year I know what I want the result to look like it ll be v 1 x y which will be evaluated as xn v x2 v x1 v x0 v y But what is v The problem with 1 04 x yis that it contains 2 verbs and a constant
164. ile was not found You can set up your verbs to catch errors instead of interrupting with a message at the console We will learn one way here and another later when we study control structures for J verbs The compound u v has infinite rank and can be applied either monadically or dyadically u v executes the verb u first monad or dyad as appropriate if u completes without error its result becomes the result ofu v but if u encounters an error v is then executed and its result becomes the result ofu v rerr 1 1 13 11 rerr y will execute 1 1 to read y if that fails it will execute the foreign 13 11 13 11 y ignores y and produces as result the error number of the last error encountered This means that if 1 1 succeeds the result of rerr will be a string while if 1 1 fails the result will be a number rerr lt system packages misc jforc ijs NB File definitions for J For C Programmers NB Copyright c 2002 Henry H Rich rerr lt c xxx yyy dat 25 You could use 3 0 to see whether the result of rerr is a string or a number 2 means string 1 4 8 or 16 means number If you want to see the error message associated with an error number there s a foreign 9 8 to give you the list of errors it ignores its y too 25 9 8 Treating a File as a Noun Mapped Files Rather than reading a file and assigning the data to a noun you can leave the data in a file and create a noun that poi
165. ill a verb performing the requested function but it is an explicit definition rather than a tacit one Note that the interpreter saw that the definition contained a reference to x so it made the verb a dyad 204 2 13 ree oye 3 12 This verb applies monad to y x times Knowing about u v you might find the tacit equivalent 2 3 12 Special Verb Forms Used in Tacit Definitions It is impossible for a tacitly defined verb to route one of its operands to an operand of a modifier inside the verb For example if we want a verb to set element x of y to 0 we can try 13 0 x y 4 0 x y but we see that the converter was unable to find a tacit form Some modifiers have exotic forms that circumvent this problem One such is the adverb which supports the form x value selector operand y This produces the same result as x value y x selector y x operand y so we could write a 13 x 0 C 0 y a O sa 9 7 5 3 1 97501 Other such forms are x m amp v yandx u amp n y which are equivalent to m amp v x y andu amp n x y respectively 205 40 Valedictory You have learned enough J to understand J programs and to put your own ideas into J I hope you will now do two things practice using J so that J becomes the language of your mind s ear the way you naturally express algorithms and read the Dictionary with care so you can learn J fully It ll be like weight training
166. ime for most y and should not be avoided According to our general principle we would expect that in dyad x held the keys and y the data Dyad is an exception to the rule y y y Y When x and y are the same you have the simple case of sorting y into ascending or descending order Match x y Match The left rank is infinite The result is 1 if x and y are the same 0 otherwise except 1 if they are numeric the comparison is tolerant 2 for some reason I don t understand if they are empty they are considered to match even if the types are different which means that getting a result of 1 from x y is no guarantee that x and y will behave identically 0 0 1 1 060 1 0 The important difference between dyad and dyad is that dyad has infinite rank so you get a single result covering the entire array and it won t fail if the shapes of x and y do not agree Monads Enfile y Enfile y consists of all the atoms of y made into a list The order is row major order 1 e all the atoms of item 0 of the original y come first followed by atoms of item 1 54 and so on within each item the ordering similarly preserves the order of subitems Examples i 23 012345 The atoms were made into a list a 2 2 3 abcdefghijk1l a abc def ghi jkl j abcdefghijkl y of any shape produces a list Recall that a single quoted character is an atom rather than a list S
167. in current month Account filename NB Journal filename acctprocess monad define ndays acctfn jourfn y NB Read files acctano openbal rdtabfile acctfn jourano jourday jouramt rdtabfile jourfn NB Verb given list of days y return days that NB each balance is a day s closing balance wt monad 1 amp gt ndays O 1 y NB Verb given an Account entry followed by the Journal NB entries for the account produce closing balance NB average daily balance ab monad wt y amp ndays 1 1 y NB Create closing balance average daily balance for NB each account Assign the start of month day 1 to the NB opening balance cavg acctano jourano ab 1 openbal jourday jouramt NB Format and print all results s Account d Opening d closing d avg d n s amp printf 1 acctano openbal cavg Let s compare the two versions The first thing we notice is that the J code is mostly commentary beginning with NB The actual processing is done in 3 lines that read the files 3 lines to perform the computation of closing and average balance and 2 lines to print the results J expresses the algorithm much more briefly The next thing we notice is that there seems to be nothing in the J code that is looping over the journal records and the accounts The commentary says create balances for each account and produce averag
168. ing it for a while If you want or not to be treated as a suffix make sure you precede it by a space 3 y defines a verb while 3 y applies the verb 3 to the noun y producing the result 3 What really happens duringm nand verb define Now that you re getting older it s time we had a little talk About verbs Up till now the verb definition sequence has been exotic and peculiar with a unique form a possessor of great power but shrouded in mystery We know only that it somehow breathes life into lines of text name verb define verb definition or name verb verb definition leaving name a verb With a layer of cosmetics scrubbed off it is more approachable In both cases the conjunctionm n is at work m must be a number and n may be either a character string or the special number 0 a boxed list is also possible but we consider it a curiosity m n executes the conjunction with the arguments m and n the result of this execution is an entity which can be any of the primary parts of speech noun adverb conjunction verb and m indicates which one m 0 means noun 1 means adverb 2 means conjunction 3 means verb 4 means dyadic verb 10 13 have meanings we will learn later You can remember the numbers 0 2 because nouns take no operands adverbs 1 and conjunctions 2 verbs are last in the precedence order so they get number 3 and dyadic verbs with their extra operand get number 4 If n is a character string
169. ing readability 201 introduction 183 parsing and execution 187 parsing table 190 summary of execution 190 trains 185 Tail verb 56 Take verb 51 Taylor series 166 Tie conjunction 88 Transpose verb 55 178 type 18 U u iv 113 u v 101 ud n 165 uD n 165 uD n 166 u dyad 80 uf gh y 205 u monad 79 ut 166 uT n 166 ut 166 u n 31 u amp v 104 u amp v 103 u amp v 105 u amp n dyad 205 u amp n monad 70 u amp v 105 u dyad 81 u monad 80 u dyad 173 u monad 58 u dyad 127 u v dyad 68 u v dyad 69 u monad for performance 109 u _1 obverse 103 un 85 u v 86 uv 88 undefined names 195 Unicode verb u 181 V valence 6 101 verb definition 99 W window driver 181 word 3 4 X x dyad 170 x monad 170
170. interpreter as it alternately parses and executes sentences from the bottom up Since the understanding of parsing and execution that you have developed during your work so far is probably a bit inaccurate we will work through examples of increasing complexity 9 3 5 NB Do this once to select simplified display amp amp With only one word there are no operands and nothing to execute so the result of the sentence is the word itself the conjunction amp SPEA SHE ces The result of executing amp 1i e executing amp with and as operands is an anonymous verb This anonymous verb will execute according to the definition of amp given its operands and i e amp ywillbe y Note that the conjunction amp is executed without reference to the operand of the anonymous verb indeed in this case there is no such operand and the anonymous verb is the result of the sentence We could assign the anonymous verb to a name in which case it would no longer be anonymous e g sqrt amp without such an assignment we will refer to it here by the nickname av The value of av is the verb described by amp amp 16 4 We know that this is executed as 16 let s see how that happens Conjunctions are executed before verbs so first amp will be executed to produce the anonymous verb we called av Then avis executed with the operand 16 av operates according to the defin
171. is a new verb which has the rank r This distinction will become important presently as we discuss nested loops 42 Consider the verb v 1 2 which is parsed as v 1 2 Ifv x changed the rank of v it would follow that v 1 2 would be v with the rank changed to 1 and then to 2 i e identical to v 2 But it is not actually v 1 2 applies v 1 on the 2 cells of the operand while v 2 applies v on those same cells and we have seen that v and v 1 are very different verbs 1 2 i 234 6 22 38 54 70 86 2 i 234 12 15 18 21 48 51 54 57 Summing the 2 cells 2 is not the same as summing the 1 cells within each 2 cell 1 2 Make sure you see why Ah you may say but 1 2 is equivalent to 1 You are right for the monadic case but not for the dyadic i 3 4 1 2 1 23 4 0 2 4 6 8 10 12 14 16 18 20 22 12 14 16 18 20 22 24 26 28 30 32 34 1 3 4 1 i 234 length error 1 3 4 1 1 2 3 4 Dyad 1 2 is executed as 1 2 i e it has rank 2 So there is only one 2 cell of the left operandi 3 4 and that cell is replicated to match the shape of the right operand The operands then agree and the 1 cells can be added Trying to add the 1 cells directly with 1 fails because the frames of the operands with respect to 1 cells do not agree The situation becomes even more complicated if the assigned left and right ranks are not the same My advice to you is simple remember that u r is a new verb that exec
172. is every programmer s dread a function that will have to be rewritten when the going gets tough The J version in contrast will have execution time proportional to A J log A J We did nothing meritorious to achieve this better behavior we simply expressed our desired result and let the interpreter pick an implementation Because we think big we treat the entire Journal and Accounts files as units we give the interpreter great latitude in picking a good algorithm In many cases the interpreter makes better decisions than we could hope to because it looks at the characteristics of the data before it decides on its algorithm For example when we sort an array the interpreter will use a very fast method if the range of numbers to be sorted is fairly small where fairly small depends on the number of items to be sorted The interpreter takes great care in its implementation of its primitives greater care than we can normally afford in our own C coding In our example it will use a high speed method for matching journal entries with accounts Calculating Chebyshev Coefficients This algorithm for calculating coefficients of the Chebyshev approximation of a function is taken verbatim from Numerical Recipes in C I have translated it into J just so you can see how compact the J representation of an algorithm can be Again the J code will be gobbledygook for now but it s concentrated gobbledygook 15 IIl Program to calculate Chebyshev
173. is the noun s name defnouns x y You are free to use other nouns in expressions but they will be replaced by their values With these definitions you can explore J s evaluations x plus y Sean a aa x plus y The result of the evaluation is a description of the evaluation that was performed The result is displayed in a box so that a sentence with multiple evaluations shows each in its proper place SumItems 1 i 2 3 SumItems 0 1 2 SumItems 3 4 5 Here we see that SumI tems was applied twice once on each 1 cell fndisplay cannot produce a valid result in cases where the rank of a verb is smaller than the rank of the result cells of the preceding verb because then the operation would be performed on part of a result cell and the result cell is just a descriptive string which cannot be meaningfully subdivided In this book if we give an example that starts with defverbs it is implied that load fndisplay and setfnform J have been executed If you prefer to see the order of evaluation expressed in functional form like that used in C you may issue setfnform math before you execute your sentences setfnform math 1 plus 2 plus y Negative Verb Rank Recall that we defined the _1 cell of a noun n to be the cells with rank one less than the rank of n and similarly for other negative ranks Ifa verb is defined with negative rank r it means as
174. ition of amp it produces the same result as 16 but it may use a different algorithm than executing 16 directly It appears that amp was executed twice once to create av and then again during the execution of av No it was executed only once to create av av operates according to the definition of amp but it is av that is executing not amp The confusion arises because of the way the interpreter displays av There is no better way to show a verb that performs the function amp than to show the way the verb was created i e with the characters amp but you should think of this as an exhibition of the pedigree of 187 av and an assurance of its good behavior rather than a list of functions to be executed In fact part of the reason for J s good performance comes from its recognizing functions that can be combined efficiently and providing customized routines to handle anonymous verbs that call for those combinations Confusion between a conjunction and the anonymous verb it produces is most likely when the conjunction is one you wrote using conjunction define or2 n In most cases the text of the conjunction actually describes a derived verb and it is natural for you to say the conjunction C is executed with operands u v and y rather than the more accurate the anonymous verb created by the application of C to u and v is executed with u and v available during the interpretation
175. ives to make this process easy The value produced by any entity when it is applied to its operand s is called its result returned value How Names Identifiers Get Assigned Assignment in J is performed by expressions of the form name entity NB private and name entity NB public Names assigned by public assignment are visible outside the entity in which they are defined names assigned by private assignment usually are not we will learn the details when we discuss modular code The difference between the two forms of assignment is in the character following the Just as in C the assignment expression is considered to produce as its result the value that was assigned so expressions like a l1l be 5 are legal J calls and copulas Just as in C the entity that is assigned to the name can be the result of evaluating an expression There are a number of additional capabilities of J assignment that you can read about in the Dictionary One that has no counterpart in C is that the name being assigned can itself be a variable i e you can calculate the name that you want to assign the value to The value assigned can be a noun object verb function adverb or conjunction the name then becomes whatever part of speech was assigned to it even if it was previously defined as a different part of speech For example n 5 creates a noun and v verb define x y creates a verb more below Note t
176. jsocket_ ifthe operation cannot be immediately completed When there is a change in the status of a nonblocking socket the interpreter executes 122 socket handler and it is up to you to have a socket_handler defined that will use sdselect to see what sockets are ready for transfers and then execute those transfers Names and IP Addresses Sockets are addressed by IP address and port number To translate a domain name to an IP address use sdgethostbyname domain_name for example sdgethostbyname www jsoftware com 0 2 216 122 139 159 where the result is return_code address_family address If the domain is unknown an address of 255 255 255 255 is returned sdgethostbyaddr address family address will translate an IP address back to a domain name The name of your machine is given by sdgethostname O 1qfe3 result is return_code name so you can get your own IP address by sdgethostbyname 1 sdgethostname 0 2 65 80 203 8 If you have a socket with an active connection you can get some information about the machine on the other end with sdgetpeername socket_number 0 2 64 58 76 229 80 where the result is return_code address_family remote _IP_addr remote port You can get information about your own end of a connected socket with sdgetsockname sock
177. l they can do Dyads Selection x y From 49 No nothing was left out is not paired with it is a verb and stands by itself The left rank is 0 the result is item number x of y One of the great insights of J is the realization that selection which in most other languages requires a special syntax like C s y X is really just a dyadic verb like any other Examples 2 314159 4 y has rank 1 its items have rank 0 item number 2 is 4 24 314159 45 The left frame is 2 each atom of x selects an item of y each an atom and the results are made into a 2 item list 1 i 33 345 y has rank 2 its items have rank 1 item number 1 is3 4 5 21 453 3 67 8 345 The left frame is 2 each atom of x selects an item of y each a list and the results are made into a 2x3 item array There are many variations on the format of x providing for multidimensional indexing where each index can be a list You will have to wait a bit to learn them but I will note that a negative index counts back from the end of the array 14314159 9 x y Copy Left rank is 1 If x is a list whose items are all 0 or 1 the result has the same rank as y and contains just the items of y for which the corresponding item of x is 1 For example 101000 314159 3 4 OOl i 3 4 8 9 10 11 x y Remove Left rank is infinite Any items of x that match items of y are removed 12345 43 24 135 3 The verb is defined with y operating on
178. lar is the same thing as a l item list No I mean no NO They are not I say this with the same resignation as when I tell my kids not to rollerblade too fast down our street knowing that only painful experience will drive the message home How can you tell them apart What we need is a way to see the shape of a noun That way is monad Theresult of y is the shape of y always a numeric list For example 1234 4 A 4 item list has the shape 4 Did you forget that1 2 3 4 isa single list rather than 4 separate numbers You can ask the interpreter how it splits a line into words by using monad 7 S 12 3 4 I 1 2 3 4 The words are shown in boxes The list1 2 3 4 is recognized as a single word 21 6 The shape of a scalar is a 0 length list as we have seen 1 2 1 Remember all verbs execute right to left so thisis 1 2 which gives the shape of the 1 item list When a verb can be invoked dyadically it is so the last is executed as a monad not as a dyad 0 2 2 Here we get the shape of the scalar an empty list a A single character is an atom whose shape is the empty list abc 3 A 3 item list one item for each character vy 0 is the empty character string which you will see a lot of Because it is easy to type it is the value most often used when an empty list of any type will do Executing monad twice gives the rank y is the rank of
179. laying a list of 5 items each of whichis 2 Let s see how this result matches the definition y is a scalar so it has one item which is also a scalar Therefore the result has the shape 5 x i e 5 concatenated with the shape of an item of y the shape of a scalar item of y is the empty list 5 concatenated with an 20 empty list is a list with the single element 5 The scalar is repeated to fill the 5 items of the result J displays a 1 cell on a single line as shown 5 ab ababa Now y is a list but its items are still scalars with rank 0 and shape empty so the result still has the shape 5 The 5 items come from the items of y cyclically 4 4 There is a tide in the affairs of men Ther e is at ide The items of y are still scalars with rank 0 and shape empty the result has the shape 4 4 The 16 items come from the items of y cyclically Not all items of y are used J displays a rank 2 array as a sequence of lines one for each 1 cell 0 2 The display is a single blank line Just like 5 2 but the resulting list has 0 items i e it is an empty list 1 2 2 Similarly a 1 item list 0 2 2 2 Here 0 2 produces an empty list as we saw above and that is the x to the second The items of y are still scalars so the result has shape empty an empty list concatenated with an empty list i e it is a scalar The displays of a scalar and a 1 item list are identical Does that mean that a sca
180. le contains an account number the day of the month for a transaction and the amount of the transaction positive if money goes into the account negative if money goes out The records in the Journal file are in order of date but not in order of account We are to match each journal entry with its account and print a line for each account giving the starting balance ending balance and average daily balance which is the average of each day s closing balance The number of days in the month is an input to the program as are the filenames of the two files I will offer C code and J code to solve this problem To keep things simple I am not going to deal with file I O errors or data with invalid format or account numbers in the Journal that don t match anything in the Accounts file C code to perform this function might look like this include lt stdio h gt define MAXACCT 500 II Program to process journal and account files printing II start end avg balance Parameters are days in current II month filename of Accounts file filename of Journal file void acctprocess int daysinmo char acctfn char jourfn FILE fid int nacct acctx float acctno openbal xactnday xactnamt struct float ano account number float openbal opening balance float prevday day number of last activity float currbal balance after last activity float weightbal weighted balance sum of closing balances acct MAXACCT 12 Il
181. lem 1 How many hours did each employee work The C code for this is void emphours hrs int hrs result hrs i is hours for employee i int i j for i 0 i lt nemp i for j 0 hrs i 0 j lt 31 j hrs i hoursworked i j The first step in translating this into J is to write the loops but without loop indexes instead indicate what elements will be operated on for each employee for each day take the sum of hoursworked Now figure out what ranks the operands have The hoursworked items that are added are scalars so we will be looping over a list of them that means we want the sum of items of a rank 1 list So the inner loop is going to be 1 What about the outer loop The information for each employee has rank 1 each employee is represented in hoursworked by a single row so a verb applied to each employee should have rank 1 Note that we don t worry about the actual shape of hoursworked once we figure out that our verb is going to operate on cells we let J s implicit looping handle any additional axes We build up the loops by applying the rank conjunction for each one so we have the inner loop 1 and the outer loop of rank 1 combined they are 1 1 The 1 1 can be changed to a single 1 and we get the final program emphours monad 1 hoursworked Problem 2 How much did each employee earn in wages The C code is void empearnings earns int earns result earns i is wages for employee
182. length frames is when the shorter frame is of zero length in other words when one of the operands has only one cell In that case the single cell is replicated to match every cell of the longer operand An easy way to force an operand to be viewed as a single cell is to make the verb have infinite rank for that operand This is not a special case the behavior follows from the rules already given but it s worth an example 40 abc _ 0 defg abcd abce abcf abcg The verb dyad _ 0 has left rank _ and the left operand has rank 1 so the operation will be applied to 1 cells of the left operand The verb has right rank 0 and the right operand has rank 1 so the operation will be applied to 0 cells of the right operand The left frame is empty the right frame is 4 The common frame is empty with length 0 so take each operand in its entirety abc defg The cells of the operands are paired abc defg The longer frame operand the right one is broken up into operand 0 cells each being paired with a copy of the shorter frame operand cell The operation dyad is performed on each pair of cells Since each result is a 1 cell abcd with length 4 and the abce longer frame is 4 the result abcf is an array with shape 4 4 abcg Figure 10 Execution of abc _ 0 defg defverbs comma abc comma
183. ls of its operand In this chapter we will learn a couple of adverbs that apply verbs to subsets of cells chosen according to simple rules in later chapters we will learn how to form irregular subsets u y has infinite rank and applies u to successive prefixes of y It applies u to the first item of y i e u 1 y to produce the first item of the result it then applies u to the first 2 items of y i e u 2 y to produce the second item of the result and so on with the last item of the result beingu y y Example 9876 1234 This just gives the length of each prefix not a terribly edifying result The details can be seen using fndisplay defverbs tally tally 9 8 7 6 tally 9 tally 9 8 tally 9 8 7 tally 9 8 7 6 Note that u is always applied to lists because always produces a list In particular the first item of the result comes from applying u to a 1 item list conversely even if y is a scalar y y isa l item list Note also that ifthe applications of u produce results of different shapes framing fills are added to bring the results to a common shape just as if they were the results from applying a verb to different cells i 3 0 0 12 The result is the prefixes themselves assembled as items of a rank 2 array e O 0 0 0 u is most often used when u is of the form u i e as u Then the verb u
184. lt will have leading axes of length 1 added as needed to bring all the results up to the same rank e g if one result has shape 2 5 and another has shape 5 the second will be converted to shape 1 5 leaving the data unchanged Then if the lengths of the axes are not identical the interpreter will extend each axis to the maximum length found at that axis in any result this requires adding atoms called fills which are always 0 for numeric results and for literal results Example i 0 0 1 2 3 000 NB original result was empty list 3 fills added 000 NB original result was 0 2 fills added 010 NB original result was 0 1 1 fill added 012 NB this was the longest result no fill added fndisplay A Utility for Understanding Evaluation J contains a script that we will use to expose the workings of evaluation You define verbs which instead of operating on their operands accumulate character strings indicating what operations were being performed This gives you a way of seeing the operations at different cells rather than just the results Start by loading the script 34 load system packages misc fndisplay ijs Then select the type of display you want We will be using setfnform J Then give the names of the verbs you want to use If you want to assign a rank you may do so by appending r to the name defverbs SumItems plus 0 Optionally define any names you want to use as nouns The value assigned to the noun
185. m the search but there is a little problem each search must go through a column of hoursworked giving the hours worked by each employee on that day but the column is not an item of hoursworked each item is a row giving hours worked on each day by one employee The solution is to transpose hoursworked by hoursworked making the items correspond to days rather than employees Then we match up the resulting1 cells with the 0 cells of the maximum found by loop 1 and find the index of each maximum using i 1 O or the equivalent i _1 Loop 3 isa simple application of dyad The final code is 63 dailydrudge monad define hoursworked i _1 gt hoursworked empno Problem 6 Order the employees by the amount of profit produced This is a bagatelle for J and we won t even bother with C code We have a verb that returns the profit for each employee so we call it and use the result as the keys for sorting the employee numbers into descending order Note that a verb must be given an argument when it is executed we use 0 as a convenient value The final J code is producers monad empno empprofit 0 which makes use of the verb we defined earlier empprofit monad define billrate payrate 1 hoursworked Problem 7 is similar Order the clients by the amount of profit produced It requires more ingenuity and the C code would be more than I want to show so let s try to write in J directly We start with
186. mes the result of the sentence 190 To follow the parsing we need to know what patterns at the top of the stack contain an executable fragment The parsing table below gives the complete list More than one symbol in a box means that any one of them matches the box name means any valid variable name and C A V and N stand for conjunction adverb verb and noun respectively leftmost stack word other stack words anything 6 Hook Adverb Se anything Is anything 8 Paren The lines in the parsing table are processed in order If the leftmost 4 words on the stack match a line in the table the fragment those words on the stack which are in boldface in the parsing table is executed and replaced on the stack by the single word returned Because the fragment is always either two or three words long it is officially known as a bident or trident The last column of the parsing table gives a description of what execution of the fragment entails You will have an easier time following the parsing if you note that the leftmost word in the executable pattern is usually one of A V N This means that you can scan from the right until you hit a word that matches one of those before you even start checking for an executable pattern If you find one of A V Nandit doesn t match an executable patte
187. ming tests because they perform the operation item by item replacing a C loop that does a test for each atom They have rank 0 and produce a scalar in each cell so the result has the shape of the larger operand x gt y the greater of x andy x lt y the lesser of x andy Arithmetic Monads These verbs have rank 0 gt y increment y 1 lt y decrement y 1 lt y the largest integer not greater than y floor function gt y the smallest integer not less than y ceiling function y absolute value of y y signum of y _1 ify is negative 0 if y is tolerantly close to 0 1 if y is positive x o y trigonometric function Think of dyad o as a monad selecting the function based onx For example 1 o yissin y and_3 o y isarctan y The functions are O sqrt 1 sqrt y 1 sin y 2 cos y 3 tan y 4 sqrt y v 1 5 sinh y 6 cosh y 7 tanh y 8 sqrt 1 y y 9 Rey 10 Mag y 11 Im y 12 Angle y x o is the inverse of x o with the exceptions _8 sqrt 1 y y __9 y _10 Conj y _11 j y _12 j y y l y 48 Boolean Monad y rank 0 Negate y This is simply the boolean interpretation of l y Operations on Arrays These verbs operate on entire arrays they have infinite right rank so they look at the entire y and they have left rank as appropriate for the operation performed I am going to give a highly simplified definition of the functions of these verbs consult the Dictionary to see al
188. miniscent of mathematical epsilon meaning element of Take and Drop x x 51 y Take y Drop The left rank is 1 but the verb handles scalars also we will consider only the case where x is a scalar x y take takes the first x items of y i e it produces a result which consists of the first x items of y x y drop discards the first x itemsofy Ifxisnegative x y takes the last x itemsofy andx y discards the last x items remember x is the absolute value of x The rank of the result is always the same as the rank of y and in all cases the order of items is unchanged Examples 2 314159 3 1 2 314159 4159 2 3 141 59 59 1 i 33 1 4 On F 0 3 x y always gives you as many items as you asked for If you overtake by asking for more than y items J will create extra ones filling them with O or as appropriate 5 314 31400 _5 abc abc NB Negative overtake fills added at front We have met fills before they were added to bring the results from different cells of a verb up to a common shape so that they could be made into an array The fills added by overtake are different they are part of the execution of the verb itself We will distinguish the two types of fill calling the ones added by the verb itself verb fills and the ones added to make cell results compatible framing fills Framing fills are always O or but you can specify the value to use for a verb
189. n i 223 0 1 2 3 4 5 6 7 8 9 10 11 lt 0 1 1 1O LIL11 lt O 1 1 i 22 3 4 If not all axes are specified the selectors are applied starting with the leading axis and any axes left over at the end are taken in full lt O 1 1 2 2 3 345 Each of the selectors may contain either a scalar or a list If a selector contains a scalar the corresponding axis will disappear from the shape of the result as in the examples above Ifa selector contains a list even a list with only one item the 93 corresponding axis will remain in the shape of the result its length will be the length of the selection list lt O 1 0 i 2 2 3 345 012 We select two rows of item number 0 The rows stay in the order we requested them and the result has rank 2 lt 0 1 1 0 2 i 2 2 3 5 2 11 8 Understand where each number came from We are taking a 2x2 array of 1 cells but only item 2 from each I cell That leaves a 2x2 result lt O 1 1 2 2 3 345 lt 0 1 2 22 3 3 lt 0O 1 i 2 2 3 1 3 In the last example we are requesting a list of 1 cells even though the list has only one item its axis remains in the result If a selector contains a box rather than the usual numeric it calls for complementary selection along that axis the contents of that box i e the contents of the contents of the selector indicate the indexes to exclude from the selection and all other ind
190. n a T block 148 28 Modular Code Separating your code into independent modules boils down to segmenting the space of variable names into subspaces that overlap to the extent you want We will discuss how J handles namespaces and then see how this corresponds to the classes and objects provided by C Locales And Locatives Every public named entity in J is a member of a single locale Each locale has a name which is a list of characters A locative is a name containing a simple name the only kind of name we have encountered so far and an explicit locale in one of the two forms simplename_localename_and simplename _ var Inthe form simplename_localename_ localename is the name of the explicit locale in the form simplename _ var the variable var must be a scalar boxed string whose opened contents provide the name of the explicit locale Examples abc _z_ is simple name abc and locale z vv lt lname def__vv is simple name def and locale Lname Note that a simple name may contain an underscore it may not end with an underscore or contain two underscores in a row Note J makes a distinction between named locales whose names are valid J variable names not including an underscore and numbered locales whose names are strings representing nonnegative decimal integers with no leading zeroes The difference between the two is small and we will ignore it The current locale is a value kept by J and used to influence the processi
191. n one the result is a list Array creating verbs Now that we know how to talk about arrays we might as well create a few and see what they look like As mentioned earlier every J verb can be used to create an array there are no special declaration verbs but we will start with a couple that do little else The J lines are taken from an interpreter session you can type them into your own session and get the same results The indented lines were typed into J and the unindented ones are J s responses Dyad Shape and monad hape Of The verb dyad is invoked asx y The result of dyad has the frame x relative to the rank of the items of y and is made up of the items of y repeated cyclically as needed It follows that the shape of this result is x concatenated with the shape of an itemofy We will have to work together on this Confronted with a definition like that you might a decide that J must be a language for tax accountants and give up b decide the definition is Greek and go on hoping it will make sense later c try a few examples to get an idea for what the definition means d read it over and over again until you understand it I hope you will eschew a and b and settle for no less than full understanding For my part I will offer a few useful examples that you can compare against the definition Not everything in J will be as abstract as this 5 522 22222 The simplest case creating and disp
192. n res as well we can extract the desired result simply as 140 2 1 res 100 100 50 50 100 50 25 25 50 25 Once you have defined the verb and the initial value you can use LoopWithInitial to solve problems of this kind You may skip the rest of this chapter if you are not curious about how LoopWithInitial works It performs the following steps LoopWithInitial conjunction define by lt _1 y NB 1 box items of y ry by NB 2 reverse order for ey ry lt n NB 3 append initial value r u amp gt ey NB 4 apply u in boxes er r NB 5 remove initial value rr er NB 6 put in original order gt rr NB 7 remove boxing Since the initial value is going to be appended to the y operand and they have dissimilar shapes it is necessary to box each item of y as well as the initial value Once the items of y are boxed they are put into reverse order because as we have seen u is much faster than u Then the initial value is appended to the reversed boxed y With that preparation complete the operation can be performed u amp gt applies u amp gt in words unbox each operand apply u and rebox starting at the end of the reversed y The first application of u amp gt will be between the original first element of y and the initial value the next application will be between the original second element of y and the result of the first application and so on The results f
193. nd i c i i amp 15 0 amp 0 4 end To begin with let s make sense of this odd sequence adverb define The adverb define defines an adverb but what s the Simple it s the left argument 157 to the adverb that was defined the adverb is executed with u setto The result of that execution of the adverb is what gets assigned to playsound So what happens when the adverb is executed The adverb calls the foreign 9 12 to see what operating system is running and executes a selected line that contains the definition of a compound verb Since that line is the last one executed it becomes the result of the adverb so the result of the adverb is the selected verb and that is what is assigned to playsound On my system this leaves playsound defined as a single compound verb playsound winmm dll PlaySound i c i i amp 15 0 amp 0 4 Lovely No check for operating system needs to be made when I invoke playsound the check was made when playsound was defined Example The LoopWithInitial Conjunction The conjunction LoopWithInitial that we learned about earlier can be written as LoopWithInitial 2 u amp gt amp amp lt v amp amp lt _1 It s just one application of amp after another We can use it to illustrate a subtlety about modifiers that you should be aware of Consider an invocation of LoopWithInitial vb init 45 vb LoopWithInitial init wb amp gt amp
194. nd locatives used to refer to entities look just like names appearing as targets of assignments but there is an additional level of complexity for references Each locale has a search path usually called simply the path which is a list of boxed locale names The path is set and queried by the foreign 18 2 which goes by the alias copath Examples 150 locl loc2 z 18 2 lt loc3 Sets the path for locale loc3 to loc1 followed by loc2 and the obligatory z copath lt loc3 locl loc2 z Queries the path for loc3 Every reference to a name implicitly uses a path A reference to a locative looks for the simple name in the explicit locale if the name is not found there the locales in the path of the explicit locale are examined one by one until the simple name is found if the name is not found in any locale in the path it is an undefined name A reference to a simple name is similar but first the private namespace of the executing explicit definition if any is searched and only if that search fails are locales searched starting in the current locale and continuing if necessary in the locales in the current locale s path Note that only the path of the starting locale either the current locale or the explicit locale specifies the search order No other paths are used Examples of references locl loc2 z 18 2 lt loc3 a_locl_ a a_loc2_ b c_loc3_ c c_lo
195. nd n operands but its text is not interpreted until the derived verb which consists of u n and the text of InLocales gets its x and y operands When InLocales is supplied with u and n it is executed to produce a verb which consists of the text of InLocales along with u and the value ofn This derived verb is hidden inside the interpreter where it waits to be applied to a y and possibly x When the derived verb is given its operands it starts interpreting the text of InLocales which was unusable until the time that y and x could be given values and initializes u andn from the values that were saved whenu InLocales n was executed Thus the text of InLocales describes a verb operating ony and x Your modifiers should refer tox and y only if necessary Ifany from the previous section could have been written Ifany 1 u y y which would produce exactly the same result as the other definition but it would usually be slower because the text could not be interpreted until x and y could be defined If the conjunction happens to be used in a verb of low rank the result could be soporific Here s a puzzle that may be of interest to those readers whose eventual goal is Full Guru certification Why did InLocales save and restore the current locale Didn t we say that completion of any named entity restores the original locale Let s see what happens when we don t restore using a simple testcase 161 t 1 0 cocurrent
196. nd the dyad s right rank If two ranks are given the first is the dyad s left rank and the second is used for the dyad s right rank and the rank of the monad If there is only one item in the list it is used for all ranks So v O 1 has monad rank 1 dyad left rank 0 and dyad right rank 1 As usual J primitives themselves have the ranks shown in the Dictionary Processing of a dyad follows the same overall plan as for monads except that with two operands there are two cell sizes call them r and rr and two frames call them f and rf If the left and right frames are identical the operation can be simply described the r cells of the left operand match one to one with the rr cells of the right operand the dyad is applied to those matched pairs of cells producing a result for each pair those results are collected as an array with frame lf We illustrate this case with an example 36 1 202 22 22 0 2 4 6 The verb has left rank 0 and the left operand has rank 2 so the operation will be applied to 0 cells of the left operand The verb has right rank 0 and the right operand has rank 2 so the operation will be applied to 0 cells of the right operand The left frame is 2 2 the right frame is 2 2 Think of the left operand as a 2x2 array of 0 cells 0 1 0 1 and the right operand as a 213 2 2x2 array of 0 cells The corresponding left and right operand cells are paired
197. nds To improve performance the interpreter recognizes a great many combinations of verb operand shape and type handling each with separate code In most cases the interpreter produces its result in accordance with the rules given above but in a few exotic cases it deviates You are quite unlikely to encounter these cases in practice the most important one is T1 gt 0Sa 0 where the rules given above would predict a shape of 0 0 Enough applications rely on shape 0 to keep this deviation in the system at least as of release 5 01 Empty cells As we discussed above a cell like any array is called empty if it has a 0 in its shape Whether the cells of an operand are empty is independent of whether the operand itself is empty How a verb handles an empty cell is entirely up to the verb the fill cell processing we discussed above does not apply The J primitives generally preserve the type of empty lists that are data but ignore the type of empty lists that are control information So even though characters are not allowable left operands of dyad i 5 produces the same result as 0 0 i 5 because the rotation count is control information In contrast 3 produces a 3 character string while3 0 0 produces a 3 item numeric list because the right operand of dyad is data The distinction between control information and data is not clear cut but in all cases the interpreter does what
198. nds but until you do the old operand is unchanged and both it and the result of the verb are available for further use u 1 y partitions y by finding the items of y that match the first item of y i e O y each such item called a fret is the start of an interval of y which runs from the fret to the last item before the next fret the last interval runs to the end of y Monad u is applied to each interval which is always a list of items of y and the results of u become the items of the overall resultofu 1 y lt 1 a list of words a list of words Each character even the one at the end started a new interval We are not restricted to boxing the intervals 1 a list of words 25361 Here we report the length of each word u _1 yislikeu 1 y except that the fret itself is omitted from the interval The interval could be empty in that case 129 lt _1 a list of words la list of words _1 a list of words 14250 u 2 yandu 2 yarelikeu 1 yandu _1 y except that the frets are those items of y that match its last item and each marks the end of an interval lt 2 Mississippi Mi ssi ssi ppil Use u 2 y to split a list when you know the ending marker for the intervals ri 7 2 0 0 Fred 500 Joe 200 Fred 50 Sam 30 Fred 500
199. nes of J to be compressed still further Programming In J 2 Preliminaries Notation C code is set in Arial font like this for I 0 1 lt 10 1 p I q J code is set in Courier New font like this p 10 q When J and C use different words for the same idea the J word is used The first few times the C word may be given in parentheses in Arial font verb function When a word is given a formal definition it is set in bold italics verb Terminology To describe the elements of programming J uses a vocabulary that will be familiar though possibly frightening the vocabulary of English grammar We will speak of nouns verbs and the like Don t worry you re not going to have to write a book report Use of this terminology is not as strange as it may seem Take verb for example an idea that corresponds to the C function or operator Why not just say operator Well that word is also used in mathematics and physics with a meaning quite different from C s Even a C function is not a true mathematical function it can return different values after invocations with the same arguments J avoids imprecise usage by choosing a familiar set of words and giving them entirely new meanings Since J is a language the vocabulary chosen is that of English grammar It is hoped that the familiarity of the words will provide some mnemonic value but as long as you learn the J meanings you are free to forget the grammatical ones
200. ng file y is overwritten y is a boxed character string or a file number Test Data CR LF 1 2 lt c Temp temp dat Dyad 1 3 AppendFile hasrank_ 0 x 1 3 y appends the character string x to the of file y creating the file if it does not exist Line 2 CR LF 1 3 lt c Temp temp dat Monad 1 55 EraseFile has rank 0 1 55 y erases the file y with no prompting Be careful the power of J makes it possible for you to delete every file on your system with one sentence The special file number 2 sends output to the screen The verb monad Display in jforc ijs uses file number 2 to display its operand which must be a character string on the screen you can put Display sentences in a long verb to see intermediate results In most cases you will prefer to use printf described below There are many other 1 n foreigns to manage file locks query directories handle index read write and do other useful things The Dictionary describes them and I will add only that the description of 1 12 is misleading the length to be written is implied 112 by the string argument x and must not be included in y therefore item of y is a boxed scalar rather than a boxed list of 2 integers as for 1 11 Error Handling u v 13 11 and 9 8 When you deal with files you have to expect errors which will look something like s 1 1 lt c xxx yyy dat file name error s 1 1 lt c xxx yyy dat indicating that the f
201. ng of names We will learn what causes it to change The current locale is the name of a locale When J starts the current locale is set to base Assignment An assignment is private if it is of the form simplename value and is executed while an explicit definition is running an explicit definition is the result of the conjunction for example a verb defined by 3 text ora modifier defined by adverb define An entity assigned by private assignment is not part of any locale and is accessible only by sentences executed by the explicit definition that was running when the entity was assigned The idea is this there is a pushdown stack of namespaces in which private entities are assigned When an explicit definition E is executed it starts with a new empty namespace that will be destroyed when E finishes Any private assignments made while E is running are made in this private namespace Any private variables referred to by E or by tacitly defined entities invoked by E are taken from this private namespace If E invokes another explicit definition F F starts off with its own private namespace and has no access to elements of E s private namespace When F 149 finishes returning control to E F s private namespace is destroyed E is said to be suspended while F is running Assignments that are not private because they assign to a locative use or are executed when no explicit definition is running are public Assignment to a locativ
202. ngth bytes The result is the address of the memory area as an integer It is 0 if the allocation failed mema has infinite rank You must box the memory address before using it as an operand to cd Do not box the address for use as an operand to memf memw ormemr Free memory memf memf address frees the memory area pointed to by address address must be a value that was returned by mema Result of O means success 1 means failure memf has infinite rank Write Into a Memory Area memw data memw address byteoffset count type causes data to be written starting at an offset of byteoffset from the area pointed to by address for a length of count items whose type is given by type type is 2 for characters 4 for integers 8 for floating point numbers 16 for complex numbers if omitted the default is 2 If type is 2 count may be one more than the length of data to cause a string terminating NUL 0 to be written after the data Read From a Memory Area memr memr address byteoffset count type produces as its result the information starting at an offset of byteoffset from the area pointed to by address for a length of count items whose type is given by type 119 type is 2 for characters 4 for integers 8 for floating point numbers 16 for complex numbers if omitted the default is 2 The result is a list with count items of the type given by type If type is 2 count may be _1 which causes the read to be terminated be
203. nks at which individual primitives are applied because of an important feature of J called integrated rank support When a verb with integrated rank support is used as the u in u n the resulting verb runs with a single primitive start and the application of the verb on the proper cells is handled within the primitive So 100 Ts a a 0 0623949 4 19501e6 100 Ts a 0 a 0 248846 4 19526e6 100 Ts a 1 a 0 0681035 4 19526e6 100 Ts a 2 a 0 0626361 4 1952e6 All these forms produce identical results The weak dependence of the speed on the rank is typical of a verb with integrated rank support Fastest execution is achieved when the verb is used alone but the form u n still runs fast and the higher the rank the less the loop control overhead The Special Code page referred to in the previous section includes the long list of the primitives with integrated rank support The practical effect of integrated rank support is that you don t need to worry much about using the largest possible rank for primitives In compounds and verbs that you write you do need to keep the rank high Ts lt a 1 a 0 00939758 525568 Ts 1 1 a 0 00952329 525184 Integrated rank support in dyad gives the two forms equal performance Look what happens when we replace the by a user defined verb with the same function 110 from Ts lt a 1 from a 0 00953335 525760 Ts 1 from 1 a 0 365966 525696 from lacks integrated rank suppor
204. ntf displays a line while sprintf produces a string result The population of s is d n printf Raleigh 240000 The population of Raleigh is 240000 s The total of j is d n sprintf 1 2 3 1 2 3 s The total of 1 2 3 is 6 You need to execute load printf to get the printf verbs defined J s printf contains a few features beyond C s such as the j field type seen above You should run the Lab Formatting with printf for details One feature with no analogue in C is qprintf which produces handy typeout for debugging a 345 b abc def 5 c 1 3 3 qprintf a bc a 3 4 5 b Jabc def 5 c 012 3 45 67 8 qprintf is described in the Formatting with printf lab Convert Character To Numeric Dyad Dyad has infinite rank x y looks for numbers in the words of y which must be a character array Words representing valid numbers are converted to numbers words not representing valid numbers are replaced by x If 1 cells of y have differing numbers of numbers x is used to pad short rows Valid numbers to dyad are anything you could type into J as a number and more the negative sign may be rather than _ numbers may contain commas which are ignored and a number may start with a decimal point rather than requiring a digit before the decimal point With the relaxed rules you can import numbers from a spreadsheet into J using x amp to convert them to numb
205. nts to the data The data will be read only when the noun is referred to This is called mapping the file to a noun J s facilities for mapping files are described in the Lab Mapped Files A quick example of a mapped file is 113 require jmf JCHAR map _jmf_ text system packages misc jforc ijs after which the noun text contains the data in the file 45 text NB File definitions for J For C Programmers Moreover if you assign a new value to text the file will be modified If you are dealing with large files especially read only files or files that don t change much mapping the files can give a huge performance improvement because you don t have to read the whole file You must be very careful though if you map files to nouns because there are unexpected side effects If we executed temp text temp abcdefgh we would find that the value of text had changed too If you try this use a file you don t mind losing The assignment of text to temp did not create a copy of data of text and when temp was modified the change was applied to the shared data Ifa file is mapped to a noun you have to make sure that the noun or any copy made of the noun in any verb you pass the noun to is not changed unless you are prepared to have some or all of the other copies changed This topic is examined in greater depth under Aliasing of Variables in the chapter on DLLs If the faster execution is enticement enough for you to take
206. nvoking a Gerund m 6 Sometimes you want to use a noun rather than a verb to designate a verb to be called An asynchronous socket handler is an example the socket handler will have many transfers going on at once each one with a callback to be executed when the transfer is complete The callbacks must be put into a table along with other information about the transfer in other words the callbacks must be nouns We have already met nouns that carried verbs we called them gerunds We found that gerunds were created by the conjunction and executed by m v While these tools are adequate to allow verbs to be passed as arguments some simplifications are available that we will discuss now A gerund created by u v is always a list each element of which as we learned is an atomic representation of a verb which we will treat as untouchable Even if there is only one verb the result of is a list vt O It makes sense for the gerund created by to be a list since it contains a list of verb representations one of which is selected for execution by m v But when we are passing a single verb as noun as an argument it is OK for it to be a scalar box And it can be invoked using the conjunction m 6 converts the gerund m into a verb So in our example we pass the callback as one box in a parameter list and then we select it turn it into a verb with m 6 and execute it as we can see in a stripped down example The
207. o the result of applied between the members of a sequence of verbs is a list of special nouns each of which is the atomic representation of a verb We are not concerned with the format of the atomic 87 representation nor will we create or modify an atomic representation that s Advanced Guru work we will be content to use the values produced by An example is x x x N I 1 1 1 I I 1 11 1 1 II I II 14 411 What makes the result of special is not the boxing but the fact that what s in the boxes is not just any old data but data in the format that can be used to recover the original verbs Once created the result of can be operated on like any other noun a 3 a I5 00101i a 11 1 1 I IE Vitti I Ii 4 1 In English grammar a gerund is a form of a verb that is used as a noun for example the word cooking in Cooking is fun The result of in J is also called a gerund and we can see that the name is apt a gerund in J is a set of J verbs put into a form that can be used as a J noun It has the latent power of the verbs put into a portable form like nitroglycerine that has been stabilized by kieselguhr to become dynamite The blasting cap that sets it off is The Agenda switch conjunction m v m v either monad or dyad uses the result of v to select a verb from the list of ver
208. o each 1 cell rather than to each 0 cell Contrast this result with the result of the verb monad which means multiply by 2 2T a 0202 2002 Evidently the verbs themselves have some attribute that affects the rank of cell they are applied to It s time for us to stop experimenting and learn what that attribute is The Concept of Verb Rank Every verb has a rank the rank of the cells to which it is applied If the rank of the verb s operand is smaller than the rank of the verb the verb is applied to the entire operand and it is up to the author of the verb to ensure that it produces a meaningful result in that case Dyads have a rank for each operand not necessarily the same A verb s rank can be infinite _ in which case the verb is always applied to the operand in its entirety In other words if a verb has infinite rank for an operand that operand is always processed as a single cell having the rank of the operand If you don t know the rank of a verb you don t know the verb Using a verb of unknown rank is like wiring in a power supply of unknown voltage it will do something when you plug it in it might even work but if the voltage is wrong it will destroy what it s connected to Avoid embarrassment Know the rank of the verbs you use The definition page of each J verb gives the ranks of the verbs defined on the page right at the top of the page after the name of the verb Since most pages define both a monad an
209. o the file s data and that pointer is never changed even if a value is assigned to the variable the whole point of mapping the noun to the file is to cause changes in the noun to be reflected in the file so any assignment to the noun causes the data to be copied into the area that is mapped to the file In addition when b is a noun mapped to a file and is assigned to another noun as with a b the noun a which is aliased to b also inherits the mapped to file attribute This behavior is necessary to make mapped files useful because assignments tox and y are implicit whenever a verb is invoked and it would defeat the whole purpose of mapping if the data of the file had to be copied every time the mapped noun was passed to a verb The combination of aliasing and mapping means that any assignment to a mapped noun also changes the values of all other nouns which share the same mapping for example if you pass a mapped noun a as the right operand of a verb that modifies its y y a and the data in the file will all be modified Keeping track of the aliasing is the price you pay for using mapped files If you need to copy a noun making sure you get a fresh unmapped data block you must not assign 120 the mapped noun directly but instead assign the result of some verb that creates its output in a new data block For example as of release 5 01 the assignment a 1 amp b will create a new data block containing the data of b and a will
210. of boxing in J before we try to relate it to structures in C because the ideas are just different enough to cause confusion The box is an atomic data type in J along with number and character As with the other types a single box is a scalar with rank 0 and empty shape Just as a numeric or character atom has a value so a boxed atom has a value called its contents The box is special in that its contents can be an array while the box itself is an atom The boxing protects the contents and allows them to be treated as an atom Arrays of boxes are allowed and as always all the atoms in an array must have the same type if any element is boxed all must be boxed Various verbs create boxes Monad lt has infinite rank and exists for the sole purpose of boxing its operand lt y creates a box whose contents are y for example When a box is displayed the contents are surrounded by the boxing characters as seen in the examples Only certain primitives can accept boxes as operands generally you cannot perform arithmetic on boxes but you can do other things like monad and dyad monad and dyad and other primitives that do not perform arithmetic The significant exception to this rule is that you can use monad and dyad and to order boxed arrays Comparison for equality between two atoms is not strictly an arithmetic operation you can compare two characters or a character and a number for equality for example and it is allowed on
211. of their derived verb as x or y come in for special treatment A simple example is the conjunction defined by 2 u v y 2 Tus v y Weexecute2 u v y and the result is an anonymous conjunction that we ll call ac4 You can t tell it from the display but ac4 is a special kind of conjunction Because it refers to y the text of ac4 can be executed only as a verb only then are x and y meaningful The stored ac4 makes note of this fact 2 u v y 5 _10 When ac4 itself is executed as 2 u v y here since ac4 isa conjunction it is executed before its result is applied to the noun operand 5 the text of ac4 is not interpreted as it was in our other examples Instead the new anonymous verb av3 is created av3 contains the defining text of ac4 along with the operands that were given to ac4 and here When the verb av3 is executed as in the line above the text of ac4 is finally interpreted with the operands of ac4 and here available asu and v and the noun operands of av3 5 here available as y and x if the invocation is dyadic the result is the resultof 5 189 36 Parsing and Execution IT Now that you understand what an anonymous verb adverb conjunction is you are ready to follow parsing and execution word by word We will finally abandon all shortcuts and process sentences exactly as the interpreter does In any compiled language a program is broken into words tokens
212. omm n stores nas a list of characters Fe C22 UE es amp We execute ac2 with left operand of and right operand of The result is an anonymous conjunction that we ll call ac3 ac3 is a conjunction because its value amp the last sentence executed by ac2 is a conjunction Yes amp by itself can be a result 188 modifiers can return any primary part of speech but try to return a conjunction from a verb and you will get an error Note that this is not the same as u amp v that would also be a valid return value from a conjunction but u and v would be substituted and amp would be executed to make the returned value an anonymous verb with the description u v Make sure you see why the and disappeared First the conjunction was executed with operands 2 and amp that produced a conjunction ac2 which was then executed with operands and but the defining text of ac2 does not look at its operands it simply produces the value amp So the operands to ac2 disappear without a trace and the result of the whole phrase is a conjunction with the value amp 2 Re 2 eae tye 2 amp Continuing the example we execute ac3 which was just the conjunction amp with left operand 2 and right operand The result is the anonymous verb av2 which will execute as 2 amp 2 FS 2 a Tar S 5 10 Finally we execute av2 with the operand 5 and get the result 10 Explicit modifiers that refer to the operands
213. oo but would require a conversion 118 When the function returns its returned value will be boxed and appended to the front of the boxed list that was the right operand of cd to produce the result of the execution of cd You may use this returned value as you see fit Any box that contained an array may have had its contents modified by the function you may open the box to get the changed value If J was unable to call the DLL the cd verb fails with a domain error You can then execute the sentence cder which will return a 2 element list indicating what went wrong The User Guide gives a complete list of errors the most likely ones are 4 0 the number of arguments did not match the number of declarations 5 x declaration number x was invalid the count starts with the declaration of the returned value which is number 0 and 6 x argument number x did not match its declaration the count starts with the first argument which is number 0 and must match the second declaration Memory Management Passing arrays into the called function is adequate only for simple functions If the function expects an argument to be a structure possibly containing pointers to other structures you will have to allocate memory for the structures fill the structures appropriately and free the memory when it is no longer needed J provides a set of verbs to support such memory management Allocate memory mema mema length allocates a memory area of le
214. oopless code we first learned about J s implicit looping which we can use to replace loops in which the same function is performed on each cell then we learned monad which lets us accumulate an operation across all the items of a noun and and which apply verbs to certain regular subsets of a noun We now examine cases in which the operations on the cells are different but where there is no sharing of information between cells In such cases the J solution is to create an array that contains control information describing the difference between the cells and then create a dyadic operation that produces the desired result when given a cell of control information and a cell of data Writing code this way can seem ingeniously clever or awkwardly roundabout depending on your point of view we will simply accept it as a necessary part of coding in J and we will learn to be effective with it What follows is a hodgepodge of tricks to treat cells individually If we were writing in C we would use if statements but since if by necessity involves a scalar comparison we will avoid it in J To add one to the elements of y whose values are even y 0 2 y To double all the elements of y whose values are even y 1 0 2 y To create an array whose even numbered elements come from y and whose odd numbered elements come from x x sv y sv y 10 which homely as it is is a standard idiom in J This expression works only for numeric
215. orms you should understand why you are going to the trouble of learning how to write programs that hide their operands First they are extraordinarily compact The explicit definitions we have written so far are laconically terse by the standards of most computer languages but they will seem positively windy compared to the tacit forms Second the shorter definitions are easier to combine with other verbs and with the modifiers that add so much power to J expressions Third the tacit definitions are parsed when they are defined in contrast to explicit definitions in which each line is parsed as it is executed we reduce interpretive overhead by using tacit forms Fourth in learning to write tacit verbs you are also learning to write tacit adverbs and conjunctions with which you will be able to craft your own private toolkit of modifiers that you can use to combine verbs in ways that are useful to your application In what follows Nx will represent a noun Vx a verb Cx a conjunction and Ax an adverb where x is any suffix We begin by observing that the rules we have learned so far give no meaning to some combinations of words Consider three verbs in a row with no noun to operate on as in the sequence VO V1 v2 where each Vn represents a verb an example would be Without some special rules we have no way to interpret this sequence Such sequences of words that cannot immediately be executed to produce a result are called
216. out declarations how does the computer know that a name represents an array For that matter how do know that a name represents an array The answer affords a first glimpse of the power of J Every J verb whether a primitive operator or a user written verb function accepts arguments that can be arrays even multidimensional arrays How is this possible Like this Suppose you write a verb that works with 2 dimensional arrays Part of your verb definition will indicate that fact If your verb is executed with an argument that is a 3 dimensional array J will automatically split the 3 dimensional array into a sequence of 2 dimensional arrays call your verb and put the pieces back together into an array of results We will very soon go into this procedure in great detail For now you should learn the vocabulary J uses to deal with arrays What C calls an n dimensional array of rank ixjx xk is in J an array of rank n with axes of length i k Every noun variable or object has a shape which is the array of rank 1 made by concatenating the lengths of all its axes For example if q is the array corresponding to the C array defined by the declaration int q 4 5 6 its shape is the array 4 5 6 As youcan see the number of items in the shape is exactly the rank Note a sequence of numbers written with no intervening punctuation defines a numeric array of rank 1 i e a list You may have to use parentheses if you have adja
217. performed on the paired operand cells and collected using the longer frame Maybe some examples will help 39 100 200 i 2 3 100 101 102 203 204 205 The verb dyad has left rank 0 and the left operand has rank 1 so the operation will be applied to 0 cells of the left operand The verb has right rank 0 and the right operand has rank 2 so the operation will be applied to 0 cells of the right operand The left frame is 2 the right frame is 2 3 The common frame is 2 with length 1 so think of each operand 100 200 012 345 as a listof2 _1 cells The _1 cells of the operands are 1001012 3001345 paired The longer frame operand the right one is broken up into operand 0 cells each being paired with a copy of the shorter frame operand cell Each paired _1 cell becomes a row of paired operand cells The operation dyad is performed on each pair of cells Since each result is an atom and 100 101 102 the longer frame is 2 3 the result 203 204 205 is an array with shape 2 3 Figure 9 Execution of 100 200 i 2 3 defverbs plus 0 100 200 plus i 2 3 100 plus 0 100 plus 1 100 plus 2 1200 plus 3 200 plus 4 200 plus 5 The simplest and most common case of different
218. plies by 2 its obverse divides by 2 2 _ 1 10 In the dyadic case the obverse of x amp u is applied The obverse of 2 amp is 3 amp 2 Not all verbs have obverses you can see the obverse of a verb v if there is one by typngv b 1 62 b 1 amp 2 65 61 b 1 amp 1 5 amp b 1 domain error There is no obverse for monad Some of J s obverses are very ingenious and you can have a pleasant afternoon experimenting to find them Most of them are listed in the Dictionary under the conjunction Apply Under Transformation u amp v and u amp v Using its ability to apply the obverse of a verb J provides a feature of great elegance and power with the conjunction u amp v monadic or dyadic u amp v all of whose ranks are the ranks of monad v executes u after a temporary change in representation given by monadv Formally u amp v yisv 1 u v y informally x u amp v yis v _1 v x u v y applied to each cell of x and y with the results collected as usual we will learn a formal notation for this later The idea is that you change the 103 operands using the transformation given by v then do your work with u then invert the transformation with v _1 Examples lt 1 amp gt 4 5 6 1516171 We add x to y the transformation is that we remove the boxing before we add and put it back after we finish The verb amp gt has rank 0 since monad gt has rank 0
219. possible And the value of a noun in a definition is the value at the time the definition was parsed If a parsed definition refers to a verb it does so by name so the value of a verb is its value when it is executed The remaining examples are curiosities to show you that it s worth your trouble to learn the intricacies of parsing Sentence a a 5 ata 5 a a 5 result is 5 a is assigned 5 0 5 5 result is 10 2 10 a is assigned a value just before that value is pushed onto the stack Sentence 2 2 amp 5 unprocessed word list 24 2 amp 5 24 2 amp 5 result is acl defined as 2 amp 4 2 Cacl 5 result is acl 8 2 acl 5 result is ac2 defined as amp 4 2 ac2 5 result is av defined as 2 amp 4 av5 result is 10 0 10 Look at what happens when we omit the parentheses Sentence 2 2 amp 5 unprocessed word list 2 4 2 amp 5 2 2 amp 5 result is 1 l 2 2 amp 1 result is acl defined as 2 amp 2 acl 1 result is ac2 defined as amp gt A 2 ac21 domain error 2 amp 1 is illegal 194 The omission produces a subtle but fatal change in the parsing order As the Dictionary says it may be necessary to parenthesize an adverbial or conjunctival phrase that produces anything other
220. r case sdrecv will return with data if it has any if the length of data is 0 that means that the connection was closed by the peer and all data sent by the peer has been received When you have finished all you data transfers for a socket you must close it with sdclose socket which has as result an unboxed return_code Listening If you want to wait for a remote machine to get in touch with you use the sdbind sdlisten sdaccept sequence instead of sdconnect The sequence is sdbind socket address_family IP_addr port to establish a connection between the socket and the address given by address _family IP_addr port If IP_addris the socket can be used to listen for a connection to any address on the machine If port is 0 the system will assign a port number The result of sdbind is an unboxed return_code sdlisten socket connection_limit causes the operating system to start listening for connections to the address bound to socket The result is an unboxed return_code A maximum of 124 connection_limit connections can be queued awaiting sdaccept Whena connection is made to a listening socket s address the socket is shown in sdselect as ready to read and you should issue sdaccept socket which will return return_code clone_socket where clone_socket is anew socket with all the attributes of socket but additionally with a connection to the remote host You should direct all your sdsend sdrecv and sdclose operations to t
221. r code simply by entering it at the keyboard and seeing what it does If you stick with it J won t just help the way you code it ll help the way you think C is a computer language it lets you control the things the computer does J is a language of computation it lets you describe what needs to be done without getting bogged down in details but in those details the efficiency of its algorithms is extraordinary Because J expressions deal with large blocks of data you will stop thinking of individual numbers and start thinking at a larger scale Confronted with a problem you will immediately break it down into pieces of the proper size and express the solution in J and if you can express the problem you have a J program and your problem is solved Unfortunately it seems to be the case that the more experience you have as a C programmer the less likely you are to switch to J This may not be because prolonged exposure to C code limits your vision and contracts the scope of your thinking to the size of a 32 bit word though studies to check that are still under way and it might be wise for you to stop before it s too late but because the better you are at C the more you have to lose by switching to J You have developed a number of coding habits for example how to manage loops to avoid errors at extreme cases how to manage pointers effectively how to use type checking to avoid errors None of that will be applicable to J J will take
222. r means multiplication x y Similarly when a J verb function or operator is executed with only one operand i e without a noun or phrase that evaluates to a noun on its left we say its invocation is monadic unary if there is a noun or noun phrase on its left that noun becomes a second operand to the verb and we say that the invocation is dyadic binary In the case of programmer defined verbs functions the versions handling the two cases are defined independently We use the term valence to describe the number of operands expected by a verb definition a verb definition has monadic valence if it can be applied only monadically dyadic valence if it can be applied only dyadically and dual valence if it can be applied either way Since the definitions of the monadic and dyadic forms of a verb can be wildly different when we name a verb we will be careful to indicate which version we are talking about monad dyad i Note that it is impossible to invoke a verb with no operands In C we can write func but in J we always must give an operand Note also that the syntax of J limits verbs functions to at most two operands When you need a verb with more than two operands you will represent it as a monad or dyad in which one of the verb s syntactic operands is an aggregate of the actual operands the verb will use during its execution The first thing the verb will do is to split its operand into the individual pieces J has primit
223. r names the part of speech is that of the entity assigned to the name The parts of speech are noun verb conjunction adverb punctuation and copula The primary parts of speech are noun verb conjunction and adverb Partition A selection of possibly noncontiguous items of an array brought together as the items of a new array to be operated on by a verb Path See search path Primitive A word whose meaning is assigned by the J language Private Of a name assigned in the namespace of an explicit definition and accessible only within the explicit definition in which it was assigned Public Of a name assigned in a locale and accessible from any locale via a locative Punctuation A part of speech Punctuation is not executed but it affects the execution of other words Punctuation in J comprises NB and control words Rank Of a noun the number of axes along which the atoms of the noun are arranged the number of items in the shape Of a verb the highest rank of the noun operands that the verb can operate on Scalar See atom Script A file containing J sentences Search Path Of a locale the list of the names of locales that will be searched to find the definition of a name originally sought but not found in Sentence An entire executable line Shape The list of the lengths of the axes of a noun Simple Name A list of letters numbers and underscores beginning with a letter used to refer to an entity Train A
224. r of hours worked Problem 4 Find the amount to bill a given client C code int billclient cno int cno number of client we are looking up int i j temp total total 0 for i 0 i lt nemp tti if emp_client i J cno for j 0 temp 0 j lt 31 j temp hoursworked i j total billrate i temp return total The function is implemented in C by looping over the employees and picking the ones that are working for the specified client In J we deal with entire arrays rather than with elements one at a time and the general plan is get the mask of employees billed to the requested client select the hoursworked records for the applicable employees for each employee 1 for each day accumulate total hours 2 for each employee multiply hours by billing rate for each employee get total billing This example is the first one in which an argument is passed into the J verb Within the verb the right argument is referred to as y and the left argument if the verb is a dyad is referred toas x As usual in C we might use fewer big loops but in J we stick to small loops The mask of employees billed to the client is given by 62 emp client y which is a mask with 1 for the selected employees O for the others remember that is a test for equality not an assignment We can select the hoursworked items for the specified client by emp_client y hoursworked then the sum for each day will be a sum
225. r than x Examples 80 _2 100 2 110 6 120 8 130 3 100 2 110 6 120 8 130 3 This is a convenient way to reshape a list to have 2 items per row when you don t know how many rows there will be 2 10 8 6 4 2 2222 2 10 8 6 4 2 2 2 2 2 Applying between each pair of items with adjacent pairs overlapping takes the backward difference of each pair To take the forward difference subtract the first from the second using 3 gt 12382315431232 38883555 12 12 12 This takes the maximum over a rolling window 3 items wide x u yissimilarto x u y but it operates on outfixes which contain all of y except the corresponding infix Its uses are few The interpreter treats empty operands with care so that you don t have to worry about them as special cases If you simply must know the details here they are If u oru is applied where there are no applicable subsets of y either because y is empty or because it is too short to muster even a single infix u is applied to a list of fills and the result has 0 items each with the shape and type of the result ofu The items of f have the shape of items of y and the length of is 0 for monad u or u or the length of an infix or outfix for dyad u or u For example 2 i 13 02 3 We were looking for infixes of length 2 each one would have had shape 2 3 but with only one item in y we have insufficient data for an infix So we create a 2x3 array of fills
226. r than the noun Fragment 2 or 3 words on the execution stack in a context that makes them executable Frame The frame of a noun with respect to k cells is the shape of the noun with the last r items removed where r is the rank of a k cell of the noun When a noun is viewed as an array of k cells the frame with respect to k cells is the shape of the array of k cells Framing Fill A fill added when the results from applying a verb on its operand cells are being joined into an array Framing fills are always 0 ora depending on the type of the result Fret A marker indicating the start or end of an interval Functional Programming A method of writing programs that describes only the operations to be performed on the inputs to the programs without the use of temporary variables to store intermediate results J s tacit programs are an example of functional programming Aficionados of functional programming consider it to be a purer statement of an algorithm than the usual statement in a procedural language as the expert J programmer Randy MacDonald has said If you re not programming functionally you re programming dysfunctionally Global Of a name accessible as a simple name by any verb In the J Dictionary the word global has the meaning we have assigned to the word public Index In an array an integer indicating position along an axis The index of the first atom along an axisisO Index List A list of integers with t
227. rand 1100 1 amp 1 x abc abxxc Choose From Lists Item By Item monad m Suppose you have two arrays a and b and a Boolean list m and you want to create a composite list from a and b using each item of m to select the corresponding item of either a if the item of mis 0 orb if 1 You could simply write 172 m 1a b and have the answer There s nothing wrong with that but J has a little doodad that is faster and uses less space as long as you want to assign the result to a name You write name m a b assignment with works too This form does not create the intermediate result from dyad If name is the same as a or b the whole operation is done in place More than two arrays may be merged this way using the form name mp a b in which each item ofm selects from one of a b c The operation is not done in place but it avoids forming the intermediate result Recursion In tacit verbs recursion can be performed elegantly using the verb which stands for the longest verb phrase it appears in that is the anonymous verb created by parsing the sentence containing the whose execution resulted in executing the Recursion is customarily demonstrated with the factorial function which we can write as factorial factorial lt 18 amp lt factorial 4 24 factorial n is defined as n factorial n 1 except that factorial 1 is 1 Here we just wrote out the rec
228. rate i Again we create a new loop to calculate the list of profit for each employee for each employee for each day take the sum of hoursworked for each employee take billing_rate wage_rate for each pair of profit and sum multiply the pair The profit is clearly a difference of scalars applied to two lists therefore it will be 0 or equivalently simply The program then is empprofit monad define billrate payrate 1 hoursworked 46 7 More Verbs Before we can write more complex programs we need to learn some more verbs We will group them into classes and give a few mnemonic hints Before we start I should point out a convention of J if a dyadic verb is asymmetric you should think of x as operating on y i e x is control information and y is data We will note the exceptions to this rule and e and the reasons for the exception Arithmetic Dyads All these verbs have rank 0 and produce a scalar result so if they are applied to two operands of equal shape the result will also have that shape if applied to two operands that agree the result has the shape of the larger operand x y addition x y subtraction y operates on x to match the mathematical definition x y multiplication x y division Note that the slash has another use so is division You don t have to worry about division by zero it produces _ infinity or __ negative infinity except for 030 wh
229. rb can have catastrophic consequences To prevent it use 101 dyadi i dyadi 100 110 120 130 valence error dyadi dyadi 100 110 120 130 100 130 150 dyadi 100 110 120 130 0331 also has a special meaning when it appears in a fork which we will encounter later Multi Line Comments Using 0 0 We know that all comments in J start with NB and go only to the end of the line There is no way to extend a comment to more than one line but there is a way to put a series of non executing lines into a script without having to have NB in each of them You simply define a noun using 0 0 and put your comment inside the noun 0 0 This is the first line of a comment The lines of the comment will become a noun but Since the noun is not assigned to anything it simply disappears Final Reminder Remember make sure the verbs you define have the proper rank 102 18 u 1l u amp v andu v The Obverse u zL What would it mean to apply a verb a negative number of times Applying a verb _1 times should undo the effect of applying the verb once In J u n applies the verb u n times and u _1 is defined as the verb that undoes the effect of applying u once u _ 1 is called the obverse in J and where possible it is the same as the mathematical inverse of u or x amp u in the dyadic case Examples gt a 5 Seen 1v 5 i Monad gt adds 1 its obverse subtracts 1 amp 2 1 10 F Monad amp 2 multi
230. recover But if he replies I see a list of 3 items each of which is a finger you can expect to start receiving emails that look like they ve been encrypted You can offer as a temporary palliative J s mechanism for converting an explicit definition to a tacit one You request the conversion by using a left operand to in the range 11 to 13 insteadoflto4 9 3 5 NB Do this once to select simplified display Sif ee cay 3 Tsy We defined a verb and since we didn t assign it to anything we see its value which is just what we defined it to be PS Oy by using 13 instead of 3 we ask the interpreter to try to find a tacit equivalent which it did Here is another way to define the verb enclose from the previous chapter 13 gt x y gt x M gt f TD 1 gt t I The interpreter likes to use in its tacit definitions Note that you use 13 to get an equivalent for both monadic and dyadic verbs there is no 14 I recommend that you use 13 as your first choice for defining tacit verbs It will find all tacit equivalents that can be generated systematically and the explicit definition is much easier to read than a tacit definition You will still have to use your tacit definition skills for irregular cases such as 13 4 x yi A PL Sea a y Ifx ory is used as an operand to a conjunction as in this example the tacit converter is not going to be able to make a tacit equivalent The result of 13 is st
231. reign 9 11 y and queried by 9 10 y initially it is 6 as a quick override you may use the fit conjunction d to specify the value of d for a single use of If an atom of y is an integer a field descriptor of 0 is applied if the atom has significance in the first 4 decimal places a field descriptor of 03d is applied otherwise a field descriptor of 0j_ 3 is applied Trailing zeros below the decimal point are omitted 91 10 6 0 0 01 0 001 0 000015 0 12345678 2 0 0 01 0 001 1 5e 5 0 123457 2 In spite of all the detail I gave about the default formatting in practice you just apply monad to any numeric operand and you get a good result Monad accepts character arguments as well and leaves them unchanged 115 7 amp gt Today is 2002 1 24 Today is 2002 1 24 We went inside the boxes with amp gt and formatted each box s contents with monad this made all the contents strings and we could run them together using monad Monad also converts boxed operands to the character arrays including boxing characters that we have seen in the display of boxed nouns Format binary data 3 n If you need to write numbers to files in binary format you need something that will coerce your numbers into character strings without changing the binary representation J provides the foreigns 3 4 and 3 5 for this purpose Each is a family of conversions invoked as a dyad where the left operand selects the con
232. ression in its own right See why this produces the 4x4 array of characters shown Remember that a is the alphabet of all ASCII characters in order An even more elegant way to produce the same result would be i 4 4 amp amp a amp i a 00 2 2 0 a ab ef Starting corner 0 0 lengths 2 2 result is a 2x2 subarray left unchanged by monad 1 2 3 2 1730 2 gh kl op Starting corner 1 2 lengths 3 2 result is a 3x2 subarray left unchanged by monad You get the idea 132 If an item of s is negative the corresponding axis of the subarray extends backward from the starting point and its absolute value is used as the starting position 2 2 22 4 0 jk no Starting corner 2 2 the character k lengths 2 2 but running backward in axis 1 result is a 2x2 subarray left unchanged by monad Note that the axis extends backward but the items retain their normal order you have merely specified the interval by its endpoint rather than its beginning point If you want to reverse the order of the axis you can do that too by making the corresponding length negative 2 2 2 2 0a kj on Starting corner 2 2 the character k lengths 2 2 but running backward in axis 1 result is a 2x2 subarray with axis reversed left unchanged by monad The subarray must end at the limits of the array If the length requests more than that the subarray will be shorter than was requested A length of _ or __ will always fe
233. result of the parsing rules is that conjunctions and adverbs just like verbs should be read right to left If I wanted to spend half of my uncle s money the amount I would be left with is 0 5 1 046 x Y No parentheses are needed because 1 04 is still not preceded by a conjunction and so1 04 amp x y is still evaluated before that value is multiplied by 0 5 Once you get the hang of it you will be able to understand and build composite verbs of great power That will be a useful skill to develop because you never know when you are going to want to make some sequence of functions the operand of a modifier and then you re going to have to be able to express the sequence in a single compound verb It will take a lot of practice as well as coding techniques to break a gristly mass of conjunction bound words into digestible pieces we ll learn them later For the time being be content if you can understand the simple examples shown above 71 11 Boxing structures The nouns we have encountered so far have all had items with identical shapes and with all atoms of the same type either numeric or character You may be afraid that such regular arrays are all that J supports and that you will have to forgo C structures and you may wonder how J will fare in the rough and tumble of the real world where data is not so regular This chapter will put those fears to rest I think we should get a formal understanding
234. rn keep looking for the next occurrence Note that the leftmost stack word in the parsing table is never a conjunction This is the ultimate source of our long noted rule that conjunctions associate left to right a conjunction can be executed when it appears in the third stack position but if another conjunction is in the leftmost position then the stack will always be pushed down to examine that conjunction s left argument Examples Of Parsing And Execution We will now follow a few sentences through parsing We will represent anonymous entities by names in italics with an indication of how the anonymous entity was created Up till now in this book we have scarcely noticed that the term verb was used both for an entity that can be applied to a noun to produce a noun result and also for the name of that entity This ambiguity will continue being precise would be too cumbersome but you should be aware of it When we say the result is av defined as that means that an anonymous verb was created whose function is described as and the nickname we are giving it the word that is that goes on the execution stack to betoken this verb is av 191 Sentence 2 a whereais1 2 3 unprocessed wordlist 2 a 2 123 not executable 2 123 not executable 2 123 not executable 2 123 result 2 46 2 246 result av defined as 3 av246 result 12 0 12 T
235. roblems The example problem is a simulation In a certain church most of the worshippers are devout but the front pew is always packed with knaves Collection is taken every Sunday Each knave brings two coins to throw in but being a knave he first removes the two most valuable coins from the plate before adding his own Given p the contents of the plate when it reaches the front row a list of coin values and k the coins brought by the knaves an nx2 array what is the final contents of the plate and what does each knave make off with For test data we will use p 100 25 100 50 5 10 giving the contents of the plate as it enter the knaves row and k _2 50 50 100 25 5 10 25 50 25 10 50 50 100 25 5 10 25 50 25 10 as the coins to be thrown in by the greedy knaves After trying for a clever solution for a little while we give up and decide we are going to have to simulate the action of each knave We start by writing a verb knave to perform a knave s action The design of this verb requires a little careful joinery to make it useful later we will invoke it as x knave y where x is an item of k i e the input for one knave and y is the result from applying knave for the previous knave the result of knave must have the same format as the y operand of knave finally the result of knave must include whatever we want as the final solution of the problem The trick is that the result of knave which will be the input to the nex
236. rom the applications of u amp gt are collected in an array Finally the preparation steps are undone discarding the initial value reversing the array into original order and unboxing the result The actual implementation of LoopWithInitial is a bit more elegant than that schematic representation Observe that step 7 is the inverse of step 1 step 6 of 2 and step 5 of 3 each pair is an opportunity to use u amp v and the actual verb produced by LoopWithInitial is knave LoopWithInitial inity knave amp gt amp amp lt 25 10 100 25 100 50 5 10 amp amp lt _1 which performs all the steps of the longer form We will examine this conjunction in a later chapter You may object that since the left operand of LoopWithInitial is applied to each item it still has to be interpreted for each item so nothing is gained by avoiding the loop An astute observation but in the second part of this book we will learn how to write verbs that are not reinterpreted for each item 141 Finally you may observe that the temporary vector which only needs to be a list turns into a rank 2 array by the time we have passed it through each item Doesn t that waste a lot of space Yes it does and if your problem is big enough that the space matters you may have a valid application for a loop An alternative would be to make the temporary vector a public variable accessed by knave in the same way that temporary variables would be used in C
237. rsseesse 127 vi Find Unique Items Monad and Monad 0 0 eeeeeesecseeeeeeeeceseceneeneeeneeeees 127 Apply On Subsets Dyad u esssessessssesssssersssseseesessesestssessessesseseesessesesssseesessesee 127 Apply On Partitions Monad u 1 and u 2 sesessesesssssesssssesseseesesseseesssseesessesee 129 Apply On Specified Partitions Dyad u 1 and u 2 nesesessseesessesersessresessesee 130 Find Sequence Of Items Dyad Bis 9c osanecasngoenecada ans cocnapeoexeues eaegoaautadan amen 131 Apply On Subarray yada sO cies ac si sesGce passives tant say aeesaiis ov cutee seats 132 Apply On All Subarrays Dyad u 3 and U7 3 veces cssesetscnense reese 133 24 When Pr grams Are Ate acs 053 esa sisssseceuitsacthaa ideas sania cach eeeagse h aman aiG 135 Calling a Published Namie srir ein iiei e a a seed A 135 Using the Argument To a Modifier ssnssesessseesseeseesseeseeseosseesresresseessesersseesseses 135 Invoking Gerund m Grenn oen eR aa EE A RE O a R 136 Passing the Definition Of a Verb 128 2 Apply ssssssssessssesssssssssessessrsssesse 137 Passing an Executable Sentence Monad and 5 5 ssssssssesssssssesessseessessrssee 138 235 OO pless Cod VI rarae aeai ee ane A A EAE AAE EE eee EEA Ai ia RESA 139 26 Modifying an array M sesssssesssesssssesssesseesrrsseessesrrsstesesrsstessesresstessesstsstessessessresseest 143 Modificato In Plac s ia ve e e aa a a E a ate 144 Dep Control SUC UGS a ennan eS ae a a
238. s The sequence elseif T block do block may be repeated as often as desired but you will be surprised to learn that once you code an elseif you are not allowed to use else in the same control structure use elseif 1 instead My antipathy for for and while has scarcely been concealed but I harbor no ill will toward if As long as you don t apply it in a loop if makes the structure of code obvious and you may use it without remorse Examples are legion the form if name sentence to create an array code to process name which is now known to have items end is the most common in my code used to guarantee that a block of code is executed only when a certain noun has a nonzero number of items When we first learned about verb definitions we said that the result of a verb was the result of the last sentence executed Now we must emend that statement sentences in T blocks do not affect the result of the verb try catch end and catcht throw The simplest form see the Dictionary for others is try block1 catch block2 end try catch end is the control structure equivalent ofu v block2 is executed only if there was an error during the execution of block1 If you want to signal an error execute the foreign 13 8 y where y is the error number you want to signal You can use this ina try block to transfer execution to the corresponding catch block throw when executed ina try block or in a function execute
239. s an array of cells the shape of that array is called the frame of the noun relative to the chosen rank of cell It follows that the frame concatenated with the shape of the cells will be equal to the shape of the noun The frame itself like all shapes is an array of rank 1 The diagram illustrates cells of different ranks Note that the twenty 6 atom 1 cells are arranged in a 4x5 array this is the meaning of the frame of the 1 cells The four 5x6 2 cells are arranged as a vector of 2 cells this is the meaning of their frame A selected cell is analogous to the subarray selected by indexing in C Using q as defined above in C q 3 is a 5x6 array i e a 2 cell q 1 0 is a 6 element vector i e a 1 cell q 2 0 3 is a scalar 0 cell The noun q we have been using as an example can be thought of in any of the following 4 ways 18 Frame as an array of Length Value Rank Choosing Axis Order Because J verbs operate on cells of nouns you should choose an order of axes that makes the cells meaningful for your application Referring to the figure we can see that the groups of items that fall into horizontal strips 1 cells or horizontal slabs 2 cells will be easy to operate on individually Vertical strips or slabs will not correspond to cells and so will not be accessible individually to work on them we may have to reorder the axes of the noun to make them correspond to cells Such reordering of axes is e
240. s set and the result is return_code option_value Examples sdsetsockopt sk SOL SOCKET jsocket_ SO DEBUG jsocket_ 1 sdsetsockopt sk SOL SOCKET jsocket_ SO LINGER _jsocket_ 1 66 sdioctl socket option value reads or sets control information result is return_code value Examples 125 sdioctl sk FIONBIO jsocket_ 0 NB set blocking sdioctl sk FIONBIO jsocket_ 1 NB set non blocking sdioctl sk FIONREAD jsocket_ 0 NB count ready data Housekeeping sdgetsockets The result is return code list of all active socket numbers sdcleanup ie ee p All sockets are closed and the socket system is reinitialized The result is O 126 23 Loopless Code V Partitions The adverb operated on subsets of the operand y that were taken in a regular way Now we will take the next step and operate on irregular subsets of y This will finally give us the chance to do interesting work with character strings Find Unique Items Monad and Monad Monad has infinite rank y is y with duplicate items removed and is called the nub of y The items of y are in the order of their first appearance in y Green grow the lilacs Gren gowthliacs ja 2 011201231312 01 12 01 23 1 3 12 a 01 12 23 1 3 y can have any rank and y gives the unique items Monad has infinite rank and tells you which items monad would pick y is a Boolean list where each element is 1 if the corresponding item of y would have
241. s the same result as y ints where ints is the list of all nonnegative integers in order Examples a 5 0123 4 A list of 5 items the items are ascending integers i 2 3 012 345 A rank 2 result 24 2 6 0 m 12 13 14 15 16 17 18 19 20 21 22 23 A rank 3 result i 0 A list of 0 items i 5 43210 If the argument is negative the absolute value is used for the shape but the items run in reverse order 25 5 Loopless Code I Verbs Have Rank Most J programs contain no loops equivalent to while and for in C J does contain while and for constructs but they carry a performance penalty and are a wise choice only when the body of the loop is a time consuming operation You are just going to have to learn to learn to code without loops I think this is the most intimidating thing about learning J more intimidating even than programs that look like a three year old with a particular fondness for periods and colons was set before the keyboard You have developed a solid understanding of loops and can hardly think of programming without using them But J is a revolutionary language and all that is solid melts into air you will find that most of your loops disappear altogether and the rest are replaced by small gestures to the interpreter indicating your intentions Come let us see how it can be done I promise if you code in J for 6 months you will no longer think in loops and if you stay with it for 2 years you
242. s to the bktmin j 1 reference in the C code We could create earned by replicating the earnings for each employee to form a 1 cell for each bracket the code for this would be bktmin 0 empearnings but it s not necessary to create that array explicitly we just use the implicit looping to cause each cellofempearnings to be replicated during the comparison with bracket _top Making all these substitutions noting that all the operations have rank 1 and summing at the end over the items within each 1 cell we get the final code renderuntocaesar monad define bktmin 0 10000 20000 bktrate 0 05 0 10 0 20 t 1 _ bktmin lt 1 0 empearnings 1 bktmin 1 bktrate 1 0 gt t We used the temporary variable t so our sentences would fit on the page Let s write a program to count the lines and words in a file This is a simple task and in C it might look like 65 int 2 wc f char f pointer to filename FILE fid int ct 2 words lines char c fid fopen f while EOF c fgetc fid if c ct 0 if c LF ct 1 return ct Rather than loop for each character in J we process files by reading the whole file into a list of characters and then operating on the list The monad ReadFile defined in the script j orc ijs has infinite rank it takes a filename y and yields as result the contents of the file as a list of characters Once the file is a lis
243. s to write some yourself It will be a frustrating experience for the first few weeks as you struggle to jigsaw your verbs into pieces that have one or two operands and yet fit together to perform a complex function It s not a necessary skill you can get along acceptably in J writing mostly explicit definitions with an occasional fork thrown in where obvious but it is a useful honorable and satisfying one Learning to write tacit verbs is like learning to walk with a book balanced on your head it will slow you down at first but in the end you ll stand taller for it The book J Phrases which is part of the J release has dozens of interesting examples of tacit programs which you can use as exercises Tacit and Compound Adverbs Adverbs as well as verbs can be defined tacitly Any sequence of adverbs is also an adverb and applies the component adverbs one by one to its left operand For example onprefixes defines an adverb that applies followed by as can be seen by onprefixes 1 2 3 4 1 3 6 10 A conjunction with one operand also defines an adverb A conjunction needs two operands but if you supply one the combination is treated as an adverb that attaches its operand to the empty side of the conjunction For example 2 amp is an adverb and 2 amp is equivalent to 2 amp For another example the J startup scripts define 199 each amp gt so that gt each 1 2 3 is equivalent to gt amp gt 1 2 3
244. self denial rather than self mutilation for it retains the vigor to express any algorithm It is as evanescent as the breeze and as powerful as the hurricane It is a sublime creation We begin our exploration with this simple program The first step toward enlightenment is to realize that something so simple is a program You may object But it can t be a program It has no parameters It has no name How would I invoke it It s a primitive maybe Ora verb Or a typographical error But not a program Let me make the case that it is indeed a program I say that a bit of text deserves the title of program if it produces a systematic result when executed with suitable arguments And I say that the program satisfies this definition Certainly I can supply it with arguments 5 7 2 35 and get the expected result The program does not refer to its operands explicitly but as long as we make the agreement that the arguments to a program appear as nouns to its left and right the program has no doubt about how to find its arguments Of course we have been using this convention for verbs all along I can give this program a name minus 5 7 minus 2 35 minus can be used in place of anywhere When we look at the value of minus we see what it contains 9 3 5 NB Do this once to select simplified display minus minus is equivalent to In J we call it a verb but it has all the essential features of a program In fact minus is t
245. spire no higher Change History 2002 6 18 Add chapters on mathematics in J and section on Symbols minor changes to wording bring text up to J Release 5 01 2002 8 16 Minor additions added section on aliasing added chapter on sockets 2002 9 26 Added sections on fndisplay integrated rank support and ordering of implied loops ill Contents OPS WOE rcpt saad N RR at aceeh Gata sttae N aie Sula ana a a thie il ALCKMOW ECO MICIIES p22 meras eer e a a a E a a a a anr iii CAINS History aoier erea aa E E EE es sats Gea ene E E EE ES iii lo Introductions eene a A E web ae E A Sees acaatashae tase paaes sees 1 Programming In J 2 2 Preliminaries etnies aat ak re aac ele each ema a Mian Gli Gaius oo dae Gl G atta alee 4 3 TNA IOI foe cee ete AEE EA E EEE E 3 PPM OY 25 cist arnt stan aa EE sadudassen EEA AESA EAT Rai 3 Sentences statements isiiieiisioisiisicn iiinis ii eiris iiai 4 Word Formation tokenizing rules ccscesecsseesceescssecesecssceeseseeceaecsesseessesenees 4 IN Ni sae saa ete CE aah a aaah ha cai nla ae ste eden oan E ate Jan urd tee 5 ETETE Gc gee AAEE AE RRR CREA SORA E AE EE EE OI 5 Valence of Verbs Binary and Unary Operators cceccescesseeseeeeeeeeeteeeaes 5 How Names Identifiers Get Assigned ccccccccesseesscetecsseesceseesseceseeneeeseeeseeeeees 6 Order of FV AVANT ae siie ae e a duce aude a ae dcuadei aus lt p SEENE 7 What a verb FUNCTION looks like eee eececsccssecessecesecees
246. t even though it is defined to have the same function as and it suffers when it is applied to each 1 cell This is a good reason for you to learn the J primitives and not replace them with mnemonic equivalents 111 20 Input And Output Foreigns In J all file operations are handled by foreigns which are created by the foreign conjunction The foreigns are a grab bag of functions and you would do well to spend some time glancing over their descriptions in the Dictionary so you ll have an idea what is available All the foreigns defined to date in J are of the form m n with numeric m and n and they are organized with m identifying the class of the foreign For example the foreigns for operations on files have m 1 File Operations 1 n Error Handling The foreigns 1 n perform file operations For ease of use I have given several of them names in jforc ijs To see the details of what they do read the Dictionary Monad 1 1 ReadFile has rank O 1 1 y takes a filename as y and produces a result that is a list of characters containing the text of the file In our examples the filename is a boxed character string y can be a file number but we won t get into that s 1 1 lt system packages misc jforc ijs s NB File definitions for J For C Programmers NB Copyright c 2002 Henry H Rich Dyad 1 2 WriteFile hasrank _ 0 1 2 y writes to the file y using the character string x to provide the contents Any existi
247. t it is trivial to compare each character against space and linefeed yielding for each a list of 1s at each position filled by the character then summing the list gives the number of each character J code to do this is NB y is string filename result is words lines wc monad define 1 LF 0 1 ReadFile y Suppose our user complains that we needs improvement specifically it should also return the number of characters in the file should not count multiple consecutive whitespace characters including TAB as separate words and should treat the trailing LF as a word delimiter as well as a line delimiter In C we would respond by adding a few more tests and flags but in J we realize that a major change to the program s function calls for a thorough rethinking To ignore multiple whitespace characters we need to define what an ignored whitespace is without using flags and loops This part of the problem often calls for creative thought here we realize that a whitespace character is to be ignored if the previous character is whitespace That s easy then we just calculate a Boolean vector 1 if the character is whitespace and then use shift and Boolean AND to ignore multiple whitespace The code to do this is worth looking at it is sw 1 w w fe TAB LF where f is the contents of the file Note that we assign a value to w just before we right shift w This is legal in J sentences are processed r
248. t item of x behind y enclose gt gt enclose xyz xyz enclose abc abc To produce 1 if all the items of y are equal 0 if not allitemsequal 1 amp allitemsequal 11111 allitemsequal 11121 To extend x to the length of y where the items of y supply default values for the corresponding omitted items of x 198 default abc 2 default Defname 0 Deftype 100 abc 2 Deftype 100 abc 2 xyz 30 default Defname 0 Deftype 100 Jabc 2 xyz 30 The verb which we met as a way to cause an error has a special meaning in a fork As the leftmost verb of the fork means ignore the left branch So Nx V1 V2 NyisV1 Nx V2 Nyand V1 V2 NyisV1 V2 Ny In both cases V1 V2 is equivalent toV1 V2 Almost always the choice between one form or the other is a matter of taste Do not fear that the extra word in the fork leads to slower execution the is not executed tt is recognized by the parser when it creates the anonymous verb for the fork As a final example here is a definition of the word counting example we wrote earlier See if you can convince yourself that it is equivalent to we2 WS TAB LF wc3 0 e amp WS LF amp ReadFile I could go on with pages more of hooks and forks for you to study but what you really need i
249. t i for a dyadic verb v is that value of i such that i v y is y for any choice of y For example the identity element for is 0 because 0 yisalwaysy The identity element for is 1 and for lt is _ Ifthere is no identity element for a verb v for example has no identity element you will get a domain error if you apply v to an empty list Examples 0 0 0 0 0 1 Empty list result is the identity element 4 13 635 7 357 There is 1 1 cell so the result is that cell This result has shape 3 not1 3 036 0 000 There are 0 1 cells so the result is a cell of identity elements Note that even though there are no cells the cell still has a shape which is made visible by 0 0 domain error 0 0 If you don t want to figure out what an identity element for a verb v is you can ask the interpreter by typing v 0 0 Before we move on you should note that since v 1 2 3 is equivalent to 1 v 2 v 3 2 v 3 is evaluated first the operation starts at the end of the list and moves toward the beginning The adverb is an adverb Like all adverbs it has a monadic and dyadic form The dyadic form x u y is equivalent to y u x in other words the operands of u are reversed The 60 ranks of dyad u are the same as those of dyad u but with left and right rank interchanged For advanced J dyad is indispensable even in ordinary use it can save time and obviate the need for parentheses 10 2 3
250. t invocation of knave must carry all the information needed by the next invocation of knave this is the way information is passed from knave to knave The main design decision is to figure out what the format of the y operand of knave will be Obviously we need to know the contents of the plate as it is given to each knave Also the purpose of knave is to calculate what coins are removed so those coins should be part of the result We decide that the y operand of knave will consist of those two things in the order coins removed plate contents and we already know that the x operand will have the format of an item of k i e the knave s two coins Now we are ready to code knave 139 It should look at the plate contents portion of its right argument sort it into order of size take the two biggest values as the result of interest and use the rest with the knave s own coins added as the plate contents to be passed to the next knave The coins that were removed are put into their place at the beginning of the result vector In J this will be knave dyad define xlen x NB number of coins added removed splate xlen y NB extract plate contents sort xlen splate xlen splate x NB build result Let s test it The y operand to the first invocation of knave will have a couple of placeholders in the place where the coins removed by the previous knave would be followed by the initial contents of the pl
251. t it produces This approach carried almost to an extreme would yield semiperimeter factors semiperimeter 0 product factors triarea product Combining the two approaches you can find a comfortable level of complexity Flatten a Verb Adverb f Splitting the definition into many lines has the unfortunate side effect that all the names referred to by triarea must be defined when triarea is executed triarea product triarea refers to product which refers to factors which refers to semiperimeter Ifyou define many tacit verbs this way the result is pollution of the namespace To leave a smaller footprint use private assignment for all the names except the name that will be public and use the adverb which replaces names in its operand with their definitions semiperimeter factors semiperimeter 0 product factors triarea triarea 3 0 If these verbs are run from a script the temporary verbs will disappear since they were assigned by private assignment leaving only the main verb triarea product f f is also used in initialization of objects as you can learn in the Lab for Object Oriented Programming 201 Note that has no effect on explicit definitions Using to improve performance Flattening a verb has two beneficial effects on performance The first is easy to see by comparing the difference bet
252. tch from the starting point to the limit of the array Dyad u 0 is a great way to pick out a subarray to work on Always consider using it whenever you find yourself using more than one application of and to select a portion of an array Even if you are working with a list using dyad u 0 is a good way to work on a portion of the list without copying any part you aren t working on Apply On All Subarrays Dyad u 3 andu 3 x u _3 y applies u to subarrays of y as specified by x The operation is similar to dyad u infix but dyad u slides a one dimensional window across y to select sequences of items of y while dyad u _3 moves a multidimensional window throughout y to select multidimensional regions When the window has only one dimension the operation is like dyad u but with a little extra control over the positions of the window The second item of x actually 1 x gives the size of the window if an item is negative the corresponding axis is reversed before u is applied to the window an infinite value means take all the way to the end of the array The first item of x 0 x is the movement vector the window is positioned at every point at which each item of the window position is an integral multiple of the corresponding item of the movement vector as long as the window fits inside y If an item of the movement vector is smaller than the corresponding item of the size the windows will overlap along that axis
253. th list is a single blank line proper as there is one list and it has no items The display of a rank 2 array with no items is different here we have zero lists so we should expect no lines at all This is indeed what happens 0 o0o 0 there is no blank line This is the result you should produce if you want a function to display nothing Here are two exercises Once you can explain each result you will be well on your way to becoming a J programmer What will each of these sentences produce answer on the next page 31 25651 10 2 3 25 1 10 15 23 Solutions 318525651 10 110 1 10 21 10 1 10 1 10 110 1 10 1 23 25 1 10 15 1 10 15 1 10 15 1 10 15 1 1 10 15 1 10 15 1 10 15 1 1 10 15 1 10 15 1 10 15 1 Monad Tally The result of y is a scalar the number of items in y This is the first item in the list y except that if y is an atom y is empty but y is 1 because remember an atom has one item itself If y is a list y is the length of the list Quiz question What is the difference between the results of 1 2 3 and 1 2 3 123 3 123 3 Answer the result of monad is always a list but the result of monad is a scalar 12 3 1 12 3 y like y gives the rank of y Since monad produces a scalar rather than a list y is usually preferred Just remember that the length of the shape is the rank Monad i Integers Monad i has infinite rank and creates an array i y produce
254. than a noun or verb Now you see why Undefined Words If you try to execute a nonexistent verb you get an error z 5 value error z z5 However that error occurs during execution of the name not during its parsing During parsing an undefined name is assumed to be a verb of infinite rank This allows you to write verbs that refer to each other and relieves you from having to be scrupulous about the order in which you define verbs For example a z This produces a verb a which when executed will execute z z a 5 5 With z defined a executes correctly Of course it s OK to assign z to another verb too b z b 5 Now can you tell me what b 1 2 3 will do Take a minute to figure it out Hint note that I used rather than b 1 2 3 12 3 Because b has rank 0 b also has rank zero so the summing is applied to atoms individually and we get a list result Do you think a 1 2 3 will have the same result a 1 2 3 6 Even though a has the same value as b its rank is different a s rank was assigned when it was parsed and at that time z was assumed to have infinite rank b s rank was assigned when it was parsed too but by that time z had been defined with rank 0 You can win a lot of bar bets with this one if you hang out with the right crowd 195 37 Forks Hooks and Compound Adverbs Now that you understand execution and in particular how anonymous entities are created and then executed you
255. that trouble you can consult the Lab to get all the details Format Data For Printing Monad And Dyad Since J reads and writes all files as character strings you will need to be able to convert numbers to their character representation Dyad has rank1 _ y maybe numeric or boxed of any rank we will not consider boxed y here If x is a scalar it is replicated to the length ofa l cellofy If y is numeric each 1 cell of y produces a character list result with each atom in the 1 cell of y being converted to a character string as described by the corresponding atom of x and with the results of adjacent 0 cells being run together the results from all 1 cells are collected into an array using the frame with respect to 1 cells That sounds like the description of a verb with rank 1 almost so but not quite because looks at the entire y and adds extra spaces as necessary to make all the columns line up neatly A field descriptor in x needs two numbers wand d These are represented as the real and imaginary parts of a complex number so that only a single scalar is needed to hold the field descriptor The real part of the field descriptor is w and it gives the width of the field in characters If the result will not fit into the field the entire field is filled with asterisks to prevent this you may use a w of 0 which will cause the interpreter to make the field as wide as necessary The imaginary part of the field descriptor is d givin
256. the Boolean function 9b 01 10 01 u 0 1 is the function table with x values running down the left and y values running along the top 9 is 1001 binary in J 2b1001 and the function table of 9 b is 1 0 O 1 if you enfile it into a vector Similarly 14 b 01 1110 You can usem b in place of combinations of Boolean verbs Unfortunately comparison verbs like gt and lt have better performance than m b so you may have to pay a performance penalty if you write for example 2 b instead of gt even though they give the same results on Booleans gt 01 00 1 0 J verb equivalents for the cases ofm b are 0 0 0 1 2 gt 3 0 4 lt 5 0 6 7 8 9 10 0 11 gt 12 Q 0 13 lt 14 151 0 174 Bitwise Boolean Operations on Integers When m is in the range 16 31 dyadm b specifies a bitwise Boolean operation in which the operation m 16 b is applied to corresponding bits of x and y Since 6 b is exclusive OR 22 b is bitwise exclusive OR 5 22 b 7 2 The XOR operation is performed bit by bit Dyad 32 b is bitwise left rotate bits shifted off the end of the word are shifted into vacated positions at the other end Dyad 33 b is bitwise unsigned left shift x is the number of bits to shift y positive x shifts left negative x shifts right in both cases zeros are shifted into vacated bit positions 2 33 b 5 20 Dyad 34 b is bitwise signed left shift it differs from the unsigned
257. the list of clients clientlist the list that tells which client each employee worked for emp client and the profit per employee given by empprofit 0 For each client we need to find the employees that worked for the client and add up the profit they brought in For this kind of problem we want an array with employees for one axis and clients for the other with a 1 in the positions where the employee is assigned then we cando array empprofit 0 and do some suitable summing of the result Let s work out what array must look like Since dyad has rank 0 array empprofit 0 is going to replicate 0 cells of the shorter frame argument empprofit 0 which means that a single profit value is going to multiply an entire 1 cell of array So we see that the leading axis of array must be clients and the second axis must be employees each item will have the client mask for one employee The way to create that is by clientlist 1 0 emp client which will compare the entire client list against each 0 cell of emp_client and form the results into an array Then clientlist 1 0 emp_client empprofit 0 will have one item per employee each item having the amount billed to each client we sum these items to produce a list with the profit from each client and use that to order the client numbers Solution custbyprofit monad define clientlist clientlist 1 0 emp client empprofit 0 For our final problem in the payroll department consider how to
258. ting to string form adds overhead that might steer you towards using a gerund or 128 2 instead I think C programmers are likely to overlook opportunities to use monad because it is so foreign to their experience It is equivalent to compiling and executing a code segment as part of the running program t s just unthinkable But in J it s commonplace If you are going to use monad you will face the problem of converting your nouns to string form Here s the display of a noun what character string would you execute to produce a noun with that value a 0 2 25 4 5 6 75 9 11 25 13 5 15 75 abc de fgh Fortunately you don t have to worry about it The foreign monad 5 5 takes as ya boxed name not a value and produces a string which when executed has the same value as the variable named by y So 5 5 lt a lt lt 1 abc de fgh lt 2 25 i 8 and if you tell me you came up with the same string I m not going to believe you 138 25 Loopless Code VI The algorithms that are hardest to put into loopless form are the ones that chug through the items of an array modifying a set of temporary variables that control the operation at each step I am going to show you how to put one example of such an algorithm into loopless form The resulting structure can be used for many such p
259. tning after doing so the coroner will find it was suicide Using locale names as data allows for dynamic separation of namespaces For example the processing of forms in J requires definition of verbs for a great many events You may let these verbs all share the same locale but if you want to segregate them the Window Driver will remember what locale was running when each form was displayed and direct events for a form to the locale handling the form 155 29 Writing Your Own Modifiers If you find that you are coding recurring patterns of operations you can write a modifier that represents the pattern You will find that reading your code is easier when the patterns are exhibited with names of your choosing You write a modifier like you write a verb using conjunction define or adverb define or2 norl1 n for one liners When you assign the modifier to a name that name becomes a conjunction or adverb and it will be invoked as u name v y monad orx u name v y dyad if it is a conjunction oru name y monad or x u name y dyad if it is an adverb When a modifier is invoked the lines of the modifier are executed one by one just as when a verb is invoked and the result of the last sentence executed becomes the result of the modifier The u and v for conjunctions operand s of the modifier are assigned to the local names u and v when the modifier starts execution in addition if u is a noun it is assigned to the local name m
260. ty 4 Antilog verb 83 Apply At Level conjunction L 175 Apply On Subsets adverb 127 Apply Under Transformation conjunction amp 103 Apply Under Transformation conjunction amp 104 Apply verb 128 2 137 array 17 assignment 6 multiple 92 private 149 public 150 atom 18 atomic representation 87 axis 19 order 19 B Behead verb 56 bitwise Boolean functions 175 bitwise left rotate 175 bitwise signed left shift 175 bitwise unsigned left shift 175 Boolean Functions verb m b 174 Box verb lt 72 boxing 72 C belittled ii derided 44 J equivalent for class 149 structure equivalent in J 75 C dyad 171 C monad 170 C 2 monad 171 calculus 165 polynomials 166 calling a DLL 118 cd verb 118 cell 18 character constants 5 20 comments in code 102 comparison tolerance 177 complex numbers 163 compound adverbs 199 compound verb 68 Constant verb m n 57 control structures 146 assert 148 catcht throw 147 for do end 146 if do else elseif end 146 return 148 select case end 147 try catch end 147 while whilst do end 146 copula 6 Copy verb 50 Curtail verb 56 Cycle verb C 170 D Deal verb 167 defnouns 34 defverbs 34 derivative 165 approximate 166 partial 165 derived entity 4 derived verb 68 Do verb 138 DoWhile conjunction u v _ 87 Drop verb 51
261. u y lt abc t 0 0 18 5 Jabe Sure enough the current locale was changed But see what happens when we give a name to the verb created by the execution of t cocurrent lt base tt lt abc t tt 0 0 18 5 base The current locale was restored What causes the difference The answer is that in the sentence lt abc t 0 the named adverb t is executed when it is given its operand lt abc The result of that execution is the derived verb lt abc t which has no name When the derived verb is executed with the operand O the text is interpreted causing a change to the current locale and when the derived verb finishes the current locale is not restored because the derived verb is anonymous If we give that derived verb a name tt here it restores the current locale on completion The observed behavior reinforces the point that the text of a modifier that refers to x or y is not interpreted when the modifier is executed it is interpreted only when the derived verb is executed 162 30 Applied Mathematics in J Complex Numbers All the mathematical functions can be applied to complex numbers A complex constant is written with the letter j separating the real and imaginary parts e g 0j1 is the square root of _1 Alternatively a constant can be written in polar form in which the letters ar or ad separate the magnitude of the number from the angle in r
262. u amp n y has infinite rank and is equivalent toy u n Examples 2 amp 012 3 124 8 amp 2 45 6 23 4 Now we can solve our original problem which was to put1 04 x y into the formx v y v should be 1 04 amp which we can verify using fndisplay as defverbs plus 0 times 0 defnouns x y x 1 04 amp times plus y and to get my total savings after receiving all payments I use monad to apply that verb for each payment giving the expression 1 046 x Y Let s take a moment to understand the parsing and execution of 1 04 amp The left operand of a modifier includes all preceding words up to and including the nearest noun or verb that is not immediately preceded by a conjunction This is a precise way of saying that modifiers associate left to right In the phrase 1 04 amp the 1 04 is not preceded by a conjunction so it is the beginning of 70 all the conjunctions left operands and the verb is parsed as if it were written 1 04 amp Note that left to right association of modifiers corresponds to right to left execution When the derived verb monad 1 04 amp is executed it performs according to the definition of monad u with 1 04 amp as the u execution of monad u inserts u between items so 1 04 amp is executed between items at each such execution the dyad is executed first followed by the monad 1 04 amp Fortunately the
263. u were expecting the 3 to be repeated remember that you can look at the words as J sees them z 2L 2 3 0 i 3 1 2 3 O i 13 1 2 3 isasingle word Le OS hea oo For a few special values namely the integers _9 through 9 infinity _ and negative infinity __ you can create an infinite rank verb to produce the value by following the constant value with This is equivalent to value _ 3 abc 3 The operand abc was ignored and the result was 3 8 Loopless Code II Adverbs and The monad which sums the items of its operand is a special case of the use of the adverb It is time to learn about adverbs and other uses of this one Modifiers An adverb is a modifier It appears to the right a noun or verb the prototype isu a where u is the noun or verb and a is the adverb The compound u a is a new entity and not necessarily the same part of speech as u When the compound u a is executed it performs the function given by the definition of a and has access to u during its execution Ifu ais a verb then it also has access to the operands of the verb during its execution the verb u a will then be invoked asu a yifmonadicorx u a yif dyadic You will note that I didn t have to writex u a y While J gives all verbs equal precedence and executes them right to left it does give modifiers adverbs and conjunctions higher precedence than verbs in the same way that C and stand
264. ur code If you run the J orC script with load system packages misc jforc ijs it will define the verb Ts Ts stands for time and space and it tells you how long it takes to run a given J sentence and how much space the interpreter used during the execution For example a3 i 1000 Ts a3 4 358le 5 5248 We define a noun a3 and we calculate the running total of its items It took 0 00004 seconds to create the 1000 item running total and used 5248 bytes We could have done the whole operation in one line with Ts i 1000 but the monad i uses time and space too so if we want to find out only what is used by we make sure that s all we measure We can use Ts to start to understand what can make J programs slow Let s define a verb to do the addition operation sum dyad x y 0 sum is an exact replacement for dyad having the same rank and function Replacing with sum does not change the result of a sentence i 7 013 6 10 15 21 sum i 7 013 6 10 15 21 But the performance is quite different as we can measure a1l0 i 10 1000 Ts al0 2 68191e 5 1280 Because is so fast we give Ts a left argument to report the average time over 1000 runs If we just ran the sentence once the result would be corrupted by small timing variations introduced by the operating system sum is not so fast so we run it only once Ts sum al0 0 00181867 3648 Quite a difference in this running tot
265. ursion by referring to factorial by name Using we can recur without a name lt 1 amp lt 4 24 stands for the whole verb containing the namely lt 1 amp lt Make a Table Adverb dyad u x u yisx u lu _ y where luis the left rank ofu Thus each cell of x is applied to the entire y The definition is simplicity itself and yet many J programmers stumble learning it I think the problem comes from learning dyad u by the example of a multiplication table The key is to note that each cell of x is applied to the entire y cell not item or atom The rank of a cell depends on the left rank ofu The multiplication table comes from a verb with rank 0 123 12 3 WN EF OAN ov W You can control the result by specifying the rank ofu 173 i 2 2 1 8 9 0189 2389 i 22 0 89 089 189 289 389 These results follow directly from the definition of dyad u fndisplay shows the details defverbs comma i 2 2 comma 1 8 9 I O 1 comma 8 9 2 3 comma 8 9 i 2 2 comma 0 89 I0 comma 8 9 1 comma 8 9 2 comma 8 9 3 comma 8 9 Boolean Functions Dyad m b Functions on Boolean operands I will just illustrate Boolean dyadm b by example m b isa verb with rank 0 m when in the range 0 15 selects
266. utes u on r cells 43 6 Starting To Write In J It s time to write some simple J programs Writing code without loops will be a shock at first Many accomplished C programmers give up because they find that writing every program is a struggle and they remember how easy it was in C I hope you will have more persistence If you do you will soon stop thinking in loops translating each one into J instead you will think directly about operand cells and the code will flow effortlessly In C when you are going to operate on some array say x 3 4 5 you write the code from the outside in You know you are going to need 3 nested loops to touch all the cells you write the control structure for each loop possibly after thinking a bit about the order of nesting finally you fill in code at whatever nesting level it fits Even if all your work is in the innermost loop you have to write all the enclosing layers just to be able to index the array When I was writing in C I made sure my output was measured in lines of code so that I could call this productivity In J you grab the heart of the watermelon rather than munching your way in starting at the rind You decide what rank of operand cell you are going to work on and you write the verb to operate on a cell You give the verb the rank of the cells it operates on and then you don t care about the shape of the operand because J s implicit looping will apply the verb to all the cells no
267. valuation implies that the result of the previous suffix can always be used as the right operand to each application of monad sum without needing any knowledge of associativity Let me tell you this interpreter is nothing if not clever and that s just what it does All we have to do is to convert our sum into a variant of sum The way to do that is simple we reverse the order of the items apply sum and reverse the order again sum amp i 7 013 6 10 15 21 This arises enough to be a standard J idiom use it whenever you need to apply an associative verb on prefixes It s much faster Ts sum amp a400 0 014805 59264 Still not as fast as but the suffix version uses time proportional to the number of items rather than the square of the number of items Compounds Recognized by the Interpreter The interpreter recognizes a great many compounds and has special code to perform the compound functions For example we have learned that u v_y gives the same result as u v y but it does not follow that the two forms are identical y is faster than y How do know what forms are handled with special code An appendix to the Dictionary gives a list of special code in the interpreter press F1 to bring up help then click on Dic at the top of the page to bring up the Contents page the appendices are listed at the end of the contents There we see that there is special code for so we know to use that
268. version to be performed 2 3 4 y converts the numeric list y to a character string representing each item of y by 4 characters whose values come from the binary values of the low order 32 bits of y 16631424334 826426164 This is an integer whose value is 0x31424344 We can convert it to a character string 2 3 4 826426164 4CB1 The 4 characters with values 0x34 0x43 0x42 0x31 correspond to the bits of the number 826426164 in little endian order We can usea i to look at the codes for each character in case they are not printable a i 2 3 4 1000 100000 232 3 0 0 160 134 1 0 232 3 0 0 corresponds to 1000 and 160 134 1 0Oto100000 _2 3 4 yistheinverseof2 3 4 y converting a character string to integers 2 3 4 4CB15CB1 826426164 826426165 The other integer conversions are similar 1 3 4 y converts the low order 16 bits of each item of y to 2 characters and _1 3 4 y converts back to integers 2 characters per integer O 3 4 yislike 1 3 4 y but the integers are unsigned i e in the range 0 65535 The floating point conversions are analogous 2 3 5 y converts each item of y to 8 characters representing long floating point form and _2 3 5 y converts back 1 3 5 yand_1 3 5 y use 4 character short floating point form 116 printf sprintf and qprintf When you need formatted lines for printing you may feel at home using printf and sprintf which work like their C counterparts pri
269. ween the two sequences a i 100000 absO 3 y O 6 2 absO a 3 17136 abs1 3 y 6 2 abs1 0 a 3 54908 To be sure in each case we have committed the crime of applying an explicitly defined verb at a low rank 0 a executes in time 0 006 but that is not the point Why is abs1 0 slower than abs0 Each one reinterprets its verb for each atom ofa The answer is that when abs1 0 is executed the definition of abs1 must be looked up for every atom of a for all the interpreter knows abs1 might be redefined during its execution The time spent doing this lookup accounts for the difference in time between abs0 and abs1 0 If we eliminate the lookup of the name abs1 that time is saved 6 2 absl 0 a 3 00941 The important lesson to learn from this is that you should define your verbs with the proper rank That will eliminate superfluous name lookups In exceptional cases you may use to avoid name lookups during execution of a complex verb The second case where can improve performance is useful only for those users who feel compelled to redefine the J primitives with mnemonic names This is a practice that I strongly deprecate and if you don t heed my advice the interpreter stands ready to punish you See the disaster that can result when the primitives are replaced by mnemonic names tally a 100000 i 6 b i 100000 10 6 2 a b 0 0157688 6 2 a tally b 0 154675 What happ
270. will see that looping code was an artifact of early programming languages ready to be displayed in museums along with vacuum tubes delay lines and punched cards Remember in the 1960s programmers laughed at the idea of programming without gotos You are not used to classifying loops according to their function but I am going to do so as a way of introducting J s primitives We will treat the subject of loopless iteration in 6 scattered chapters showing how to replace different variants of loops 1 Loops where each iteration of the loop performs the same operation on different data 2 Loops that apply an operation between all the items of the array for example finding the largest item 3 Loops where the operation to be performed on each cell is different 4 Loops that are applied to regularly defined subsets of the data 5 Loops that are applied to subsets of the data defined irregularly 6 Loops that accumulate information between iterations of the loop The simplest case is the most important and we start with a few experiments Examples of Implicit Loops 2 345 567 The verb dyad is addition and we have our first example of an implicit loop the left argument 2 was added to each atom in the right argument 26 123 456 579 And look If each operand is a list the respective items are added We wonder if the behavior of 2 3 4 5 was because items of the shorter operand are repeated cyclically 12 456 Jlength error
271. within each 1 cell resulting in a list of hours for each selected employee The line 1 emp_client y hoursworked performs the functions of loops 1 and 2 in the pseudocode loop within each cell and loop 2 in the implied loop over the cells Then it is a simple matter to select the billing rates for the employees multiply each billing rate by the number of hours worked and take the total over all employees billed to the customer The solution using a temporary variable to hold the mask is billclient monad define mask emp client y mask billrate 1 mask hoursworked Problem 5 For each day find the worker who billed the most hours C code int dailydrudge drudges int drudges 31 result empno of worker with most hours each day int i j highhours for i 0 i lt 31 i highhours 1 for j 0 j lt nemp j if hoursworked j i J gt highhours drudges i empnojj highhours hoursworked j i We note that the inner loop which records the employee number of any employee who worked more than the previous high does not correspond to any of our J verbs So we break this loop into operations that do correspond to J verbs for each day find the maximum number of hours worked 1 for each day find the index of the employee who worked that much 2 for each day translate the index to an employee number 3 Loop 1 is simply gt hoursworked Loop 2 calls for dyad i to perfor
272. without the application of the obverse It is just a way of saving a few keystrokes x u amp v yislikex u amp v_ y except that its ranks are both the same as the rank of monad v As with and you are advised to stick to u amp v unless you are sure you need u v The monadic forms u v y and u amp v_ y are equivalent tou v yandu v y respectively Avoid them 105 An observation about dyadic verbs We noted earlier that usually ifa dyad x v y is not symmetric i e if x and y are treated differently the x operand is the one which is more like control information and the y operand is the one more like data We can see now that this is a consequence of the definition of x u v y _ the verb that is applied repeatedly or the verb whose obverse is taken is x amp u only the value of y changes between executions ofu When you define dyadic verbs you should take care to follow the same rule in assigning the left and right operands 106 19 Performance Measurement amp Tips J lets you express your ideas tersely but it is up to you to make sure they are good ideas Since each keystroke in a J sentence can summon up billions of machine cycles you must make sure that you don t force the interpreter into a bad choice of algorithm This will be hard at first when you are struggling to figure out what the interpreter is doing never mind how it is doing it fortunately the interpreter gives you tools to measure the performance of yo
273. wo programs because it can be executed monadically as well 183 minus 6 8 6 8 The point is that every J verb even the primitives and the compound verbs is a program in the usual sense Verbs like minus that do not mention their operands by name but instead apply them according to J s parsing rules are called tacit verbs Verbs created bym n likedyad y that mention their operands by name are called explicit verbs The compound verbs we have learned already are examples of tacit verbs Some of the verbs that we have had occasion to define so far can be written in tacit form addrow monad y 1 could be rewritten as addrow 1 and v dyad 1 04 x y is equivalent to v 1 04 amp as we have seen We have already encountered tacit definitions without noticing dotprod Q 1 1 2 3 dotprod 1 2 3 14 and we can certainly define more sortup defines a verb that sorts its operand into ascending order sortup 31 41 59 26 53 26 31 41 53 59 Some of the verbs we have encountered seem too complex for a compound verb For example in mean monad y y we need to perform two operations on y and combine their results something that is beyond the capabilities of the compound verbs we have encountered so far We will next learn how to produce a tacit equivalent for mean and a great many other verbs 184 34 First Look At Forks Before we learn the rules for making tacit f
274. x To get a 1 character list use monad Vy 1 The official name for monad is the quaint but unedifying ravel meaning separate or undo the texture of I prefer the equally quaint but more descriptive enfile which means arrange in a line as if on a string Reverse and Transpose y Reverse The items of y are put into reverse order J 2 5 43210 y Transpose The axes of y are reversed This is difficult to visualize for high ranks but easy for the most common case rank 2 55 Take and Drop Single Item w m y Head y Tail y Behead y Curtail The operations performed are simple the biggest problem is remembering which primitive does what Remember that means take and means drop and that means beginning and means end So y is the first item of y y is the last item of y y is all of y except the first item y is all of y except the last item 3 45 3 Sh 34 8 9 10 11 3 45 45 y is identical to 1 y and y is identical to _1 y y is not identical to 1 y because 1 y has the same rank as y while y has the rank of an item of y Grade Create Ordering Permutation y Grade Up y Grade Down y creates a numeric list with y items such that i y is the index of the ith largest item ofy y y gives y sorted into ascending order y isa permutation vector i e it contains e
275. x because of the analogy with If x and y are sets x y is the set difference Indexing i e x i y Index Of The left rank is infinite x i y looks through the items of x to find one that matches y the result is the item number of the first match Examples 3141591 5 4 i 43 i 678 2 It may occur to you that for y to match an item of x the rank of y must be the same as the rank of an item of x call that rix which is one less than the rank of x namely x unless x is an atom formally rixis 0 gt x 1 or more cleverly x If y is of higher rank each rix cell of y is matched against items of x Formally x i yisequivalenttox i _ rix y Example 3141594 15 c 14 Gin 33 3 in i23 01 If an rix cell of y matches none of the items of x the result value for that cell is x i e one more than the largest valid item number of x 50 x 3141591 84 1 6 2 6 To be hair splittingly accurate we must say thatx i y is equivalent to x i _ rix _ y because the rank of dyad i is infinite This distinction will matter eventually e y Element Of The left rank is infinite x e y isa lightweight version ofy i x the result is 1 if x matches an item of y 0 if not The verb is applied to riy cells of x where riy is the rank of an item of y Formally x e y is the same as y y i x Dyade isan exception to the rule that x is control information and y is data It was defined to be re
276. y as a single item list I suggest you not read on until you understand why Resuming our inquiries into dyad we have 25 1 10 110 1 10 1 10 1 10 1 10 Again y is a list so the items of y are scalars The shape of the result is 2 5 and the items of y are repeated to fill that shape Note that the corresponding atoms in each cell are aligned in the display 15 1 10 1 10 1 10 1 Similarly but now the result has shape 1 5 This is not the same as a 5 item list which has shape 5 Again monad shows the shape 15 1 10 i5 When y is not a scalar or a list its items are not scalars 22 3 15 1 10 1 10 1 10 1 1 10 1 10 1 1 10 1 10 1 Remember this is processedas 3 1 5 1 10 The parenthesized part produces an array of shape 1 5 since this has rank 2 its items are its 1 cells each with shape 5 The shape of the overall result is x concatenated with the shape of an item of y to wit 3 5 This is populated with the cells of y of which there is only 1 3 2 5 There is a tide in the affairs of men There isa There You should be able to explain where each line came from 22 25 1 10 110 1 10 1 10 1 10 1 10 110 1 10 1 10 110 1 10 22 256 51 10 225 Now the shape of the result is 2 2 5 arank 3 array J displays the 2 cells with a blank line in between Similarly a rank 4 array is displayed as all the 3 cells with 2 blank lines in between and so on for higher ranks We have seen that the display of a zero leng
277. y gives a picture of what is happening 85 defverbs incr 0 incr 3 5 x u n y also has infinite rank It evaluatesx u x u nm times y A simpler way to say this is to say that it is equivalent to x amp u n y since x amp u yis equivalenttox uy 2 2 2 2 5 80 2 4 5 80 u v yandx u v y are defined similarly first v is evaluated monadically or dyadically as appropriate and then result is used as n Formally u v y is u v y yandx u v yisx u x v y y With dyad u v it will be rare that x and y both make sense as an operand into both u and v and you will usually use and to cause u or v to operate on only one operand For example to coalesce the x 1 leading axes of y into one axis you could use x y 1 i 223 0 1 2 3 4 5 6 7 8 9 10 11 2 i 223 012345678 91011 This is hardly a commonplace usage but let s analyze it since conjunctions are still new to us The verb is parenthesized so the first thing executed is u v where vis x y isjust x so x is going to tell us how many times to apply x amp Now x amp y isjust the sameas y because the purpose of the is to ignore the left argument that was put on by x amp We remember monad from our discussion of monad u it combines the 2 leading axes of y So x y will combine the 2 leading axes of y x times in other words combine the x 1 leading axes ofy
278. y x Both the left and right ranks ofu D n are the monadic rank of u so you can specify different step sizes for different atoms of y if x is a scalar it is used for the step size at all atoms of y You can do calculus on polynomials by manipulating the polynomial forms without having to create the verbs that operate on those forms p y rank 1 takes polynomial y in either coefficient or multiplier root form and produces the coefficient form of the derivative ofy x p y rank0O 1 produces the coefficient form of the integral of y with constant term x Taylor Series t t and T With derivatives available Taylor series are a natural next step u t y isthe y Taylor coefficient of u expanded about 0 and x u t y evaluates that term at the point x in other words x u t yisx y u t y Allranksofu t are 0 u T nisa verb which is the n term Taylor approximation to u expanded about 0 u t yis y ut y Hypergeometric Function with H The generalized hypergeometric function is specified by two lists of numbers a numerator list and a denominator list The generalized hypergeometric function is the sum over all k of the infinite generalized hypergeometric series which is a power series in which the coefficient of the y term is the product of the rising factorials of length k of the numerator items divided by a similar product for the denominator items and then divided by k 166 The conjunction H is use
279. you d want it to and you should experiment if you need reassurance If Fill Cells Are Not Enough Sometimes executing a verb on a cell of fills simply won t do maybe your verb produces a side effect or maybe it will go berserk if its operand is O In those cases you must take steps to ensure that it is not executed on an empty list To help you out with the most common case in which the only way a list can be empty is to have no items that is another way of saying that the first item of the shape is 0 I offer you a set of adverbs and conjunctions which you can have by executing load system packages misc jforc ijs u Ifany applies the verb u which may be monadic or dyadic provided y has items if y has 0 items the resultofu Ifanyisy i 0 4 0 Ifany i 0 4 0 4 Sincei 0 4 has no items Ifany caused it to be passed through unchanged x u Ifanyx y producesx u y if x has items or y if x has no items The conjunction u Butifnull n can be monadic or dyadic it applies u if y has items if y has no items it produces a result ofn 5 Butifnull 6 0 5 5 Butifnull 6 0 0 6 x u Butifxnull n yproducesx u y if x has items or n if x has no items 78 13 Loopless Code I1I Adverbs and We have learned about ordinary verbs that operate on the cells of an operand and we have learned u which operates between all the cells of its operand In between those extremes are verbs that operate on subsets of the cel
280. you have to tell it what to call The simplest way to do this is to change the verb into a modifier then when it executes it has access to its operands which can be verbs So instead of f verb definition you write f adverb define monadic definition dyadic definition and within the definition of you can refer to the left operand of the adverb which goes by the name u If you decide to write a conjunction instead its right operand is v 135 The noun operands of the derived verb u foru f v arex and y as usual We will discuss user defined modifiers in a later chapter In Chapter 3 this technique was used to calculate the Chebyshev coefficients of a function The function to be approximated is one of the inputs to this calculation so we write an adverb and let its left operand be the function With chebft defined as in the example an example of its use is 10 2 amp 0 chebft 0 1 1 64717 0 232299 0 0537151 0 00245824 0 000282119 Here the function to be approximated is the cosine function 2 amp 0 evaluated for 10 Chebyshev coefficients over the interval 0 1 When you define a modifier you have no way to specify that you are defining only the dyadic case as you can for a verb with dyad define Instead you use the form given above in which the monadic definition if any is separated from the dyadic by a line containing a single character chebft did this since its derived verb is always dyadic I
281. you think you d like to use them Random Numbers Monad has rank 0 yisarandom element ofi y An example use is 3 3 1000 755 458 532 218 47 678 679 934 383 Dyad has rank 0 x y isa list of x items selected without repetition from i y asifthe listi y were shuffled and the first x elements were taken 5 52 24 8 48 46 22 The verbs use a multiplicative congruential random number generator Plot J includes a great package for making 2 D and 3 D plots Check out the Lab named Plot Package to see how to use it For a quick preview enter 167 load plot numeric trig surface plot sin steps 0 3 30 to see how easy it is to get a plot of sin x y Computational Addons The web site at www jsoftware com has several addons that you can download These are executable libraries along with J scripts to call functions in them that offer efficient implementations of often used functions Two of interest in applied mathematics are the LAPACK addon and the FFT addon If you want a fast implementation of the singular value decomposition referred to earlier install the LAPACK addon then you can use require addons lapack lapack require addons lapack dgesvd dgesvd_jlapack_ yourmatrix which will quickly return the desired singular values and singular vectors Useful Scripts Supplied With J The directory yourJdirectory system packages contains a number of subdirectories full of useful scripts The math an

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