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Text File | 1997-09-16 | 63KB | 770 lines

XNÉIO 0 1 ° XNÉCT 0 1E²13 ° XCÉFC 1 6 .,*0_² ° XNÉRL 0 1190959745 ° XCÉPR 1 1 ° XCÉLX 1 5 TEACH ° XAA╜1-ÉIO-∞6 ° *(1996 4 6 15 59 48 592) ° FARABIC ÉFX 'ARABIC' 'Σ Origin of the arabic notation' ° '''The symbols representing numeric values (1234567890) can be traced' ° '' ° '''to a very old Hindu astronomical text called SINDHIND. In ancient'' ° ' ° '''times there were many competing forms for representing numbers. In' ° '' ° '''the period from the 7th to 13th century, the Greek, Roman and Hindu ° ''' ° '''notations were used extensively by different groups. The Hindu nota ° tion''' ° '''eventually proved superior because it was based on only 10 symbols, ° ''' ° '''including the zero, and could be used to represent arbitrarily larg ° e''' '''numbers.'',r' ° '''Eventually the Hindu notation was adopted by Arabic scholars. When' ° '' ° '''the European translators of Arabic texts opened the doors to ancien ° t''' ° '''learning in the 13th and 14th centuries, the gradually evolving Hin ° du''' ° '''notation became known as Arabic. By the 16th century, the evolution ° ''' ° '''was essentially complete as illustrated in the published works by'' ° ' '''Durer.'',r' ° '''Interestingly enough, the Arabs continued to evolve their own''' ° '''notation, so that modern Arabic numerals are quite distinctive from ° X''' '''those used in the Western World.''' 'endd' ° XCB 1 4 TEXT ° XNB1 1 4 0 1 0 1 ° XNB2 1 4 0 0 1 1 ° XAC╜3 3µ('A')('B')(1)(2)(3)('A')('B')(1)(2) ° *(1997 9 14 12 30 44 504) ° XFDISCLAIMER ÉFX 'DISCLAIMER' 'Σ Copyright statement' 'disclaimer' ° *(1996 4 6 15 59 48 592) ° FDISPLAY ÉFX 'D╜S DISPLAY A;ÉIO;R;C;HL;HC;HT;HB;VL;VB;V;W;N;B' ° 'Σ DISPLAY A GENERAL ARRAY IN PICTORIAL FORM' ° 'Σ NORMAL CALL IS MONADIC. DYADIC CALL USED ONLY IN' ° 'Σ RECURSION TO SPECIFY DISPLAY RANK, SHAPE, AND DEPTH.' 'ÉIO╜0' ° '»(0=ÉNC ''S'')/''S╜µA''' 'R╜╞µ,S Σ PSEUDO RANK.' ° 'C╜''┐┌└┘'' Σ UR, UL, LL, AND LR CORNERS.' ° 'HL╜''─'' Σ HORIZONTAL LINE.' ° 'HC╜HL,''Θ╕'',HL,''~+ε'' Σ HORIZONTAL BORDERS.' ° 'HT╜HC[(0<R)⌡1+0<╞²1╞,S]' 'ΣW╜,0╧■╞0µΓ(1⌐µA)╞A' ° 'HB╜HC[3+3╛(''2⌡~A╧«A'' ÉEA ''1+╞ε0⌡(1⌐⌡/µA)╞,A'')+3⌡1<µµS]' ° 'VL╜''│'' Σ VERTICAL LINE.' ° 'VB╜VL,''Φ╟'' Σ VERTICAL BORDER.' ° 'V╜VB[(1<R)⌡1+0<²1╞²1╟,S]' ° '»(0εµA)/''A╜(1⌐µA)µΓ╞A'' Σ SHOW PROTOTYPE OF EMPTIES.' ° '╕(1<╧A)/GEN' '╕(2<µµA)/D3' ° 'D╜«A Σ SIMPLE ARRAYS.' 'W╜1╞µD╜(²2╞1 1,µD)µD' ° 'N╜²1+1╟µD' '╕(0=µµA)/SS' ° 'D╜(C[1],V,((W-1)µVL),C[2]),((HT,NµHL),[0]D,[0]HB,NµHL),C[0],(WµVL),C[ ° 3]' '╕0' 'SS:HB╜((0 '' '')=╞0µΓA)/'' -''' ° 'D╜'' '',('' '',[0]D,[0]HB,Nµ'' ''),'' ''' '╕0' ° 'GEN:D╜«DISPLAY■A Σ ENCLOSED ...' 'N╜Dδ.⌠'' ''' ° 'D╜(Nδ~1ΦN)≡D' 'D╜(δ≡~'' ''╤D)/D' 'D╜((1,µS)µS)DISPLAY D' ° '╕(2≥µ,S)╟D3E,0' 'D3:D╜0 ²1╟0 1╟«ΓA Σ MULT-DIMENSIONAL ...' ° 'W╜1╞µD' 'N╜²1+1╟µD' ° 'D╜(C[1],V,((W-1)µVL),C[2]),((HT,NµHL),[0]D,[0]HB,NµHL),C[0],(WµVL),C[ ° 3]' 'D3E:N╜²2+µ,S' ° X 'V╜C[Nµ1],[0]VB[1+0<²2╟,S],[0](((²3+╞µD),N)µVL),[0]C[Nµ2]' 'D╜V,D' ° *(1996 4 6 15 59 48 592) ° FEXIT ÉFX 'EXIT' 'Σ Exit from function' '''To log off type: )OFF''' ° X '╕' ° *(1996 4 6 15 59 48 592) ° FGO ÉFX 'GO;T;E;B' 'Σ Expression driver' 'L0:B╜E╜''''' 'æ╜'' ''' ° 'T╜æ' '╕(^/'' ''=T)/L0' '╕((^/'')OFF ''=5╞6╟T)doif ''EXIT'')/0' ° '╕(('':''εT)doif ''B╜evaldd (+/^\'''' ''''=T)╟T'')/L0' ° '''E╜ÉEM'' ÉEA T' '╕(0=µ,E)/L0' '╕B/L0' ° '''This is not a valid APL2 expression''' 'æ╜''*''' ° X '╕(''?''⌠╞1╟æ)/L0' 'E' '╕L0' ° *(1997 8 22 12 47 29 540) ° FHELP ÉFX 'HELP;N;I;T' 'Σ Help to student' '''WHAT TO DO'',r' ° ''' ° To get out of the lesson''' ''' ENTER: EXIT''' ° ''' ° To log off APL2''' ° ''' FIRST, ENTER: EXIT THEN ENTER: )OFF''' ° ''' ° To get help''' ''' ENTER: HELP''' ° ''' ° When you see a prompt on a blank line''' ° ''' ENTER AN APL2 EXPRESSION - OR JUST PRESS: ENTER''' 'do' ° ''' ° If you get this line'',r' ° '''This is not a valid APL2 expression'',r,''*'',r' ° ''' YOU CAN EITHER''' ''' A. PRESS: ENTER''' ° ''' B. PRESS: ? and then ENTER to see what was incorrect''' ° ''' in your expression causing that response'',r' 'do' ° '''HINTS'',r' ° '''This lesson is made up of '',(«N╜8),'' components named TEACHx, whe ° re''' '''the x stands for a digit:'',r' 'I╜0' ° 'L0:T╜''TEACH'',(«I╜I+1)' 'T,'' '',1╟notb(ÉCR T)[2;]' '╕(N>I)/L0' ° 'do' ° '''You may review either of these components separately. To do that,'' ° ' ° '''first enter EXIT (and RETURN), then enter the name of the lesson.'' ° ' '''component (e.g. TEACH4).'',r' ° '''To re-start the lesson, just enter TEACH'',r' ° '''When the screen fills up, it is a good idea to move the cursor to'' ° ' '''the start of a convenient paragraph, and press ENTER'',r' ° '''You may also wish to press PAGE UP to review the prevous pages.''' ° 'do' '''RESOURCES'',r' ° '''You may also enter ONE of the following words at a time'',r' ° X '''ARABIC Origin of the Arabic numerals''' 'endd' ° *(1997 9 16 12 37 49 560) ° FRI ÉFX 'u╜RI w;t;b' 'ΣDD' 'b╜(t╜w=4)/''u╜''''IV''''''' '╕(t doif b)/0' ° X 'u╜wµ''I''' ° *(1997 9 16 12 37 31 488) ° FRIX ÉFX 'u╜RIX w;t;b' 'ΣDD' 'b╜(t╜9=w)/''u╜''''IX''''''' ° X '╕(t doif b)/0' 'u╜RV w' ° *(1997 9 16 12 36 49 556) ° FROMAN ÉFX 'u╜a ROMAN w;t;b' 'ΣDD' ° 'b╜(t╜39<w)/''u╜''''This number is too large''''''' '╕(t doif b)/0' ° X 'u╜RX w' ° *(1997 5 31 11 59 20 572) ° FRULE ÉFX 'V RULE W' 'Σ Display rules' '╕(2=ÉNC ''V'')/L0' ° X '2 rule W ╪ ╕0' 'L0:V rule W' ° *(1997 9 16 12 37 44 540) ° FRV ÉFX 'u╜RV w;t;b' 'ΣDD' 'b╜(t╜4<w)/''u╜''''V'''',RV w-5''' ° X '╕(t doif b)/0' 'u╜RI w' ° *(1997 9 16 12 37 13 416) ° FRX ÉFX 'u╜RX w;t;b' 'ΣDD' 'b╜(t╜9<w)/''u╜''''X'''',RX w-10''' ° X '╕(t doif b)/0' 'u╜RIX w' ° *(1997 8 11 12 43 22 452) ° FTEACH ÉFX 'TEACH' 'Σ Start fourth lesson in APL2 by Zdenek V JIZBA' ° 'exit ''TEACH''' 'initialize' 'TEACH1 Σ Catenation and ravel' ° 'TEACH2 Σ Compression' 'TEACH3 Σ Replication' ° 'TEACH4 Σ N-wise reduction' 'TEACH5 Σ Logical primitive functions' ° 'TEACH6 Σ Indexing; indexing function and bracket indexing' ° 'TEACH7 Σ Vector specification and selective specification' ° X 'TEACH8 Σ From arabic to roman notation' 'problems' ° *(1997 5 31 11 48 47 636) ° FTEACH1 ÉFX 'TEACH1' 'Σ Review previous lesson' 'exit ''TEACH1''' ° '''In the previous lesson we learned about direct definition. Although ° ''' ° '''this form of programming is not supported by APL2, it is a useful'' ° ' ° '''tool in studying the language. Direct definition is supported here' ° '' ° '''by built in software, and we will use it to study primitive functio ° ns,''' '''operators, and other constructs of the language.'',r' 'do' ° '''In this lesson we will learn about those features of APL2 that are' ° '' ° '''most useful in development of algorithms. We will find out what can ° ''' ° '''be done with CATENATION, BOOLEAN VECTORS, LOGICAL FUNCTIONS and''' ° '''INDEXING.'',r' ° '''Suppose we wanted to combine two words into one. Let us assume that ° ''' ° '''the two words are IN and TENT. Let us try some experiments:'',r' ° 'show ''''''IN'''' ''''TENT'''''' ''''''IN''''''''TENT''''''' ° '''If our goal was to generate the word INTENT, we have clearly failed ° .''' 'do' ° '''Let us see what these two expressions actually generate.''' ° 'show ''DISPLAY ''''IN'''' ''''TENT'''''' ''DISPLAY ''''IN''''''''TENT ° ''''''' ° '''In the first example we merely created a two element vector contain ° ing''' ° '''the two words ''''IN'''' and ''''TENT''''. In the second example, t ° he double''' ° '''quote between IN and TENT was merely converted to a single quote.'' ° ' 'do' '''This provides us with a new rule:'',r' '0 RULE 9' ° '''The reason for this rule is to distinguish the end of a character'' ° ' '''string quote from a quote that is PART of the string.''' 'do' ° '''But we still do not know how to combine strings into single strings ° .''' '''The comma (,) serves that purpose'',r' ° 'show ''''''IN'''',''''TENT'''''' ''DISPLAY ''''IN'''',''''TENT''''''' ° '''The process of combining vectors with a comma is called CATENATION. ° ''' '''It works with numbers as well'',r' ° 'show ''V1╜?3µ10'' ''V2╜?3µ10'' ''DISPLAY V1 V2'' ''DISPLAY V1,V2''' ° 'do' ° '''Is there a monadic version of the comma? Yes, and it is called RAVE ° L.''' ° '''We will study it in lesson 6, but right now we can state two''' ° '''properties of ravel:'',r' ° ''' 1. The ravel of a scalar is a vector of length 1'',r' ° 'show ''µ,5''' ° ''' 2. The ravel of a vector leaves the vector unchanged'',r' ° X 'show ''µ3 4 5'' ''µ,3 4 5''' 'endd' ° *(1997 4 29 12 20 30 448) ° FTEACH2 ÉFX 'TEACH2;VECTOR;CONS' 'Σ Compression' 'exit ''TEACH2''' ° '''In lesson 2 (TEACH2) we learned about reduction, and about the''' ° '''special symbol called an OPERATOR. With the reduction operator''' ° '''we are able to do with a single keystroke what in other programming ° ''' ° '''languages requires special constructs usually called DO loops.'',r' ° '''An operator certainly does not solve the problem any more efficient ° ly''' ° '''than a do loop. What it does, is to remove from you, the programmer ° ''' ° '''the task of dealing with operational details that SHOULD be handled ° ''' '''by the smart computer.''' 'do' ° '''Since the reduction operator provides such a NOTATIONAL advantage,' ° '' ° '''it should be no surprise that its versatility has been extended.'', ° r' ° '''We have already seen that BOOLEAN vectors contain only values zero' ° '' ° '''and one. The zero values stand for denial (no, false) while the one ° ''' ° '''values stand for affirmation (yes, true). It therefore makes sense' ° '' '''to apply a boolean vector as a left argument to the reduction''' ° '''operator'',r' 'show ''1 0 0 1 0 0 0 0 1/''''REDUCTION''''''' ° '''The boolean vector has the same length as the character string. Onl ° y''' ° '''those letters are returned for which the corresponding boolean valu ° e''' ° '''is TRUE (1). This process of using boolean vectors as left argument ° ''' ° '''to the reduction operator is called COMPRESSION. Unlike reduction,' ° '' '''compression does not reduce the rank of the result'',r' ° 'show ''µµ''''COMPRESSION'''''' ''µµ1 1 1 0 0 0 0 0 0 1 1/''''COMPRESS ° ION''''''' '''It works just as well for numbers'',r' ° 'show ''0 1 1/3 4 5'' ''(10µ1 0)/∞10''' ° '''Note that in the second example the left argument is an APL2 expres ° sion''' ° '''that returns a boolean vector. To see the great power that this giv ° es''' ° '''us, recall that we can use any primitive functions that produce a'' ° ' '''boolean result.'',r' ° 'show ''VECTOR╜?20µ10'' ''VECTOR'' ''(VECTOR<5)/VECTOR''' ° '''The last expression is an algorithm to extract only those elements' ° '' ° '''in the variable VECTOR, with a value less than 5. The next example' ° '' '''shows how we can use this feature with character vectors'',r' ° 'show ''(''''MISSISSIPPI''''ε''''AEIOU'''')/''''MISSISSIPPI''''''' ° '''To extract consonants, we could proceed in the following way:'',r' ° 'show ''CONS╜''''BCDFGHJKLMNPQRSTVWXZ''''''' ° 'show ''(''''MISSISSIPPI''''εCONS)/''''MISSISSIPPI''''''' ° '''Of course one could always use'',r' ° 'show ''(1-''''MISSISSIPPI''''ε''''AEIOU'''')/''''MISSISSIPPI''''''' ° '''Later in this lesson we will see yet another way to remove wowels'' ° X' '''without using CONS.''' 'endd' ° *(1997 5 31 11 50 39 612) ° FTEACH3 ÉFX 'TEACH3' 'Σ Replication' 'exit ''TEACH3''' ° '''The early versions of APL supported reduction, but it had been''' ° '''pointed out that the process could be generalized. After all,''' ° '''the boolean vector can be interpreted as saying: "Return the right' ° '' ° '''element the number of times indicated by the corresponding value''' ° '''in the left argument."'',r' ° '''Here we no longer think of the boolean vector as a YES NO switch,'' ° ' ° '''but rather as a kind of multiplier or replicator. APL2 indeed does' ° '' '''support this more generalized notion of reduction, calling it''' ° '''REPLICATION'',r' ° 'show ''1 1 2 0 0 0 0 1 0 0 2 0 0 1 2 0 1 0 0 0 0/''''MISER FINDS TIP ° IN IT''''''' '''Let us write the rule for replication:'',r' ° '0 RULE 10' ° '''Naturally, replication will also work on numeric vectors, and even' ° '' '''on mixed vectors'',r' ° 'show ''+/(∞5)/∞5'' ''1 2 0 3/1 2,''''AB''''''' ° '''Why did we have to use a comma in the second example? Also explain ° why''' '''the first example produces the same result as'',r' ° X 'show ''+/(∞5)*2''' 'endd' ° *(1997 9 16 12 42 1 388) ° FTEACH4 ÉFX 'TEACH4;VECTOR;PRICE;VOLATILITY' 'Σ N-wise reduction' ° 'exit ''TEACH4''' ° '''Since the reduction operator is so useful, can we improve on it?''' ° '''APL as a language is not static. As new demands for problem solving ° ''' ° '''arise, the APL community continues to seek ever more powerful ways' ° '' '''to express complex processes with simple notation.'',r' ° '''In many fields of science as well as in economics and statistics''' ° '''there is frequent need to calculate what is called "moving ''' ° '''averages". This is a process where given some integer, say N''' ° '''we want to calculate the average for N consecutive values of a''' ° '''vector. So for a vector of length M (naturally M≥N) we want to''' ° '''produce a vector of (1+M-N) average values.''' 'do' ° ''' N-WISE REDUCTION'',r' ° '''In older versions of APL this calculation required a fairly simple' ° '' ° '''but still somewhat inconvenient expression. For one, the APL progra ° mmer''' ° '''had to remember the expression, derive it, or remember the name of' ° '' ° '''a user defined function that produced the desired result. Such a''' ° '''mental effort distracts from the problem solving task at hand.'',r' ° '''In APL2 there is a form of the reduction operator that solves this' ° '' '''class of problems in a very simple way.'',r' 'show ''2+/∞10''' ° '''This is equivalent to doing the following:'',r' ° '(∞9),''+'',(1+∞9),''='',[1.1]2+/∞10' 'do' ° '''We could just as well have used other primitives. Consider ⌡ ⌐ and ° ╛'',r' ° 'show ''3⌡/∞10'' ''VECTOR╜?20µ10'' ''VECTOR'' ''5⌐/VECTOR'' ''5╛/VECTO ° R''' ° '''One might ask if the idioms N⌐/VECTOR and N╛/VECTOR have any''' ° '''practical applications. Consider the stock broker who is intersted' ° '' ° '''in the volatility of a stock over a period of say 5 weeks. Then if' ° '' ° '''VECTOR represents the closing price of a stock at the end of a''' ° '''week for a period of 20 weeks, we can define volatility in the''' ° '''following way:'',r' ° '1 show ''PRICE╜20+VECTOR'' ''VOLATILITY:(α⌐/∙)-α╛/∙'' ''5 VOLATILITY ° PRICE''' ° '''The point of any generalized APL2 primitive is that it has many''' ° '''more application than those originally conceived. Consider the''' ° '''following, which is an algorithm for finding double letter pairs''' ° X '''in text'',r' 'show ''2=/''''MISSISSIPPI''''''' 'endd' ° *(1997 5 1 12 24 15 296) ° FTEACH5 ÉFX 'TEACH5' 'Σ Logical primitives' 'exit ''TEACH5''' ° '''As we have seen, boolean variables are useful in logical operations ° .''' ° '''If you have studied logic, you know of primitive logic functions''' ° '''called NOT, AND and OR. Suppose that we have the following two bool ° ean''' '''vectors'',r' 'show ''B1╜0 1 0 1'' ''B2╜0 0 1 1''' ° '''Now let us consider the primitive logical operations on these vecto ° rs.''' ° '''A logical NOT means negation so that what was true becomes false, a ° nd''' ° '''what was false becomes true. We certainly can do that with'',r' ° 'show ''1-B1'' ''1-B2''' ° '''There is, however, a simpler way with the logical primitive NOT (~) ° '',r' 'show ''~B1'' ''~B2''' ° '''For simple expressions this may not seem as much of an advantage, b ° ut''' ° '''as logical expressions become more complex, the special symbol ~ is ° ''' ° '''a big help. Furthermore, people working with logic are quite used t ° o''' '''this notation. (For an even better reason, see below.)''' 'do' ° '''In the same way we could use the primitives ⌐ and ╛ to simulate AND ° ''' '''and OR'',r' 'show ''B1⌐B2'' ''B1╛B2''' ° '''The first expression will return true if either the left or the rig ° ht''' ° '''argument returns true. The second expression will return true only ° if''' ° '''both arguments are true. Despite this, APL2 supports the two logica ° l''' '''primitives AND ^, and OR δ'',r' 'show ''B1δB2'' ''B1^B2''' ° '''There is a good reason for having these special primitives. If ⌐ or ° ╛''' ° '''are applied to arbitrary numeric vectors, APL2 will always produce ° a''' ° '''result. However ~, δ and ^ will work ONLY with boolean vectors. If ° you''' ° '''were to try an expression such as ^/∞5, you will get an ERROR MESSA ° GE.''' 'do' ° '''By using the logical primitives you allow APL2 to help you in the'' ° ' ° '''development of logical algorithms. It is amazing how frequently one ° ''' ° '''produces a "logical" vector and be dismayed to find it not to be''' ° '''boolean. (At least that is my experience.)''' 'do' ° '''Notice that ~ is monadic, while δ and ^ are dyadic. There are no''' ° '''monadic δ and monadic ^. There is a dyadic ~, but it is not related ° ''' ° '''to simple logical operations. We will study dyadic ~ in a later''' ° '''session.'',r' ° '''We should, however mention two other logical primitives: NOR τ and' ° '' '''NAND σ.''' 'show ''B1τB2'' ''B1σB2''' ° '''You may have noticed that the four element vectors B1 and B2 show'' ° ' ° '''all possible combinations for pairs of true-false. Yet, there is''' ° '''a total of 16 distinct patterns for a four character vector''' ° '''containing 0 or 1. As an exercise, generate all 16 patterns using'' ° ' ° '''any primitive functions that you have studied so far, and the vecto ° Xrs''' '''B1 and B2. (Hint: start with 0^B1, B1^B2)''' 'endd' ° *(1997 5 31 11 50 5 476) ° FTEACH6 ÉFX 'TEACH6' 'Σ Indexing;VECTOR' 'exit ''TEACH6''' ° ''' INDEXING'',r' ° '''We have been doing a lot with vectors as single entities. But suppo ° se''' '''we wish to study individual components of a vector'',r' ° 'show ''VECTOR╜?10µ5'' ''VECTOR''' ° '''How do we access the second element of VECTOR? There is an APL2 pri ° mitive''' '''called the INDEXING FUNCTION. It uses the ╙ symbol.'',r' ° 'show ''2╙VECTOR''' ° '''The ╙ symbol is sometimes also called "squad". The indexing functio ° n''' ° '''is new to APL2. Although we introduce it here, a more detailed''' ° '''discussion requires concepts not yet mentioned. The indexing functi ° on''' ° '''is meant to suplement an older notation called BRACKET INDEXING.'', ° r' ° '''Bracket indexing uses two symbols called square brackets. The left' ° '' ° '''square bracket [ and the right square bracket ]. To access the seco ° nd''' ° '''element of VECTOR, we would write the following expression'',r' ° 'show ''VECTOR[2]''' ° '''Notice that we do not refer to the bracket notation as a function.' ° ',r' '''To access more than one element of VECTOR, we would write'',r' ° 'show ''VECTOR[1+∞4]''' ° '''The length of the result is equal to the length of the vector insid ° e''' '''brackets'',r' 'show ''VECTOR[5µ2]'' ''µVECTOR[123µ1]''' ° '''In lesson 2 we learned that vectors can have zero length. Let us''' ° '''see what happens if we place such a null vector inside brackets'',r ° ' 'show ''VECTOR[∞0]'' ''µVECTOR[∞0]''' ° '''Now let us try to place NOTHING inside brackets.'',r' ° 'show ''VECTOR[]''' ° '''Clearly NOTHING is not the same as NULL. Furthermore nothing behave ° s''' '''as if the brackets were not there.''' 'do' ° '''Since the numbers inside brackets point to specific elements of''' ° X '''the vector, we have the following rule'',r' '0 RULE 11' 'endd' ° *(1997 5 4 13 8 23 280) ° FTEACH7 ÉFX 'TEACH7;VECTOR;A;B;C' 'Σ Specification' 'exit ''TEACH7''' ° '''We have seen in lesson 1 that specification is the process of placi ° ng''' ° '''data into a VARIABLE by means of an assignment arrow. This process' ° '' ° '''has been extended in APL2. Many problems, that in the older version ° s''' ° '''of APL required complex expressions, can now be solved with a much' ° '' '''simpler notation.''' 'do' ''' VECTOR SPECIFICATION'',r' ° '''More than one variable can be assigned with a single specification' ° '' ° '''expression. The variables to the left of the assignment arrow shoul ° d''' ° '''be placed in parentheses. This is called VECTOR SPECIFICATION'',r' ° 'show ''(A B C)╜(∞6) ''''TEXT'''' (3 3µ''''AB'''',∞3)'' ''A'' ''B'' '' ° C''' ''' SELECTIVE SPECIFICATION'',r' ° '''In older versions of APL, only bracket indexing was allowed. In APL ° 2,''' ° '''the expressions allowed to the left of the assignment arrow have be ° en''' ° '''extended. The expanded process is called SELECTIVE SPECIFICATION. T ° he''' ° '''expression including the variable into which the right argument is ° to''' '''be placed must be in parentheses.''' ° '''Here are some examples'',r' ° 'show ''VECTOR╜∞9'' ''(3╙VECTOR)╜0'' ''VECTOR''' ° 'show ''(3╟VECTOR)╜∞6'' ''VECTOR'' ''((9µ1 0)/VECTOR)╜''''X'''''' ''VE ° XCTOR''' '''More examples will be shown in later lessons.''' 'endd' ° *(1997 8 11 13 12 58 476) ° FTEACH8 ÉFX 'TEACH8' 'Σ ROMAN exercise' 'exit ''TEACH7''' ° '''Earlier this lesson, we stated that the notation and primitive''' ° '''functions described here, can be used in APL2 programming. As you'' ° ' ° '''learn about APL, try to think about specific problems of your own'' ° ' ° '''knowledge. Then try to develop APL solutions to these problems.''' ° 'do' ° '''To give you some idea on how one may proceed, the following example ° ''' ° '''illustrates a very important feature of APL, namely its recursive'' ° ' '''nature.'',r' ° '''The ancient Romans did not use what we now call the Arabic notation ° ''' ° '''to describe numbers. The Roman notation consisted of the usage of'' ° ' ° '''the capital letters I V X L C D and M. These letters represented th ° e''' ° '''numbers 1 5 10 50 100 500 and 1000. To display a number, the letter ° s''' ° '''were placed with decreasing value from left to right, but with nota ° ble''' ° '''exceptions. For example 4 was represented as IV, and 6 was represen ° ted''' ° '''as VI. Let us now develop an algorithm for converting integers into ° ''' '''the Roman notation.''' 'do' ° '''To begin with, let us for simplicity go only as high as 39. (As we' ° '' '''learn new primitives we will eventually do much better).'',r' ° '1 show ''ROMAN:RX ∙:39<∙:''''This number is too large''''''' ° '''Function ROMAN will return the message if we try to convert a''' ° '''number greater than 39'',r' 'show ''ROMAN 40''' ° '''Next we define function RX'',r' ° '1 show ''RX:RIX ∙:9<∙:''''X'''',RX ∙-10''' ° '''Function RX calls on function RIX, so we have to define that one'', ° r' '1 show ''RIX:RV ∙:9=∙:''''IX''''''' ° '''Function RIX takes care of the special case when ∙ is 9'',r' ° '1 show ''RV:RI ∙:4<∙:''''V'''',RV ∙-5''' ° '''And finally we have to define RI'',r' ° '1 show ''RI:∙µ''''I'''':∙=4:''''IV''''''' ° '''Now we can test the ROMAN algorithm'',r' ° 'show ''ROMAN 39'' ''ROMAN 17''' ° '''The method we have used to solve the conversion of Arabic numerals' ° '' ° '''to Roman is called a PRODUCTION SYSTEM. The algorithms in functions ° ''' ° '''RX RV and RI are PRODUCTION RULES, and the set of possible branches ° ''' ° '''is a TREE STRUCTURE called DISCRIMINATION NET. This terminology''' ° '''is not related in any way to APL. It comes from a discipline''' ° '''called Artificial Intelligence. This APL2 solution is an example of ° ''' ° '''the RECURSIVE nature of the language, since both RX and RV may call ° ''' '''on themselves with a reduced argument.''' 'do' ° '''As an exercise try to define functions in the same manner as ROMAN, ° ''' ° '''to convert Roman notation to Arabic. (For the origin of Arabic nota ° tion''' '''enter ARABIC.'',r' ° '''In a later chapter we will re-define ROMAN to evaluate numbers up'' ° ' ° '''to 3999. (Beyond that the Romans used a different notation with''' ° X '''special symbols Γ and π.)''' 'endd' ° XNV1 1 3 9 10 1 ° XNV2 1 3 10 6 6 ° XNVECTOR 1 10 1 1 3 5 1 3 3 5 4 3 ° XCa 1 6 ° *(1997 7 13 12 28 49 504) ° Faddquote ÉFX 'u╜addquote w' ° 'Σ Put quotes around a string, and double existing quotes' ° X 'u╜ÉAV[40],((1+w=ÉAV[40])/w),ÉAV[40]' ° XCalpha 1 26 ABCDEFGHIJKLMNOPQRSTUVWXYZ ° *(1997 7 24 13 20 38 476) ° Fav ÉFX 'av;A;N;I;ÉIO' 'Σ Display characters in ÉAV' 'ÉIO╜0' ° 'A╜22 78µ'' ''' 'N╜3 0«φ12 22µ1+∞356' 'A[;,(6⌡∞12)°.+2 3 4]╜N' ° 'A[;6+6⌡∞12]╜φ12 22µÉAV' 'ΣA[8 10 13;6]╜'' ''' 'A[13;6]╜'' ''' ° X 'A[14+∞8;68 69 70 72]╜'' ''' 'A' ° XCb 1 6 u╜'IV' ° Ac╜('R0')('(∙[1]µ''M''),(''CDM'' R1 ∙[2]),(''XLC'' R1 ∙[3]),''IVX'' R1 ° X∙[4]')('ΣDD ') ° XCd 0 ½ ° *(1991 11 11 8 25 13 316) ° Fdate ÉFX 'u╜date w' 'Σ Format date and time of day' 'u╜«6╞w' ° X 'u╜('' ''⌠u)Γu' 'u╜εu,■''-- .. ''' ° XCdig 1 10 1234567890 ° *(1997 9 9 13 0 45 372) ° Fdisclaimer ÉFX 'disclaimer' 'Σ Copyright statement' ° '(10µ'' ''),''Copyright, Z. V. Jizba, 1995,1996,1997'',r' ° ''' This and subsequent workspaces labelled TEACHxx are made available ° ''' ° '''at no cost to anyone who desires to learn how to use effectively''' ° '''the IBM/OS2 version of APL2.'',r' ° '''This software is provided "AS IS" with no WARRANTY of any kind, eit ° her''' ° '''express or implied. Any risk in its use resides with you, the user ° of''' '''these tutorials.'',r' ''' ACKNOWLEDGEMENTS'',r' ° ''' In writing these tutorials, I am greatly indebted to Roy Sykes, wh ° ose''' ° '''excellent lectures increased my understanding of the language.''' ° '''Discussions with the late Harry Bertucelli clarified a number of''' ° '''concepts and caused me to change some improper terminology that was ° ''' ° '''used in previous versions of these tutorials. Mr. Benjamin Archer'' ° ' ° '''kindly checked out a nearly final version, bringing to my attention ° ''' '''some ommisions, misspellings, and invalid terminology.'',r' ° X '''(PRESS ENTER to continue)''' ° *(1997 7 13 12 28 50 508) ° Fdo ÉFX 'do;T;E' 'Σ Expression driver' 'E╜''''' 'æ╜'' ''' 'T╜æ' ° '╕(^/'' ''=T)/0' ° '╕(('':''εT)doif ''B╜evaldd (+/^\'''' ''''=T)╟T'')/2' ° '''E╜ÉEM'' ÉEA T' '╕(0=µ,E)/2' ° '''This is not a valid APL2 expression''' 'æ╜''*''' '╕(''?''⌠╞1╟æ)/2' ° X 'E' '╕2' ° *(1997 7 13 12 28 50 508) ° Fdoif ÉFX 'U╢╜V╢ doif W╢;t╢' 'Σ Rule' '╕(^/~U╢╜V╢)/0' ° X '''U╢╜V╢ doif■ W╢'' ÉEA ''»V╢/W╢''' ° *(1997 9 9 12 50 14 444) ° Fendd ÉFX 'endd' 'Σ end of special feature' '20µ''²'' ╪ ╕(4<µÉLC)/0' ° X 'do' ° *(1997 8 11 13 14 21 336) ° Ferase ÉFX 'erase;T;EXIT;GO;HELP;DISPLAY;ARABIC;RULE;DISCLAIMER' ° 'Σ Erase all global functions and variables' 'T╜ÉNL 3' ° 'T╜(~T^.εlc,'' '')≡T' 'T╜ÉEX(~T[;∞5]^.=''TEACH'')≡T' 'T╜ÉNL 2' ° X 'T╜(~T^.εlc,'' '')≡T' ° *(1997 7 27 13 47 41 608) ° Fevaldd ÉFX 'u╜evaldd w;c;n' 'Σ Evaluate direct definition' 'u╜0' ° 'n╜(w∞''Σ'')-1' 'c╜(((n╞w)⌠'':'')Γn╞w),Γ''ΣDD '',(n+1)╟w' ° '╕((1 label╞c)doif ''''''Invalid label'''''')/0' ° '╕((2=µc)doif ''u╜showdd 1╙c'')/0' ° '╕((3=ÉNC╞c)doif ''u╜⌡µÉ╜(╞c),'''' is already defined.'''''')/0' ° '╕((3=µc)doif ''u╜simdd c'')/0' 'c╜(Γ''α∙ aw'')replace■c' ° 'u╜ε''u╜'',((''a''εεc[2 3 4])/''a ''),(╞c),'' w;t;b''' ° 'u╜u(5πc)(''b╜(t╜'',(3πc),'')/'',addquote ''u╜'',4πc)' ° X 'u╜u,''╕(t doif b)/0''(''u╜'',2πc)' 'u╜╧ÉFX u' ° *(1997 7 25 13 27 52 564) ° Fexit ÉFX 'V exit W;T' 'Σ Exit if too many suspended functions' ° '╕(0⌠ÉNC ''V'')/L0 ╪ V╜10' 'L0:╕(V>µÉLC)/0' ° '''There are too many suspended functions''' '''Please enter '',W' ° X '╕' ° *(1997 7 26 12 33 39 536) ° Fget ÉFX 'U╜V get W;t;T;ÉPR' 'Σ Prompt for response from keyboard' ° 'ÉPR╜T╜ÉAV[ÉIO+255] ╪ ╕(0⌠ÉNC ''V'')/L0 ╪ V╜1' 'L0:V╜V╧1' 'æ╜W ╪ t╜æ' ° 'U╜(+/^\t=T)╟t' '╕(''╕''⌠╞U)/L1 ╪ ╕' 'L1:╕V/0' 't╜(U⌠'' '')ΓU' ° X 'U╜(µt),(ΓU),t' ° *(1997 7 28 13 33 8 424) ° Fglobals ÉFX 'globals' 'Σ Initialize useful global variables' ° 'uc╜''ABCDEFGHIJKLMNOPQRSTUVWXYZ''' ° 'lc╜''abcdefghijklmnopqrstuvwxyz''' 'dig╜''1234567890''' ° X 'r╜ÉAV[13+ÉIO]' 'q╜''''''''' ° *(1997 7 3 12 47 6 368) ° Finitialize ÉFX 'initialize;T' 'Σ Initialize workspace' ° '''AT ALL TIMES, TO CONTINUE, PRESS RETURN!'',r' ° '''To see disclaimers enter:'',r,'' disclaimer''' 'do' 'erase' ° 'globals' ° '''Make sure the CAP LOCK light on your keyboard (upper right) is ON!' ° X'' 'endd' ° *(1997 7 27 13 14 33 444) ° Flabel ÉFX 'u╜v label w' ° 'Σ Return 1 if label w does not begin with a cap' ° '╕(0⌠ÉNC ''v'')/L0 ╪ v╜0' 'L0:v╜v╧1 ╪ w╜εw ╪ ╕v/L1 ╪ ╕(u╜0⌠ÉNC w)/0' ° X 'L1:╕(u╜~^/wεlc,uc,dig)/0' 'u╜w[1]εlc,dig' ° XClc 1 26 abcdefghijklmnopqrstuvwxyz ° XNn 0 59 ° *(1997 7 13 12 28 55 528) ° Fnon ÉFX 'non;T;RC;ET;R' 'Σ Ignore keyboard entry' 'æ╜'' ''' 'T╜æ' ° '╕(0=µ(T⌠'' '')/T)/0' '(RC ET R)╜ÉEC T' '╕(0=RC)/2' ° X '╕((1=RC)doif ''R'')/2' '╕2' ° *(1997 7 13 12 28 55 528) ° Fnotb ÉFX 'u╜notb w' 'Σ Remove trailing blanks' ° '╕((1<╧w)doif ''u╜notb■ w'')/0' '╕((1<µµw)doif ''u╜πnotb Γ[2]w'')/0' ° X 'u╜(1-(,'' ''⌠Φw)∞1)╟w' ° XCp 1 2 " ° *(1997 7 27 12 55 6 496) ° Fproblems ÉFX 'problems' 'Σ Problems' ° '''That is all for this lesson. Remember, if you want to practice,''' ° '''and plan to use direct definitions, be sure to first enter GO.''' ° '''Direct definitions will then be accepted. To exit GO, enter EXIT.'' ° ,r' ° '''To erase a previously defined DIRECT DEFINITION FUNCTION, enter'',r ° ' ''' )ERASE functionname'',r' ° X '''WARNING! do not use )ERASE on other labels.'',r' ° XCq 0 ' ° XCr 0 ° *(1997 7 13 12 28 56 532) ° Freplace ÉFX 'u╜v replace u;i;r;s' 'Σ Replace elements in v in u' ° 'i╜Γ∞µu' 's╜2πv╜(v⌠'' '')Γv' 'i╜⌡r╜i⌡■Γ[1]u°.=╞v' ° X 'u[(εi)/εr]╜s[(εi)/εi⌡■∞µs]' ° *(1997 7 13 12 28 56 532) ° Fround ÉFX 'U╜V round W' 'Σ Half adjust to V th decimal' ° X 'U╜(╛0.5+W⌡10*V)÷10*V' ° *(1997 5 31 12 3 40 432) ° Frule ÉFX 'V rule W' 'Σ Additional remarks on rules' ° '╕(2=ÉNC ''V'')/L ╪ V╜0' 'L:╕(~(╞W)ε∞8)/LA ╪ V rulea╞W ╪ ╕L0' ° 'LA:╕((╞W)=8+∞3)/L9,L10,L11' '''This rule number is undefined'' ╪ ╕' ° 'L9:╕(V>1)/L9X' ° ''' RULE 9: To produce a quote inside a character string, enter''' ° ''' the quote twice.'',2µr' '╕(0=V)/L0' ° 'L9X:''This is used in most other programming languages.''' '╕L0' ° 'L10:╕(V>1)/L10X' ° ''' RULE 10: In replication the left argument must contain only''' ° ''' integers. The length of the left argument must be''' ° ''' the same as the length of the right argument.'',2µr' ° '╕(0=V)/L0' ° 'L10X:''The left argument may also have negative integers. Only the nu ° mber''' ° '''of positive integers and zeros must add up to the length of the rig ° ht''' ° '''argument. Replicate with negative integers is not discussed here be ° cause''' ° '''the explanation requires concepts that have not yet been mentioned. ° '',r' 'show ''1 1 2 ²1 0 1 1 1 ²1 1 0 1 1/''''CALIFORNIAN''''''' '╕L0' ° 'L11:╕(V>1)/L11X' ° ''' RULE 11: The expression inside square brackets must return''' ° ''' positive integers in the range of the length of''' ° ''' the indexed vector. If the expression is MISSING,''' ° ''' the result is as if the brackets were missing.'',2µr' ° '╕(0=V)/L0' ° 'L11X:''The expression in brackets need not be a vector. It could be a ° n''' '''array as in the next example'',r' ° 'show ''''''CALIFORNIA''''[3 3µ?9µ10]'' ''(∞4)[3 3µ∞4]''' ° '''For rules not described in this lesson, see workspace INDEX''' ° X 'L0:do' '╕(0=µW╜1╟W)/0' '╕L' ° *(1997 5 31 11 58 30 608) ° Frulea ÉFX 'V rulea W' 'Σ Additional remarks on rules' ° '╕(2=ÉNC ''V'')/L ╪ V╜0' 'L:╕(W=∞8)/L1,L2,L3,L4,L5,L6,L7,L8' ° '''For rules not described in this lesson, see workspace INDEX''' ° '╕0' 'L1:╕(V>1)/L1X' ° '''RULE 1. For arithmetic operations a vector must be matched with a'' ° ' ''' scalar, or another vector of the same length.'',2µr' ° '╕(0=V)/0' ° 'L1X:''This is the first of many rules dealing with the concept of''' ° '''CONFORMABILITY. This means that elements to be processed by APL2''' ° '''must in some ways match.''' '╕L0' 'L2:╕(V>1)/L1X' ° '''RULE 2. The reduction operator places the function, (specified to'' ° ' ° ''' its left), between all pairs of the vector. It then evaluat ° es''' ''' the resulting expression.'',2µr' '╕(0=V)/0' ° 'L2X:''There is more to this, but at this stage the rule is essentiall ° y''' '''correct.''' '╕L0' 'L3:╕(V>1)/L1X' ° '''RULE 3. All APL2 expressions are evaluated from right to left'',2µr ° ' '╕(0=V)/0' ° 'L3X:''This rule is valid in APL, which is the earlier version of APL2 ° .''' ° '''For APL2, the rule has been restated in a totally different way.''' ° '''However until we introduce some concepts that are new to APL2, the' ° '' '''newer form reduces to RULE 3 as stated.''' '╕L0' 'L4:╕(V>1)/L1X' ° '''RULE 4. Vectors in APL2 can have lengths of zero and one.'',2µr' ° '╕(0=V)/0' ° 'L4X:''This is the first concept that begins to introduce the idea tha ° t no''' ° '''matter what the APL expression, if it is valid, it must produce a'' ° ' ° '''valid APL2 object. (Shades of which came first, the chicken or the ° egg)''' '╕L0' 'L5:╕(V>1)/L1X' ° '''RULE 5. The shape of a variable is a vector. The shape of a vector' ° '' ''' is a vector of length 1. The shape of a scalar is a''' ° ''' vector of length zero.'',2µr' '╕(0=V)/0' ° 'L5X:''The left argument of RESHAPE is by definition a vector. If APL2 ° is''' ° '''to be internally consistent, the inverse operation to RESHAPE (whic ° h''' '''is indeed SHAPE) MUST return the same type of object.''' '╕L0' ° 'L6:╕(V>1)/L1X' ° '''RULE 6. Reduction of a scalar returns that scalar REGARDLESS of''' ° ''' the function to the left of the reduction operator.'',2µr' ° '╕(0=V)/0' ° 'L6X:''Reduction BY DEFINITION reduces the RANK by one. The exception ° is''' ° '''for objects whose rank is zero. Since there is no such thing as''' ° '''negative rank, reduction is undefined for scalars, and therefore''' ° '''the result is to leave the APL2 object unchanged.''' '╕L0' ° 'L7:╕(V>1)/L1X' ° '''RULE 7. The reduction of a null vector produces the identity''' ° ''' element of the function.'',2µr' '╕(0=V)/0' ° 'L7X:''A null vector may be empty, but it still has rank equal to one. ° ''' ° '''Therefore its reduction MUST be a scalar. Since there is no data''' ° '''in a null vector, the scalar must be such that it has no effect''' ° '''on the operation performed by the function: Zero added to anything' ° '' ° '''has no effect in addition; one multiplied by anything has no effect ° ''' '''in multiplication; and so on.''' '╕L0' 'L8:╕(V>1)/L1X' ° '''RULE 8. Parentheses can (and should) be used to modify the right''' ° ''' to left rule to improve efficiency'',2µr' '╕(0=V)/0' ° 'L8X:''Parentheses are used in APL2 for other purposes as well, but fo ° r''' ° '''now, we will use them just to modify the right to left rule (RULE 3 ° ).''' ° '''If you place parentheses where they are not needed, APL2 will ignor ° Xe''' '''them.''' 'L0:do' '╕(0=µW╜1╟W)/0' '╕L' ° XCs 1 2 " ° *(1997 7 13 12 28 57 536) ° Fshow ÉFX '╢V show ╢W;╢T;╢B' 'Σ Display and execute ╢W' ° '╢T╜(0=ÉNC ''╢V'')doif ''╢V╜0''' ° '╕((0=╧╢W)doif ''show ╢W,'''' '''''')/0' ° '╕((1<╧╢W)doif ''╢V show ■╢W'')/0' ''' '',╢W' ° X '╕((╢V^'':''ε╢W)doif ''╢T╜evaldd ╢W'')/L0' '''ÉEM'' ÉEA ╢W' 'L0:do' ° *(1997 7 13 12 28 57 536) ° Fshowdd ÉFX 'u╜showdd w;a;b;c;r' ° 'Σ Display a direct definition function' ° '╕((1=╧w)doif ''u╜showdd Γw'')/u╜0' ° '╕((3⌠ÉNC╞w)doif ''(ε╞w),'''' is not a function'''''')/0' ° 'c╜Γ[2]ÉCR╞w' 'c╜notb(2╞c),(Γ''aw α∙'')replace■2╟c' ° '╕((~''ΣDD''╧3╞2πc)doif ''''''Not a direct definition function'''''')/ ° 0' 'u╜1' 'b╜('' ''⌠╞c)Γ╞c' 'a╜'' ''' 'r╜2╟3πc' ° '╕((3=µc)doif ''a,(╞w),'''':'''',r,(3<µ2πc)/'''' Σ'''',3╟2πc'')/0' ° 'a╜a,(╞w),'':'',(2╟5πc),'':''' 'b╜(+\r=''('')-+\r='')''' 'b╜b∞0' ° X 'a╜a,(²3╟(b-1)╞3╟r),'':'',2╟»(b+2)╟r' 'a,(3<µ2πc)/'' Σ'',3╟2πc' ° *(1997 7 13 12 28 57 536) ° Fshowfn ÉFX 'U╜V showfn W;F;N;T;ÉIO' 'Σ Simulate ╖W[É]' ° 'T╜(0=ÉNC ''V'')doif ''V╜0''' 'ÉIO╜0' ° 'U╜r,'' '',''╖'',W,''[É]'',(╞V)╞''╖''' 'N╜1╞µF╜ÉCR W' 'N╜«∞N' ° 'N╜(N⌠'' '')ΓN' 'F╜(π''['',■N,■Γ''] ''),F' ° 'T╜(1<µ,V)doif ''F╜F[1╟V;]'' ''U╜''''''''''' 'U╜²1╟U,r,,F,r' ° X 'U╜((-+/^\'' ''=ΦU)╟U),('' ╖'')[╞V],r' ° *(1997 7 13 12 28 58 540) ° Fsimdd ÉFX 'u╜simdd w;e' 'Σ Direct definition mode' 'u╜0' ° '╕((0⌠ÉNC╞w)doif ''''''Already defined'''''')/0' 'e╜''α''ε2πw' ° 'w[2]╜Γ''u╜'',''α∙ aw'' replace 2πw' 'w╜w[1 3 2]' ° X 'w[1]╜Γε''u╜'',(e/''a ''),w[1],'' w''' 'u╜╧ÉFX w' ° XNt 0 1 ° *(1992 6 3 9 59 17 424) ° Ftab ÉFX 'U╜V tab W;T;A;B;C;D;E;F;G;M;ÉPW' 'Σ Tabulate list W' ° 'T╜(0=ÉNC ''V'')doif ''V╜0''' 'M╜''Invalid data for tabulation''' ° 'V╜4╞V' 'ÉPW╜130╛30⌐G╜V[2]+79⌡V[2]=0' ° 'L1:╕((1<╧W)doif ''''''W╜∞0'''' ÉEA ''''W╜πW'''''')/L1' ° '╕(((0=µεW)δ2<µµW)doif ''U╜(~V╧4╞0)/M'')/0' ° 'T╜(1≥µµU╜«W)doif ''U╜πW╜(U⌠'''' '''')ΓU''' ° 'T╜(0<V[1])doif ''U╜(«(Φ1,╞µW)µ(V[3]µ'''' ''''),∞(╞µW)-V[3]),'''' '''' ° ,U''' '╕(G<30)/0' 'T╜(F╜µεV[4])+C╜1╟B╜µA╜(V[3],0)╟U' ° 'T╜⌐(1╞B)÷1⌐╛(ÉPW+F)÷T' 'U╜(E╜(V[3],C)╞U),[1](B╜T,1╟B)╞A' ° '''D╜εV[4]'' ÉEA ''D╜ÉAV[εV[4]+33⌡V[4]=0]''' 'L0:A╜(T,0)╟A' ° X '╕(0=1╞µA)/0' 'U╜U,(((T+V[3]),µD)µD),E,[1]B╞A' '╕L0' ° *(1997 7 13 12 28 59 544) ° Ftest ÉFX 'U╜V test W;P' ° 'Σ Describe problem in W, (correct answer in V)' 'U╜2' 'L1:W' ° 'É╜'' ''' 'P╜æ' '''╕L0'' ÉEA ''P╜»P''' '╕(V╧P)/0' ° X 'L0:╕(0=U╜U-1)/0' '''Incorrect. Try again''' '╕L1' ° XCu 1 2 IV ° XCuc 1 26 ABCDEFGHIJKLMNOPQRSTUVWXYZ ° XNw 0 4 ° XCx 0 ! °