Fig.25: the Role of LEX
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1 The Lnguge for Specifying Lexicl Anlyzer We shll now study how to uild lexicl nlyzer from specifiction of tokens in the form of list of regulr expressions The discussion centers round the design of n existing tool clled LEX, for utomticlly generting lexicl nlyzer progrm A LEX source progrm is specifiction of lexicl nlyzer, consisting of set of regulr expressions together with n ction for ech regulr expression The ction is piece of code which is to e executed whenever token specified y the corresponding regulr expression is recognized The output of LEX is lexicl nlyzer progrm constructed from the LEX source specifiction Unlike most progrmming lnguges, source progrm for LEX does not supply ll the detils of the intended computtion Rther, LEX itself supplies with its output progrm tht simultes finite utomton This progrm tkes trnsition tle s dt The trnsition tle is tht portion of LEX's output tht stems directly from LEX's input The sitution is shown in Fig25Where the lexicl nlyzer L is the trnsition tle plus the progrm to simulte n ritrry finite utomton expressed s trnsition tle Only L is to e included in the compiler eing uilt LEX Source LEX Compiler Lexicl Anlyzer L Input Text Lexicl Anlyzer L Sequence of tokens Fig25: the Role of LEX A LEX source progrm consists of two prts, sequence of uxiliry definition followed y sequence of trnsltion rules Auxiliry Definitions The uxiliry definitions re sttements of the form: 41
2 D 1 =R 1 D 2 =R 2 D n =R n Where ech D i is distinct nme, nd ech R i is regulr expression whose symol re chosen from {D 1, D 2, D i-1 }, ie, chrcters or previously defined nmes The D i 's re shorthnd nmes for regulr expressions is our input symol lphet Exmple: We cn define the clss of identifiers for typicl progrmming lnguge with the sequence of uxiliry definitions Letter = AB Z Digit = 01 9 Identifier = Letter (LetterDigit)* Trnsltion Rules The trnsltion rules of LEX progrm re sttements of the form:- P 1 {A 1 } P 2 {A 2 } P m {A m } Where ech P i is regulr expression clled pttern, over the lphet consisting of nd the uxiliry definition nmes The ptterns descrie the form of the tokens Ech A i is progrm frgment descriing wht ction the lexicl nlyzer should tke when token P i is found The A i 's re written in conventionl progrmming lnguge, rther thn ny prticulr lnguge, we use pseudo lnguge To crete the lexicl nlyzer L, ech of the A i 's must e compiled into mchine code 42
3 The lexicl nlyzer L creted y LEX ehves in the following mnner: L red its input, one chrcters t time, until it hs found the longest prefix of the input which mtches one of the regulr expressions, P i Once L hs found tht prefix, L removes it from the input nd plces it in uffer clled TOKEN (Actully, TOKEN my e pir of pointers to the eginning nd end of the mtched string in the input uffer itself) L then executes the ction A i It is possile, tht none of the regulr expressions denoting the tokens mtches ny prefix of the input In tht cse, n error hs occurred, nd L trnsfers control to some error hndling routine It is lso possile tht two or more ptterns mtch the sme longest prefix of the remining input If tht is the cse, L will rek the tie in fvor of tht token which cme first in the list of trnsltion rules Exmple: Let us consider the collection of tokens defined in Fig7, LEX progrm is shown in Fig26 AUXILIARY DEFINITION Letter= AB Z Digit= 01 9 TRANSLATION RULES BEGIN {return 1} END {return 2} IF {return 3} THEN {return 4} ELSE {return 5} letter(letterdigit)* {LEX VAL:= INSTALL( ); return 6} digit + {LEX VAL:= INSTALL( ); return 7} < {LEX VAL := 1; return 8} <= {LEX VAL := 2; return 8} = {LEX VAL := 3; return 8} 43
4 < > {LEX VAL := 4; return 8} > {LEX VAL := 5; return 8} >= {LEX VAL := 6; return 8} Fig26: LEX Progrm Suppose the lexicl nlyzer resulting from the ove rules is given input BEGIN followed y lnk Both the first nd sixth pttern mtches BEGIN, nd no pttern mtches longer string Since the pttern for keyword BEGIN precedes the pttern for identifier in the ove list, the conflict is resolved in fvor of the keyword For nother exmple, suppose <= re the first two chrcters red While pttern < mtches the first chrcter, it is not the longest pttern mtching prefix of the input Thus LEX's strtegy tht the longest prefix mtching pttern is selected mkes it esy to resolve the conflict etween < nd <=, y choosing <= s the next tokens Implementing the Lexicl Anlyzer The LEX cn uild from its input lexicl nlyzer tht ehves roughly like finite utomton The ide is to construct NFA Ni for ech tokens pttern Pi in the trnsltion rules, then links these NFA's together with new strt sttes s shown in Fig27 Next we convert this NFA to DFA Fig27: NFA recognize severl tokens simultneously 44
5 There re severl nunce in this procedure of which the reder should e wre First, there re in the comined NFA severl different "ccepting sttes" Tht is, the ccepting stte of ech N i indictes tht its own token, P i hs een found When we convert to DFA, the susets we construct my include severl different finl sttes Moreover the finl sttes lose some of their significnce, since we re looking for the longest prefix of the input which mtches some ptterns After reching finl stte, the lexicl nlyzer must continue to simulte the DFA until it reches stte with no next stte for the current input symol Upon reching termintion, it is necessry to review the sttes of the DFA which we hve entered while processing the input Ech such stte represents suset of the NFA's sttes, nd we look for the lst DFA stte which includes finl stte for one of the pttern-recognizing NFA's N i Tht finl stte indictes which token we hve found If none of the sttes which the DFA hs entered includes ny finl stte of the NFA, then we hve n error condition If the lst DFA stte to include finl NFA stte in fct includes more thn one finl stte, then the finl stte for the pttern listed first hs priority Exmple: Suppose we hve the following LEX progrm AUXILIARY DEFINITION (none) TRANSLATION RULES {} /* ctions re omitted here*/ {} * + {} The three tokens ove re recognized y the simple utomt of Fig28 We my convert the NFA's of Fig28 into one NFA s descried erlier The result is shown in Fig29 Then this NFA my e converted to DFA using the lgorithm nd the trnsition tle is shown in Fig30, where the sttes of the DFA hve een nmed y lists of the sttes of the NFA The lst column in Fig30 indictes the token which will e recognized if tht stte is the lst stte entered tht recognizes ny token t ll In ll cses ut the lst line, stte 68, the token recognized is the only token whose finl stte is included mong the NFA sttes forming the DFA stte For exmple, mong NFA sttes 2, 4, nd 7, 45
6 only 2 is finl, nd it is the finl stte of the utomton for regulr expression '' in Fig28 Thus, DFA stte 247 recognizes token '' In the cse of DFA stte 68, oth 6 nd 8 re finl sttes of their respective nondeterministic utomt Since the trnsltion rules of our LEX progrm mention efore * +, NFA stte 6 hs priority, nd we nnounce tht hs een found in DFA stte Fig28: Three tokens re recognized y the simple utomt Fig29: NFA recognizing three different tokens 46
7 Stte Token found none * none * Fig30: Trnsition Tle for DFA Suppose tht the first input chrcters re The DFA of Fig30 strts off in stte 0137 On input '' it goes to stte 247 Then on input '' it progresses to stte 58, nd on input '' it hs no next stte We hve thus reched termintion, progressing through the DFA sttes 0137, then 247, then 58 The lst of these includes the finl NFA stte 8 from Fig28 Thus the ction for stte 58 of the DFA is to show tht the token * + hs een recognized nd to select, the prefix of the input tht led to stte 58, s TOKEN If the DFA stte 58, tht is the lst stte entered efore termintion not include finl stte of some NFA, then we would consider the DFA stte previously entered, tht is stte 247 nd recognize the token '', which stte 247 cll for The prefix '' would in tht cse e TOKEN In Fig31 we see NFA s for the ptterns of Fig26 We use here simple NFA for identifiers nd constnts 47
8 0 1 B E G I N E N D I 12 F 13 T 15 H 16 E 17 N E 20 L 21 S 22 E Letter 25 Letter or digit 26 Digit 27 Digit 28 < < = = < > > > = Fig31: Comined NFA for Tokens 48
9 The DFA constructed from Fig31 is shown in Fig32 Ech stte of the deterministic utomt is suset of the sttes of the nondeterministic utomton Stte S 0 is the set of sttes {0, 1, 7, 11, 14, 19, 24, 26, 28, 30, 33, 35, 38, 40} S 0 # # # # # B E G I N 2,25 3,25 4,25 5,25 6,25 E # # # N D 8,20,25 9,25 10,25 # # # L S E 21,25 22,25 23,25 I # # F 12,25 13,25 # # # # T H E N 15,25 16,25 17,25 18,25 Letter other tht B,E,I,T 25 Letter or Digit Digit 27 Digit < = 29,31,36 32 = 34 > 37 Fig32: DFA constructed From Fig31 > = 39,
10 The sttes mrked # hve trnsition to stte {25} on ll letters nd digit for which no trnsition is shown On input such s 'THE' followed y non letter or digit, the DFA rek in stte {17, 25}, which contin ccepting stte 25 of the nondeterministic utomton Tht stte signls tht n identifier hs een found nd the pproprite ction is tken On input '<A', the lexicl nlyzer goes to stte {29, 31, 36} on '<', from which no trnsition on 'A' is possile On the three stte of NFA, only stte 29 is ccepting nd it indictes tht the token '<' hs een found, nd tke the specified ction If we wnt to minimize the DFA, we must use one of the methods for minimiztion, nd fter tht we see tht the result DFA is the sme 50
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