1 Recursive Descent (LL(1) grammars)

Transcription

1 UNIVERSITY OF CALIFORNIA Dparmn of Elcrical Enginring and Compur Scincs Compur Scinc Division CS 164 Fall 2001 R. J. Faman CS 164: Programming Languags and Compilrs: Parsing Prviously, I 1 discussd wha i mans o hav a drivaion of a snnc according o a grammar, and how on can us a drivaion (onc found) o guid ha applicaion of smanic acions ha compu a smanic valu (i.., maning or ranslaion) of a snnc (whr snnc can man an nir program). In his handou, w urn o h qusion of how on finds such a drivaion. 1 Rcursiv Dscn (LL(1) grammars) Conx-fr grammars rsmbl rcursiv programs: hy ar crainly rcursivly dfind, and on can rad a rul A BCD as oparsana in h inpu, firs pars a B, hna C, andhnad. This insigh lads o a rahr inuiiv form of parsing known as rcursiv dscn. 1.1 From Grammars o Programs Considr h following grammar, wih augmns wrin in a kind of parn languag. Grammar 1. s \ra \of \ra 2 \= $$ = (addtrm $1 $2) 2 \ra + \> $$ = $2 2 \ra $\psilon$ \> $$ = nil \ra ( ) \> $$ = $2 \ra i \> $$ = $1 Hr,i, +, (, ),and $ (ndoffil)arrminalsymbols,andhsmanicvaluofani is, w ll say, a laf nod of an absrac synax r. W dfin (dfun addtrm (T1 T2) ;; ak wo rs, rurn sum as a r (if (null T2) T1 (lis + T1 T2))) 1 Takn in larg par from h Spring 1999 nos of Prof. P. Hilfingr. 1

2 L s assum ha hr is a funcion nxtokn() ha rurns h synacic cagory of h nx okn of h inpu (on of i, +, (, ), and $ ), and a funcion poptokn() ha rmovs ha okn. W somims combin hs ino a funcion a ha opionally aks an argumn: if h nx okn dos no qual h argumn, h procdur maks a fuss, xiing wih an rror. Givn no argumn, a jus pops h okn off h inpu. W can dfin boh vrsions a onc in Lisp: (dfun a( \&opional (x nil supplid?)) (cond ((no supplid?)(poptokn)) ((qual x (nxtokn))(poptokn)) ( (rror)))) Our sragy will b o produc a s of funcions, on for ach non-rminal symbol. Th body of ach funcion will dircly ranscrib h grammar ruls for h corrsponding nonrminal. To sar wih, w ll ignor h smanic acions: (dfun S () (E) a ( \of ) (dfun E () (T)(E2)) (dfun E2 () (cond ((qual (nxtokn) +) (a) (E)) ((qual (nxtokn \of))) ;$\psilon$ ( (rror)))) (dfun T () (cond ((qual (nxtokn) \( ) (a)(e) (a \) )) ((qual (nxtokn) i) (a)) ( (rror)))) If you xamin his closly, you should s ach grammar rul is ranscribd ino program x. Nonrminals on h righ-hand sids urn ino funcion calls; rminals urn ino calls o a. To pars a program (o sar hings off), you simply call (S). If you rac h xcuion of his program for a givn snnc and look a h ordr in which calls occur, comparing i o h pars r for ha snnc, you will s ha h program ssnially prforms a prordr walk (also calld op down ) of h pars r, corrsponding o a lfmos drivaion of h r. Adding in smanic acions complicas hings only a lil. Now w mak h funcions rurn h smanic valu for hir r: (dfun S () (l ((1 (E))) (a \of ) 1)) 2

3 (dfun E () (l ((1 (T)) (2 (E2))) (addtrm 1 2))) (dfun E2 () (cond ((qual (nxtokn) +) (a) (E)) ((qual (nxtokn) \of)) ( (rror)))) (dfun T () (cond ((qual (nxtokn) \( ) (a)(prog1 (E)(a \) ))) ((qual (nxtokn) i) (a)) ( (rror))) Programming no: In lisp, (prog1 a b c.. ) is a consrucion ha valuas a, b, c in ordr, and hn rurns a. Jus lik (l ((mp a)) b c... mp). 1.2 Choosing a branch: FIRST and FOLLOW I sill havn old you whr h if samns cam from. In gnral, you ll b facd wih svral ruls for a givn nonrminal l s say A α 1, A α 2,c.,whrachα i is a sring of rminal and nonrminal symbols. Ths rcursiv dscn parsrs work by choosing ( prdicing ) which of h α i o pursu basd on h nx, as y unscannd inpu okn. Assuming firs ha non of h α i can produc h mpy sring, w can choos h branch of h funcion for A ha corrsponds o rul A α i if h nx symbol of inpu is in FIRST(α i ), which (whn α i dos no produc h mpy sring) is dfind as h s of rminal symbols ha can bgin a snnc producd from α i. As long as hs ss of symbols do no ovrlap, w can unambiguously choos which branch o ak. Suppos on of h branchs, say α k, can produc h mpy sring, in which cas w dfin FIRST(α k ) o conain h mpy sring as wll as any symbols ha can bgin α k.w should choos h α k branch if ihr h nx inpu symbol is in FIRST(α k ) or h nx inpu symbol is in FOLLOW(A), which is dfind as h s of rminal symbols ha can com immdialy afr an A in som snnial form producd from h sar symbol. Clarly, w r in roubl if mor han on α i can produc h mpy sring, so for his ranslaion o rcursiv dscn o work, w mus insis ha a mos on branch can produc h mpy sring. If hr is no ovrlap in any of h ss of rminal srings producd by h procdur abov, hn w say ha h grammar is LL(1), maning ha i can b parsd Lf o righ o giv a Lfmos drivaion, looking ahad a mos 1 symbol of inpu. 3

4 1.3 Daling wih non-ll(1) grammars. You will hav noicd, no doub, ha h grammar abov is a bi odd, compard o a normal xprssion grammar. For on hing, i looks rahr conord, and for anohr, i groups xprssions o h righ rahr han h lf (i ras a+b+c as a+(b+c) ). If I ry o wri a mor naural grammar, howvr, I run ino roubl: Grammar 2. A. s \ra \of B. \ra + C. \ra D. \ra ( ) E. \ra i Th problm is ha h s o drmin whhr o apply h firs or scond rul for braks down: h sam symbols can sar an as can sar a. Anohr problm is ha h grammar is lf rcursiv: from, on can produc a snnial form ha bgins wih ; in a program his causs an infini rcursion. Boh of hs caus h grammar o b non-ll(1). Mos books go ino a gra dal of hair o g around problms lik his. Frankly, I ak a mor pracical sanc. Th parn abov is qui common. So much so ha h grammar is ofn wrin as A. s \ra \of B. \ra + D. \ra ( ) E. \ra i whr h bracs {} indica opionally zro or mor ims. This is asily dal wih by mans of a loop: (dfun E() (l ((1 (T))) (whil (qual (nxinpu) +) (a) (sf 1 (lis + 1 (T)))) 1)) ;; for fans of h Lisp do, hr s anohr vrsion (dfun E() (do ((1 (T)(lis + 1 (T)))) ((no (qual (nxinpu) +)) 1) (a))) 2 Boom-up Parsing You can hink of h difficulis wih LL(1) parsing rflcd in h nd o mung h grammars or cha wih spcial loops as arising from hir prdiciv naur. A pur 4

5 rcursiv dscn parsr chooss which righ-hand sid o us on h basis of jus h firs rminal symbol machd by ha righ-hand sid. If w could insad wai unil w had rad an nir producion, w migh do a br job. This is in fac h cas, bu h rsuling parsrs (calld boom-up parsrs) ffcivly rquir mchanical aids o b usful. Forunaly such aids xis, in h form of ools such as Yacc or Bison. I could simply nd h sory hr, bu I bliv you should hav a las on look a how your ools work bfor saring o ra hm as black magic. 2.1 Shif-rduc Parsing L s again considr Grammar 2 from abov, and look a a rvrs drivaion of h sring i+(i+i)$ : 1. i + ( i + i ) \of 2. + ( i + i ) \of 3. + ( i + i ) \of 4. + ( + i ) \of 5. + ( + i ) \of 6. + ( + ) \of 7. + ( ) \of 8. + \of 9. \of 10. p Rad from h boom up, his is a sraighforward righmos drivaion, bu wih a mysrious gap in h middl of ach snnial form. Th gap marks h posiion of h handl in ach snnial form h porion of h snnial form up o and including h symbols producd (rading upwards) by applying h nx producion or (rading downwards) h symbols abou o b rducd by applying h nx rvrs producion. Rading his downwards, you s ha h gap procds hrough h inpu (i.., h snnc o b parsd) from lf o righ. W call h symbols lf of h gap h sack (righ symbol on op) and h symbol jus o h righ of h gap h lookahad symbol. To add smanic acions, w jus apply h ruls aachd o a givn producion ach im w us i o rduc, aaching h rsuling smanic valu o h rsuling nonrminal insanc. For xampl, suppos ha h smanic valus aachd o h hr i s in h prcding xampl ar laf nods 1, 2, and 3, rspcivly. Thn, using x : E o man smanic valu E is aachd o symbol x, w hav h following pars 1. i:1 + ( i:2 + i:3 ) \of 2. :1 + ( i:2 + i:3 ) \of 3. :1 + ( i:2 + i:3 ) \of 4. :1 + ( :2 + i:3 ) \of 5. :1 + ( :2 + i:3 ) \of 6. :1 + ( :2 + :3 ) \of 7. :1 + ( :(+ 2 3) ) \of 8. :1 + :(+ 2 3) \of 9. :(+ 1 (+ 2 3)) \of 10. p 5

6 Iniially, only h rminal symbols hav smanic valus (as supplid by h lxr). Each rducion compus a nw smanic valu for h nonrminal symbol producd, as dircd by h grammar. Wih or wihou smanic acions, h procss illusrad abov is calld shif-rduc parsing. Each sp consiss ihr of shifing h lookahad symbol from h rmaining inpu (righ of h gap) o h op of h sack (lf of h gap), or of rducing som symbols (0 or mor) on op of h sack o a nonrminal according o on of h grammar producions (and prforming any smanic acions). Each lin in h xampls abov rprsns on rducion, plus som numbr of shifs. For xampl, lin 5 rprsns h rducion of o, followd by h shif of + and i. In all hs grammars, i is convnin o hav h nd-of-fil symbol ( $ ) and h sar symbol (in h xampls, p ) occur in xacly on producion. This firs producion hn has no imporan smanic acion aachd o i. This mans ha as soon as w shif h nd-of-fil symbol, w hav ffcivly accpd h sring and can sop. 2.2 Rcognizing Possibl Handls: h LR(0) machin W can complly mchaniz h procss of shif-rduc parsing as long as w can drmin whn w hav a handl on h sack, and which handl w hav. Th algorihm hn bcoms whil (\of{} no y shifd) if (handl is on op of h sack) rduc h handl; ls shif h lookahad symbol; I urns ou, inrsingly nough, ha alhough conx-fr languags canno b rcognizd in gnral by fini-sa machins, hir righmos handls can b rcognizd. Tha is, w can build a DFA ha allows us o prform h handl is on op of h sack s by pushing h sack hrough h DFA from boom o op (lf o righ in h diagrams abov). This DFA will also ll us which producion o us o rduc h handl. To do his, I will firs show how o consruc a handl grammar a grammar ha dscribs all possibl handls. Th rminal symbols of his grammar will b all h symbols (rminal and nonrminal) of h grammar w ar rying o pars. I will hn show how o convr h handl grammar ino an NFA, afr which h usual NFA-o-DFA consrucion will finish h job. Th nonrminals of h handl grammar for Grammar 2 ar H p, H,andH. H p mans a handl ha occurs during a righmos drivaion of a sring from p. Likwis, H mans a handl ha occurs during a righmos drivaion of a sring from, and so forh. L s sar wih H p. Thr ar wo cass: ihr h sack consiss of h handl $ and w ar rady for h final rducion, or w ar sill in h procss of forming h and havn gon around o shifing h $ y in ohr words, w hav som handl ha occurs during a drivaion of som sring from. Such a handl is supposd o b dscribd by H.This givs us h ruls: $H_p$ \ra \of $H_p$ \ra $H_$ (Again, h symbols and $ ar boh rminal symbols hr; H p is h nonrminal.) Now l s considr H. From h grammar, w s ha on possibl handl for is. I is also possibl ha w ar par way hrough h procss of rducing o his, so ha w hav h wo ruls 6

7 $H_$ \ra $H_$ \ra $H_$ Likwis, w also s ha anohr possibl handl is +. I is hrfor possibl o hav + on h sack, followd by a handl for an as-y-incompl, or finally, i is possibl ha h bfor h + is no y compl. Ths considraions lad o h following ruls for H : $H_$ \ra + $H_$ \ra + $H_$ $H_$ \ra $H_$ (Th las rul is uslss, bu harmlss). Coninuing, h full handl grammar looks lik his: $H_p$ \ra \of $H_$ $H_$ \ra $H_$ $H_$ \ra + + $H_$ $H_$ $H_$ \ra i $H_$ \ra ( ) ( $H_$ This grammar has a spcial propry: h only plac ha a nonrminal symbol appars on a righ-hand sid is a h nd (h nonrminals in h handl grammar ar H p, H,and H ). This is significan bcaus grammars wih his propry can b convrd ino NFAs vry asily. Considr, for xampl, h grammar A \ra x B B \ra y A B \ra z Th NFA in Fig. 1 rcognizs his grammar. W simply ransla ach nonrminal ino a sa, and ransfr o ha sa whnvr a righ-hand sid calls for rcognizing h corrsponding nonrminal. Th ranslaion in h figur uss som psilon ransiions whr hy rally could b avoidd, bcaus his will b convnin in h producion of a machin for h handl grammar. Whn w us his chniqu o convr h handl grammar ino a NFA, w g h machin shown in Fig. 2. I hav pu labls in h sas ha hin a why hy ar prsn. For xampl, h sa labld. + is supposd o man h sa of bing par way hrough a handl for h producion + jus bfor h +. Ths labls (producions wih a do in hm) ar known as LR(0) ims. Som of h sas hav h sam labls; howvr, if you xamin hm, you will s ha any sring ha rachs on of hm also rachs h ohr, so ha h idnical labls ar appropria. Thr ar no final sas mniond, bcaus all h informaion w ll nd rsids in h labls on h sas. Th final sp is o convr h NFA of Fig. 2 ino a DFA (so ha w can asily urn i ino a program). W us h s-of-sas consrucion ha you larnd prviously. Th labls on h rsuling sas ar ss of LR(0) ims; I lav ou h labls H p, H,andH, sinc hy urn ou o b rdundan. You can vrify ha w g h machin shown in Fig. 3. 7

8 x A y B z Figur 1: Exampl of convring a rgular grammar ino a NFA. 2.3 Using h Machin W may rprsn h LR(0) machin from Fig. 3 as a sa-ransiion abl: Acion Goo Sa i + ( ) $ 0. s2 s s6 s5 2. re re re re re 3. rc rc rc rc rc 4. s2 s ACCEPT 6. s2 s s6 s9 8. rb rb rb rb rb 9. rd rd rd rd rd Th numric nris in his abl (prcdd by s (shif) in h acion abl and apparing plain in h goo abl) com from h sa ransiions (arcs) in Fig. 3. Th r (rduc) nris com from LR(0) ims wih a do a h righ (indicaing a sa of bing o h righ of a ponial handl. ) Th lrs afr r rfr o producions in Grammar 2. To s how o us his abl, considr again h sring i+(i+i)$. Iniially, w hav h siuaion 1. $_0$ i + ( i + i ) \of Hr, h sparas h sack on h lf from h unprocssd inpu on h righ; h lookahad symbol is righ afr. Th subscrip 0 indicas ha h DFA sa ha 8

9 H p p ->. - p ->. - - p -> -. H ->. ->. -> +. ->. + -> > +. ->. + -> > +. H ->. i ->. ( ) i -> i. ( -> (. ) -> (. ) ->. ( ) ( ) -> (. ) -> ( ). Figur 2: NFA from h handl grammar. 9

10 5 0 p ->. - ->. ->. + ->. i i 1 - ->. + + p ->. - i 2 -> i. -> > +. ->. i ->. ( ) 8 -> + ->. ( ) 3 ( + ( ->. i ( 4 -> (. ) ->. ->. + ->. i 7 -> (. ) ->. + ) 9 -> ->. ( ) Figur 3: DFA consrucd from Fig. 2: Th canonical LR(0) machin. 10

11 corrsponds o h lf nd of h sack is sa 0. W us h abl, saring in sa 0. Thr is nohing on h sack, and row 0 of h abl lls us ha hr is no rducion possibl, bu ha w could procss an i okn if i wr on h sack. Thrfor, w shif h i okn from h inpu, giving 2. $_0$i$_2$ + ( i + i ) \of (Th subscrip 2 shows h DFA s sa afr scanning h i on h sack). Again, w sar in sa 0 and scan h sack, using h ransiions in h abl. This lavs us in sa 2. Row 2 in h abl lls us ha no shifs ar possibl, bu w may rduc ( r ) using producion E ( i). W hrfor pop h i off h sack, and push a back on, giving 2. $_0$$_3$ + ( i + i ) \of Running h machin ovr his nw sack lands us in sa 3, which says ha no shifs ar possibl, bu w can us rducion C ( ), which givs 3. $_0$$_1$ + ( i + i ) \of Now h machin nds up in sa 1, whos row lls us ha ihr a + or an $ could b nx on h sack, so ha w can shif ihr of hs, lading o 3a. $_0$$_1$ +$_6$ ( i + i ) \of Th sa 6 nry lls us ha w can shif (, and hn h sa 4 nry lls us w can shif i, giving 3b. $_0$$_1$ +$_6$($_4$ i$_2$ + i ) \of whrupon w s, again from h sa 2 nry, ha w should rduc using producion E: 4. $_0$$_1$ +$_6$ ($_4$ $_3$ + i ) \of and h sa 3 nry lls us o rduc using producion C: 5. $_0$$_1$ +$_6$ ($_4$ $_7$ + i ) \of and so forh. In gnral, hn, w rpadly prform h following sps for ach shif and rducion h parsr aks: FINDSTATE: sa = 0; for ach symbol, s, on h sack, sa = abl[sa][s]; FINDACTION: if abl[sa][lookahad] is s$n$ push h lookahad symbol on h sack; advanc h inpu; ls if abl[sa][lookahad] is r$k$ L $A \ra x_1\cdos x_m$ b producion $k$; pop $m$ symbols from h sack; push symbol $A$ on h sack; ls if abl[sa][lookahad] is ACCEPT nd h pars; 11

12 Th FINDACTION par of his fragmn aks a consan amoun of im for ach acion. Howvr, h im rquird for FINDSTATE incrass wih h siz of h sack. W can spd up h parsing procss wih a bi of mmoizaion. Rahr han sav h sack symbols, w insad sav h sas ha scanning hos symbols rsuls in (h subscrips in my xampls abov). Each parsing sp hn looks lik his: FINDACTION: if abl[op(sack)][lookahad] is s$n$ push $n$ on h sack; advanc h inpu; ls if abl[op(sack)][lookahad] is r$k$ L $A \ra x_1\cdos x_m$ b producion $k$; pop $m$ sas from h sack; // Rmindr: op(sack) is now changd! push abl[op(sack)][$a$] on h sack; ls if abl[sa][lookahad] is ACCEPT nd h pars; and our sampl pars looks lik his: 0. 0 i + ( i + i ) \of ( i + i ) \of ( i + i ) \of ( i + i ) \of 3a ( i + i ) \of 3b i + i ) \of 3c i ) \of i ) \of i ) \of 5a i ) \of 5b ) \of ) \of ) \of 7a \of \of \of 9a ACCEPT I s imporan o s ha all w hav don wih his chang is o spd up h pars. 2.4 Rsolving conflics Grammar 2 is calld an LR(0) grammar, maning ha is LR(0) machin has h propry ha ach sa conains ihr no rducion ims (ims wih a do a h far righ) or xacly on rducion im and nohing ls. In ohr words, an LR(0) grammar is on ha can b parsd from Lf o righ o produc a Righmos drivaion using a shif-rduc parsr ha dos no consul h lookahad characr (uss 0 symbols of lookahad). Fw grammars ar so simpl. Considr, for xampl, 12

13 0 p ->. - ->. ->. v ->. f ( ) v ->. i f ->. i v i f 1 6 p ->. - - p -> >. 3 v -> v. ->. v 4 ->. f ( ) v -> i. f -> i. i v ->. i f ->. i f ( ) -> f. ( ) -> f (. ) -> f ( ). Figur 4: LR(0) machin for Grammar 3. Grammar 3. A. program \ra \of B. C. \ra D. \ra f ( ) E. \ra v F. f \ra i G. v \ra i which givs us h DFA in Fig. 4. As you can s from h figur, hr ar problms in sas #2 and #4. Sa #2 has an LR(0) shif/rduc conflic: i is possibl boh o rduc by rducion C or o shif h In his paricular cas, i urns ou ha h corrc hing o do is o shif whn h lookahad symbol and o rduc ohrwis; ha is, rducing will always caus h pars o fail lar on. Sa #4 has an LR(0) rduc/rduc conflic: i is possibl o rduc ihr by rducion F or G. In his cas, h corrc hing o do is o rduc using F if h nx inpu symbol is ( and by G ohrwis. W nd up wih h following parsing abl: 13

14 Acion Goo Sa ( ) $ f v 0. s s6 2. rc s7 rc rc rc 3. re re re re re 4. rg rg rf rg rg 5. s8 6. ACCEPT 7. s s10 9. rb rb rb rb rb 10. rd rd rd rd rd Bcaus h choic bwn rducion and shif, or bwn wo rducions, dpnds on h lookahad symbol (in conras o Grammar 2), w say Grammar 3 is no LR(0). Howvr, sinc on symbol of lookahad suffics, w say ha i is LR(1) parsabl from Lf o righ producing a Righmos drivaion using a shif-rduc parsr wih 1 symbol of lookahad. In fac, Grammar 3 is wha w call LALR(1), h subclass of LR(1) for which h parsing abl has h sam sas and columns as for h LR(0) machin, and w mrly hav o choos h nris proprly o g h dsird rsul. (LALR mans Lookahad LR. Sinc LR parsrs do look ahad anyway, i s a rribl nam, bu w r suck wih i.) Yacc and Bison produc LALR(1) parsrs. Th class LR(1) is biggr, bu fw pracical grammars ar LR(1) wihou bing LALR(1), and LALR(1) parsing abls ar considrably smallr. Unforunaly, i is no clar from jus looking a h machin ha w hav filld in h problmaic nris corrcly. In paricular, whil h choic bwn rducions F and G in sa #4 is clar in his cas, h gnral rul is no a all obvious. As for h LR(0) shifrduc conflic in sa #2, i is obvious ha is h lookahad symbol, hn shifing has o b accpabl, bu prhaps his is bcaus h grammar is ambiguous and ihr h shif or h rducion could work, or prhaps if w lookd wo symbols ahad insad of jus on, w would somims choos h rducion rahr han h shif. On sysmaic approach is o us h FOLLOW ss ha w usd in LL(1) parsing. Facd wih an LR(0) rduc/rduc conflic such as f i vs. v i in sa #4, w choos o rduc o f if h lookahad symbol is in FOLLOW(f), choos o rduc v if h lookahad symbol is in FOLLOW(v), and choos ihr on ohrwis (or lav h nry blank). Likwis, w can assur ha h LR(0) shif-rduc conflic in sa #2 is proprly rsolvd in favor of as long dos no appar in FOLLOW(), as in fac i dosn. Whn his simpl mhod rsolvs all conflics and lls us how o fill in h LR(0) conflics in h abl, w say ha h grammar is SLR(1) (h S is for Simpl ). Grammar 3 happns o b SLR(1). Howvr, hr ar cass whr h FOLLOW ss fail o rsolv h conflic bcaus hy ar no snsiiv o h conx in which h rducion aks plac. Thrfor, ypical shifrduc parsr gnraors go a sp furhr and us h full LALR(1) lookahad compuaion. This works similarly o h FOLLOW compuaion dscribd in h xbook (pg. 189 of ASU ). W aach a s of lookahad symbols o h nd of ach LR(0) im, giving wha is calld an LR(1) im. Think of an LR(1) im such as \ra. v, 14

15 0 p ->. - ->., L1 L1 ->. v, L2 ->. f ( ), L2 v ->. i, L2 f ->. i, L3 v i 1 - p -> >., L1 3 -> v., L2 v 4 i v -> i., L2 6 9 p -> -. L3 7 L1 ->., L1 L1 ->. v, L2 ->. f ( ), L2 v ->. i, L2 f f -> i., L3 f ->. i, L3 5 f 8 ( -> f. ( ), L2 -> f (. ), L2 Lgnd: L1 = - ) 10 L2 = -> f ( )., L2 L3 = ( Figur 5: LALR(1) machin for Grammar 3. as maning w could b a h lf nd of a handl for, and afr ha handl, w xpc o s ihr an $ or W add hs lookahads o h LR(0) machin in Fig. 4 by applying h following wo opraions rpadly unil nohing changs, saring wih mpy lookahad ss: If w s an im of h form A α.bβ, L 1 inasa(whrl is a s of lookahads, B is a nonrminal, and α and β ar squncs of 0 or mor rminal and nonrminal symbols), hn for vry ohr im in ha sam sa of h form B.γ, L 2, add h s of rminal symbols FIRST(βL 1 )ohsl 2. (Hr, w dfin FIRST(L 1 )o b simply L 1. Thrfor, FIRST(βL 1 ) is simply FIRST(β) ifβ dos no produc h mpy sring, and ohrwis i is FIRST(β) L 1 ɛ). If w s an im of h form A α.xβ, L 1 in a sa, hn find h ransiion from ha sa on symbol X and find im A αx.β, L 2 in h arg of ha ransiion. Add h symbols in L 1 o L 2. Applying hs opraions o h machin in Fig. 4 givs h LALR(1) machin in Fig