Bottom-Up Parsing. Canonical Collection of LR(0) items. Part II

Transcription

1 2 ottom-up Prsing Prt II 1 Cnonil Colletion of LR(0) items CC_LR(0)_I items(g :ugmented_grmmr){ C = {CLOURE({ })} ; repet{ foreh(i C) foreh(grmmr symol X) if(goto(i,x) && GOTO(I,X) C) C = C {GOTO(I,X)}; }until(no new sets of items re dded to C) return C; }

2 4 LR(0) utomton G : ugmented grmmr LR(0) utomton for G Q, q0, GOTO: Q (T G N G ) Q, F where: Q = F = items(g ), q0 = CLOURE({ }) 3 Constrution of the LR(0) utomton for the expression grmmr: E E E E + T T T T* F F F (E) id

3 E E E E + T T T T* F F F (E) id 5 E E E E + T T T T* F F F (E) id id I 0 E >.E E >. E+T E >.T T >.T*F T >.F F >.(E) F >.id ( F E I + I 6 T 1 E > E+. T E > E. T >. T*F E > E. + T T >.F F F >.(E) F >.id ( T I 2 E > T. T > T.*F * I 7 T > T*.F F >.(E) F >. id id I 5 I 4 F ( I 9 E > E+T. T > T.*F I 3 I 10 T > T*F. * I 7 I 3 T > F. id I 5 I 4 I 5 F > id. id I 4 F > (. E ) E >. E + T E >.T T >. T * F T >. F F >. ( E ) F >. id E F T I 8 F > ( E. ) E > E. + T I 2 ) + I 6 I 11 F > ( E ). ( I 3 6

4 8 E E E E + T T T T* F F F (E) id F E + T I 0 I 1 I 6 T id F $ ( id ept ( I 9 * I 2 * id I 7 ( F I 10 I 5 + I 8 ) I 11 T I 4 id E ( F I 3 7 hift-redue prsing using LR(0) utomton Push(I0); repet{ //egin to sn the input from left to right Ii = top() nd next input symol ; if (Ij = GOTO(Ii,)) then shift nd Ij; // Push() // push(ij). else if ( β Ii) then{ perform redue y β ; go to the stte Ij = GOTO(I,) nd push nd Ij into the stk //where I is the stte on the top_of stk //fter removing β nd the orresponding sttes; } _exit(rejet if none of the ove n e done); _exit(report onflits if more thn one n e done); } until EOF is seen

5 MODEL OF N LR PRER input 1... i... n $ s m s m-1 : : $ LR Prsing Progrm output stk CTION GOTO 9 I0,...,In enumertion of items(g') I0= CLOURE({ʼ }) Q = {0,...,n} s Is GOTO(s,X)=s' GOTO(Is,X)= Is' Proposition: if GOTO(s',X)=s nd GOTO(s",Y)=s then X=Y s Q we ssoite unique symol X s T G N G to s 0 if GOTO(s',X)=s then X s = X. Moreover, X 0 represents the symol $ (ottom of the stk) Q, 0, GOTO: Q (TGʼ NGʼ) Q, F Fix for G' finite ijetive enumertion p: {1,...,n} G'-produtions, s.t. p(1)= ' 10

6 12 Construt the LR(0) utomton for the grmmr C Ce C df 11 C Ce C df ept

7 C Ce C df!"#$%&'()*+,--,./01!"#$%&'&()*+&,-..-/012 3%4%#*&5)*("&)+*6()7&$6#)7%89 (:&;& &α<β&*6%)&#==&,>31?0β2@ε&*+&,-..-/0<2 (:&;& &α<&*6%)&#==&,-..-/0;2&*+&,-..-/0<2 (:&;& &α<β&#)=&ε&(8&()&,>31?0β2&*6%)&#==&,-..-/0;2& *+&,-..-/0<2 Follow 1 ()={$} Follow 1 ()={f,$} Follow 1 ()={f,$} Follow 1 (C)={e,f,$} 13 C Ce C df!"#$%&'()*+,--,./01!"#$%&'&()*+&,-..-/012 3%4%#*&5)*("&)+*6()7&$6#)7%89 (:&;& &α<β&*6%)&#==&,>31?0β2@ε&*+&,-..-/0<2 (:&;& &α<&*6%)&#==&,-..-/0;2&*+&,-..-/0<2 (:&;& &α<β&#)=&ε&(8&()&,>31?0β2&*6%)&#==&,-..-/0;2& *+&,-..-/0<2 Follow 1 ()={$} Follow 1 ()={f,$} Follow 1 ()={f,$} Follow 1 (C)={e,f,$} 14

8 15 terminls nd $ PRING TLE non terminls tte CTION GOTO i = CTION[i,] G = GOTO[j,] j G CTION[i,]= s j (shift symol j (nmely Xj)) CTION[i,]= r k (redue prodution p(k)) CTION[i,]= ept CTION[i,]= error 16

9 LR-PRER CONFIGURTION onfigurtion stk remining input ( 0 s 1... s m, i i+1... n $ ) top right-sententil form X0 Xs 1... Xs m i i+1... n 17 EHVIOR OF THE LR PRER ( s 0 s 1... s m, i i+1... n $ ) CTION[sm,i] = shift s symol Xs ( s 0 s 1... s m s, i+1... n $ ) ( s 0 s 1... s m, i i+1... n $ ) CTION[sm,i] = redue k ( s 0 s 1... s m-r s, i i+1... n $ ) p(k)= β β = r s=goto[s m-r, ] symol Xs 18

10 20 EHVIOR OF THE LR PRER ( s 0 s 1... s m, i i+1... n $ ) CTION[sm,i] = ept end of prsing ( s 0 s 1... s m, i i+1... n $ ) CTION[sm,i] = error error reovery routine 19 LR-prsing lgorithm

11 F E + T T id F $ ( id ept ( * 2 7 F id ( + ) 5 8 id T E 4 ( F 3 9 * X 1 = E X 2 = T X 3 = F X 4 = ( X 5 = id X 6 = + X 7 = * X 8 = E X 9 = T X 10 = F X 11 = ) 21 X 1 = E X 2 = T X 3 = F X 4 = ( X 5 = id X 6 = + X 7 = * X 8 = E X 9 = T X 10 = F X 11 = ) id id + id 22

12 24 X 1 = E X 2 = T X 3 = F X 4 = ( X 5 = id X 6 = + X 7 = * X 8 = E X 9 = T X 10 = F X 11 = ) 23 1.G : ugmented grmmr, with enumertion p of produtions; 2.onstrut the LR(0) utomton (nonil item set C={I0,...,In}, I0=CLOURE[ ] GOTO funtion); 3.foreh stte i: () if ( α β Ii && GOTO[Ii,]=Ij) dd shift j to CTION[i,]; () if ( α Ii && ) foreh ( FOLLOW()) dd redue p -1 ( α) to CTION[i,]; () if ( Ii) dd ept to CTION[i,$] 4. foreh (stte i,k && symol ) if (GOTO[Ij,]=Ik) dd k to GOTO[s,] 5. if (CTION[i,] is undefined) dd error to CTION[i,] 6. if (GOTO[i,] is undefined) dd error to GOTO[i,]

13 26 25 LR-prsing lgorithm

14 ' C C Ce df Exerise: Construt the PRING TLE 27 ' C q2 '!" q8 C!d"f!"!" C Ce df Prse tle omplete. Press "prse" to use it. q2 '!" q0 '!"!"!"!" q8 C!d"f!"!" q0 '!"!"!"!" q4 d!"ce!"!"c q10 FIRT C!d"f q1!" q4 d!"ce!"!"c q10 C!d"f q1!" f q5!" f q5!" q11 C!df" q11 C!df" q3!" FOLLOW {, } { f, $ } { } { f, $ } C { d } { f, e, $ } {, } { $ } q3!" C q6!"ce C!"df!"C q7!c"!c"e C q6 e q7!c"!c"e!"ce C!"df!"C q9!ce" e q9!ce" 28

15 30 ' C C Ce df q2 '!" q8 C!d"f!"!" q0 '!"!"!"!" d q1!" q4!"ce!"!"c q5!" f q11 q10 C!df" C!d"f q3!" C q6 q7!c"!c"e!"ce C!"df!"C e q9!ce" Prse tle omplete. Press "prse" to use it. FIRT FOLLOW {, } { f, $ } { } { f, $ } C { d } { f, e, $ } {, } { $ } q1 29 q2 q0!" q3!" '!" q8 C!d"f!"!" '!"!"!"!" q4 d!"ce!"!"c q5!" Construting n LR-prsing Tle I0=CLOURE[ ] GOTO funtion); q11 q10 C!df" C!d"f C q6 q7!c"!c"e!"ce C!"df!"C e q9!ce" 1.G : ugmented grmmr, with enumertion p of produtions; 2.onstrut the LR(0) utomton (nonil item set C={I0,...,In}, 3.foreh stte i: () if ( α β Ii && GOTO[Ii,]=Ij) dd shift j to CTION[i,]; () if ( α Ii && ) foreh ( FOLLOW()) dd redue p -1 ( α) to CTION[i,]; () if ( Ii) dd ept to CTION[i,$] 4. foreh (stte i,k && symol ) if (GOTO[Ij,]=Ik) dd k to GOTO[s,] 5. if (CTION[i,] is undefined) dd error to CTION[i,] 6. if (GOTO[i,] is undefined) dd error to GOTO[i,]

16 ' Prse tle omplete. Press "prse" to use it. FIRT FOLLOW { } {, } {, } { } { } { $ } q0!" '!" q1 '!" q9!" q4 q0!" '!"!"!"!"!" q1 '!" q9!" q0 q2!" '!"!"!"!" q1 q7 '!"!" q8 q4!"!"!"!"!"!"!"!" q7!" q3!"!"!" q4 q9!"!"!"!" q2!" q12!" q8!" q8 q3!"!"!"!" q12!" q2!"!"!" q7!" q6!"!"!" q6!"!"!" q3!"!"!" q12!" q5!" q10!" q11!" q5!" q6 q10!"!"!"!" q11!" q5!" q10!" q11!" 31 Do eleted Do tep Do ll Next Prse ' Prse tle omplete. Press "prse" to use it. FIRT $ FOLLOW { } {, } 0{, } s2 { } 1 { } 1 { $ } 2 s4 3 3 s6 s7 5 4 s4 s9 8 q0 q2 q3 q5 5 s10!"!"!"!" rse tle omplete. Press "prse" to use it. 6 s6 s7 11 '!"!"!" 7 r5 FIRT FOLLOW!"!" Next Prse tle omplete. to use it. } 8 s12 rse Press { }"prse" {, } {, { } 9 r3 r3 q10 { } {, } { } {, } { $ } { } q7 10 r1 { } { $ } q4!" q9 11 r4!" q6 12!" r2 r2!" q1!"!" q8 q12 q11!"!" q0 q2 q3 q5 '!"!"!"!"!"!"!"!" '!" q0!" '!" q1 '!" q9!" $ 0 s s4 3 3 s6 s7 5 q4!"!"!" q7!" 4 s4 s9 8!"!"!" q6 q10!" 32

17 it is not miguous Item set? 33 Item set : Prsing tle? 34

18 36 Item set : = if ( α Ii && ) foreh ( FOLLOW()) dd redue p -1 ( α) to CTION[i,]; prsing tle shift 6 CTION[2,=] symol = FOLLOW(R) nd therefore: redue R L CTION[2,=] onflit! 35 Vile prefix prefix of right-sententil forms tht n pper on the top of the stk Not ll prefixes of right-sententil forms n pper on the stk, however, sine the prser must not shift pst the hndle Exmple: TCK: (, (E, nd (E), ut not (E)* (E) is hndle (the prser must redue it to F efore shifting *)

19 38 Vile prefix: prefix of right-sententil forms tht n pper on the top of the stk nd tht does not ontinue pst the right end of the rightmost hndle of tht sententil form. prefix γ of αβ, where ' * αw αβw rm rm β 1 β 2 is vlid for vile prefix αβ 1 if ' * αw αβ 1 β 2 w rm rm 37 β 1 β 2 is vlid for vile prefix αβ 1 if ' * αw αβ 1 β 2 w rm rm if β 2 ε then shift if β 2 =ε then redue β 1 possile onflits Two vlid items my tell us to do different things for the sme vile prefix. ome of these onflits n e resolved y looking t the next input symol THEOREM: the set of vlid items for vile prefix γ is extly the set of items rehed from the initil stte long the pth leled γ in the LR(0) utomton for the grmmr

20 Item set : = Prsing tle shift 6 CTION[2,=] symol = FOLLOW(R) nd therefore: redue R L CTION[2,=] Conflit! 39 LR(1) PRING Grmmr: L = R R L R id R L shift 6 CTION[2,=] symol = FOLLOW(R) nd therefore: redue R L CTION[2,=] there is no right-sententil form of the grmmr tht egins R =... Thus stte 2, whih is the stte orresponding to vile prefix L only, should not relly ll for redution of tht L to R $L $R 40

21 In LR(1) prsing, if α i, nd FOLLOW(), then we perform the redution α However, it is possile tht when stte i is on the top of the stk, we hve the vile prefix βα on the top of the stk, nd β nnot e followed y. In this se, we nnot perform the redution α. 41 LR(1) items n LR(1) item is in the form of [ α β,] 1.the first field α β is n LR(0) item 2.the seond field is terminl elonging to suset (possily proper) of FOLLOW[]. Intuition: perform redution sed on n LR(1) item [ α, ] only when the next symol is. Insted of mintining FOLLOW sets of vile prefixes, we mintin FIRT sets of possile future extensions of the urrent vile prefix. [ α β, ] is vlid for vile prefix γ if there exists derivtion where γ = δα w = w' or w=ε nd =$. ' * δw δαβw rm rm γ 42

22 Exmples of LR(1) items Grmmr: = = rm rm vile prefix n reh [, ] = = rm rm vile prefix n reh [,$] [ α β, ] is vlid for vile prefix γ if there exists derivtion ' * δw δαβw rm where γ = δα w = w' or w=ε nd =$. rm 43 Finding ll LR(1) items Ides: redefine the losure funtion. uppose [ α β,] is vlid for vile prefix γ δα. In other words, = δ ω = δ αβ ω. rm rm γ ω is ɛ or sequene of terminls. Then for eh prodution η, ssume βω derives the sequene of terminls eω. γ γ γ = rm δα βω = rm δα eω = rm δαη eω Thus [ η,] is lso vlid for γ for eh FIRT(β). Note is terminl. o FIRT(β) =FIRT(βω). [ α β, ] is vlid for vile prefix γ if there exists derivtion ' * δw δαβw rm rm γ where γ = δα w = w' or w=ε nd =$. 44

23 46 set_of_items CLOURE(I: set_of_items){ J=I; repet{ foreh ( [ α β,] J) foreh ( γ G) foreh ( FIRT[β]) J = J {[ γ,]}; }until (no more new items re dded to J) return J; } 45 set_of_items GOTO(I set_of_items, X:symol){ J= ; foreh([ α Xβ,] I) J = J {[ αx β,]}; return CLOURE(J); }

24 48 CC_LR(1)_I items(g :ugmented_grmmr){ C = {CLOURE({[,$]})} ; repet{ foreh(i C) foreh(grmmr symol X) if(goto(i,x) && GOTO(I,X) C) C = C {GOTO(I,X)}; }until(no new sets of items re dded to C) return C; } 47 1.G : ugmented grmmr, with enumertion p of produtions; 2.onstrut the LR(1) utomton (nonil item set C={I0,...,In}, I0=CLOURE[[,$]] GOTO funtion); 3.foreh stte i: () if ([ α β,] Ii && GOTO[Ii,]=Ij) dd shift j to CTION[i,]; // is terminl () if ([ α,] Ii && ) dd redue p -1 ( α) to CTION[i,]; () if ([,$] Ii) dd ept to CTION[i,$] 4. foreh (stte i,k && symol ) if (GOTO[Ij,]=Ik) dd k to GOTO[s,] 5. if (CTION[i,] is undefined) dd error to CTION[i,] 6. if (GOTO[i,] is undefined) dd error to GOTO[i,]

25 50 Exmple for onstruting LR(1) losures Grmmr: CC C C d losure 1 ({[, $]}) = {[, $], [ CC, $], [C C, /d], J=I; [C d, /d]} Note: repet{ FIRT(ɛ$) = {$} FIRT(C$) = {, d} [C foreh C, /d] ( mens γ G) set_of_items CLOURE(I: set_of_items){ } foreh ( [ α β,] J) [Cforeh C, ] ( nd FIRT[β]) J = J {[ γ,]}; [C C, d]. }until (no more new items re dded to J) return J; tion 1 GOTO 1 stte d $ C 0 s3 s ept 2 s6 s7 5 3 s3 s4 8 4 r3 r3 5 r1 6 s6 s7 9 7 r3 8 r2 r2 9 r2 49 Exmple for onstruting LR(1) losures Grmmr: CC C C d losure 1 ({[, $]}) = {[, $], [ CC, $], [C C, /d], [C d, /d]} Note: FIRT(ɛ$) = {$} FIRT(C$) = {, d} [C C, /d] mens [C C, ] nd [C C, d]. set_of_items CLOURE(I: set_of_items){ J=I; repet{ foreh ( [ α β,] J) foreh ( γ G) foreh ( FIRT[β]) J = J {[ γ,]}; }until (no more new items re dded to J) return J; }

26 52 Exmple for onstruting LR(1) losures Grmmr: CC C C d losure 1 ({[, $]}) = {[, $], [ CC, $], [C C, /d], [C d, /d]} Note: FIRT(ɛ$) = {$} FIRT(C$) = {, d} [C C, /d] mens [C C, ] nd [C C, d]. set_of_items CLOURE(I: set_of_items){ J=I; repet{ foreh ( [ α β,] J) foreh ( γ G) foreh ( FIRT[β]) J = J {[ γ,]}; }until (no more new items re dded to J) return J; } 51 set_of_items CLOURE(I: set_of_items){ J=I; repet{ foreh ( [ α β,] J) foreh ( γ G) foreh ( FIRT[β]) J = J {[ γ,]}; }until (no more new items re dded to J) return J; }

27 54 Exmple for onstruting LR(1) losures Grmmr: CC C C d losure 1 ({[, $]}) = {[, $], [ CC, $], [C C, /d], [C d, /d]} Note: FIRT(ɛ$) = {$} FIRT(C$) = {, d} [C C, /d] mens [C C, ] nd [C C, d]. tion 1 GOTO 1 stte d $ C 0 s3 s ept 2 s6 s7 5 3 s3 s4 8 4 r3 r3 5 r1 6 s6 s7 9 7 r3 8 r2 r2 9 r2 53 LLR(1) prser Lookhed LR The method tht is often used in prtie. Most ommon syntti onstruts of progrmming lnguges n e expressed onveniently y n LLR(1) grmmr [DeRemer 1969]. LR(1) nd LLR(1) lwys hve the sme numer of sttes. Numer of sttes is out 1/10 of tht of LR(1). imple oservtion: n LR(1) item is of the form [ α β,] We ll α β the first omponent. Definition: in n LR(1) stte, set of first omponents is lled its ore.

28 56 1.G : ugmented grmmr, with enumertion p of produtions; 2.onstrut the LR(1) utomton (nonil item set C={I0,...,In}, I0=CLOURE[[,$]] GOTO funtion); 3.foreh stte i: () if ([ α β,] Ii && GOTO[Ii,]=Ij) dd shift j to CTION[i,]; // is terminl () if ([ α,] Ii && ) dd redue p -1 ( α) to CTION[i,]; () if ([,$] Ii) dd ept to CTION[i,$] 4. foreh (stte i,k && symol ) if (GOTO[Ij,]=Ik) dd k to GOTO[s,] 5. if (CTION[i,] is undefined) dd error to CTION[i,] 6. if (GOTO[i,] is undefined) dd error to GOTO[i,] 55 Intuition for LLR(1) grmmrs In n LR(1) prser, it is ommon thing tht severl sttes only differ in lookhed symols, ut hve the sme ore. To redue the numer of sttes, we might wnt to merge sttes with the sme ore. If I 4 nd I 7 re merged, then the new stte is lled I 4,7. fter merging the sttes, revise the GOT O 1 tle ordingly. Merging of sttes n never produe shift-redue onflit tht ws not present in one of the originl sttes. I 1 = {[ α,],...} For I 1, one of the tions is to perform redue when the lookhed symol is. I 2 = {[ β γ,],...} For I 2, one of the tions is to perform shift on input. Merging I 1 nd I 2, the new stte I 1,2 hs shift-redue onflits. However, we merge I 1 nd I 2 euse they hve the sme ore. Tht is, [ α,] I 2 nd [ β γ,d] I 1. The shift-redue onflit lredy ours in I 1 nd I 2. Merging of sttes n produe new redue-redue onflit.

29 LLR(1) trnsition digrm d I 0!>., $!>. CC, $ C!>. C, /d C!>.d, /d C I 3 C!>.C, /d C!>.C, /d C!>.d, /d I 2!> C.C, $ C!>.C, $ C!>.d, $ I 1!>., $ C d I 5!> CC., $ I 6 C!>.C, $ C!>.C, $ C!>.d, $ C d C d I 8 C!> C., /d I 9 C!> C., $ I 4 C!> d., /d I 7 C!> d., $ 57 Possile new onflits from LLR(1) My produe new redue-redue onflit. For exmple (textook pge 267, Exmple 4.58), grmmr: d f e e The lnguge reognized y this grmmr is {d, e, d, e}. You my hek tht this grmmr is LR(1) y onstruting the sets of items. You will find the set of items {[,d], [,e]} is vlid for the vile prefix, nd {[,e], [,d]} is vlid for the vile prefix. Neither of these sets genertes onflit, nd their ores re the sme. However, their union, whih is {[, d/e], [, d/e]}, genertes redue-redue onflit, sine redutions y oth nd re lled for on inputs d nd e. 58

30 60 How to onstrut LLR(1) prsing tle Nive pproh: Construt LR(1) prsing tle, whih tkes lots of intermedite spes. Merging sttes. pe nd/or time effiient methods to onstrut n LLR(1) prsing tle re known. Construting nd merging on the fly. 59 ummry LLR(1) LLR(1) LL(1) LR(1) LR(1) LR(1) LR(1) LR(0) LR(1) nd LLR(1) n lmost express ll importnt progrmming lnguges issues, ut LLR(1) is esier to write nd uses muh less spe. LL(1) is esier to understnd nd uses muh less spe, ut nnot express some importnt ommon-lnguge fetures. My try to use it first for your own pplitions.

31 62 miguous grmmrs re not too d... miguous grmmrs unmiguous grmmrs LR(1) miguous grmmrs often provide shorter, more nturl speifition thn their equivlent unmiguous grmmrs. 61 miguity from preedene nd ssoitivity Preedene nd ssoitivity re importnt lnguge onstruts. Exmple: G 1 : E E + E E E (E) id miguous, ut esy to understnd nd mintin! G 2 : E E + T T T T F F F (E) id Unmiguous, ut diffiult to understnd nd mintin!

32 64 LR(0) sttes input: id+id id the prser enters stte 7 fter proessing id+id 63 miguity from dngling-else Grmmr: ttement Other ttement if Condition then ttement if Condition then ttement else ttement When seeing if C then else there is shift/redue onflit, we lwys fvor shift. Intuition: fvor longer mth. Need mehnism to let user speify the defult onflithndling rule when there is shift/redue onflit.