A redefinition of Embedded Push-Down Automata

Transcription

1 Workshop TAG+5, Pris, My 2000 A redefinition of Embedded Push-Down Automt Miguel A. Alonso y, Eric de l Clergerie z nd Mnuel Vilres y y Deprtmento de Computción, Universidd de L Coruñ Cmpus de Elviñ s/n, L Coruñ (Spin) {lonso,vilres}@dc.fi.udc.es z INRIA, Domine de Voluceu Rocquecourt, B.P. 105, Le Chesny (Frnce) Eric.De_L_Clergerie@inri.fr Abstrct A new definition of Embedded Push-Down Automt is provided. We prove this new definition preserves the equivlence with tree djoining lnguges nd we provide tbultion frmework to execute ny utomton in polynomil time with respect to the length of the input string. 1. Introduction Embedded Push-Down Automt (EPDA) were defined in (Vijy-Shnker, 1988) s n extension of Push-Down Automt tht ccept exctly the clss of Tree Adjoining Lnguges. They cn lso be seen s level-2 utomt in progression of liner iterted pushdowns involving nested stcks (Weir, 1994). An EPDA consists of finite stte control, n input tpe nd stck mde up of non-empty stcks contining stck symbols. A trnsition cn consult the stte, the input string nd the top element of the top stck nd then chnge the stte, red chrcter of the input string nd replce the top element by finite sequence of stck elements to give new top stck, nd new stcks cn be plced bove nd below the top stck. EPDA cn describe prsing strtegies for tree djoining grmmrs in which djunctions re recognized top-down. The sme kind of strtegies cn be described in strongly-driven 2-stck utomt (de l Clergerie & Alonso Prdo, 1998) nd liner indexed utomt (Nederhof, 1999), which hs ssocited tbultion frmeworks llowing those utomt to be executed in polynomil time with respect to the size of the input string. In this pper we propose redefinition of EPDA in order to provide tbultion frmework for this clss of utomt. 2. EPDA without sttes Finite-stte control is not fundmentl component of push-down utomt, s the current stte in configurtion cn be stored in the top element of the stck of the utomton (Lng, 1991). Finite-stte control cn lso be eliminted from EPDA, obtining new definition tht considers EPDA s tuple (V T ; V S ; ; $ 0 ; $ f ) where V T is finite set of terminl symbols, V S is finite set of stck symbols, $ 0 2 V S is the initil stck symbol, $ f 2 V S is the finl stck symbol nd is finite set of six types of trnsition: SWAP: Trnsitions of the form C 7?! F tht replce the top element of the top stck while scnning. The ppliction of such trnsition on stck B returns the stck C. This reserch ws prtilly supported by the FEDER of EU (Grnt 1FD C04-02) nd Xunt de Glici (Grnt PGIDT99XI10502B).

2 Miguel A. Alonso, Eric de l Clergerie & Mnuel Vilres PUSH: Trnsitions of the form C 7?! C F tht push F onto C. The ppliction of such trnsition on stck C returns the stck CF. POP: Trnsitions of the form C F 7?! G tht replce C nd F by G. The ppliction of such trnsition on CF returns the stck G. WRAP-A: Trnsitions wrp-bove of the form C 7?! C; F tht push new stck F on the top of the utomton stck. The ppliction of such trnsition on stck C returns the stck C F. WRAP-B: Trnsitions wrp-below of the form C 7?! C; F tht store new stck C just below the top stck, nd chnge from C to F the top element of the top stck. The ppliction of such trnsition on stck C returns the stck C F. UNWRAP: Trnsitions of the form C; F 7?! G tht delete the top stck F nd replce the new top element by G. The ppliction of such trnsition on stck C F returns the stck G. where C; F; G 2 V S, 2 ( V S, 2 V, ) S 2 V T [ fg nd 62 V S is new symbol used s stck seprtor. It cn be proved tht trnsitions of EPDA with sttes cn be emulted by trnsitions in nd vice vers. An instntneous configurtion is pir (; w), where represents the contents of the utomton stck nd w is the prt of the input string tht is yet to be red. A configurtion (; w) derives configurtion ( 0 ; w), denoted (; w) ( 0 ; w), if nd only if there exists trnsition tht pplied to gives 0 nd scns from the input string. We use to denote the reflexive nd trnsitive closure of. An input string is ccepted by n EPDA if ( $ 0 ; w) ( $ f ; ). The lnguge ccepted by n EPDA is the set of w 2 VT such tht ( $ 0; w) ( $ f ; ). 3. Compiling TAG into EPDA We consider ech elementry tree of TAG s formed by set of context-free productions P(): node N nd its g children N : : : N 1 g re represented by production N! N : : : N 1 g. The elements of the productions re the nodes of the tree, except for the cse of elements belonging to V T [ f"g in the right-hnd side of production. Those elements my hve no children nd cn not be djoined, so we identify such nodes lbeled by terminl with tht terminl. We use 2 dj(n ) to denote tht tree my be djoined t node N. If djunction is not mndtory t N, then nil 2 dj(n ). We consider the dditionl productions >! R, >! R nd F!? for ech initil tree 2 I nd ech uxiliry tree 2 A, where R is the root node of nd R nd F re the root node nd foot node of, respectively. After disbling > nd? s djunction nodes the genertive cpbility of the grmmr remins intct. Figure 1 shows the generic compiltion schem from TAG to EPDA, where symbols r hve r;s been introduced to denote dotted productions. The mening of ech compiltion rule is grphiclly shown in figure 2. This schem is prmeterized by?! N, the informtion propgted top-down w.r.t. the node N, nd by N?, the informtion propgted bottom-up. When the schem is used to implement top-down strtegy?! N = N nd N? =, where is fresh stck symbol. A bottom-up strtegy requires?! N = nd N? = N. For Erley-like prsing strtegy,?! N = N nd N? = N, where N nd N re used to distinguish the top-down prediction from the bottom-up propgtion of node. We cn observe in figure 1 tht ech stck stores pending djunctions with respect to the node plced on the top of the stck in top-down tretment of djunctions: when n djunction node

3 A redefinition of Embedded Push-Down Automt [INIT] $ 0 7?! $ 0 r 0;0 2 I [CALL] r r;s 7?! r r;s???! 62 spine(); nil 2 dj( ) [SCALL] r r;s 7?! r r;s ;???! 2 spine(); nil 2 dj( ) [SEL]??! N r;0 7?! r r;0 [TAB] r r;nr 7?!?? N r;0 [RET] r r;s ;??? 7?! r r;s+1 62 spine(); nil 2 dj( ) [SRET] r r;s ;??? 7?! r r;s+1 2 spine(); nil 2 dj( ) [SCAN]??! N r;0 7?! N?? r;0 N r;0! [ACALL-] r r;s 7?! r r;s ; r;s dj( ) 6= fnilg?! [ACALL-b] r;s 7?! r;s > 2 dj( )? [ARET] r r;s ; > 7?! r r;s+1 2 dj( ) [FCALL-] r 7?! f;0 r ; f;0? N f;0 = F [FCALL-b] r;s? 7?!???! [FRET] r ; f;0??? 7?! r f;1 N f;0 = F ; 2 dj( ) [FINAL] $ 0 r 0;1 7?! $ f 2 I Figure 1: Generic compiltion schem from TAG to EPDA γ ACALL ARET β SCALL SRET CALL RET FCALL FRET Figure 2: Mening of compiltion rules

4 Miguel A. Alonso, Eric de l Clergerie & Mnuel Vilres Trnsition EPDA L-LIA SWAP C 7?! F C[] 7?! F [] PUSH C 7?! CF C[] 7?! F [C] POP CF 7?! G F [C] 7?! G[] WRAP-A C 7?! C; F C[] 7?! C[] F [ ] WRAP-B C 7?! C; F C[] 7?! C[ ] F [] UNWRAP C; F 7?! G WRAP-B+PUSH C 7?! C; X F C[] 7?! C[ ] F [X] WRAP-B+POP X C 7?! C; F C[X] 7?! C[ ] F [] Figure 3: Equivlence between EPDA nd L-LIA is reched, the djunction node is stored on the top of the stck ([ACALL-]) nd the trversl of the uxiliry tree is strted ([ACALL-b]); the djunction stck is propgted through the spine ([SCALL]) down to the foot node, where the trversl of the uxiliry tree is suspended to resume the trversl of the subtree rooted by the djunction node ([FCALL-]), which is eliminted of the stck ([FCALL-b]). To void confusion, we store r;s insted of r r;s to indicte tht n djunction ws strted t node. A symbol cn be seen s symbol r witing n djunction to be completed. 4. EPDA nd Left-oriented Liner Indexed Automt Left-oriented Liner Indexed Automt (L-LIA) is clss of utomt defined by Nederhof (1999) tht cn be used to implement prsing strtegies for TAG in which djunctions re recognized in top-down wy. Given EPDA, the equivlent L-LIA is obtined by mens of simple chnge in the nottions: if we consider the top element of stck s stck symbol, nd the rest of the stck s the indices list ssocited to them, we obtin the correspondence shown in figure 3. This chnge in nottion is lso useful to show tht EPDA ccept exctly the clss of tree djoining lnguges. Tht tree djoining lnguges re ccepted by EPDA is shown by the compiltion schem defined previously. To prove tht the lnguges ccepted by EPDA re tree djoined lnguges, we exhibit procedure tht, given n EPDA A = (V T ; V S ; ; $ 0 ; $ f ), builds liner indexed grmmr (Gzdr, 1987) G = (V T ; V N ; V I ; S; P ) tht recognizes the lnguge ccepted by A. Non-terminls in V N re pirs ha; Bi, where A; B 2 V S, nd V I = V S. Productions in P re obtined from trnsitions in s follows: For ech trnsition C hf; Ei[] is creted. 7?! F nd for ech E 2 V S, production hc; Ei[]! For ech trnsition C 7?! CF nd for ech E 2 V S, production hc; Ei[]! hf; Ei[C] is creted. For ech trnsition C F hg; Ei[] is creted. 7?! G nd for ech E 2 V S, production hf; Ei[C]!

5 A redefinition of Embedded Push-Down Automt For ech pir of trnsitions C 7?! b C; F 0 nd C; F 7?! G, nd for ech E 2 V S, production hc; Ei[]! b hf 0 ; F i[ ] hg; Ei[] is creted. For ech pir of trnsitions C 7?! b C; F 0 nd C; F 7?! G, nd for ech E 2 V S, production hc; Ei[]! b hf 0 ; F i[] hg; Ei[ ] is creted. For ech E 2 V S, production he; Ei[ ]! is creted. The xiom of the grmmr is S = h$ 0 ; $ f i. Applying induction in the length of derivtions, we cn prove tht hc; Ei[] ) w if nd only if ( C; w) ( E; ). 5. Tbultion The direct execution of EPDA my be exponentil with respect to the length of the input string nd my even loop. To get polynomil complexity, we must void duplicting computtions by tbulting trces of configurtions clled items. The mount of informtion to keep in n item is the crucil point to determine to get efficient executions. The tbultion of EPDA using PUSH nd POP trnsitions without restrictions seems to be difficult. By studying the compiltion schem of figure 1, we observe tht the compiltion rules [ACALL-] nd [ACALL-b] cn be combined to form single rule [ACALL] generting trnsitions WRAP-B+PUSH of the form C 7?! C; X F : [ACALL] r r;s 7?! r r;s ; r;s > such tht 2 dj( ). The [FCALL-] nd [FCALL-b] cn be combined to form single rule generting trnsitions WRAP-B+POP of the form X C 7?! C; F : [FCALL] r;s r f;0 7?! r f;0 ; such tht N f;0 = F nd 2 dj( ). In this section, we consider the tbultion of subset of EPDA consisting of trnsitions SWAP, WRAP-A, WRAP-B, UNWRAP, WRAP-B+PUSH nd WRAP-B+POP. In order to define items nd ttending to the form of the trnsitions, we clssify derivtions of EPDA into the following types: Cll derivtions. Correspond to the propgtion of stck by mens of WRAP-B, WRAP-B+PUSH nd WRAP-B+POP trnsitions: ( A; h+1 : : : n ) ( A 1 XB; i+1 : : : n ) ( A 1 XC; j+1 : : : n ) where A; B; C; X 2 V S, 2 V S nd ; 1 2 ( V S ). The two occurrences of denote the sme stck in the sense tht is neither consulted nor modified through the derivtion. These derivtions re independent of nd, so they cn be represented by items

6 Miguel A. Alonso, Eric de l Clergerie & Mnuel Vilres Return derivtions. Correspond to the bottom-up propgtion of unitry stck by mens of UNWRAP trnsitions: ( A; h+1 : : : n ) ( A 1 XB; i+1 : : : n ) ( A 1 B 2 D; p+1 : : : n ) ( A 1 B 2 E; q+1 : : : n ) ( A 1 C; j+1 : : : n ) where A; B; C; D; E; X 2 V S, 2 V, S ; 1; 2 2 ( V S nd is pssed unffected ) through derivtion. These derivtions re independent of but not with respect to the subderivtion ( D; p+1 : : : n ) ( E; q+1 : : : n ), so they re be represented in compct form by items Specil point derivtions. When X = we hve prticulr cse of previous derivtions: ( B; i+1 : : : n ) ( C; j+1 : : : n ) where B; C 2 V S, nd 2 ( V S ). These derivtions cn be represented by items [?;? j B; i;?; C; j;? j?;?;?;?] To combine items, we use the set of inference rules shown in figures 4 nd 5. Ech rule is of the form 1;:::; k trns, mening tht if ll ntecedents i re tbulted items nd 0 there exist the trnsitions trns, then the consequent item 0 should be creted. In order to simplify the inference rules, but without loss of generlity, we hve considered tht scnning is only performed by SWAP trnsitions. The computtion strts with the initil item [?;? j $ 0 ; 0;?; $ 0 ; 0;? j?;?;?;?]. An input string 1 : : : n hs been recognized if the finl item [?;? j $ 0 ; 0;?; $ f ; n;? j?;?;?;?] is present. It cn be proved tht hndling items with the inference rules is equivlent to pplying the trnsitions on the whole stcks. To illustrte the reltion between EPDA nd L-LIA, figures 4 nd 5 show the trnsitions of both models of utomt tht must be considered to pply given inference rule. Therefore, the proposed tbulted technique cn be lso pplied to L-LIA working with trnsitions SWAP, WRAP-A, WRAP-B, UNWRAP, WRAP-B+PUSH nd WRAP-B+POP. 6. Conclusion Embedded Push-Down Automt hve been redefined: finite-stte control hs been eliminted nd severl kinds of trnsition hve been defined. We hve lso shown tht the new definition preserves the equivlence with tree djoining lnguges nd tht tbultion techniques re possible to execute these utomt in polynomil time with respect to the length of the input string. References DE LA CLERGERIE E. & ALONSO PARDO M. (1998). A tbulr interprettion of clss of 2-Stck Automt. In COLING-ACL 98, 36th Annul Meeting of the Assocition for Computtionl Linguistics nd 17th Interntionl Conference on Computtionl Linguistics, Proceedings of the Conference, volume II, p , Montrel, Quebec, Cnd: ACL.

7 A redefinition of Embedded Push-Down Automt Rule EPDA trnsition L-LIA trnsition [A; h j B; i; X; F; k; X j?;?;?;?] where k = j if = nd k = j + 1 if 2 V T [A; h j B; i; X; F; k;? j D; p; E; q] where k = j if = nd k = j + 1 if 2 V T [?;? j F; j;?; F; j;? j?;?;?;?] [?;? j F; j;?; F; j;? j?;?;?;?] C C 7?! F C[] 7?! F [] 7?! F C[] 7?! F [] C 7?! C; F C[] 7?! C[] F [ ] C 7?! C; F C[] 7?! C[] F [ ] [A; h j F; j; X; F; j; X j?;?;?;?] [?;? j F; j;?; F; j;? j?;?;?;?] C 7?! C; F C 7?! C; F C[] 7?! C[ ] F [] C[] 7?! C[ ] F [] [C; j j F; j; X 0 ; F; j; X 0 j?;?;?;?] [C; j j F; j; X 0 ; F; j; X 0 j?;?;?;?] C 7?! C; X 0 F C[] 7?! C[ ] F [X 0 ] C 7?! C; X 0 F C[] 7?! C[ ] F [X 0 ] [M; m j N; t; X 0 ; A; h; X 0 j?;?;?;?] [M; m j F; j; X 0 ; F; j; X 0 j?;?;?;?] [M; m j N; t; X 0 ; A; h;? j D; p; E; q] [?;? j F; j;?; F; j;? j?;?;?;?] XC 7?! C; F XC 7?! C; F C[X] 7?! C[ ] F [] C[X] 7?! C[ ] F [] Figure 4: Tbultion rules GAZDAR G. (1987). Applicbility of indexed grmmrs to nturl lnguges. In U. REYLE & C. ROHRER, Eds., Nturl Lnguge Prsing nd Linguistic Theories, p D. Reidel Publishing Compny. LANG B. (1991). Towrds uniform forml frmework for prsing. In M. TOMITA, Ed., Current Issues in Prsing Technology, p Norwell, MA, USA: Kluwer Acdemic Publishers. NEDERHOF M.-J. (1999). Models of tbultion for TAG prsing. In Proc. of the Sixth Meeting on Mthemtics of Lnguge (MOL 6), p , Orlndo, Florid, USA. VIJAY-SHANKER K. (1988). A Study of Tree Adjoining Grmmrs. PhD thesis, University of Pennsylvni. Avilble s Technicl Report MS-CIS LINC LAB 95 of the Deprtment of Computer nd Informtion Science, University of Pennsylvni. WEIR D. J. (1994). Liner iterted pushdowns. Computtionl Intelligence, 10 (4), p

8 Miguel A. Alonso, Eric de l Clergerie & Mnuel Vilres Rule EPDA trnsition L-LIA trnsition [?;? j F 0 ; j;?; F; k;? j?;?;?;?] [A; h j B; i; X; G; k; X j?;?;?;?] C 7?! C; F 0 C[] 7?! C[] F 0 [ ] [?;? j F 0 ; j;?; F; k;? j?;?;?;?] [A; h j B; i; X; G; k;? j D; p; E; q] C 7?! C; F 0 C[] 7?! C[] F 0 [ ] [A; h j F 0 ; j; X; F; k;? j D; p; E; q] [A; h j B; i; X; G; k;? j D; p; E; q] C 7?! C; F 0 C[] 7?! C[ ] F 0 [] [?;? j F 0 ; j;?; F; k;? j?;?;?;?] [A; h j B; i; X; G; k;? j D; p; E; q] C 7?! C; F 0 C[] 7?! C[ ] F 0 [] [C; j j F 0 ; j; X 0 ; F; k;? j D; p; E; q] [A; h j D; p; X; E; q;? j O; u; P; v] [A; h j B; i; X; G; k;? j O; u; P; v] C 7?! C; X 0 F 0 C[] 7?! C[ ] F 0 [X 0 ] [C; j j F 0 ; j; X 0 ; F; k;? j O; u; P; v] [?;? j O; u;?; P; v;? j?;?;?;?] [A; h j B; i; X; G; k;? j D; p; E; q] C 7?! C; X 0 F 0 C[] 7?! C[ ] F 0 [X 0 ] [M; m j F 0 ; j; X 0 ; F; k;? j D; p; E; q] [M; m j N; t; X 0 ; A; h; X 0 j?;?;?;?] [A; h j B; i; X; G; k;? j F 0 ; j; F; k] XC 7?! C; F 0 C[X] 7?! C[ ] F 0 [] [?;? j F 0 ; j;?; F; k;? j?;?;?;?] [M; m j N; t; X 0 ; A; h;? j D; p; E; q] [A; h j B; i; X; G; k;? j F 0 ; j; F; k] XC 7?! C; F 0 C[X] 7?! C[ ] F 0 [] Figure 5: Tbultion rules