Parsing Directed Acyclic Graphs with Range Concatenation Grammars

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1 Prsing Directed Acyclic Grphs with Rnge Conctention Grmmrs Pierre Boullier nd Benoît Sgot Alpge, INRIA Pris-Rocquencourt & Université Pris 7 Domine de Voluceu Rocquencourt, BP Le Chesny Cedex, Frnce {PierreBoullier,BenoitSgot}@inrifr Astrct Rnge Conctention Grmmrs (RCGs) re syntctic formlism which possesses mny ttrctive properties It is more powerful thn Liner Context-Free Rewriting Systems, though this power is not reched to the detriment of efficiency since its sentences cn lwys e prsed in polynomil time If the input, insted of string, is Directed Acyclic Grph (DAG), only simple RCGs cn still e prsed in polynomil time For non-liner RCGs, this polynomil prsing time cnnot e gurnteed nymore In this pper, we show how the stndrd prsing lgorithm cn e dpted for prsing DAGs with RCGs, oth in the liner (simple) nd in the non-liner cse 1 Introduction The Rnge Conctention Grmmr (RCG) formlism hs een introduced y Boullier ten yers go A complete definition cn e found in (Boullier, 2004), together with some of its forml properties nd prsing lgorithm (qulified here of stndrd) which runs in polynomil time In this pper we shll only consider the positive version of RCGs which will e revited s PRCG 1 PRCGs re very ttrctive since they re more powerful thn the Liner Context-Free Rewriting Systems (LCFRSs) y (Vijy-Shnker et l, 1987) In fct LCFRSs re equivlent to simple PRCGs which re suclss of PRCGs Mny Mildly Context- Sensitive (MCS) formlisms, including Tree Adjoining Grmmrs (TAGs) nd vrious kinds of Multi-Component TAGs, hve lredy een 1 Negtive RCGs do not dd forml power since oth versions exctly cover the clss PTIME of lnguges recognizle in deterministic polynomil time (see (Boullier, 2004) for n indirect proof nd (Bertsch nd Nederhof, 2001) for direct proof) trnslted into their simple PRCG counterprt in order to get n efficient prser for free (see for exmple (Brthélemy et l, 2001)) However, in mny Nturl Lnguge Processing pplictions, the most suitle input for prser is not sequence of words (forms, terminl symols), ut more complex representtion, usully defined s Direct Acyclic Grph (DAG), which correspond to finite regulr lnguges, for tking into ccount vrious kinds of miguities Such miguities my come, mong others, from the output of speech recognition systems, from lexicl miguities (nd in prticulr from tokeniztion miguities), or from nondeterministic spelling correction module Yet, it hs een shown y (Bertsch nd Nederhof, 2001) tht prsing of regulr lnguges (nd therefore of DAGs) using simple PRCGs is polynomil In the sme pper, it is lso proven tht prsing of finite regulr lnguges (the DAG cse) using ritrry RCGs is NP-complete This ppers ims t showing how these complexity results cn e mde concrete in prser, y extending stndrd RCG prsing lgorithm so s to hndle input DAGs We will first recll oth some sic definitions nd their nottions Afterwrds we will see, with slight modifiction of the notion of rnges, how it is possile to use the stndrd PRCG prsing lgorithm to get in polynomil time prse forest with DAG s input 2 However, the resulting prse forest is vlid only for simple PRCGs In the non-liner cse, nd consistently with the complexity results mentioned ove, we show tht the resulting prse forest needs further processing for filtering out inconsistent prses, which my need n exponentil time The proposed filtering lgorithm llows for prsing DAGs in prctice with ny PRCG, including non-liner ones 2 The notion of prse forest is reminiscent of the work of (Lng, 1994) 254 Proceedings of the 11th Interntionl Conference on Prsing Technologies (IWPT), pges , Pris, Octoer 2009 c 2009 Assocition for Computtionl Linguistics

2 2 Bsic notions nd nottions 21 Positive Rnge Conctention Grmmrs A positive rnge conctention grmmr (PRCG) G = (N, T, V, P, S) is 5-tuple in which: T nd V re disjoint lphets of terminl symols nd vrile symols respectively N is non-empty finite set of predictes of fixed rity (lso clled fn-out) We write k = rity(a) if the rity of the predicte A is k A predicte A with its rguments is noted A( α) with vector nottion such tht α = k nd α[j] is its j th rgument An rgument is string in (V T ) S is distinguished predicte clled the strt predicte (or xiom) of rity 1 P is finite set of cluses A cluse c is rewriting rule of the form A 0 ( α 0 ) A 1 ( α 1 ) A r ( α r ) where r, r 0 is its rnk, A 0 ( α 0 ) is its left-hnd side or LHS, nd A 1 ( α 1 ) A r ( α r ) its right-hnd side or RHS By definition c[i] = A i ( α i ), 0 i r where A i is predicte nd α i its rguments; we note c[i][j] its j th rgument; c[i][j] is of the form X 1 X nij (the X k s re terminl or vrile symols), while c[i][j][k], 0 k n ij is position within c[i][j] For given cluse c, nd one of its predictes c[i] surgument is defined s sustring of n rgument c[i][j] of the predicte c[i] It is denoted y pir of positions (c[i][j][k], c[i][j][k ]), with k k Let w = 1 n e n input string in T, ech occurrence of sustring l+1 u is pir of positions (w[l], w[u]) st 0 l u n clled rnge nd noted lu w or lu when w is implicit In the rnge lu, l is its lower ound while u is its upper ound If l = u, the rnge lu is n empty rnge, it spns n empty sustring If ρ 1 = l 1 u 1, nd ρ m = l m u m re rnges, the conctention of ρ 1,, ρ m noted ρ 1 ρ m is the rnge ρ = lu if nd only if we hve u i = l i+1, 1 i < m, l = l 1 nd u = u m If c = A 0 ( α 0 ) A 1 ( α 1 ) A r ( α r ) is cluse, ech of its surguments (c[i][j][k], c[i][j][k ]) my tke rnge ρ = lu s vlue: we sy tht it is instntited y ρ However, the instntition of surgument is sujected to the following constrints If the surgument is the empty string (ie, k = k ), ρ is n empty rnge If the surgument is terminl symol (ie, k + 1 = k nd X k T ), ρ is such tht l + 1 = u nd u = X k Note tht severl occurrences of the sme terminl symol my e instntited y different rnges If the surgument is vrile symol (ie, k + 1 = k nd X k V ), ny occurrence (c[i ][j ][m], c[i ][j ][m ]) of X k is instntited y ρ Thus, ech occurrence of the sme vrile symol must e instntited y the sme rnge If the surgument is the string X k+1 X k, ρ is its instntition if nd only if we hve ρ = ρ k+1 ρ k in which ρ k+1,, ρ k re respectively the instntitions of X k+1,, X k If in c we replce ech rgument y its instntition, we get n instntited cluse noted A 0 ( ρ 0 ) A 1 ( ρ 1 ) A r ( ρ r ) in which ech A i ( ρ i ) is n instntited predicte A inry reltion clled derive nd noted G,w is defined on strings of instntited predictes If Γ 1 nd Γ 2 re strings of instntited predictes, we hve Γ 1 A 0 ( ρ 0 ) Γ 2 Γ 1 A 1 ( ρ 1 ) A m ( ρ m ) Γ 2 G,w if nd only if A 0 ( ρ 0 ) A 1 ( ρ 1 ) A m ( ρ m ) is n instntited cluse The (string) lnguge of PRCG G is the set L(G) = {w S( 0 w w ) + G,w ε} In other words, n input string w T, w = n is sentence of G if nd only there exists complete derivtion which strts from S( 0n ) (the instntition of the strt predicte on the whole input text) nd leds to the empty string (of instntited predictes) The prse forest of w is the CFG whose xiom is S( 0n ) nd whose productions re the instntited cluses used in ll complete derivtions 3 We sy tht the rity of PRCG is k, nd we cll it k-prcg, if nd only if k is the mximum 3 Note tht this prse forest hs no terminl symols (its lnguge is the empty string) 255

3 rity of its predictes (k = mx A N rity(a)) We sy tht k-prcg is simple, we hve simple k-prcg, if nd only if ech of its cluse is non-comintoril: the rguments of its RHS predictes re single vriles; non-ersing: ech vrile which occur in its LHS (resp RHS) lso occurs in its RHS (resp LHS); liner: there re no vriles which occur more thn once in its LHS nd in its RHS The suclss of simple PRCGs is of importnce since it is MCS nd is the one equivlent to LCFRSs 22 Finite Automt A non-deterministic finite utomton (NFA) is the 5-tuple A = (Q, Σ, δ, q 0, F ) where Q is non empty finite set of sttes, Σ is finite set of terminl symols, δ is the ternry trnsition reltion δ = {(q i, t, q j ) q i, q j Q t Σ }, q 0 is distinguished element of Q clled the initil stte nd F is suset of Q whose elements re clled finl sttes The size of A, noted A, is its numer of sttes ( A = Q ) We define the ternry reltion δ on Q Σ Q s the smllest set st δ = {(q, ε, q) q Q} {(q 1, xt, q 3 ) (q 1, x, q 2 ) δ (q 2, t, q 3 ) δ} If (q, x, q ) δ, we sy tht x is pth etween q nd q If q = q 0 nd q F, x is complete pth The lnguge L(A) defined (generted, recognized, ccepted) y the NFA A is the set of ll its complete pths We sy tht NFA is empty if nd only if its lnguge is empty Two NFAs re equivlent if nd only if they define the sme lnguge A NFA is ε-free if nd only if its trnsition reltion does not contin trnsition of the form (q 1, ε, q 2 ) Every NFA cn e trnsformed into n equivlent ε-free NFA (this clssicl result nd those reclled elow cn e found, eg, in (Hopcroft nd Ullmn, 1979)) As usul, NFA is drwn with the following conventions: trnsition (q 1, t, q 2 ) is n rrow lelled t from stte q 1 to stte q 2 which re printed with surrounded circle Finl sttes re douly circled while the initil stte hs single unconnected, unlelled input rrow A deterministic finite utomton (DFA) is NFA in which the trnsition reltion δ is trnsition function, δ : Q Σ Q In other words, there re no ε-trnsitions nd if (q 1, t, q 2 ) δ, t ε nd (q 1, t, q 2 ) δ with q 2 q 2 Ech NFA cn e trnsformed y the suset construction into n equivlent DFA Moreover, ech DFA cn e trnsformed y minimiztion lgorithm into n equivlent DFA which is miniml (ie, there is no other equivlent DFA with fewer sttes) 23 Directed cyclic grphs Formlly, directed cyclic grph (DAG) D = (Q, Σ, δ, q 0, F ) is n NFA for which there exists strict order reltion < on Q such tht (p, t, q) δ p < q Without loss of generlity we my ssume tht < is totl order Of course, s NFAs, DAGs cn e trnsformed into equivlent deterministic or miniml DAGs 3 DAGs nd PRCGs A DAG D is recognized (ccepted) y PRCG G if nd only if L(D) L(G) A trivil wy to solve this recognition (or prsing) prolem is to extrct the complete pths of L(D) (which re in finite numer) one y one nd to prse ech such string with stndrd PRCG prser, the (complete) prse forest for D eing the union of ech individul forest 4 However since DAGs my define n exponentil numer of strings wrt its own size, 5 the previous opertion would tke n exponentil time in the size of D, nd the prse forest would lso hve n exponentil size The purpose of this pper is to show tht it is possile to directly prse DAG (without ny unfolding) y shring identicl computtions This shring my led to polynomil prse time for n exponentil numer of sentences, ut, in some cses, the prse time remins exponentil 31 DAGs nd Rnges In mny NLP pplictions the source text cnnot e considered s sequence of terminl symols, ut rther s finite set of finite strings As 4 These forests do not shre ny production (instntited cluse) since rnges in prticulr forest re ll relted to the corresponding source string w (ie, re ll of the form ij w) To e more precise the union opertion on individul forests must e completed in dding productions which connect the new (super) xiom (sy S ) with ech root nd which re, for ech w of the form S S( 0 w w) 5 For exmple the lnguge ( ) n, n > 0 which contins 2 n strings cn e defined y miniml DAG whose size is n

4 mentioned in th introduction, this non-unique string could e used to encode not-yet-solved miguities in the input DAGs re convenient wy to represent these finite sets of strings y fctorizing their common prts (thnks to the minimiztion lgorithm) In order to use DAGs s inputs for PRCG prsing we will perform two generliztions The first one follows Let w = t 1 t n e string in some lphet Σ nd let Q = {q i 0 i n} e set of n + 1 ounds with totl order reltion <, we hve q 0 < q 1 < < q n The sequence π = q 0 t 1 q 1 t 2 q 2 t n q n Q (Σ Q) n is clled ounded string which spells w A rnge is pir of ounds (q i, q j ) with q i < q j noted p i p j π nd ny triple of the form (q i 1 t i q i ) is clled trnsition All the notions round PRCGs defined in Section 21 esily generlize from strings to ounded strings It is lso the cse for the stndrd prsing lgorithm of (Boullier, 2004) Now the next step is to move from ounded strings to DAGs Let D = (Q, Σ, δ, q 0, F ) e DAG A string x Σ st we hve (q 1, x, q 2 ) δ is clled pth etween q 1 nd q 2 nd string π = qt 1 q 1 t p q p Q (Σ Q) is ounded pth nd we sy tht π spells t 1 t 2 t p A pth x from q 0 to f F is complete pth nd ounded pth of the form q 0 t 1 t n f with f F is complete ounded pth In the context of DAG D, rnge is pir of sttes (q i, q j ) with q i < q j noted q i q j D A rnge q i q j D is vlid if nd only if there exists pth from q i to q j in D Of course, ny rnge pq D defines its ssocited su-dag D pq = (Q pq, Σ pq, δ pq, p, {q}) s follows Its trnsition reltion is δ pq = {(r, t, s) (r, t, s) δ (p, x, r), (s, x, q) δ } If δ pq = (ie, there is no pth etween p nd q), D pq is the empty DAG, otherwise Q pq (resp Σ pq ) re the sttes (resp terminl symols) of the trnsitions of δ pq With this new definition of rnges, the notions of instntition nd derivtion esily generlize from ounded strings to DAGs The lnguge of PRCG G for DAG D is defined y L (G, D) = f F {x S( q 0 f D ) + G,D ε} Let x L(D), it is not very difficult to show tht if x L(G) then we hve x L (G, D) However, the converse is not true (see Exmple 1), sentence of L(D) L (G, D) my not e in L(G) To put it differently, if we use the stndrd RCG prser, with the rnges of DAG, we produce the shred prse-forest for the lnguge L (G, D) which is superset of L(D) L(G) However, if G is simple PRCG, we hve the equlity L(G) = D is DAG L (G, D) Note tht the suclss of simple PRCGs is of importnce since it is MCS nd it is the one equivlent to LCFRSs The informl reson of the equlity is the following If n instntited predicte A i ( ρ i ) succeeds in some RHS, this mens tht ech of its rnges ρ i [j] = kl D hs een recognized s eing component of A i, more precisely their exists pth from k to l in D which is component of A i The rnge kl D selects in D set δ kl D of trnsitions (the trnsitions used in the ounded pths from k to l) Becuse of the linerity of G, there is no other rnge in tht RHS which selects trnsition in δ kl D Thus the ounded pths selected y ll the rnges of tht RHS re disjoints In other words, ny occurrence of vlid instntited rnge ij D selects set of pths which is suset of L(D ij ) Now, if we consider non-liner PRCG, in some of its cluses, there is vrile, sy X, which hs severl occurrences in its RHS (if we consider top-down non-linerity) Now ssume tht for some input DAG D, n instntition of tht cluse is component of some complete derivtion Let pq D e the instntition of X in tht instntited cluse The fct tht predicte in which X occurs succeeds mens tht there exist pths from p to q in D pq The sme thing stnds for ll the other occurrences of X ut nothing force these pths to e identicl or not Exmple 1 Let us tke n exmple which will e used throughout the pper It is non-liner 1-PRCG which defines the lnguge n n c n, n 0 s the intersection of the two lnguges n c n nd n n c Ech of these lnguges is respectively defined y the predictes n c n nd n n c ; the strt predicte is n n c n 257

5 1 2 3 Figure 1: Input DAG ssocited with c n n c n (X) n c n (X) n n c (X) n c n (X) n c n (X) n c n (X) n c n (X) n c n (Xc) n c n (X) n c n (ε) ε n n c (Xc) n n c (X) n n c (X) n n (X) n n (X) n n (X) n n (ε) ε If we use this PRCG to prse the DAG of Figure 1 which defines the lnguge {, c}, we (erroneously) get the non-empty prse forest of Figure 2 though neither nor c is in n n c n 6 It is not difficult to see tht the prolem comes from the non-liner instntited vrile X 14 in the strt node, nd more precisely from the ctul (wrong) mening of the three different occurrences of X 14 in n n c n (X 14 ) n c n (X 14 ) n n c (X 14 ) The first occurrence in its RHS sys tht there exists pth in the input DAG from stte 1 to stte 4 which is n n c n The second occurrence sys tht there exists pth from stte 1 to stte 4 which is n n n c While the LHS occurrence (wrongly) sys tht there exists pth from stte 1 to stte 4 which is n n n c n However, if the two X 14 s in the RHS hd selected common pths (this is not possile here) etween 1 nd 4, vlid interprettion could hve een proposed With this exmple, we see tht the difficulty of DAG prsing only rises with non-liner PRCGs If we consider liner PRCGs, the su-clss of the PRCGs which is equivlent to LCFRSs, the 6 In this forest ovl nodes denote different instntited predictes, while its ssocited instntited cluses re presented s its dughter(s) nd re denoted y squre nodes The LHS of ech instntited cluse shows the instntition of its LHS symols The RHS is the corresponding sequence of instntited predictes The numer of dughters of ech squre node is the numer of its RHS instntited predictes c 4 stndrd lgorithm works perfectly well with input DAGs, since vlid instntition of n rgument of predicte in cluse y some rnge pq mens tht there exists (t lest) one pth etween p nd q which is recognized The pper will now concentrte on non-liner PRCGs, nd will present new vlid prsing lgorithm nd study its complexities (in spce nd time) In order to simplify the presenttion we introduce this lgorithm s post-processing pss which will work on the shred prse-forest output y the (slightly modified) stndrd lgorithm which ccepts DAGs s input 32 Prsing DAGs with non-liner PRCGs The stndrd prsing lgorithm of (Boullier, 2004) working on string w cn e sketched s follows It uses single memoized oolen function predicte(a, ρ) where A is predicte nd ρ is vector of rnges whose dimension is rity(a) The initil cll to tht function hs the form predicte (S, 0 w ) Its purpose is, for ech A 0 -cluse, to instntite ech of its symols in consistnt wy For exmple if we ssume tht the i th rgument of the LHS of the current A 0 -cluse is α ixy α i nd tht the i th component of ρ 0 is the rnge p i q i n instntition of X, n Y y the rnges p X q X, p q nd p Y q Y is such tht we hve p i p X q X = p < q = p + 1 = p Y q Y q i nd w = w w with w = p Since the PRCG is non ottom-up ersing, the instntition of ll the LHS symols implies tht ll the rguments of the RHS predictes A i re lso instntited nd gthered into the vector of rnges ρ i Now, for ech i (1 i RHS ), we cn cll predicte (A i, ρ i ) If ll these clls succeed, the instntited cluse cn e stored s component of the shred prse forest 7 In the cse of DAG D = (Q, Σ, δ, q 0, F ) s input, there re two slight modifictions, the initil cll is chnged y the conjunctive cll predicte(s, q 0 f 1 ) predicte (S, q 0 f F ) with f i F 8 nd the terminl symol cn e instntited y the rnge p q D only if (p,, q ) 7 Note tht such n instntited cluse could e unrechle from the (future) instntited strt symol which will e the xiom of the shred forest considered s CFG 8 Techniclly, ech of these clls produces forest These individul forests my shre suprts ut their roots re ll different In order to hve true forest, we introduce new root, the super-root whose dughters re the individul forests 258

6 n n c n 14 n n c n (X 14 ) n c n 14 n n c 14 n c n 14 n n c 14 n c n (X 14 ) n c n 14 n n c (X 14 ) n n 14 n c n 14 n n 14 n c n ( 13 X 33 c 34 ) n c n 33 n n ( 12 X ) n n 22 n c n 33 n n 22 n c n (ε 33 ) ε n n (ε 22 ) ε Figure 2: Prse forest for the input DAG c is trnsition in δ The vrile symol X cn e instntited y the rnge p X q X D only if p X q X D is vlid 33 Forest Filtering We ssume here tht for given PRCG G we hve uilt the prse forest of n input DAG D s explined ove nd tht ech instntited cluse of tht forest contins the rnge p X q X D of ech of its instntited symols X We hve seen in Exmple 1 tht this prse forest is vlid if G is liner ut my well e unvlid if G is non-liner In tht ltter cse, this hppens ecuse the rnge p X q X D of ech instntition of the non-liner vrile X selects the whole su-dag D px q X while ech instntition should only select sulnguge of L(D px q X ) For ech occurrence of X in the LHS or RHS of non-liner cluse, its su-lnguges could of course e different from the others In fct, we re interested in their intersections: If their intersections re non empty, this is the lnguge which will e ssocited with p X q X D, otherwise, if their intersections re empty, then the instntition of the considered cluse fils nd must thus e removed from the forest Of course, we will consider tht the lnguge ( finite numer of strings) ssocited with ech occurrence of ech instntited symol is represented y DAG The ide of the forest filtering lgorithm is to first compute the DAGs ssocited with ech rgument of ech instntited predicte during ottom-up wlk These DAGs re clled decortions This processing will perform DAG compositions (including intersections, s suggested ove), nd will erse cluses in which empty intersections occur If the DAG ssocited with the single rgument of the super-root is empty, then prsing filed Otherwise, top-down wlk is lunched (see elow), which my lso erse non-vlid instntited cluses If necessry, the lgorithm is completed y clssicl CFG lgorithm which erse non productive nd unrechle symols leving reduced grmmr/forest In order to simplify our presenttion we will ssume tht the PRCGs re non-comintoril nd ottom-up non-ersing However, we cn note tht the following lgorithm cn e generlized in order to hndle comintoril PRCGs nd in prticulr with overlpping rguments 9 Moreover, we will ssume tht the forest is non cyclic (or equivlently tht ll cycles hve previously een removed) 10 9 For exmple the non-liner comintoril cluse A(XY Z) B(XY ) B(Y Z) hs overlpping rguments 10 By clssicl lgorithm from the CFG technology 259

7 331 The Bottom-Up Wlk For this principle lgorithm, we ssume tht for ech instntited cluse in the forest, DAG will e ssocited with ech occurrence of ech instntited symol More precisely, for given instntited A 0 -cluse, the DAGs ssocited with the RHS symol occurrences re composed (see elow) to uild up DAGs which will e ssocited with ech rgument of its LHS predicte For ech LHS rgument, this composition is directed y the sequence of symols in the rgument itself The forest is wlked ottom-up strting from its leves The constrint eing tht n instntited cluse is visited if nd only if ll its RHS instntited predictes hve lredy ll een visited (computed) This constrint cn e stisfied for ny non-cyclic forest To e more precise, consider n instntition c ρ = A 0 ( ρ 0 ) A 1 ( ρ 1 ) A p ( ρ p ) of the cluse c = A 0 ( α 0 ) A 1 ( α 1 ) A m ( α m ), we perform the following sequence: 1 If the cluse is not top-down liner (ie, there exist multiple occurrences of the sme vriles in its RHS rguments), for such vrile X let the rnge p X q X e its instntition (y definition, ll occurrences re instntited y the sme rnge), we perform the intersection of the DAGs ssocited with ech instntited predicte rgument X If one intersection results in n empty DAG, the instntited cluse is removed from the forest Otherwise, we perform the following steps 2 If RHS vrile Y is liner, it occurs once in the j th rgument of predicte A i We perform rnd new copy of the DAG ssocited with the j th rgument of the instntition of A i 3 At tht moment, ll instntited vriles which occur in c ρ re ssocited with DAG For ech occurrence of terminl symol t in the LHS rguments we ssocite (new) DAG whose only trnsition is (p, t, q) where p nd q re rnd new sttes with, of course, p < q 4 Here, ll symols (terminls or vriles) re ssocited with disjoints DAGs For ech LHS rgument α 0 [i] = X1 i Xi j Xi p i, we ssocite new DAG which is the conctention of the DAGs ssocited with the symols X i 1,, Xi j, nd Xi p i 5 Here ech LHS rgument of c ρ is ssocited with non empty DAG, we then report the individul contriution of c ρ into the (lredy computed) DAGs ssocited with the rguments of its LHS A 0 ( ρ 0 ) The DAG ssocited with the i th rgument of A 0 ( ρ 0 ) is the union (or copy if it is the first time) of its previous DAG vlue with the DAG ssocited with the i th rgument of the LHS of c ρ This ottom-up wlk ends on the super-root with finl decortion sy R In fct, during this ottomup wlk, we hve computed the intersection of the lnguges defined y the input DAG nd y the PRCG (ie, we hve L(R) = L(D) L(G)) Exmple 2 c Figure 3: Input DAG ssocited with c c With the PRCG of Exmple 1 nd the input DAG of Figure 3, we get the prse forest of Figure 4 whose trnsitions re decorted y the DAGs computed y the ottom-up lgorithm 11 The crucil point to note here is the intersection which is performed etween {c, c} nd {c, } on n n c n (X 14 ) n c n 14 n n c 14 The non-empty set {c} is the finl result ssigned to the instntited strt symol Since this result is non empty, it shows tht the input DAG D is recognized y G More precisely, this shows tht the su-lnguge of D which is recognized y G is {c} However, s shown in the previous exmple, the (undecorted) prse forest is not the forest uilt for the DAG L(D) L(G) since it my contin non-vlid prts (eg, the trnsitions lelled {c} or {} in our exmple) In order to get the 11 For redility resons these DAGs re represented y their lnguges (ie, set of strings) Bottom-up trnsitions from instntited cluses to instntited predictes reflects the computtions performed y tht instntited cluse while ottom-up trnsitions from instntited predictes to instntited cluses re the union of the DAGs entering tht instntited predicte 260

8 n n c n 14 {c} n n c n (X 14 ) n c n 14 n n c 14 {c, c} {c, } n c n 14 n n c 14 {c} {} n c n (X 14 ) n c n 14 {c} n n c (X 14 ) n n 14 {c} n c n ( 12 X 24 ) n c n 24 n n c (X 13 c 34 ) n n c 13 {c} {c} {} {c} n c n 14 n c n 24 n n 14 n n c 13 {c} {c} {} {} n c n ( 23 X 33 c 34 ) n c n 33 n n ( 12 X ) n n 22 n c n (X 24 ) n c n 24 {c} n n c (X 13 ) n n 13 {} {c} n c n 24 n n 13 {} n c n ( 23 X 33 c 34 ) n c n 33 n c n 33 n c n (ε 33 ) ε n n ( 12 X ) n n 22 n n 22 n n (ε 22 ) ε Figure 4: Bottom-up decorted prse forest for the input DAG c c 261

9 right forest (ie, to get PRCG prser not recognizer which ccepts DAG s input) we need to perform nother wlk on the previous decorted forest 332 The Top-Down Wlk The ide of the top-down wlk on the prse forest decorted y the ottom-up wlk is to (re)compute ll the previous decortions strting from the ottom-up decortion ssocited with the instntited strt predicte It is to e noted tht (the lnguge defined y) ech top-down decortion is suset of its ottom-up counterprt However, when top-down decortion ecomes empty, the corresponding sutree must e ersed from the forest If the ottom-up wlk succeeds, we re sure tht the top-down wlk will not result in n empty forest Moreover, if we perform new ottom-up wlk on this reduced forest, the new ottom-up decortions will denote the sme lnguge s their top-down decortions counterprt The forest is wlked top-down strting from the super-root The constrint eing tht n instntited A( ρ)-cluse is visited if nd only if ll the occurrences of A( ρ) occurring in the RHS of instntited cluses hve ll lredy een visited This constrint cn e stisfied for ny non-cyclic forest Initilly, we ssume tht ech rgument of ech instntited predicte hs n empty decortion, except for the rgument of the super-root which is decorted y the DAG R computed y the ottomup pss Now, ssume tht top-down decortion hs een (fully) computed for ech rgument of the instntited predicte A 0 ( ρ 0 ) For ech instntited cluse of the form c ρ = A 0 ( ρ 0 ) A 1 ( ρ 1 ) A i ( ρ i ) A m ( ρ m ), we perform the following sequence: 12 1 We perform the intersection of the top-down decortion of ech rgument of A 0 ( ρ 0 ) with the decortion computed y the ottom-up pss for the sme rgument of the LHS predicte of c ρ If the result is empty, c ρ is ersed from the forest 2 For ech LHS rgument, the previous results re disptched over the symols of this 12 The decortion of ech rgument of A i( ρ i) is either initilly empty or hs lredy een prtilly computed rgument 13 Thus, ech instntited LHS symol occurrence is decorted y its own DAG If the considered cluse hs severl occurrences of the sme vrile in the LHS rguments (ie, is ottom-up non-liner), we perform the intersection of these DAGs in order to leve single decortion per instntited vrile If n intersection results in n empty DAG, the current cluse is ersed from the forest 3 The LHS instntited vrile decortions re propgted to the RHS rguments This propgtion my result in DAG conctentions when RHS rgument is mde up of severl vriles (ie, is comintoril) 4 At lst, we ssocite to ech rgument of A i ( ρ i ) new decortion which is computed s the union of its previous topdown decortion with the decortion just computed Exmple 3 When we pply the previous lgorithm to the ottom-up prse forest of Exmple 2, we get the top-down prse forest of Figure 5 In this prse forest, ersed prts re lid out in light gry The more noticle points wrt the ottom-up forest re the decortions etween n n c n (X 14 ) n c n 14 n n c 14 nd its RHS predictes n c n 14 nd n n c 14 which re chnged oth to {c} insted of {c, c} nd {c, } These two chnges induce the indicted ersings 13 Assume tht ρ 0[k] = pq D, tht the decortion DAG ssocited with the k th rgument of A 0( ρ 0) is D pq = (Q pq, Σ pq, δ pq, p, F pq ) (we hve L(D pq ) L(D pq )) nd tht α 0[k] = α 1 kxα 2 k nd tht ij D is the instntition of the symol X in c ρ Our gol is to extrct from D pq the decortion DAG D ij ssocited with tht instntited occurrence of X This computtion cn e helped if we mintin, ssocited with ech decortion DAG function, sy d, which mps ech stte of the decortion DAG to set of sttes (ounds) of the input DAG D If, s we hve ssumed, D is miniml, ech set of sttes is singleton, we cn write d(p ) = p, d(f ) = q for ll f F pq nd more generlly d(i ) Q if i Q Let I = {i i Q pq d(i ) = i} nd J = {j j Q pq d(j ) = j} The decortion DAG D ij is such tht L(D ij ) = S i I,j J {x x is pth from i to j } Of course, together with the construction of D ij, its ssocited function d must lso e uilt 262

10 n n c n 14 {c} n n c n (X 14 ) n c n 14 n n c 14 {c} {c} n c n 14 n n c 14 n c n (X 14 ) n c n 14 {c} n n c (X 14 ) n n 14 {c} n c n ( 12 X 24 ) n c n 24 n n c (X 13 c 34 ) n n c 13 {c} {c} n c n 14 n c n 24 n n 14 n n c 13 {c} {} n c n ( 23 X 33 c 34 ) n c n 33 n n ( 12 X ) n n 22 n c n (X 24 ) n c n 24 {c} n n c (X 13 ) n n 13 {} {c} n c n 24 n n 13 {} n c n ( 23 X 33 c 34 ) n c n 33 n c n 33 n c n (ε 33 ) ε n n ( 12 X ) n n 22 n n 22 n n (ε 22 ) ε Figure 5: Top-down decorted prse forest for the input DAG c c 263

11 34 Time nd Spce Complexities In this Section we study the time nd size complexities of the forest filtering lgorithm Let us consider the su-dag D pq of the miniml input DAG D nd consider ny (finite) regulr lnguge L L(D pq ), nd let D L e the miniml DAG st L(D L ) = L We show, on n exmple, tht D L my e n exponentil wrt D pq Consider, for given h > 0, the lnguge ( ) h We know tht this lnguge cn e represented y the miniml DAG with h + 1 sttes of Figure 6 Assume tht h = 2k nd consider the su-lnguge L 2k of ( ) 2k (nested wellprenthesized strings) which is defined y 1 L 2 = {, } ; 2 k > 1, L 2k = {x, x x L 2k 2 }, It is not difficult to see tht the DAG in Figure 7 defines L 2k nd is miniml, ut its size 2 k+2 2 is n exponentil in the size 2k + 1 of the miniml DAG for the lnguge ( ) 2k This results shows tht, there exist cses in which some miniml DAGs D tht define sulnguges of miniml DAGs D my hve exponentil size (ie, D = O(2 D ) In other words, when, during the ottom-up or top-down wlk, we compute union of DAGs, we my fll on these pthologic DAGs tht will induce comintoril explosion in oth time nd spce 35 Implementtion Issues Of course, mny improvements my e rought to the previous principle lgorithms in prcticl implementtions Let us cite two of them First it is possile to restrict the numer of DAG copies: DAG copy is not useful if it is the lst reference to tht DAG We shll here devel the second point on little more: if n rgument of predicte is never used in nt non-linerity, it is only wste of time to compute its decortion We sy tht A k, the k th rgument of the predicte A is nonliner predicte rgument if there exists cluse c in which A occurs in the RHS nd whose k th rgument hs t lest one common vrile nother rgument B h of some predicte B of the RHS (if B = A, then of course k nd h must e different) It is cler tht B h is then non-liner s well It is not difficult to see tht decortions needs only to e computed if they re ssocited with non-liner predicte rgument It is possile to compute those non-liner predicte rguments stticlly (when uilding the prser) when the PRCG is defined within single module However, if the PRCG is given in severl modules, this full sttic computtion is no longer possile The non-liner predicte rguments must thus e identified t prse time, when the whole grmmr is ville This rther trivil lgorithm will not e descried here, ut it should e noted tht it is worth doing since in prctice it prevents decortion computtions which cn tke n exponentil time 4 Conclusion In this pper we hve shown how PRCGs cn hndle DAGs s n input If we consider the liner PRCG, the one equivlent to LCFRS, the prsing time remins polynomil Moreover, input DAGs necessitte only rther cosmetic modifictions in the stndrd prser In the non-liner cse, the stndrd prser my produce illegl prses in its output shred prse forest It my even produce (non-empty) shred prse forest though no sentences of the input DAG re in the lnguge defined y our non-liner PRCG We hve proposed method which uses the (slightly modified) stndrd prser ut prunes, within extr psses, its output forest nd leves ll nd only vlid prses During these extr ottomup nd top-down wlks, this pruning involves the computtion of finite lnguges y mens of conctention, union nd intersection opertions The sentences of these finite lnguges re lwys sustrings of the words of the input DAG D We choose to represent these intermedite finite lnguges y DAGs insted of sets of strings ecuse the size of DAG is, t worst, of the sme order s the size of set of strings ut it could, in some cses, e exponentilly smller However, the time tken y this extr pruning pss cnnot e gurnteed to e polynomil, s expected from previously known complexity results (Bertsch nd Nederhof, 2001) We hve shown n exmple in which pruning tkes n exponentil time nd spce in the size of D The deep reson comes from the fct tht if L is finite (regulr) lnguge defined y some miniml DAG D, there re cses where su-lnguge of 264

12 0 1 2 h 1 h Figure 6: Input DAG ssocited with the lnguge ( ) h, h > k k k+2 3 Figure 7: DAG ssocited with the lnguge of nested well-prenthesized strings of length 2k L my require to e defined y DAG whose size is n exponentil in the size of D Of course this comintoril explosion is not ftlity, nd we my wonder whether, in the prticulr cse of NLP it will prcticlly occur? References Frnois Brthélemy, Pierre Boullier, Philippe Deschmp, nd Éric de l Clergerie 2001 Guided prsing of rnge conctention lnguges In Proceedings of the 39th Annul Meeting of the Assocition for Comput Linguist (ACL 01), pges 42 49, University of Toulouse, Frnce Jeffrey D Hopcroft nd John E Ullmn 1979 Introduction to Automt Theory, Lnguges, nd Computtion Addison-Wesley, Reding, Mss Bernrd Lng 1994 Recognition cn e hrder thn prsing Computtionl Intelligence, 10(4): K Vijy-Shnker, Dvid Weir, nd Arvind K Joshi 1987 Chrcterizing structurl descriptions produced y vrious grmmticl formlisms In Proceedings of the 25th Meeting of the Assocition for Comput Linguist (ACL 87), pges , Stnford University, CA Eerhrd Bertsch nd Mrk-Jn Nederhof 2001 On the complexity of some extensions of rcg prsing In Proceedings of IWPT 01, Beijing, Chin Pierre Boullier, 2004 New Developments in Prsing Technology, volume 23 of Text, Speech nd Lnguge Technology, chpter Rnge Conctention Grmmrs, pges Kluwer Acdemic Pulishers, H Bunt, J Crroll, nd G Stt edition 265