On the Applicability of Global Index Grammars Jos´eM. Casta˜no Computer Science Department Brandeis University [email protected] Abstract TALs/LILs) is able to capture up to 4 counting n n n n dependencies (includes L4 = fa b c d jn ¸ 1g n n n n n We investigate Global Index Gram- but not L5 = fa b c d e jn ¸ 1g). They were mars (GIGs), a grammar formalism proven to have recognition algorithms with time that uses a stack of indices associated complexity O(n6 ) (Satta, 1994). In general for with productions and has restricted a level-k MCSL the recognition problem is in context-sensitive power. We discuss O(n3 ¢2 k¡1 ) and the descriptive power regard- some of the structural descriptions ing counting dependencies is 2k (Weir, 1988). that GIGs can generate compared with Even the descriptive power of level-2 MCSLs those generated by LIGs. We show (Tree Adjoining Grammars (TAGs), Linear In- also how GIGs can represent structural dexed Grammars (LIGs), Combinatory Catego- descriptions corresponding to HPSGs rial Grammars (CCGs) might be considered in- (Pollard and Sag, 1994) schemas. sufficient for some NL problems, therefore there have been many proposals3 to extend or modify 1 Introduction them. On our view the possibility of modeling coordination phenomena is probably the most The notion of Mildly context-sensitivity was in- crucial in this respect. troduced in (Joshi, 1985) as a possible model In (Casta˜no,2003) we introduced Global In- to express the required properties of formalisms dex Grammars (GIGs) - and GILs the corre- that might describe Natural Language (NL) sponding languages - as an alternative grammar 1 phenomena. It requires three properties: a) formalism that has a restricted context sensitive constant growth property (or the stronger semi- power. We showed that GIGs have enough de- linearity property); b) polynomial parsability; scriptive power to capture the three phenomena c) limited cross-serial dependencies, i.e. some mentioned above (reduplication, multiple agree- limited context-sensitivity. The canonical NL ments, crossed agreements) in their generalized problems which exceed context free power are: forms. Recognition of the language generated by multiple agreements, reduplication, crossing de- a GIG is in bounded polynomial time: O(n6 ). 2 pendencies. We presented a Chomsky-Sch¨utzenberger repre- Mildly Context-sensitive Languages (MCSLs) sentation theorem for GILs. In (Casta˜no,2003c) have been characterized by a geometric hierar- we presented the equivalent automaton model: chy of grammar levels. A level-2 MCSL (eg. LR-2PDA and provided a characterization the- 1See for example, (Joshi et al., 1991), (Weir, 1988). 2However other phenomena (e.g. scrambling, Geor- 3There are extensions or modifications of TAGs, gian Case and Chinese numbers) might be considered to CCGs, IGs, and many other proposals that would be be beyond certain mildly context-sensitive formalisms. impossible to mention here. orems of GILs in terms of the LR-2PDA and mitted” only to one non-terminal. As a con- GIGs. The family of GILs is an Abstract Fam- sequence they are semilinear and belong to the ily of Language. class of MCSGs. The class of LILs contains L4 The goal of this paper is to show the relevance but not L5 (see above). of GIGs for NL modeling and processing. This A Linear Indexed Grammar is a 5-tuple should not be understood as claim to propose (V; T; I; P; S), where V is the set of variables, GIGs as a grammar model with “linguistic con- T the set of terminals, I the set of indices, S tent” that competes with grammar models such in V is the start symbol, and P is a finite set as HPSG or LFG. It should be rather seen as of productions of the form, where A; B 2 V , a formal language resource which can be used ®; γ 2 (V [ T )¤, i 2 I: to model and process NL phenomena beyond a. A[::] ! ® B[::] γ b. A[i::] ! ® B[::] γ context free, or beyond the level-2 MCSLs (like c. A[::] ! ®B[i::] γ those mentioned above) or to compile grammars Example 1 L(G ) = fwcw jw 2 fa; bg¤g, created in other framework into GIGs. LIGs wcw Gww = (fS; Rg; fa; bg; fi; jg; S; P ) and P is: played a similar role to model the treatment of the SLASH feature in GPSGs and HPSGs, and 1.S[::] ! aS[i::] 2.S[::] ! bS[j::] to compile TAGs for parsing. GIGs offer addi- 3.S[::] ! cR[::] 4.R[i::] ! R[::]a 5.R[j::] ! R[::]b 6. R[] ! ² tional descriptive power as compared to LIGs or TAGs regarding the canonical NL problems 2.2 Global Indexed Grammars mentioned above, and the same computational cost in terms of asymptotic complexity. They GIGs use the stack of indices as a global con- also offer additional descriptive power in terms trol structure. This formalism provides a global of the structural descriptions they can generate but restricted context that can be updated at for the same set of string languages, being able any local point in the derivation. GIGs are a to produce dependent paths.4 kind of regulated rewriting mechanisms (Dassow This paper is organized as follows: section 2 and P˘aun,1989) with global context and his- reviews Global Index Grammars and their prop- tory of the derivation (or ordered derivation) as erties and we give examples of its weak descrip- the main characteristics of its regulating device. tive power. Section 3 discusses the relevance The introduction of indices in the derivation is of the strong descriptive power of GIGs. We restricted to rules that have terminals in the discuss the structural description for the palin- right-hand side. An additional constraint that drome, copy and the multiple copies languages is imposed on GIGs is strict leftmost derivation fww+jw 2 Σ¤g. Finally in section 4 we discuss whenever indices are introduced or removed by how this descriptive power can be used to en- the derivation. code HPSGs schemata. Definition 1 A GIG is a 6-tuple G = (N; T; I; S; #;P ) where N; T; I are finite pair- 2 Global Index Grammars wise disjoint sets and 1) N are non-terminals 2.1 Linear Indexed Grammars 2) T are terminals 3) I a set of stack indices 4) S 2 N is the start symbol 5) # is the start stack Indexed grammars, (IGs) (Aho, 1968), and symbol (not in I,N,T ) and 6) P is a finite set of Linear Index Grammars, (LIGs;LILs) (Gazdar, productions, having the following form,5 where 1988), have the capability to associate stacks of indices with symbols in the grammar rules. IGs 5The notation in the rules makes explicit that oper- are not semilinear. LIGs are Indexed Grammars ation on the stack is associated to the production and neither to terminals nor to non-terminals. It also makes with an additional constraint in the form of the explicit that the operations are associated to the com- productions: the stack of indices can be “trans- putation of a Dyck language (using such notation as used in e.g. (Harrison, 1978)). In another notation: a.1 4For the notion of dependent paths see for instance [y::]A ! [y::]®, a.2 [y::]A ! [y::]®, b. [::]A ! [x::]a ¯ (Vijay-Shanker et al., 1987) or (Joshi, 2000). and c. [x::]A ! [::]® x 2 I, y 2 fI [ #g, A 2 N, ®; ¯ 2 (N [ T )¤ and there is only one stack affected at each deriva- a 2 T . tion step, with the consequence of the semilin- a.i A ! ® (epsilon) ² earity property of LILs. GIGs share this unique- a.ii A ! ® (epsilon with constraints) ness of the stack with LIGs: there is only one [y] stack to be considered. Unlike LIGs and IGs the b. A ! a ¯ (push) x stack of indices is independent of non-terminals c. A ! ® a ¯ (pop) x¯ in the GIG case. GIGs can have rules where the Note the difference between push (type b) and right-hand side of the rule is composed only of pop rules (type c): push rules require the right- terminals and affect the stack of indices. Indeed hand side of the rule to contain a terminal in the push rules (type b) are constrained to start the first position. Pop rules do not require a termi- right-hand side with a terminal as specified in nal at all. That constraint on push rules is a (6.b) in the GIG definition. The derives def- crucial property of GIGs. Derivations in a GIG inition requires a leftmost derivation for those are similar to those in a CFG except that it is rules ( push and pop rules) that affect the stack possible to modify a string of indices. We de- of indices. The constraint imposed on the push fine the derives relation ) on sentential forms, productions can be seen as constraining the con- which are strings in I¤#(N [T )¤ as follows. Let text sensitive dependencies to the introduction ¯ and γ be in (N [ T )¤, ± be in I¤, x in I, w be of lexical information. This constraint prevents ¤ GIGs from being equivalent to a Turing Machine in T and Xi in (N [ T ). as is shown in (Casta˜no,2003c). 1. If A ! X1 :::Xn is a production of type (a.) ¹ 2.2.1 Examples (i.e. ¹ = ² or ¹ = [x], x 2 I) then: The following example shows that GILs con- i. ±#¯Aγ ) ±#¯X1 :::Xn γ tain a language not contained in LILs, nor in the ¹ family of MCSLs. This language is relevant for ii. x±#¯Aγ ) x±#¯X1 :::Xn γ ¹ modeling coordination in NL.