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 = {a b c d |n ≥ 1} n n n n n We investigate Global Index Gram- but not L5 = {a b c d e |n ≥ 1}). 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 ∈ V , a formal language resource which can be used α, γ ∈ (V ∪ T )∗, i ∈ I: to model and process NL phenomena beyond a. A[..] → α B[..] γ b. A[i..] → α B[..] γ context free, or beyond the level-2 MCSLs (like c. A[..] → αB[i..] γ those mentioned above) or to compile grammars Example 1 L(G ) = {wcw |w ∈ {a, b}∗}, created in other framework into GIGs. LIGs wcw Gww = ({S, R}, {a, b}, {i, j}, 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 {ww+|w ∈ Σ∗}. 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 ∈ 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..]α, a.2 [y..]A → [y..]α, b. [..]A → [x..]a β (Vijay-Shanker et al., 1987) or (Joshi, 2000). and c. [x..]A → [..]α x ∈ I, y ∈ {I ∪ #}, A ∈ N, α, β ∈ (N ∪ T )∗ and there is only one stack affected at each deriva- a ∈ T . tion step, with the consequence of the semilin- a.i A → α (epsilon) ² earity property of LILs. GIGs share this unique- a.ii A → α (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 β (push) x stack of indices is independent of non-terminals c. A → α a β (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 β and γ be in (N ∪ T )∗, δ 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.) µ 2.2.1 Examples (i.e. µ = ² or µ = [x], x ∈ I) then: The following example shows that GILs con-

i. δ#βAγ ⇒ δ#βX1 ...Xn γ tain a language not contained in LILs, nor in the µ family of MCSLs. This language is relevant for ii. xδ#βAγ ⇒ xδ#βX1 ...Xn γ µ modeling coordination in NL. Example 2 (Multiple Copies) . 2. If A → aX1 ...Xn is a production of type + ∗ µ L(Gwwn ) = {ww | w ∈ {a, b} } (b.) or push: µ = x, x ∈ I, then: Gwwn = ({S, R, A, B, C, L}, {a, b}, {i, j}, S, #,P ) and where P is: S → AS | BS | CC → RC | L δ#wAγ ⇒ xδ#waX1 ...Xn γ µ R → RAR → RBR → ² ¯i ¯j [#] A → a B → b L → La | a L → Lb | b 3. If A → X1 ...Xn is a production of type (c.) i j ¯i ¯j µ The derivation of ababab: or pop : µ =x, ¯ x ∈ I, then: #S ⇒ #AS ⇒ i#aS ⇒ i#aBS ⇒ ji#abS ⇒ xδ#wAγ ⇒ δ#wX1 ...... Xn γ µ ji#abC ⇒ ji#abRC ⇒ i#abRBC ⇒ #abRABC ⇒ #abABC ⇒ i#abaBC ⇒ ji#ababC ⇒ ji#ababL ⇒ The reflexive and transitive closure of ⇒ is i#ababLb ⇒ #ababab ∗ denoted, as usual by ⇒. We define the language The next example shows the MIX (or Bach) ∗ of a GIG, G, L(G) to be: {w|#S ⇒ #w and w language. (Gazdar, 1988) conjectured the MIX is in T ∗} language is not an IL. GILs are semilinear, The main difference between, IGs, LIGs and (Casta˜no,2003c) therefore ILs and GILs could GIGs, corresponds to the interpretation of the be incomparable under set inclusion. derives relation relative to the behavior of the Example 3 (MIX language) .L(Gmix ) = stack of indices. In IGs the stacks of indices are ∗ {w|w ∈ {a, b, c} and |a|w = |b|w = |c|w ≥ 1} distributed over the non-terminals of the right- Gmix = ({S, D, F, L}, {a, b, c}, {i, j, k, l, m, n}, S, #,P ) hand side of the rule. In LIGs, indices are asso- where P is: S → FS | DS | LS | ² F → c F → b F → a ciated with only one non-terminal at right-hand i j k D → aSb | bSa D → aSc | cSa D → bSc | cSb side of the rule. This produces the effect that ¯i ¯j k¯ D → aSb | bSa D → aSc | cSa D → bSc | cSb on the graph-structured stack of indices are per- l m n L → c L → b L → a formed at a constant time where the constant is ¯ m¯ n¯ l determined by the size of the index vocabulary. The following example shows that the family O(n6 ) is the worst case; O(n3 ) holds for gram- of GILs contains languages which do not belong mars with state-bound indexing (which includes to the MCSL family. unambiguous indexing)6; O(n2 ) holds for unam- Example 4 (Multiple dependencies) biguous context free back-bone grammars with n n n + state-bound indexing and O(n) for bounded- L(Ggdp) = { a (b c ) | n ≥ 1}, state7 context free back-bone grammars with Ggdp = ({S, A, R, E, O, L}, {a, b, c}, {i}, S, #,P ) and P is: state-bound indexing. S → AR A → aAE A → a E → b i 3 GIGs and structural description R → b L L → OR | CC → c C | c i ¯i (Gazdar, 1988) introduces Linear Indexed O → c OE | c ¯i Grammars and discusses its applicability to Nat- The derivation of the string aabbccbbcc shows ural Language problems. This discussion is ad- five dependencies. dressed not in terms of weak generative capac- #S ⇒ #AR ⇒ #aAER ⇒ #aaER ⇒ i#aabR ⇒ ity but in terms of strong-generative capacity. ii#aabbL ⇒ ii#aabbOR ⇒ i#aabbcOER ⇒ #aabbccER ⇒ i#aabbccbR ⇒ ii#aabbccbbL ⇒ Similar approaches are also presented in (Vijay- ii#aabbccbbC ⇒ i#aabbccbbcC ⇒ #aabbccbbcc Shanker et al., 1987) and (Joshi, 2000) (see (Miller, 1999) concerning weak and strong gen- 2.3 GILs Recognition erative capacity). In this section we review some The recognition algorithm for GILs we presented of the abstract configurations that are argued for in (Casta˜no,2003) is an extension of Earley’s al- in (Gazdar, 1988). gorithm (cf. (Earley, 1970)) for CFLs. It has to 3.1 The palindrome language be modified to perform the computations of the R stack of indices in a GIG. In (Casta˜no,2003) a CFGs can recognize the language {ww |w ∈ ∗ graph-structured stack (Tomita, 1987) was used Σ } but they cannot generate the structural de- to efficiently represent ambiguous index opera- scription depicted in figure 1 (we follow Gazdar’s tions in a GIG stack. Earley items are modified notation: the leftmost element within the brack- adding three parameters δ, c, o: ets corresponds to the top of the stack): [δ, c, o,A → α • Aβ, i, j] [..] The first two represent a pointer to an active [a] [b,a] node in the graph-structured stack ( δ ∈ I and a [c,b,a] b [d,c,b,a] c ≤ n). The third parameter (o ≤ n) is used c [c,b,a] d [b,a] to record the ordering of the rules affecting the d [a] stack. c b [..] The O(n6 ) time-complexity of this algorithm a reported in (Casta˜no,2003) can be easily ver- ified. The complete operation is typically the Figure 1: A non context-free structural descrip- R costly one in an Earley type algorithm. It can tion for the language ww (Gazdar, 1988) be verified that there are at most n6 instances of Gazdar suggests that such configuration the indices (c1 , c2 , o, i, k, j) involved in this oper- would be necessary to represent Scandinavian ation. The counter parameters c1 and c2 , might be state bound, even for grammars with ambigu- 6Unambiguous indexing should be understood as ous indexing. In such cases the time complex- those grammars that produce for each string in the lan- guage a unique indexing derivation. ity would be determined by the CFG backbone 7Context Free grammars where the set of items in each properties. The computation of the operations state set is bounded by a constant. unbounded dependencies.Such an structure can be generated by a LIG, and can by an IG (see be obtained using a GIG (and of course a LIG). (Casta˜no,2003b) for a complete discussion and But the mirror image of that structure can- comparasion of GIG and LIG generated trees). not be generated by a GIG because it would GIGs cannot produce this structural descrip- require to allow push productions with a non tion, but they can generate the one presented in terminal in the first position of the right-hand figure 3, where the arrows depict the leftmost side. However the English adjective construc- derivation order. GIGs can also produce similar tions that Gazdar argues that can motivate the structural descriptions for the language of mul- LIG derivation, can be obtained with the follow- tiple copies (the language {ww+| w ∈ Σ∗} as ing GIG productions as shown in figure 2. shown in figure 4, corresponding to the gram- Example 5 (Comparative Construction) . mar shown in example 2. AP → AP NP AP → A¯ A¯ → AA¯ A → a A → b A → c [ ] i j k NP → a NP NP → b NP NP → c NP ¯i ¯j k¯ [b,a,b,a] [b,a,b,a] [..] AP b [a,b,a] [a,b,a] d a [a] AP NP a [b,a] [b,a] c b AP [..] [a] NP c NP b a [ ] AP NP b [c] NP

A a Figure 3: A GIG structural description for the [b,c] NP copy language A A [a,b,c] [..] A A a [ ] [a] [b,c] [b,a] [a,b,a] A b a [b,a,b,a] [c] c b a b Figure 2: A GIG structural description for the [a,b,a] [b,a,b,a] [b,a,b,a] R language ww [b,a] [c,b,a] [b,a,b,a] [a] [b,a] b [a,b,a] [b,a,b,a] [b,a,b,a] [ ] It should be noted that the operations on indices [a] a [b,a] [a,b,a] ε [b,a] [a,b,a] b [a] b follow the reverse order as in the LIG case. On b a [a] [b,a] the other hand, it can be noticed also that the a a [ ] [a] ε b introduction of indices is dependent on the pres- b [ ] a ence of lexical information and its transmission a is not carried through a top-down spine, as in Figure 4: A GIG structural description for the the LIG or TAG cases. The arrows show the multiple copy language leftmost derivation order that is required by the operations on the stack. 3.2 The Copy Language 4 GIGs and HPSGs Gazdar presents two possible LIG structural de- We showed in the last section how GIGs can scriptions for the copy language. Similar struc- produce structural descriptions similar to those tural descriptions can be obtained using GIGs. of LIGs, and others which are beyond LIGs and However he argues that another tree structure TAGs descriptive power. Those structural de- could be more appropriate for some Natural scriptions corresponding to figure 1 were corre- Language phenomenon that might be modeled lated to the use of the SLASH feature in GPSGs with a copy language. Such structure cannot and HPSGs. In this section we will show how HEAD 1 the structural description power of GIGs, is not SUBJ < > H only able to capture those phenomena but also SUBJ additional structural descriptions, compatible 2 HEAD 1 with those generated by HPSGs. This follows SUBJ < 2 > from the ability of GIGs to capture dependen- cies through different paths in the derivation. Figure 5: Head-Subject Schema There has been some work compiling HPSGs into TAGs (cf. (Kasper et al., 1995), (Becker Figure 6 shows an equivalent structural de- and Lopez, 2000)). One of the motivations scription corresponding to the GIG produc- was the potential to improve the processing tions and derivation shown in the next exam- efficiency of HPSG, performing HPSG deriva- ple (which might correspond to an intransitive tions at compile time. Such compilation process verb). The arrows indicate how the transmis- allowed to identify significant parts of HPSG sion of features is encoded in the leftmost deriva- grammars that were mildly context-sensitive. tion order, an how the elements contained in the We will introduce informally some slight mod- stack can be correlated to constituents or lexical ifications to the operations on the stacks per- items (terminal symbols) in a constituent recog- formed by a GIG. We will allow the productions nition process. of a GIG to be annotated with finite strings XP in I ∪ I¯ instead of single symbols. This does [..] [v..] not change the power of the formalism. It is a YP XP [v..] standard change in PDAs (cf. (Harrison, 1978)) [n..] to allow to push/pop several symbols from the Y X [n..] [v..] stack. Also the symbols will be interpreted rel- y x ative to the elements in the top of the stack (as a Dyck set). Therefore different derivations Figure 6: Head-Subject in GIG format might be produced using the same production according to what are the topmost elements of Example 6 (Intransitive verb) XP → YPXP XP → XYP → YX → x Y → y the stack. This is exemplified with the produc- nv¯ n tions X → x and X → x, in particular in the #XP ⇒ #YPXP ⇒ #yXP ⇒ n#YXP ⇒ nv¯ [n]v n#yX ⇒ v#yx first three cases where different actions are taken Head-Comps-Schema Figure 7 shows the (the actions are explained in the parenthesis) : tree structure corresponding to the Head- nnδ#wXβ ⇒ vnδ#wxβ (pop n and push v) nv¯ Complement schema in HPSG. nvδ¯ #wXβ ⇒ δ#wxβ (pop n andv ¯) nv¯ HEAD 1 COMP < > vnδ#wXβ ⇒ vnvnδ¯ #wxβ (pushn ¯ and v) 2 nv¯ H C C n-2 nδ#wXβ ⇒ vnδ#wxβ ( check and push) 1 HEAD [n]v 1 n COMP < 2 > 3 We exemplify how GIGs can generate similar , 3 n structural descriptions as HPSGs do, in a very Figure 7: Head-Comps Schema tree representa- oversimplified and abstract way. We will ignore tion many details and try give an rough idea on how the transmission of features can be carried out The following GIG productions generate the from the lexical items by the GIG stack, obtain- structural description corresponding to figure 8, ing very similar structural descriptions. where the initial configuration of the stack is Head-Subj-Schema assumed to be [n]: Figure 5 depicts the tree structure corre- Example 7 (transitive verb) . sponding to the Head-Subject Schema in HPSG XP → XCPCP → YCPX → x CP → ² nv¯ n¯ (Pollard and Sag, 1994). Y → y n XP The derivation: [ ] n#XP ⇒ n#XCP ⇒ nv¯ #xCP ⇒ nv¯ #xY CP ⇒ v#xyCP ⇒ v#xy XP YP [n] [n] XP [ v ] [n] Kim YP XP [nn]

CP [ v ] we X [n v] X XP [n v] [ v n ] x Y CP [ v ] know ε y YP XP [ n v n ]

Figure 8: Head-Comp in GIG format Sandy X XP The productions of example 8 (which use [ v n ] claims some of the previous examples) generate the YP XP structural description represented in figure 9, [ n v n ] corresponding to the derivation given in exam- Dana X CP [ ] ple 8. We show the contents of the stack when ε each lexical item is introduced in the derivation. hates Example 8 (SLASH in GIG format) . Figure 9: SLASH in GIG format XP → YPXPXP → XCPXP → XXP CP → YPCPX → hates CP → ² nv¯ n¯ some languages that are not in the LIL/TAL X → know X → claims n¯v¯ nv¯ v¯ family. GILs do include those languages that YP → Kim|Sandy|Dana|we n are beyond context free and might be required A derivation of ‘Kim we know Sandy claims Dana for NL modelling. The similarity between GIGs hates’: and LIGs, suggests that LILs might be included #XP ⇒ #YPXP ⇒ n#Kim XP ⇒ n#Kim Y P XP ⇒ nn#Kim we XP ⇒ in GILs. We presented a succinct comparison nn#Kim we X XP ⇒ vn¯ #Kim we know XP ⇒ of the structural descriptions that can be gen- vn¯ #Kim we know Y P XP ⇒ erated both by LIGs and GIGs, we have shown nvn¯ #Kim we know Sandy XP ⇒ nvn¯ #Kim we know Sandy X XP ⇒ that GIGs generate structural descriptions for vn¯ #Kim we know Sandy claims XP ⇒ the copy language which can not be generated vn¯ #Kim we know Sandy claims Y P XP ⇒ ∗ by LIGs. We showed also that this is the nvn¯ #Kim we know Sandy claims Dana XP ⇒ case for other languages that can be generated #Kim we know Sandy claims Dana hates by both LIGs and GIGs. This corresponds Finally the last example and figure 10 show to the ability of GIGs to generate dependent how coordination can be encoded. paths without copying the stack. We have Example 9 (SLASH and Coordination) shown also that those non-local relationships XP → YPXPXP → XCPXP → XXP that are usually encoded in HPSGs as feature CP → YPCPCP → ² X → visit [nvn¯ ]c transmission, can be encoded in GIGs using its X → talk to C → and CXP → XPCXP nv¯ n¯ stack, exploiting the ability of Global stacks to CXP → CXPX → did Y P → W ho|you n¯v¯ n encode dependencies through dependent paths and not only through a spine. 5 Conclusions Acknowledgments: We presented GIGs and GILs and showed the descriptive power of GIGs is beyond CFGs. Thanks to J. Pustejovsky for his continuous support and encouragement on this project. Many thanks also to the CFLs are properly included in GILs by def- anonymous reviewers who provided many helpful com- inition. We showed also that GIGs include ments. This work was partially supported by NLM Grant [ ] XP editors, Natural Language Parsing and Linguistic Theories, pages 69–94. D. Reidel, Dordrecht.

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