Cross Serial Dependencies Are Not Hard to Process

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C r o s s - S e r i a l D e p e n d e n c i e s A r e N o t H a r d t o P r o c e s s

Carl Vogel

Ulrike Hahn

Holly Branigan

Institute for Computational Department of Experimental Centre for Cognitive Science

Linguistics

University of Stuttgart

Azenbergstr. 12

D-70174 Stuttgart

Germany

Psychology

University of Oxford

South Parks Road

Oxford OX1 3UD

England

University of Edinburgh

2 Buccleuch Place

Edinburgh EH8 9LW

Scotland

{vogel,holly,ueh}~cogsci.ed.ac.uk

Abstract

Cross-serial dependencies in D u t d l and Swiss-German are the only k n o w n extra- context fi'ee natural language syntactic

phenonmna. Psycholinguistie evidence

suggests cross-serial orderings tend to be easier to process t, lmn nested c o n s [ , r u c -

|iions. We, argue thai; |;tie expressivity re- quirements of the corresponding formal languages do not actually entail |;hat processing reduplication languages require the worst-ease time complexity for lmi- guages of the same expressive class. We dist;inguish between context-free representability and contc, xt-free processing. We show that for any language with up

to context fl'ee

expressive power, pro- cessing cross-scriM dependencies can be accommodated without atfect;ing tmrsing complexil,y. This is relal,ed to other work on reduplication phenonmna in formal models of computation.

1 I n t r o d u c t i o n

The cross-serial dependencies in Dutch and Swiss- German are the only known constituent-h;vel syn- tacl;ic phenomena whMl make natural languages not representable in con|,ext fi'ee languages (Gaz- dar, 198,5; Gazdar mid Pullum, 1985). Psycholin- guistic s~,udy of the cross-scriM dependencies re- veals thai; tim cross-serial orderings tend to be preferred over nested constructions (Bach eL al., 1986)} Bach et al. argue Dora this d m t tim pushdown stack cannot be the universal basis of the human parsing mechanism (since the pushdown a u t o m a t o n is essentially a context free recognil, ion device whidt cannot represent cross-serial dependencies). Stabler (1994), on the other hand, con- siders t;tm findings of Bach el; al. (1986) as evidence for finite hunian sentence processing capac- ity. In l, his paper, we dist, inguish between conl,ext- fi'ee representability and context-fl'ee processing.

1Nested constructions are a quintessentially context free phenomenon.

We show that for any language with up to context fi'ee expressive power, processin9 cross-serial dependencies can be accommodated without af- fecting parsing complexity. While this does essentially inflme the language with indexed expressivity, it does so while allowing us to rc~ain eollt,ext

free (or even regular) parsing eonqJexity. Ess~'n- tially, il; is possible t,o carve oul; a cross-section of l,he expressivity hierarchy with dm dcaircd processing complexity. The result is based oil the simple observation that t,he cross-serial dependencies m'e idealized by the string duplication language (whereas the nested dependencies m'e idealized by the palindrome language), and t h a t it is t, rivial to provide a context-free (or regular) language parse

for half of the st;ring, followed by a Lest: of equal- il,y for the remaining half of the string. This is consis(,ent; with tindings that cross-serial dependencies are not; hard to process, but qualilies the interpret;ation that Bach el; al. give to their re- suits and l,he implications on the human parsing niechanism, hi parl;icular, this suggests thai, with ml addil, ional operation |tie pushdown stack can be adequate for processing h u m a n lasiguages. It, also suggests an explanation for die finding that Dutch cross-serial dependencies arc easier to process than Gernlan nested dependencies. We out-

line fllrliher consequences of our proposal in terms of patterns of disfhiencies that are likely to occur in languages that admit cross-serial dependencies and propose a strate.gy R~r emtfirical investigation.

2 P r e l i m i n a r i e s

To calibrate our discussion, we quickly review t,h~, salient terminology from formal langm~ge theory and the current undersl,anding of dm import; tor natural language.s.

2.1 T e r m i n o l o g y

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tion rules. £1 is the class of languages generated by context sensitive granlmars--the sole restriction on production rules in this type of grammar is t h a t the right hand side (RHS) of each rule is at least as long as the left hand side (LHS). £1.5 denotes the class of languages generated by indexed grammars. Gazdar (1985) provides the most per- spicuous notation for the restricted forms t h a t production rules m a y take in such grammars: 2

1. A[...] - - + W[...]

2. A[...] ---+ B[i, ...]

3. A[i,...] ----+ W[...]

Indexed grammars incorporate a notion of stack- ing; rules of the form in (2) describe push operations, and those of the form in (3) involve pops. Rules of the form (1) are copy operations. The elipses indicate t h a t the remainder of the stack is passed on from the LHS to each nonterminal (and only the nonterminals) on the RHS. £ 2 is the class of context free languages generated by grammars whose productions are restricted such t h a t the LHS of each is a single nonterminal symbol, and each RHS is a sequence of terminals and nonterminals. Finally, the regular languages, £3 are those produced by regular grammars, character- ized by rules t h a t have a single nonterminal symbol on the LHS and on the RHS, either a terminal symbol or a terminal and a single nonterminal.

These classes of languages can be arranged into a hierarchy based on proper containment rela- tions among them: £ 3 C £2 C £1.5 C £1 C £0 (£0 is the least restrictive, the most expressive). Aho (1968) shows the existence of languages t h a t are a proper subset of the indexed languages and a proper superset of the context free. Joshi et al. (1989) conjecture that there is actually a convergence in expressive power among the 'mildly context sensitive' (MCS) languages, but other work points out exceptions (Sav- itch, 1989; Vogel and Erjavec, 1994). Since the reduplication languages (Savitch, 1989) are cen- tral to the point of this paper we define t h e m - - the languages homomorphic to the set of strings

{ww[w 6 {a,b}*}. The string duplication languages are not context free, although they are closely related to the string reversal languages

({wwR[w 6 {a, b}*}, where the R indicates the reversal operator) which are context free. The two languages induce different dependency relation- ships which is best described as nesting in the context free case and cross-serial in the indexed case:

a b b a a b a b . I - - - - ÷ 4- . . . ÷ I

. I - - - - ÷

2The bracketed material indicates a stack of in- dices; W denotes a sequence of elements of terminals and nonterminals; A, B denote nonterminals.

An important property of the each of the language classes is t h a t it is closed under bottl in- tersection with regular languages (e.g., the inter- section of a context free language and a regular language is no more expressive t h a n a context free language) and homomorphism (e.g., an order preserving map of each symbol in a language to a single element (possibly a string) of a context free language implies t h a t the first language is also context free). It is convenient to refer to languages with homomorphismSwwR{WWRIwto E {a, b}*} ai~d

{wwIw 6 {a,b}*} as and ww, respectively. Corresponding to expressivity class and the associated model of computation is the complexity of recognition for each class. Table 1 gives an informal ranking of the language classes with their corresponding worst case recognition complexity on the standard model of computation. Thus, given a context free grammar for w w R and

a string of length n, then in the worst case it will take an amount of time proportional to the cube of the length of the string to determine whether the string is in ww R (and identify its structure). While the expressivity hierarchy is useful for dif- ferentiating classes of lmlguages in precise terms like worst-case recognition complexity, it is easy to use the hierarchy incorrectly. For instance, it is not valid to conclude that because a language is in a particular language class all subsets of that language are also included t h a t language class (e.g. ww;i is a proper subset of w, yet w 6£3

w w R 6£2). Also, in most cases the structural de- scriptions that underlie strings of a language are of more interest than the string sets themselves. For this reason it is useful to distinguish weak and

strong containment of a grammar in a language class: e.g., a grammar is weakly context free if its stringset is context free; a grammar is strongly

context free if its treeset is also context free.

2.2 Applicability to Natural Language

Pullum and Gazdar (1982) survey the arguments up to the time they wrote for the non-coritext- freeness of natural language. The most interesting were those that considered idealizations of linguistic phenomena in terms of the string duplicating language, ww. In each case they found the m'- gument flawed: the phenomena in question did not yield languages whose stringsets were homomorphic to tile duplication language. Bresnan et al. (1982) argue that Dutch is not strongly context free. Shieber (1985) provides a stringset argument about a dialect of Swiss-German, which has a class of verb phrases with cross-serial dependencies (through case marking) between NPs and their Vs, which establishes even the weak-non- context-freeness of natural language because of

homomorphism to ww. Manaster-Ramer (1987)

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Hierarchy Level ]] Language T y p e . Model of Computation Complexity

• . , , ,

0 u n r e s t r i c t e d phrase structure undecidable

g r a m m a r (=.r.e.) ..

1 context sensitive (C recursive) P S P A C E

1.5 indexed

1.75 mildly context sensitive

2 context free

3 regular

Turing Machine (TM)

Linear Bounded A u t o m a t a LBA)

ested Stack A u t o m a t a

NSA)

mbeded Pushdown

A u t o m a t a (EPDA)

Pushdown A u t o m a t a (PDA) Finite State Machines (FSM)

NP-Complete

n 7

n a

' linear " '

Table 1: Models of G r a m m a r and C o m p u t a t i o n

corrected stringset argument t h a t Dutch licences

a"b'*c '~ constructions, which are MCS. No known syntactic phenomenon requires greater t h a n indexed language expressivity.

The point of this paper is to emphasize t h a t although a particular Swiss-German dialect renders natural language syntax non-context free, it does not entail that natural languages, induding the ones t h a t license cross-serial dependencies, incur the worst case recognition complexity costs for indexed languages. In fact, we argue in the next section t h a t w w is fairly straightforward to process. Essentially, we consider languages x x homomorphic to w w , where x can be either £3 or £2, and argue t h a t the recognition for x x is no worse than worst case recognition for £3 if x E£3 and no worse than the worst case for £2 i f x E£2, even though x x is itself indexed.

3 Cross-Serial D e p e n d e n c i e s Are

N o t Hard to P r o c e s s

It is always possible to compile less restrictive grammar formalisms into more restrictive covering formalisms, allowing different constituent analy- ses and potential stringset overgeneration. Meta- grammatical techniques give an alternative t h a t preserve coverage, b u t use special purpose processing. We suggest a parsing m e t h o d for languages t h a t rely on w w which does not cost a greater complexity fec than the worst case for parsing context fi'ee grammars. T h e m e t h o d is metagrammatical and therefore akin to propos- als put forward previously for handling coordination (Dahl and McCord, 1983) with logic grammars and TAGs (Shieber, 1995) or for extraposi- tion (Milward, 1994). T h e m e t h o d is constrained enough not to augment overall processing complexity, implying t h a t ww does not require the worst case recognition complexity for its characteristic class, the MCS languages.

3.1 W h y not?

Trivially, the string duplication languages can be recognized with time complexity proportional to

the length of the string - - if the string is of even length, and its first half is identical to the second half, then this can be established in just linear time. Though trivial in the sense of being about mere recognition, this is nonetheless interesting. In particular, under the reasonable hy- pothesis t h a t humans are not in general reverse- wired a it is easier to process serial orders thml their reverse. In this trivial recognition model we could take tile serial ordering as primitive, b u t to use the same model as a recognizer for the con- text free string reversal languages would require an additional step of reversing the second tlalf of the string before checking equivalence, which means the recognition complexity is n l o g n . Thus, for trivial recognition tim string duplication languages are easier to process t h a n the string reversal lazlguagcs. This is a concrete illustration that not every language costs the worst case recognition complexity for its expressivity class.

However, in the case of natural languages, parsing is of greater interest than mere recognition. A generalization of the recognizer m e t h o d can be used inside a parsing approach as well. Suppose some i such t h a t i > 2; suppose we want a recognizer for { w w ] w E {a,b}*} where w E £ i , then we can use a parser t h a t is no worse than cubic (if i : 2) and which can be linear (if i = 3) to determine if w EEl. Thus, if we parse exactly half of the string using a processor designed for languages in £ i , and then ascertain whether the remaining half is identical, then we remain in the

[image:3.596.77.515.69.196.2]

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same processing complexity class, since the iden- tity check occurs after tile parse and only requires linear time, b u t we also have structural information a b o u t the sentence as a whole. We know the structure of the first half of the string, and the second half of tile string b u t not the structure of tile second half (the g r a m m a r for w could be ambiguous), although we can assume t h a t the second w was licensed by exactly the same tree structure as the first. This m e t h o d also preserves a relative difference between parsing w w a n d w w n , at least for £3. Since w w ~ can be represented directly within £ 2 it can be argued t h a t we should not be required to use the m e t a g r a m m a t i c a l m e t h o d of parsing it, just to keep s y m m e t r y with the duplication languages. Interestingly, if w is in £2 a n d we use the m e t a g r a m m a t i c a l parsing method, then w w ~¢ also requires more processing time t h a n w w for the same reason as the trivial case. Suppose instead t h a t we allow ww n to be parsed without using tile m e t a g r a m m a t i c a l method. In t h a t case w w is relatively even easier t.o process since it costs [wl 3 to parse with the m e t a g r a m m a t i c a l approach b u t

w w I~ will cost (2[wl) 3 in tile direct approach. It, might be claimed t h a t just as we argue w w not to require the worst case complexity for its language class (£1.5), neither need ww n for £2; but, the reversal language is a canonical example of a language t h a t makes m a x i m a l use of the stack in the PDA. In any case, the m e t a g r a m m a t i c a l m e t h o d for parsing ww costs no more t h a n just parsing strings in the characteristic language class of w.

If this were the complete story then we could only recognize languages h o m o m o r p h i c to the duplication languages. Clearly even the Ziirich dialect of Swiss-German allows other constructions, all of which we can assume are context free (Pul- lure and Gazdar, 1982). Essentially we want to be able to write a r b i t r a r y £ 3 or £2 g r a m m a r s and also be able to parse the string duplication language for whichever £ i we choose. T h e language defined by such a union is no longer £ i , b u t will not contain a r b i t r a r y £1.5 strings, and if i = 3 then the union will not even contain a r b i t r a r y context fi'ee strings. However, the situation is more involved t h a n tile basic approach since there needs to be a way to indicate where the m e t a g r a m m a t - teal a p p r o a c h is to be invoked. Add a single feature to the g r a m m a r interpreted by tile processor as 'expect a copy'. 4

1. A ---+ W B M Y

We allow context free productions of the form shown in (1), where A and B are nonterminals and W, Y are (possibly e m p t y ) sequences of terminals and nonterminals, B possibly occurring a m o n g

4 O l l c e w e admit 'interpretability by the processor' we in principle have TM power. Itowever we make quite restricted use of such interpretation. The rule format makes clear that it is less expressive than indexed grammars when interpreted directly.

the nonterminals of Y. For an ambiguous CFG, there is no g u a r a n t e e t h a t multiple instances of a nontcrminal will rewrite to through the same sequence of productions to yield the same string.

There are any n u m b e r of ways t h a t this basic n o t a t i o n can be used in a m e t a g r a m m a t i c a l approach. In the first instance, we take c to be a signal to the processor to generate an e x p e c t a t i o n for a duplicate of the terminal sequence t h a t the nonterminal it is a t t a c h e d to gets rewritten to, and t h a t this expectation m u s t be satisfied by the next nonterminal of the same n a m e a n d in the same local domain. 5 This a p p r o a c h will require t h a t the sequence of terminals rewritten from the first B in (1) will be duplicated by the terminal sequence rewritten from the first instance of B (if any) t h a t occurs in Y. T h e restriction will not hold of subsequent instances of the nonterminal marked for copying in the same local domain nor at ditferent levels in the analysis. A stronger in- t e r p r e t a t i o n could require an e x p e c t a t i o n for the same constituent analysis of the nonterminal as well. Since we do not allow the feature to stack, tile string-based m e t h o d does not yield the full expressive power of indexed languages. T h e point is just t h a t it's possible to keep a CF (or regular) g r a m m a r , and supplement the processor with a string-duplication o p e r a t o r which can be; invoked at the subsentence level. This is sufficient to yield languages thai; more closely resemhle the Ziirich dialect in having other constructions besides the duplication construction, yet remaining efficiently processable. ~

We have implemented tile interpreter in a chart parser t h a t can be used in either top-down or b o t t o m - u p fashion. Edges in the chart are m a r k e d with a category (some nonterminal or preter- minal symbol from the g r a m m a r ) , constituents, subs|ring span and expectations (along with a unique identifier for each edge). This is modi- fied to include a list of constraints, which for the present purposes is p r e s u m e d to be just duplication checks. An edge with no expectations is inactive (saturated) and one with expectations is active. In the completer step, when active edges combine with adjacent inactive edges whose category satisfies the current expectation of the active, the usual process of creating a new edge with one less expectation is a u g m e n t e d with an- other: if the current expectation has an associated copy feature, then the new edge is m a r k e d with a constraint interpreted by the parser as in- dicated above - - the nonterminal symbol and tile string spanned by the inactive edge are noted so

5We take a local domain, in tree terms, as a node and tile set of nodes that it immediately dominates.

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t h a t the next inaetive edge of the same category (if one is expected) will have to span an |dent|- eL1 string. Constraints of this form are not passed on after satisfied once, and are not passed out of the local domain. Within the same set of restric- tions the implemented constraint could have been ' e x p e c t a reversed copy'. This would require con> p u t a t i n g the string's reverse before a n n o t a t i n g the constraint list.

4 D i s c u s s i o n

Tile context; free languages have alre.ady been studied from the perspective of minimal addition

to incorporate copy languages. Savitch (1989)

does exactly t h a t by prese, nting the model of con> put;at|on required for the class of languages de- lined by augment;ing the CFLs with redut)lication: a Reduplicat;ion P D A (RPDA). An I~PDA is just a PDA which has a special type of symbol thai, can tie p u t onto the stack to nlake the machine treat the p a r t of the stack above it ms if it were a queue. Essentially, t,his obtains the reversM behavior nee, ded of a st.ack to process copy languages as well as rew',rsals. Mull,|pie instances of the special sylnbol can be placed on |.he stack. Say- itch present,s a chara.ct,erization of the languages ill te, rms of stxingsets and the requisil;e compu- Lal;ional structures. The family t h a t we charac- terized above in t e r m s of graInntars arc tn'operly a sullset of the languages recognized by R.PDA, a restrk:tion of R P D A languages which Savitch (1989) terlns simple R, P D A lanqu,.qes. T h e model of comput~ttion here is an R P D A in which only (me spe, cial symbol is allowed on the stack at any one, time. We have not In'oven the equivalence we conje(:tllre bel, we(,'tl our Inetagranunatical m e t h o d and the reduplication contex&free gram- mars (RCFC, s) t h a t Savi|,ch introduces as genera- tive of simple R P D A languages. Saviteh's (1989)

g r a m m a r s are stated in terms of rule s c h e m a t a (a tin|re set) t h a t general,e potentially infinite sets of rewril;e rules. This is the tradeoff lletwe, en doing things m e t a g r a I n m a t i e a l l y and directly.

Josh| and Rainbow

(Josh|,

1990; Rainbow and

Josh|,

1994) have also considered the perforntan('e d a t a associated with processing crossed vs. nested dependencies and present an alternative com- putal, ion model, |;tie bottom-up embedded P D A

(BEPDA), designed for a wit|an|; of tree-adjoining g r a l n m a r (it uses a stack of slacks and a more complex operation for eml)tying the stack). II,am- bow an(1 Josh| (1994) use the processing model to d e m o n s t r a t e t h a t it can account for the dilDrence between crossed and nested dependencies in terlns of the a m o u n t of time associated objects spend in the pushdown store of the B E P D A using a mildly context free language model t h a t captures dependencies directly, rather t;han m e t a g r a m m a t i c a l l y . 7

r Josh| (1990) gives a similar analysis fi)r EDPAs.

Essentially, their analysis (:oncludes (;tie satne: when judging string isomorphisnls, it; is easier to m a k e the j u d g m e n t of identic~flly ordered pairs t h a n it is to reversely ordered pairs. Thus, the cross-serial dependencies n e e d n ' t cost the worst ease complexity for parsing indexed or mildly ecru- text sensitive languages. Parsing ww languages requires, at worst, (;lie worst ease complexity of parsing w in whichever language class w is restricted to. Shieber (1985) pointed out without p r o o f t h a t (;tie nonCl,' d a t a associated ZiMch di- Nee(; is linearly parsable; our task has been to

clarify how this follows from the language (;heory.

4.1 A C a v e a t

For eilicicnt processing of ww to entail correspond~ ing eomplexity fin" n a t u r a l lmlguages t h a t license cross-serial dependencies hinges crucially on there being eflMently (:(mlputable hoinonmrphisms tm- tween the natural language, and the string dupli- eati<m languages. This is aIl open question, tIow- ever, given t h a t empirical work t h a t COlnpares processing of crossed atld nested dependencies alld concludes t h a t the

m'oss-serial

dependencies are preferred to nested ones (Bach el; al., 1986), and giw~,n (}tit' arl{un!.ent thai, cross-serial dependencies are in theory easier to process, we feel it. teas(m- able to enterta.in the asSUml)tion t h a t somel;hing such exists. This does n(~t require us l.(~ assunlo thai; ileol)le a(:Lually use conl,exl;-fl'ee g r a m m a r s and COlllp/lte holllolnort)hisills ill order 1,o itnder- stand natural languages, just thai; l:he c(mlt)ul;a- tional model should lm at least approximat.ely as eflicient as t)eoph~,.

4.2 I m I ) l i c a t i o n s

()tit' inetagralnmatical a p p r o a c h to dealing with cross serial dependencies involves the ~uSSUlnpl,ion of an operation for testing string duplication. We hinl;ed earlier t h a t we h;el there to lm sutlicient reason to believe t h a t copy-checldng is a basic cognitive

flmction,

and although we d o n ' t suppose that, people have built in production systems and processors isolnorphic to ollr chart parser aim base language, we do think t h a t t,his copy-dmeking is invoked in the processing of crossed depe.Ildencies. Our approach to accounting for the processing complexity t h a t the string duplication languages should take does make e m p M c a l predictions and these can lie teste, d. For instance, if it is t;he case t h a t such a nmchanisin exists, then p a t t e r n s of string-copy disthtency should ocellr with (lifferenl. frequency in languages t h a t lk:ense cross-serial (le- pendencies t h a n in those tha, t (t(I iI.ot. A stxing- copy dislhleney is just one t h a t involves a repeat of p a r t of the sentence, ul, t;ered so far:

1. We went to the to lhe store to buy some Jlo'.a'.

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guages t h a t do not a d m i t cross serial dependencies. These differences should be manifest in speech c o r p o r a like those t h a t are currently being accumulated (Anderson et al., 1992; Miller, 1995), b u t which n ~ d a u g m e n t a t i o n by a corpus derived from copy-language dialects. Verifying this would, for example, establish whether the copied strings need to be constituents, a n d this has a bearing on whether processing models designed for incremental i n t e r p r e t a t i o n (Milward, 1992) are the best de- scriptors of h u m a n p e r f o r m a n c e . " We do not offer a r g u m e n t s t h a t our m e t a g r a m m a t i c a l a p p r o a c h is the best description of h u m a n processing of cross- serial dependencies, just t h a t it is a n o t h e r theo- retical justification for the difference in processing nested dependencies and efficient processing of crossed dependencies.

Acknowledgements

Vogel is grateful to the SFB 340 for funding his stay S t u t t g a r t ; H a h n acknowledges the sup- p o r t of ESRC g r a n t No. R004293341442; Brani- t a n , EPSI~C research studentship No. 92315069. All would like to t h a n k Catherine Collin, Toma~ Erjavec, T s u t o m u Fujinami:, Merce P r a t , Fred P0powich, M a r k Steedman, and the anonymous reviewers.

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