A Computational Approach to Minimalism

Transcription

1 A Computational Approach to Minimalism Alain LECOMTE CLIPS-IMAG BP Grenoble, cedex 9, France Abstract The aim of this paper is to recast minimalist principles (after recent developments of Chomsky s theory) inside a logical, and therefore deductive, framework. This will allow us to implement such principles by means of logical programming. We begin by a reminder on Stabler s minimalist grammars and on Categorial Grammar. We show then that there is a bridge in between the two formalisms, and we propose a logical formulation. 1. Introduction Minimalism is now the current trend in Generative Grammar ([Chomsky, 1995]) and many computational linguists are interested in the implementation of at least some of its principles. On another hand, many researchers have outlined a seemingly obvious relationship between Minimalism and Categorial Grammars. Both theoretical approaches emphasize the role of the lexicon and the role played by very general operations, like Merge and Move in the minimalist framework, or Application and Abstraction in Categorical Grammars. In both cases, the language at hand can be described as the closure of the lexicon under these operations (or generalized transformations, as Chomsky calls them). But the main difference lies in the fact that, in Categorial Grammars, those operations have a proper logical formulation, which is not the case for Minimalism. In the Lambek calculus ([Moortgat, 1997]), they are in fact the rules for elimination and introduction of three particular connectives: /, \ and, where is a non commutative product and / and \ are its associated divisions, in order that: A B C A C/B B A\C. It does not seem in the spirit of Generative Grammar to look for a logical formulation of its transformations. Nevertheless, that could be an interesting goal if we are looking for some implementation of Minimalism using logic programming or type-theory (and therefore programming languages like Prolog or Caml, and even proof-assistants like Coq cf. [Bertot and Castéran, 2004]).

2 There have been attempts to formalize minimalist ideas, mainly from E. Stabler [Stabler, 1997, 1999, 2001], under the denomination of Minimalist Grammars. We shall try in this paper to show how to translate Minimalist Grammars into a type-theoretical framework which only uses a few rules taken from logics. By doing so, we are able to suggest a particular viewpoint on some of the main assumptions made by Chomsky, like for instance, the assumption of parameters ([Baker, 2001]). 2. Minimalist Grammars 2.1. Definitions A minimalist grammar is defined by a 4-uple <V, Cat, Lex, Φ> where : - V is a finite set (non-syntactic features) - Cat is a finite set (syntactic features) - Lex is a set of expressions built up from V and Cat (the lexicon, see below) - Φ is a set of partial functions from t-uples of expressions to expressions (generative functions) According to [Stabler, 1997], expressions are finite, binary and ordered trees with projection and labels only at leaves. A finite, binary and ordered tree with projection is defined as a binary tree (N, < ) provided with a linear order relation between nodes and a relation < on nodes such that: x<y interprets as x immediately projects over y, x<* y as : x projects over y and x< + y as : x strictly projects over y with the condition : - ( x )(( y)( x < y)) (( u)( z)( x < z u < z)) For all nodes x and y, x is a head of y iff : - y is a leaf and x = y - or there exists some node z, daughter of y, which projects over all the children of y and x is a head of z. The maximal projection of a head x is the smallest node y with head x, for the relation <, such that a < b means a is father of b. The language defined by such a grammar is the closure of the lexicon under the generative functions (L(G) = CL(Lex, Φ)). Labels conform to the following regular expression: select* (licensor) select* (base) licensee* P* I* 2.2. Generative functions Merge Let t be an expression, the head of which is H(t). Let t[f] the expression we obtain by prefixing the feature f before the features which label H(t), let t(phon(α)) the expression t in

3 which the concatenation of phonetic features is α, then for all expressions t 1, t 2 and all c Base : merge(t 1 [=c], t 2 [c]) = [< t 1, t 2 ] if t 1 Lex merge(t 1 [=c], t 2 [c]) = [> t 2, t 1 ] if not. merge(t 1 [=C], t 2 [c]) = [< t 1 (phon(t 1 )^phon(t 2 )), t 2 (phon(ε)) ] if t 1 Lex merge(t 1 [C=], t 2 [c]) = [< t 1 (phon(t 2 )^phon(t 1 )), t 2 (phon(ε)) ] if t 1 Lex Move Let t* an expression which is the maximal projection of a head t. Let, for each t 1, t 2, such that t strictly contains t 1 but does not contain t 2, t{t 1 /t 2 } the result of replacing t 1 by t 2 inside t, then : for all expression t 1 [+f] which contains only one maximal subtree t 2 [-f]* : move(t 1 [+f]) = [> t 2 *, t 1 { t 2 [-f]* /λ}] where λ is the tree with only an empty node. If we take a version of MG which assumes weak and strong licensors, this definition is refined : move(t 1 [+F]) = [> t 2 *, t 1 { t 2 [-f]* /λ}] move(t 1 [+f]) = [> t 2 (phon(ε))*, t 1 { t 2 [-f]* / t 2 * }] où t 2 is l arbre t 2 * d où tous les traits non-phonétiques ont été supprimés Weak and strong features In some versions of MG ([Stabler, 1997]), a difference is made between strong and weak licensors respectively denoted by +X and +x. The former attract all the features in the moved subtree, whereas the latter attract only phonetic features. By changing the value of a feature (strong or weak), we shall change the word order. This exemplifies parameters An example Let us consider the VP «see a movie» with the lexicon : see : =d +acc =d v /see/ a : =n d case /a/ movie : n /movie/ We can represent the derivation by the following steps : = n d case / a / n / movie / = d + acc = d v / see / [ < d case / a /, / movie /] [ < + acc = d v / see / [ < case / a /, / movie /]] [ < [],[ < = d v / see / [ < / a /, / movie /]]]

4 which consists in : - elimination of the feature n, - elimination of the first feature d - elimination of the feature acc 3. Categorial Grammars It is well known that in AB categorial grammars (AB stands for Ajduckiewicz Bar Hillel), a language is the closure of the lexicon under two reduction rules, called Forward Application and Backward Application : FA: A/B B A; BA: B B\A A. Lambek grammars (or L grammars) use more rules. In the L calculus, the category constructors / and \ are seen as mere variants of a logical implication. The point is that this implication is not the classical one, but the linear one (that means that A B consumes its premise when applied to A), and that it is also sensitive to the order of the premises. Technically speaking, the L calculus is often presented as the multiplicative fragment of non commutative intuitionistic linear logic (with the restriction that no antecedent is empty, when presented as a sequent calculus). According to this view, FA and BA are simply the elimination rules for respectively / and \. The L calculus adds the corresponding introduction rules (in order to get completion), and the L calculus with product adds the elimination and introduction rules for the product. We can write, in the Natural Deduction presentation: [e \]: B B\A [e /]: A/B B A A [i \]: Γ [A] i [i /]: [A] i Γ.... B i B/A An important property of the L-calculus is that we can associate to each rule a corresponding operation in the λ-calculus in order to associate each proof with a λ-term. Such a λ-term can serve as a representation of the semantics of the expression which has been proved to be correct. Elimination rules for / and \ are associated with application, and introduction rules of the same connectives are associated with abstraction. This fact relies on the Curry Howard isomorphism, which comes from the study of intuitionistic logic. It provides an elegant way to obtain semantic forms à la Montague (see [Moortgat, 1997] for a complete survey). B i A\B

5 4. The bridge between categorical and minimalist grammars 4.1. Rules In this paper, we shall adopt a commutative and labelled version of the Lambek calculus, only keeping the distinction between / and \ in order to distinguish left and right concatenation of strings. [e /] : [e \] : Γ x :A/B y : B y : B Γ x :B\A Γ, xy : A Γ, yx : A To use these rules assumes that the lexical entries are no longer notated with features =x, =X and X= à la Stabler, but with the searched category «under the fraction sign» either as A/B, or as B\A. We must also assume the features linked together by means of a product, the rules for which are the following ones : [i ] : [e ] : Γ x : A y : B Γ w : A B, x : A, y : B, z : C Γ, (x, y) : A B, Γ, let(x, y) = (π1(w), π2(w)) in z : C Thus, when we have : fusion(t 1 [=c], t 2 [c]) = [< t 1, t 2 ] if t 1 Lex fusion(t 1 [=c], t 2 [c]) = [> t 2, t 1 ] if not. this translates into : o t 1 [=c] = A/c for the first case, o t 1 [=c] = c\a for the second one, o t 2 [c] = c B or B c We can then perform the following reductions : A/c c B A B et B c c\a B A With these notations, we can reformulate the analysis of the VP «see a movie». We must at first change the lexicon into : see : /see/ : (acc\v) /d a : /a/ : (case d)/n movie : /movie/ : n Assuming that acc is a possible value for case and that therefore a type case t can discharge hypotheses acc (or nom or obl) and t, we obtain: /see/ : (acc\v)/d x :d x :d /a/ : (cas d)/n /movie/ : n y : acc y : acc x : d /see/ x : acc \ v /a movie/ : (cas d) y : acc, x : d y /see/ x : v π 1 (/a movie/) /see/ π 2 (/a movie/) : v

6 Definition : merge = [e \] or [e /] and move = [e ], where / and \ are the residuates of the commutative product, simply labelled differently from each other with regards to the phonetic features that they combine. Remark : the commutativity of the product entails the permutability of the features in the lexical entries, which is not an assumption of Stabler s minimalist grammars. It would then be necessary to prove that it does not matter. Intuitively, features are still consumed in some order: their call order. This order is determined by the order of slash subcategories in the selecting category. If a category has the form A (B (C ( D))), the call order of the selected or licensed categories is always A, B, C, simply because we stand inside a system without the introduction rule of Condition on admissible proofs Of course all the deductions we can draw in that systems are not proper sentence derivations. Without non commutativity, the field is open to derivations of non acceptable sentences, or sentences which have not the intended meanings (for instance Peter loves Mary as having same meaning as Mary loves Peter!). This big drawback is avoided if we put conditions on proofs. Such conditions will not be artificial, in fact they coincide with the kind of conditions we should introduce in order to make the process of deduction use the least memory as possible. The idea is to keep hypotheses in memory during the least time as possible. For that, we shall associate with each hypothesis a rank which corresponds to its order of arrival in the derivation, starting from the axioms associated with the lexical entries. By this way, we can impose that when the elimination rule is used, if there are several candidates for discharge, it is always the one of lowest rank which is chosen. We shall call that the anteriority condition inside the proofs Examples We can now deal with some examples. Let us begin by «Paul sees a movie». It is a tensed sentence: an inflection is introduced. In principle, it goes with the morphological element which affects the tensed verb, here denoted by «pres t 3ps» (present, 3 rd person, singular). The form sees is supposed to come from a head movement of the verbal head see, followed by its adjunction to the inflection head, which movement, in this particular case, triggers a morphological merge (that Chomsky calls m-merger). We will deal with this question of head movement later on. For the time being, we have: Paul : /Paul/ : cas d, prest 3ps : prest-3ps : (nom\ip)/vp see : /see/ : (d\(acc\vp))/d a : /a/ : (cas d)/n movie : /movie/ : n which leads to the following deduction : /see/ : (d\(acc\vp))/d x : d 1 x : d y :d 2 y : d x : d 1 /see/x : d\(acc\vp) z : cas 3 z : cas y :d 2, x : d 1 y/see/x : acc\vp /a movie/ : cas d z : cas 3,y :d 2, x : d 1 zy/see/x : vp y :d 2 y/see a movie/ : vp

7 and then prest-3ps : (nom\ip)/vp y :d 2 y/see a movie/ : vp u : nom 4 u : nom y :d 2 y prest-3ps/see a movie/ : nom\ip /Paul/ : cas d u : nom 4, y :d 2 uy prest-3ps/see a movie/ : ip /Paul/ prest-3ps/see a movie/ : ip We must notice that when d and case are discharged by means of the object /a movie/, there are two candidates for d : y and x. x is chosen because of its anteriority in the proof.. Here is now a case of cyclic move, about the question of interrogatives. Suppose we want to analyse in French: «quel livre tu lis?» (what book do you read? Litt. What book you read? a «fast» way of expressing a question in ordinary speech.) quel : /quel/ : (wh cas d)/n ε : (wh\cp)/ip prest-2ps : prest-2ps : (nom\ip)/vp livre : /livre/ : n tu : /tu/ : nom d lire : /lire/ : (d\(acc\vp))/d We must notice here that we are using an entry without any phonological content: it is associated with the comp position. This lexical entry provides us with a necessary link between ip and cp, the latter being needed for attracting wh. We can represent the derivation by the following tree: wh cas d 4,6 cp wh 6 cp wh\cp ip (wh\cp)/ip ip cas d 2,5 nom 5 nom\ip vp (nom\ip)/v p acc 3 [cas d 1,3 ] 4 vp d 2 acc\vp d\(acc\vp) (d\(acc\vp))/d d 1

8 4.4. Head movements Rather than to make the morphological process of m-merger «on line», like it was indicated above, it seems more practical and perhaps more efficient to assume that this merger has always been already performed in the lexicon. This allows us to associate a product type with an inflected verb instead of attributing a particular type to the verb and a particular one to the inflection. We should have then: what : /what/ : (wh case d)/n ε : (wh\cp)/ip read : /lis/ : (nom\ip)/vp (d\(acc\vp))/d book : /book/ : n you : /tu/ : nom d 4.5. Derived rule Theoretically, the product elimination steps may occur at any time (as soon as the hypotheses to be discharged have been introduced of course). The rank of such steps does not matter with regards to the final result. Practically, we shall assume that such steps occur immediately after the needed hypotheses have been introduced. This amounts actually to using a derived rule (with three premises): Derived rule : [e ]/3 : Γ w : A B x:a x :A y : B, z : C Γ, let(x, y) = (π1(w), π2(w)) in z : C 4.6. Coding projections Until now, we were rather imprecise on the definition of the projections π 1 and π 2 which intervene in the product elimination rule. We shall give now a precise definition of them. For that, we shall introduce a supplementary marker, which indicates, for a given feature, the requirement of the projection of the phonology on that feature. We also introduce the dual marker, the role of which will be to neutralize the previous one. The ternary rule of product elimination is now given together with various definitions of π 1 and π 2 according to the position of. [e ]/3 : Γ w : A B x:a x :A y : B, z : C Γ, let(x, y) = (π1(w), π2(w)) in z : C with π 1 (w) = w and π 2 (w) = ε if : - w is of type A B and y is not of type B, and x is not of type A, - or w is of type A B and y is not of type B, and x is of type A, - or w is of type A B and x is of type A, π 1 (w) = ε and π 2 (w) = w if : - w is of type A B and x is not of type A, and y is not of type B, - or w is of type A B and x is not of type A, and y is of type B,

9 - or w is of type A B and y is of type B, There is a phonological clash if: - w is of type A B and y is of type B, - or w is of type A B and x is of type A, The other cases are considered unspecified cases, we have the choice between π 1 (w) = w and π 2 (w) = ε and π 1 (w) = ε and π 2 (w) = w if w is not a variable. In any case, we shall write ξ a variable of type X and ξ a variable of type X. Such variables will be said to be sorted. The sort of the variable is inherited by merge (that means by the application of the (/ or \) elimination rule). A variable ξ which substitutes to a variable ζ (resp. ζ) becomes a variable ξ (resp. ξ). If there is any underspecification of the projection of a variable ξ, then we assume that π 1 (ξ) = π 2 (ξ) = ξ. With these conventions, we can rewrite a lexicon in the following manner: what : /what/ : (wh (cas d))/n book : /book/ : n ε : ( wh\cp)/ip you : /you/ : nom d read : /read/ : ( nom\ip)/vp (d\(acc\vp))/ d (abbreviated in I VP) with which we can analyse «what book (do) you read?» 1 U : (d\(acc\vp))/ d x : d (1) y : d U x : d\(acc\vp) (2) z :cas yu x : acc\vp (3) u : cas d z yu x : vp (4) V :( nom\ip)/vp yu u : vp (5) /read/ : I VP VyU u : : nom\ip (6) /read/y u : nom\ip (7) and then : z :cas /lis/ y u : nom\ip (8) /you/ : nom d z /lis/ y u : ip (9) ε : ( wh\cp)/ip /you read/ u : ip (10) u :wh /you read/ u : wh\cp (11) /what book/ : wh (cas d) u /you read/ u : cp (12) /what book you read/ : cp At the step (12), we remember that the variable u is of type (case d) (the type of the hypotheses is not indicated on the figure in order to simplify it, but in the true derivation, it is always there). This allows us to substitute the pair (π 1 (/what book/) = /what book/, π 2 (/what book/) = ε) to the pair ( u, u) since the marker cancels the request for a projection of the phonology on the second coordinate. 1 The insertion of «do» is let apart for sake of brevity.

10 4.7. Parameters In this model, parameters are essentially dependant on two kinds of cross-linguistics variations : the slash orientation and the position of and. In Tibetan, for instance, (but this can be applied also to other SOV Asiatic languages) the type associated with comp is ip\(cp/wh) instead of ( wh\cp)/ip and the type for inflection is vp\( nom\ip) instead of ( nom\ip)/vp. Moreover, there is a special morpheme of type wh: pä, kä or ngä (dependent on the last consonant) which is used to make a yes/no question. Finally, phonologies are projected over the component relative to case. We correctly parse the question khyerang khapar thrung pare (where were you born? litt. You where born were) with the lexicon : khyerang : /khyerang/ : nom d thrung : /thrung/ : (d\( obl\vp))/d khapar : /khapar/ : wh cas d pare: /pare/ : (ip\(cp/wh)) vp\( nom\ip) The sentence khyerang kyakar-la kye pare pä? (were you born in India? litt. you in India born were?), is correctly obtained if we add to the lexicon: pä : /pä/ : wh 5. Conclusion We presented here a way to compile some version of minimalist grammars into a fragment of a type-logical framework. This serves two purposes: - A theoretical one: it would be interesting for linguists to get some very precise formulation of their preferred theory, and it is also interesting to see in what extent the syntactic system of language is a process of resource consumption, - A practical one: such formulation enables us to envisage implementations of the theory by means of logical frameworks like the proof assistant Coq, and when this task is realized, we may envisage to build parsers based on proof search for many various languages, using only variations in the lexicons, like indicated above in the case of English and Tibetan. In a wider presentation, it would be possible to show how logical forms can also be obtained in this formalism, in a manner reminiscent of the Curry-Howard isomorphism. References [Bertot and Castéran, 2004] Bertot, Y and Castéran, P, Le Coq Art. available from to appear in 2004, [Baker, 2001] Baker, M. The atoms of language, Basic Books, New York, 2001 [Chomsky, 1995] Chomsky, N. The Minimalist Program, MIT Press, Cambridge, 1995 [Moortgat, 1997] Moortgat, M., Categorial Type Logics, in van Benthem and ter Meulen (eds) Handbook of Logic and Language, , North Holland, 1997, [Stabler, 1997] Stabler, E., Derivational minimalism, in C. Retoré (ed) Logical Aspects of Computational Linguistics, 68 95, LNCS n 1328 (LNAI series), Springer, 1997, [Stabler, 1999] Stabler, E., Minimalist Grammars and recognition, in Linguistic form and its computation, Bad Teinach, Germany, 1999, [Stabler, 2001] Stabler, E. Recognizing Head Movement, in de Groote, Morrill, Retoré (eds) Logical Aspects of Computational Linguistics, , LNCS n 2099 (LNAI series), Springer, 2001