Applicative and Combinatory Categorial Grammar

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association between rules and Curry's combinators on the one hand and ..... Reduction (rules of combinators eliminatio n). Phenotype. G e n o t y p e. C. O. M. P.

Applicative and Combinatory Categorial Grammar (from syntax to functional semantics) Ismaïl BISKRI

Jean Pierre DESCLES

ISHA - LALIC 96, bld Raspail 75006 Paris email : [email protected] tel : (1) 44 39 89 63 / 50 fax : (1) 44 39 89 51

“applicative grammar” ; (iii) The cognitive level where the meanings of lexical predicates are represented by semantic cognitive schemes.

Abstract Applicative and Combinatory Categorial Grammar is an extension of Steedman's Combinatory Categorial Grammar by a canonical association between rules and Curry's combinators on the one hand and metarules which control type-raising operations on the other hand. This model is included in the general framework of Applicative and Cognitive Grammar (Desclés) with three levels of representation : (i) phenotype (concatened expressions) ; (ii) genotype (applicative expressions) ; (iii) the cognitive representations (meaning of linguistic predicates). The aim of the paper is : (i) an automatic parsing of phenotype expressions that are underlying to sentences ; (ii) the constructing of applicative expressions. The theoritical analysis is applied to spurious ambiguity and coordination.

1. Model of Applicative Cognitive Grammar

and

Applicative and Cognitive Grammar (Desclés 90) is an extension of the Universal Applicative Grammar (Shaumyan 87). It postulates three levels of representations of languages : (i) Phenotype level (or phenotype) where the particulary characteristics of natural languages are described (for example order of words, morphological cases, etc...). The linguistic expressions of this level are concatened linguistic units, the concatenation is noted by : ‘u1-u2-...-un’ ; (ii) Genotype level (or genotype) where grammatical invariants and structures that are underlying to sentences of phenotype level are expressed. The genotype level is structured like a formal language called “genotype language” ; it is described by a grammar called

Representations of levels two and three are expressions of typed combinatory logic (Curry & Feys 58) (Shaumyan 87). We abstract operators associated with elimination and introduction inference rules like in “Gentzen calculus”. For instance, we present combinators B, C , S,  , * with the following rules (U 1, U2, U3 are typed applicative expressions) :

introduction rules

elimination rules

U 1 (U 2 U3)

B U1 U2 U3

--------------(i-B) B U1 U2 U3

----------------(e-B) U1 (U2 U 3)

U1 U2

C U2 U1 * --------------(e-C ) * U1 U2

-------------(i-C ) * C U2 U1 * U 1 U3 (U2 U3)

S U1 U2 U3 -------------------(e-S ) U1 U3 (U 2 U3)

---------------------(i-S ) S U1 U2 U3 U 1 (U 2 U4) (U 3 U4) --------------------------(i- )  U1 U2 U3 U4

These rules -expansion :

lead

 U1 U2 U3 U4 ---------------------(e- ) U1 (U2 U 4) (U 3 U4)

to

-reduction

((B U1 U 2) U3) –> (U 1 (U2 U3))

or

((C U1) U2) –> (U 2 U1) * ((S U1 U 2) U3) –> ((U 1 U 3)(U2 U3)) (( U1 U2 U 3) U4) –> (U 1 (U2 U4)(U 3 U 4))

In what it follows, we are interested in relations between the two first levels (phenotype and genotype) by implementing a system of formal analysis - called : Applicative and Combinatory Categorial Grammar (ACCG) which explicitally connects phenotype expressions to its underlain representations in the genotype1. This system consists in : (i) the syntactical analysis of concatened expressions of phenotype by using Combinatory Categorial Grammar. (ii) the constructing from the result of syntactical analysis of the functional semantic interpretation of phenotype expressions.

1.1 Categorial grammars Categorial Grammars assign syntactical categories to each linguistic unit. Syntactical categories are orientated types developped from basic types and from two constructive operators ‘/’ and ‘\’. (i) N (nominal syntagm) and S (sentence) are basic types. (ii) If X and Y are orientated types then X/Y and X\Y are orientated types2. A linguistic unit u with orientated type X will be designed by ‘[X : u]’. Both rules of application (forward and backward) are noted :

[X/Y : u 1] - [Y : u 2] ----------------------------> ; [X : (u 1 u 2)]

[Y : u 1] - [X\Y : u 2] ----------------------------< [X : (u 2 u 1)]

1 In phenotype, linguistic expressions are concatened

according to syntagmatic rules of French. In genotype, expressions are arranged according to the applicative order. 2 Here, we choose Steedman's notation (1989) : X/Y and X\Y are functional orientated types. A linguistic unit 'u' with the type X/Y (respectively X\Y) is considered as operator (or function) whose typed operand Y is positioned on the right (respectively on the left) of operator.

The premises in each rule are concatenations of linguistic units with orientated types considered as being operators or operands, the consequence of each rule is an applicative expression with orientated type. Combinatory Categorial Grammar (Steedman 89) generalizes classical Categorial Grammars by introducing the operation of type-raising and composition on functional types. The new proposed rules aim at quasi-incremental (from right to left) in order to eliminate the problem of spurious ambiguity (Haddock 87 ; Pareschi & Steedman 87).

1.2 Applicative and Categorial Grammar

Combinatory

In ACCG, we consider that the rules of Steedman’s Combinatory Categorial Grammar introduce the combinators B, C , S into the * syntagmatic sequence. This introduction makes it possible to turn one concatened structure to one applicative structure. The rules of ACCG are :

Type-raising rules : [X : u] ----------------------->T [Y/(Y\X) : (C u)] *

[X : u] ; -----------------------Tx [Y/(Y/X) : (C u)] *

[X : u] ; -----------------------B [X/Z : (B u1 u2)]

[Y\Z : u1]-[X\Y : u2] ; ------------------------S ;

[Y\Z : u1]-[(X\Y)\Z : u2] ----------------------------S x; [X\Z : (S u1 u2)]

[Y/Z : u1]-[(X\Y)/Z : u2] --------------------------T [S/(S\N) : (C John)] [(S\N)/N : loves ] * --------------------------------------------------------------->B [S/N : (B (C John) loves)] *

The first rule (>T) applied to the typed unit [N : John] turns operand to operator. It

constructs an applicative structure (C John) * whose type is S/(S\N). The introduction of combinator C illustrates in the applicative * representation the type-raising : (C John) * works like an operator with his functional type. The rule (>B) combinates the typed linguistic units [S/(S\N) : (C John)] and * [(S\N)/N : loves] with the combinator B in order to compose the two functional units (C John) * and loves. A full processing based upon Applicative and Combinatory Categorial Grammar is carried out in two main steps : (i) the first step is illustrated by the checking of the proper syntactic connection and by the constructing of predicative structures with some combinators introduced in certain positions of syntagmatic structure, (ii) the second step consists in using the -reduction rules of combinators in order to create a predicative structure that is underlying phenotype expression. The obtained expression is an applicative one and belongs to genotype language. ACCG generates processes that associate one applicative structure to one concatened expression of phenotype. What remains to be eliminated is the combinators of obtained expression in order to construct the “normal form” (in the technical meaning of -reduction) that expresses the functional semantic interpretation. This calculus is completely done in genotype. Therefore, this process that we propose takes the shape of a compilation whose steps are summed up in figure 1 :

4

P hen oty p e

Ty ped con caten ed expression C O M P I L A T I O N

Proo f on typ es with App licativ e an d Co mb in ato ry C ategorial Grammar

Ty ped app licativ e ex pressio n with combinators  R eduction (ru les of combinators eliminatio n)

Normal form (fu nctio nal semantics)

G e n o t y p e

figure 1 Let us deal with a simple example : John loves Mary 1

[N:John]-[(S\N)/N:loves]-[N:Mary]

Typed concatened structure of phenotype

2

[S/(S\N):(C John)]-[(S\N)/N:loves]-[N:Mary] * [S/N:(B (C John) loves)]-[N:Mary] * [S:((B (C John) loves) Mary)] *

(>T)

3 4

(>B)

Compilation

(>) V

5 6 7

[S : ((B (C John) loves) Mary)] * [S : ((C John) (loves Mary))] * [S : ((loves Mary) John)]

Typed applicative structure of genotype (B) (C ) *

The type raising (> T) allocating the operand John makes it possible to generate the operator (C John) that the functional rule * (>B) composes with the operator loves. The complex operator (B (C John) loves) is * applied to the operand Mary in order to form the applicative expression of genotype ((B (C John) loves) Mary). The reduction * of combinators in genotype constructs the functional semantic interpretation that is underlying to phenotype expression (input).

2. Structural reorganization

Normal form of genotype

The syntactic analysis “from left to right” raises the problem of non-determinism introduced by the presence in the language of backward modifiers that stand as operators which are applied to the whole or a part of a structure previously constructed. If, in the first case the use of a rule of application allows the analysis to be carried on 3, it is quite different for the second case 3 Let us take the example of sentence John hit Mary

yesterday where the backward modifier yesterday operates on the whole sentence John hit Mary ; yesterday whose syntactic type is S\S, in order to continue the analysis, it is enough to apply yesterday to John hit Mary by the rule ( S/(S\N) : (C u1)] * Example :John loves Mary and William Jenny ...-[CONJD : and]-[N : William]-[N : Jenny] ...-[S/(S\N) : (C William)]-[N : Jenny] *

Metarule 3 : If u2 has type N and u1 has type Y/X (u1 preceded by and), then we apply the backward type-raising ( X\(X/N) : (C u2)]. * Example :John loves Mary et William Jenny ...-[S/(S\N) : (C William)]-[N : Jenny] * ...(C William)]-[(S\N)\((S\N)/N) :(C Jenny)] * * In this case X = S\N ; Y = S.

[N : John]-[(S\N)/N : eats]-[N/N : the]-[N : apple]

5. Examples E1 : John loves Mary and hates Jenny Phenotype 1 4 5 6 7 8 9

[N:John]-[(S\N)/N:loves]-[N:Mary]-[CONJD:and]-[(S\N)/N:hates]-[N:Jenny] ... [S:((B (C John) loves) Mary)]-[CONJD:and]-[(S\N)/N:hates]-[N:Jenny] * [S:((B (C John) loves) Mary)]-[CONJD:and]-[S\N:(hates Jenny)] * [S:((C John) (loves Mary))]-[CONJD:and]-[S\N:(hates Jenny)] * [S/(S\N):(C John)]-[S\N:(loves Mary)]-[CONJD:and]-[S\N:(hates Jenny)] * [S/(S\N):(C John)]-[S\N:( and (loves Mary) (hates Jenny))] * [S:((C John) ( and (loves Mary) (hates Jenny)))] *

(>) (B) (>dec) () (>)

Genotype 10 11

[S : ((C John) ( and (loves Mary) (hates Jenny)))] * [S : (( and (loves Mary) (hates Jenny)) John)]

12

[S : (and ((loves Mary) John) ((hates Jenny) John))]

(C ) * ( )

9

E2 : John loves Mary and William Jenny Phenotype 1 4 5 6 7 8 9 10 11

[N:John]-[(S\N)/N:loves]-[N:Mary]-[CONJD:and]-[N:William]-[N:Jenny] ... [S:((B (C John) loves) Mary)]-[CONJD:and]-[N:William]-[N:Jenny] * ...-[CONJD:and]-[S/(S\N):(C William)]-[N:Jenny] * ...-[CONJD:and]-[S/(S\N):(C William)]-[(S\N)\(S/(S\N)):(C Jenny)] * * ...-[CONJD:and]-[S\(S/(S\N)):(B (C William)(C Jenny)] * * [S:((B (C John) (C Mary)) loves)]-[CONJD:and]-... * * [(S\N)/N:loves]-[S\(S/(S\N)):(B (C John) (C Mary))]-[CONJD:and]-... * * [(S\N)/N:loves]-[S\(S/(S\N)):( and (B(C John)(C Mary))(B(C William)(C Jenny)))] * * * * [S:(( and (B (C John) (C Mary))(B (C William)(C Jenny))) loves)] * * * *

(>T), M2 (Bx) (d) (T), M1 (>B) (>) (>dec) () (>)

Genotype 9 10

[S : ((B (C (the flag)) is) (and white red))] * [S : ((C (the flag)) (is (and white red)))] *

(B)

10

11

[S : ((is (and white red)) (the flag))]

Other examples and more details are provided in (Biskri 95). Analysises are implemented. Here, we do not give the details of algorithm.

6. Conclusion We think we have presented within the framework of Applicative Cognitive Grammar a model of analysis that realizes the ‘interface’ between syntax and semantic. For a lot of examples of French this model is able to realize the following aims : (i) to produce an analysis which verifies the syntactic correction of statements. (ii) to develop automatically the predicative structures that yield the functional semantic interpretation of statements. Moreover, this model has the following characteristics : 1- we do not make any calculus parallel to syntactic calculus like Montague’s one (1974). A first calculus verifies the syntactic correction, this calculus is carried on by a construction of functional semantic interpretation. This has been made possible by the introduction of combinators to some specific positions of syntagmatic order. 2- We introduce some components of functional semantic by some applicative syntactic tools (combinators). 3- We calculate the functional semantic interpretation by some applicative syntactic methods (combinators reduction). In order to sum up, we interpret by means of absolute syntactic technics. The distinction syntax/semantic should be then thought again in another perspective.

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(C ) *

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