Towards Abstract Categorial Grammars

Towards Abstract Categorial Grammars Philippe de Groote LORIA UMR no 7503 – INRIA Campus Scientifique, B.P. 239 54506 Vandœuvre l`es Nancy Cedex – France degroote@loria.fr Abstract We introduce a new categorial formal- ism based on intuitionistic linear logic. This formalism, which derives from current type-logical grammars, is ab- stract in the sense that both syntax and semantics are handled by the same set of primitives. As a consequence, the formalism is reversible and provides different computational paradigms that may be freely composed together. 1 Introduction Type-logical grammars offer a clear cut between syntax and semantics. On the one hand, lexical items are assigned syntactic categories that com- bine via a categorial logic akin to the Lambek cal- culus (Lambek, 1958). On the other hand, we have so-called semantic recipes, which are ex- pressed as typed λ-terms. The syntax-semantics interface takes advantage of the Curry-Howard correspondence, which allows semantic readings to be extracted from categorial deductions (van Benthem, 1986). These readings rely upon a homomorphism between the syntactic categories and the semantic types. The distinction between syntax and semantics is of course relevant from a linguistic point of view. This does not mean, however, that it must be wired into the computational model. On the contrary, a computational model based on a small set of primitives that combine via simple compo- sition rules will be more flexible in practice and easier to implement. In the type-logical approach, the syntactic con- tents of a lexical entry is outlined by the following patern: <atom> : <syntactic category> On the other hand, the semantic contents obeys the following scheme: <λ-term> : <semantic type> This asymmetry may be broken by: 1. allowing λ-terms on the syntactic side (atomic expressions being, after all, partic- ular cases of λ-terms), 2. using the same type theory for expressing both the syntactic categories and the seman- tic types. The first point is a powerfull generalization of the usual scheme. It allows λ-terms to be used at a syntactic level, which is an approach that has been advocated by (Oehrle, 1994). The sec- ond point may be satisfied by dropping the non- commutative (and non-associative) aspects of cat- egorial logics. This implies that, contrarily to the usual categorial approaches, word order con- straints cannot be expressed at the logical level. As we will see this apparent loss in expressive power is compensated by the first point. 2 Definition of a multiplicative kernel In this section, we define an elementary gram- matical formalism based on the ideas presented in the introduction. This elementary formalism is founded on the multiplicative fragment of linear logic (Girard, 1987). For this reason, we call it a multiplicative kernel. Possible extensions based on other fragments of linear logic are discussed in Section 5. 2.1 Types, signature, and λ-terms We first introduce the mathematical apparatus that is needed in order to define our notion of an ab- stract categorial grammar. Let A be a set of atomic types. The set T (A) of linear implicative types built …

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