We show that the relational semantics of the Lambek calculus, both nonassociative and associative, is also sound and complete for its extension with classical propositional logic. Then, using filtrations, we obtain the finite model property for the nonassociative Lambek calculus extended with classical propositional logic.

We prove the finite model property (fmp) for BCI and BCI with additive conjunction, which answers some open questions in Meyer and Ono [11]. We also obtain similar results for some restricted versions of these systems in the style of the Lambek calculus [10, 3]. The key tool is the method of barriers which was earlier introduced by the author to prove fmp for the product-free Lambek calculus [2] and the commutative product-free Lambek calculus [4].

We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an$\omega $-rule, and prove that the derivability problem in this calculus is$\Pi _1^0$-hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with (...) unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007). (shrink)

We prove that the Lambek calculus is complete w.r.t. L-models, i.e., free semigroup models. We also prove the completeness w.r.t. relativized relational models over the natural linear order of integers.

If all dependent expressions were adjacent some variety of immediate constituent analysis would suffice for grammar, but syntactic and semantic mismatches are characteristic of natural language; indeed this is a, or the, central problem in grammar. Logical categorial grammar reduces grammar to logic: an expression is well-formed if and only if an associated sequent is a theorem of a categorial logic. The paradigmatic categorial logic is the Lambek calculus, but being a logic of concatenation the Lambek calculus can only capture (...) discontinuous dependencies when they are peripheral. In this paper we present the displacement calculus, which is a logic of intercalation as well as concatenation and which subsumes the Lambek calculus. On the empirical side, we apply the new calculus to discontinuous idioms, quantification, VP ellipsis, medial extraction, pied-piping, appositive relativisation, parentheticals, gapping, comparative subdeletion, cross-serial dependencies, reflexivization, anaphora, dative alternation, and particle shift. On the technical side, we prove that the calculus enjoys Cut-elimination. (shrink)

We consider language models for the Lambek calculus that allow empty antecedents and enrich them with constants for the empty language and for the language containing only the empty word. No complete calculi are known with respect to these semantics, and in this paper we consider several trivalent systems that arise as fragments of these models? logics.

Some formal properties of enriched systems of Lambek calculus with analogues of conjunction and disjunction are investigated. In particular, it is proved that the class of languages recognizable by the Lambek calculus with added intersective conjunction properly includes the class of finite intersections of context-free languages.

We prove completeness for some language-theoretic models of the full Lambek calculus and its various fragments. First we consider syntactic concepts and syntactic concepts over regular languages, which provide a complete semantics for the full Lambek calculus \. We present a new semantics we call automata-theoretic, which combines languages and relations via closure operators which are based on automaton transitions. We establish the completeness of this semantics for the full Lambek calculus via an isomorphism theorem for the syntactic concepts lattice (...) of a language and a construction for the universal automaton recognizing the same language. Finally, we use automata-theoretic semantics to prove completeness of relation models of binary relations and finite relation models for the Lambek calculus without and with empty antecedents and \), thus solving a problem left open by Pentus. (shrink)