Automatic Concepts and Automata-Theoretic Semantics for the Full Lambek Calculus

摘要

We introduce a new semantics for the (full) Lambek calculus, which is based on an automata-theoretic construction. This automata-theoretic semantics combines languages and relations via closure operators which are based on automaton transitions. We establish the strong 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. Automata-theoretic semantics is interesting because it connects two important semantics of the Lambek calculus, namely the relational and the language-theoretic. At the same time, it establishes a strong relation between two canonical constructions over a given language, namely its syntactic concept lattice and its universal automaton.