Language Models for Some Extensions of the Lambek Calculus

TL;DR

This paper explores extensions of the Lambek calculus with additive connectives and the unit constant, revealing differences in behavior, issues with distributivity, and undecidability results, impacting language interpretation and logical properties.

Contribution

It analyzes the logical properties of extended Lambek calculi, highlighting differences in additive connectives, and introduces an undecidable calculus reflecting algebraic properties of the empty word.

Findings

Conjunction and disjunction behave differently in extended Lambek calculi.

Adding both additive connectives leads to incompleteness due to distributivity.

Undecidability is proven for systems extending the Lambek calculus with the unit constant.

Abstract

We investigate language interpretations of two extensions of the Lambek calculus: with additive conjunction and disjunction and with additive conjunction and the unit constant. For extensions with additive connectives, we show that conjunction and disjunction behave differently. Adding both of them leads to incompleteness due to the distributivity law. We show that with conjunction only no issues with distributivity arise. In contrast, there exists a corollary of the distributivity law in the language with disjunction only which is not derivable in the non-distributive system. Moreover, this difference keeps valid for systems with permutation and/or weakening structural rules, that is, intuitionistic linear and affine logics and affine multiplicative-additive Lambek calculus. For the extension of the Lambek with the unit constant, we present a calculus which reflects natural algebraic properties of the empty word. We do not claim completeness for this calculus, but we prove undecidability for the whole range of systems extending this minimal calculus and sound w.r.t. language models. As a corollary, we show that in the language with the unit there exissts a sequent that is true if all variables are interpreted by regular language, but not true in language models in general.