Infinitary Action Logic: Complexity, Models and Grammars

Abstract

Action logic of Pratt [21] can be presented as Full Lambek Calculus FL [14, 17] enriched with Kleene star *; it is equivalent to the equational theory of residuated Kleene algebras (lattices). Some results on axiom systems, complexity and models of this logic were obtained in [4, 3, 18]. Here we prove a stronger form of *-elimination for the logic of *-continuous action lattices and the $${\Pi_{1}^{0}}$$ ---completeness of the equational theories of action lattices of subsets of a finite monoid and action lattices of binary relations on a finite universe. We also discuss possible applications in linguistics.