Action Logic is Undecidable

Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. This logic involves Kleene star, axiomatized by an induction scheme. For a stronger system which uses an $omega$-rule instead (infinitary action logic) Buszkowski and Palka (2007) have proved $Pi_1^0$-completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by D. Kozen in 1994. In this article, we show that it is undecidable, more precisely, $Sigma_1^0$-complete. We also prove the same complexity results for all recursively enumerable logics between action logic and infinitary action logic; for fragments of those only one of the two lattice (additive) connectives; for action logic extended with the law of distributivity.