Action Logic is Undecidable

Stepan Kuznetsov

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arXiv · 1912.11273

Action Logic is Undecidable

Stepan Kuznetsov

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Abstract

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.

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Stepan Kuznetsov. 2019-12-24. Action Logic is Undecidable. https://arxiv.org/abs/1912.11273

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