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

中文翻译：

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动作逻辑不可判定

动作逻辑是剩余 Kleene 格的代数逻辑（不等式理论）。这种逻辑的操作之一是 Kleene 星，它被归纳方案公理化。对于一个更强大的系统，它使用 -rule 代替（无限动作逻辑），Buszkowski 和 Palka（2007）证明 - 完整性（因此，不可判定性）。动作逻辑本身的可判定性是一个悬而未决的问题，由 Kozen 在 1994 年提出。在本文中，我们表明它是不可判定的，更准确地说， -完全的。我们还证明了对于动作逻辑和无限动作逻辑之间的所有递归可枚举逻辑、仅具有两个格（加法）连接词之一的这些逻辑的片段以及使用分布定律扩展的动作逻辑的相同的不可判定性结果。