We propose a cut-free cyclic system for transitive closure logic (TCL) based on a form of hypersequents, suitable for automated reasoning via proof search. We show that previously proposed sequent systems are cut-free incomplete for basic validities from Kleene Algebra (KA) and propositional dynamic logic (\(\text {PDL}\)), over standard translations. On the other hand, our system faithfully simulates known cyclic systems for KA and \(\text {PDL}\), thereby inheriting their completeness results. A peculiarity of our system is its richer correctness criterion, exhibiting ‘alternating traces’ and necessitating a more intricate soundness argument than for traditional cyclic proofs.

我们提出了一种基于超序列形式的传递闭包逻辑（TCL）的无割循环系统，适合通过证明搜索进行自动推理。我们证明，先前提出的序列系统对于 Kleene 代数 (KA) 和命题动态逻辑 ( \(\text {PDL}\) ) 的基本有效性来说，在标准翻译上是无割不完整的。另一方面，我们的系统忠实地模拟了 KA 和\(\text {PDL}\)的已知循环系统，从而继承了它们的完整性结果。我们系统的一个特点是其更丰富的正确性标准，表现出“交替痕迹”，并且比传统的循环证明需要更复杂的稳健性论证。