Linear temporal logic (LTL) is suitable not only for infinite-trace systems, but also for finite-trace systems. In particular, LTL with finite-trace semantics is frequently used as a specification formalism in runtime verification, in artificial intelligence, and in business process modeling. The satisfiability of LTL with finite-trace semantics, a known PSPACE-complete problem, has been recently studied and both indirect and direct decision procedures have been proposed. However, the proof theory of LTL with finite traces is not that well understood. Specifically, complete proof systems of LTL with only infinite or with both infinite and finite traces have been proposed in the literature, but complete proof systems directly for LTL with only finite traces are missing. The only known results are indirect, by translation to other logics, e.g., infinite-trace LTL. This paper proposes a direct sound and complete proof system for finite-trace LTL. The axioms and proof rules are natural and expected, except for one rule of coinductive nature, reminiscent of the Gödel–Löb axiom.

中文翻译：

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有限迹线性时序逻辑：共导完备性

线性时序逻辑 (LTL) 不仅适用于无限迹系统，也适用于有限迹系统。特别是，具有有限跟踪语义的 LTL 经常用作运行时验证、人工智能和业务流程建模中的规范形式。最近研究了具有有限迹语义的 LTL 的可满足性，这是一个已知的 PSPACE 完全问题，并且已经提出了间接和直接决策程序。然而，有限迹的 LTL 证明理论并没有那么好理解。具体而言，文献中已经提出了仅具有无限或具有无限和有限迹的 LTL 的完整证明系统，但缺少直接用于仅具有有限迹的 LTL 的完整证明系统。唯一已知的结果是间接的，通过转换为其他逻辑，例如，无限跟踪 LTL。本文提出了一种用于有限迹 LTL 的直接健全和完整的证明系统。公理和证明规则是自然的和预期的，除了一个共同归纳性质的规则，让人想起哥德尔-洛布公理。