arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2209.02307cs.AIcs.LO
收藏

安全与共安全语言的一阶逻辑刻画

A first-order logic characterization of safety and co-safety languages

Fondazione Bruno Kessler(布鲁诺·凯塞勒基金会);University of Udine(乌迪内大学);Free University of Bozen-Bolzano(波尔扎诺自由大学)

Alessandro Cimatti, Luca Geatti, Nicola Gigante, Angelo Montanari, Stefano Tonetta

AI总结:

本文引入了一阶理论片段 SafetyFO 和 coSafetyFO,证明其分别精确刻画 SafetyLTL 和 coSafetyLTL,为 LTL 的安全与共安全片段提供了一阶逻辑刻画,并给出了相关表达能力的证明。

摘要:

线性时序逻辑(Linear Temporal Logic,LTL)是最受欢迎的时序逻辑之一,在计算机科学的各个分支中发挥着作用。其被广泛使用的众多原因之一在于其强大的基础特性:LTL等价于无计数 omega-自动机、无星 omega-正则表达式,并且(根据 Kamp 定理)等价于线性序的一阶理论(First-Order Theory of Linear Orders,FO-TLO)。安全语言和共安全语言——分别仅需有限前缀即可判定一个字是否不属于或属于该语言——在降低诸如 LTL 的模型检验和反应式综合等问题的复杂度方面发挥着关键作用。SafetyLTL(对应地,coSafetyLTL)是 LTL 的一个片段,仅允许使用全称(对应地,存在)时序模态,它只能识别安全(对应地,共安全)语言。本文的主要贡献是引入了 FO-TLO 的一个片段,称为 SafetyFO,以及其对偶形式 coSafetyFO,它们在表达能力上对于 LTL 可定义的安全语言和共安全语言是完全的。我们证明它们分别精确刻画了 SafetyLTL 和 coSafetyLTL,这一结果与 Kamp 定理相契合,并为用一阶语言刻画 LTL(及其片段)提供了更清晰的视角。此外,它给出了一个直接、紧凑且自洽的证明,即任何可在 LTL 中定义的安全语言同样可在 SafetyLTL 中定义。作为副产品,我们获得了关于 SafetyLTL 的弱明天算子在有限字和无限字上解释时所具表达能力的一些有趣结果。而且,我们证明,当在有限字上解释时,去除了明天算子(对应地,弱明天算子)的 SafetyLTL(对应地,coSafetyLTL)能够刻画有限字上 LTL 的安全(对应地,共安全)片段。

英文摘要:

Linear Temporal Logic (LTL) is one of the most popular temporal logics, that comes into play in a variety of branches of computer science. Among the various reasons of its widespread use there are its strong foundational properties: LTL is equivalent to counter-free omega-automata, to star-free omega-regular expressions, and (by Kamp's theorem) to the First-Order Theory of Linear Orders (FO-TLO). Safety and co-safety languages, where a finite prefix suffices to establish whether a word does not belong or belongs to the language, respectively, play a crucial role in lowering the complexity of problems like model checking and reactive synthesis for LTL. SafetyLTL (resp., coSafetyLTL) is a fragment of LTL where only universal (resp., existential) temporal modalities are allowed, that recognises safety (resp., co-safety) languages only. The main contribution of this paper is the introduction of a fragment of FO-TLO, called SafetyFO, and of its dual coSafetyFO, which are expressively complete with respect to the LTL-definable safety and co-safety languages. We prove that they exactly characterize SafetyLTL and coSafetyLTL, respectively, a result that joins Kamp's theorem, and provides a clearer view of the characterization of (fragments of) LTL in terms of first-order languages. In addition, it gives a direct, compact, and self-contained proof that any safety language definable in LTL is definable in SafetyLTL as well. As a by-product, we obtain some interesting results on the expressive power of the weak tomorrow operator of SafetyLTL, interpreted over finite and infinite words. Moreover, we prove that, when interpreted over finite words, SafetyLTL (resp. coSafetyLTL) devoid of the tomorrow (resp., weak tomorrow) operator captures the safety (resp., co-safety) fragment of LTL over finite words.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

↑