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.
翻译:线性时间逻辑(LTL)是最受欢迎的时态逻辑之一,它在计算机科学的各个领域发挥作用。在其广泛使用的众多原因中,有其基础性强的特性:LTL等价于计数自由无穷自动机,星自由无穷正则表达式,(通过坎普定理)等价于线性序的一阶理论(FO-TLO)。安全语言和协同安全语言是,在安全语言(resp.,属于该语言的单词只需要一个有限的前缀就足以确定)和协同安全语言(resp.,不属于该语言的单词只需要一个有限的前缀就足以确定)中,降低了像模型检查和响应合成等问题的复杂性。SafetyLTL(resp.,coSafetyLTL)是LTL的一个片段,只允许使用全称(resp.,存在)时态修饰语,只能识别安全语言(resp.,协同安全语言)。本文的主要贡献是介绍FO-TLO的片段——SafetyFO和它的双重形式coSafetyFO,它们与LTL可定义的安全性和协同安全性语言在表达上是完备的。我们证明它们分别正确地刻画了SafetyLTL和coSafetyLTL,这个结果与坎普定理相结合,为用一阶语言刻画(片段)LTL提供了更清晰的视角。此外,它提供了一个直接、紧凑且自包含的证明,即LTL可定义的任何安全语言也可以在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和无限单词上解释的一些有趣结果。此外,我们证明,当在有限单词上解释时,去除明天算子(resp.,弱明天算子)的SafetyLTL(resp.,coSafetyLTL)捕获了有限单词上的安全(resp.,协同安全)片段。