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：线性时序逻辑（LTL）是最流行的时序逻辑之一，应用于计算机科学的多个分支。其广泛应用的原因之一在于其强大的基础性质：LTL等价于无计数Ω自动机、无星号Ω正则表达式，以及（根据坎普定理）线性序的一阶理论（FO-TLO）。安全性语言和共安全性语言分别通过有限前缀即可判断一个词是否不属于或属于该语言，在降低LTL模型检测和反应式综合等问题的复杂度中发挥关键作用。SafetyLTL（相应地，coSafetyLTL）是LTL的一个片段，仅允许全称（相应地，存在）时序模态，仅识别安全性（相应地，共安全性）语言。本文的主要贡献是引入FO-TLO的一个片段SafetyFO及其对偶coSafetyFO，它们在表达上完全覆盖LTL可定义的安全性语言和共安全性语言。我们证明它们分别精确刻画了SafetyLTL和coSafetyLTL，这一结果补充了坎普定理，并为一阶语言对LTL（及其片段）的刻画提供了更清晰的视角。此外，它也直接、简洁且自包含地证明了LTL中可定义的任何安全性语言同样可在SafetyLTL中定义。作为副产品，我们获得了关于SafetyLTL的弱明天算子（在有限词和无限词上解释）表达能力的一些有趣结果。进一步，我们证明当在有限词上解释时，去除明天算子（相应地，弱明天算子）的SafetyLTL（相应地，coSafetyLTL）恰好捕获了LTL在有限词上的安全性（相应地，共安全性）片段。