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 one successor (S1S[FO]). 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 S1S[FO], 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：线性理论(LTL)是最流行的时空逻辑(LTL) 。 安全和共同安全语言(LTL)是一个最受欢迎的时空逻辑之一, 它在计算机科学的多种分支中发挥作用。 其广泛使用的各种原因有其强大的基础特性: LTL 相当于反自由的奥美-奥多马塔, 相当于无星的奥美-定期表达, (Kamp的理论) 相当于一个继承者的第一阶级理论( S1S[FO] ) 。 安全和共同安全语言, 其中有限的前缀足以确定一个单词是否属于或属于该语言, 在降低问题的复杂性方面发挥着关键的作用。 LTLTL( resp., cofetLTLTL) 是一个小片段, 当一个仅允许通用( resp., 存在) 时间模式, 仅承认安全( rep., 共同安全) 语言。 本文的主要贡献是引入S1S[FO] 的碎片, 安全FO, 以及其二阶- 的精确语言(LTLT) 解释, 的直立-LTLT) 的内, 的自定义- real- real- real- real- real- real- real- real- decreal- devely the the the ex freal devely) lade, lade, lade, lade, lade, 它能能能, 它能能提供一种更清楚, 它能, 它能性能性能提供一种更清楚的功能,, 它能,, 它的功能的功能的功能的功能, 它能, 它能提供一种更精确的功能,,, 它的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的功能的解, 。