Alessandro Cimatti ; Luca Geatti ; Nicola Gigante ; Angelo Montanari ; Stefano Tonetta - A first-order logic characterization of safety and co-safety languages

lmcs:10061 - Logical Methods in Computer Science, August 10, 2023, Volume 19, Issue 3 - https://doi.org/10.46298/lmcs-19(3:13)2023
A first-order logic characterization of safety and co-safety languagesArticle

Authors: Alessandro Cimatti ORCID; Luca Geatti ORCID; Nicola Gigante ; Angelo Montanari ORCID; Stefano Tonetta

    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.


    Volume: Volume 19, Issue 3
    Published on: August 10, 2023
    Accepted on: June 26, 2023
    Submitted on: September 20, 2022
    Keywords: Computer Science - Artificial Intelligence,Computer Science - Logic in Computer Science

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