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Finite-trace linear temporal logic: coinductive completeness

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Abstract

Linear temporal logic (LTL) is suitable not only for infinite-trace systems, but also for finite-trace systems. In particular, LTL with finite-trace semantics is frequently used as a specification formalism in runtime verification, in artificial intelligence, and in business process modeling. The satisfiability of LTL with finite-trace semantics, a known PSPACE-complete problem, has been recently studied and both indirect and direct decision procedures have been proposed. However, the proof theory of LTL with finite traces is not that well understood. Specifically, complete proof systems of LTL with only infinite or with both infinite and finite traces have been proposed in the literature, but complete proof systems directly for LTL with only finite traces are missing. The only known results are indirect, by translation to other logics, e.g., infinite-trace LTL. This paper proposes a direct sound and complete proof system for finite-trace LTL. The axioms and proof rules are natural and expected, except for one rule of coinductive nature, reminiscent of the Gödel–Löb axiom.

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Notes

  1. See [4] for multi-valued variants of LTL.

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Acknowledgements

We would like to warmly thank Yliès Falcone and César Sánchez for organizing the RV’16 conference, and them as well as Martin Steffen and Fred Schneider for lively discussions and debates related to the Coinduction proof rule. We also thank Moshe Vardi for referring us to recent work on finite-trace LTL published in artificial intelligence conferences [10,11,12, 25, 48]; we were not aware of these efforts when we published the RV’16 conference version of this paper [35]. Special thanks to my student Xiaohong Chen, who helped double-check the correctness of the proofs and the appropriateness of the results. Last but not least, we would like to warmly thank the anonymous reviewers for substantial suggestions on how to improve this paper. The work presented in this paper was supported in part by NSF Grants CCF-1421575 and CNS-1619275, and by an IOHK gift (http://iohk.io).

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    Grigore Roşu

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Roşu, G. Finite-trace linear temporal logic: coinductive completeness. Form Methods Syst Des 53, 138–163 (2018). https://doi.org/10.1007/s10703-018-0321-3

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