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A tool for deciding the satisfiability of continuous-time metric temporal logic

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Abstract

Constraint LTL over clocks is a variant of CLTL, an extension of linear-time temporal logic allowing atomic assertions in a concrete constraint system. Satisfiability of CLTL over clocks is here shown to be decidable by means of a reduction to a decidable Satisfiability Modulo Theories (SMT) problem. The result is a complete Bounded Satisfiability Checking procedure, which has been implemented by using standard SMT solvers. The importance of this technique derives from the possibility of translating various continuous-time metric temporal logics, such as MITL and QTL, into CLTL over clocks itself. Although standard decision procedures of these logics do exist, they have never been realized in practice. Suitable translations into CLTL over clocks have instead allowed us the development of the first prototype tool for deciding MITL and QTL. The paper also reports preliminary, but encouraging, experiments on some significant examples of MITL and QTL formulae.

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Notes

  1. Note that, in the formulae of this section, we write \(z_\delta \sim c\) as an abbreviation for \(\bigvee _{i \in \{ 0,1 \}} z^i_\delta \sim c\) (similarly for \(z_\psi \sim c\)).

  2. With slight abuse, we use rational bound \(0.5\); as customary, a formula with only integer bounds can be obtained by doubling all constants appearing in the formula.

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Acknowledgments

We would like to thank the anonymous reviewers for their comments, which have greatly helped improve the presentation of the paper.

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Authors and Affiliations

  1. Politecnico di Milano – DEIB, Piazza Leonardo da Vinci, 32, 20133, Milan, Italy

    Marcello M. Bersani, Matteo Rossi & Pierluigi San Pietro

  2. CNR IEIIT-MI, Milano, Italy

    Pierluigi San Pietro

Authors
  1. Marcello M. Bersani
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  2. Matteo Rossi
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  3. Pierluigi San Pietro
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Corresponding author

Correspondence to Marcello M. Bersani.

Additional information

This research was supported by the Programme IDEAS-ERC, Project 227977-SMScom and by PRIN Project 2010LYA9RH.

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Bersani, M.M., Rossi, M. & San Pietro, P. A tool for deciding the satisfiability of continuous-time metric temporal logic. Acta Informatica 53, 171–206 (2016). https://doi.org/10.1007/s00236-015-0229-y

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