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An In-Depth Investigation of Interval Temporal Logic Model Checking with Regular Expressions

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Software Engineering and Formal Methods (SEFM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10469))

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In the last years, the model checking (MC) problem for interval temporal logic (ITL) has received an increasing attention as a viable alternative to the traditional (point-based) temporal logic MC, which can be recovered as a special case. Most results have been obtained by imposing suitable restrictions on interval labeling. In this paper, we overcome such limitations by using regular expressions to define the behavior of proposition letters over intervals in terms of the component states. We first prove that MC for Halpern and Shoham’s ITL (HS), extended with regular expressions, is decidable. Then, we show that formulas of a large class of HS fragments, namely, all fragments featuring (a subset of) HS modalities for Allen’s relations meets, met-by, starts, and started-by, can be model checked in polynomial working space (MC for all these fragments turns out to be PSPACE-complete).

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  1. 1.

    All the results we prove in the paper hold for the strict semantics as well.

  2. 2.

    As shown in [4], this is not the case in general: the computation-tree-based semantics of [10,11,12] is subsumed by the state-based one of [14] and follow-up papers.

  3. 3.

    The factor 2 in front of \(|\psi '|\) is needed as the small-model requires a formula in NNF.


  1. Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)

    Article  Google Scholar 

  2. Bozzelli, L., Molinari, A., Montanari, A., Peron, A.: An in-depth investigation of ITL MC with regular expressions. Technical report 2, University of Udine, Italy (2017).

  3. Bozzelli, L., Molinari, A., Montanari, A., Peron, A., Sala, P.: Interval temporal logic model checking: the border between good and bad HS fragments. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 389–405. Springer, Cham (2016). doi:10.1007/978-3-319-40229-1_27

    Chapter  Google Scholar 

  4. Bozzelli, L., Molinari, A., Montanari, A., Peron, A., Sala, P.: Interval vs. point temporal logic model checking: an expressiveness comparison. In: FSTTCS (2016)

    Google Scholar 

  5. Bozzelli, L., Molinari, A., Montanari, A., Peron, A., Sala, P.: MC the logic of Allen’s relations meets and started-by is P\(^{\rm NP}\)-C. In: GandALF, pp. 76–90 (2016)

    Google Scholar 

  6. Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: The dark side of interval temporal logic: marking the undecidability border. Ann. Math. Artif. Intell. 71(1–3), 41–83 (2014)

    Article  MathSciNet  Google Scholar 

  7. Esparza, J., Hansel, D., Rossmanith, P., Schwoon, S.: Efficient algorithms for model checking pushdown systems. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 232–247. Springer, Heidelberg (2000). doi:10.1007/10722167_20

    Chapter  Google Scholar 

  8. Halpern, J.Y., Shoham, Y.: A propositional modal logic of time intervals. J. ACM 38(4), 935–962 (1991)

    Article  MathSciNet  Google Scholar 

  9. Kupferman, O., Piterman, N., Vardi, M.Y.: From liveness to promptness. Formal Methods Syst. Des. 34(2), 83–103 (2009)

    Article  Google Scholar 

  10. Lomuscio, A., Michaliszyn, J.: An epistemic HS logic. In: IJCAI, pp. 1010–1016 (2013)

    Google Scholar 

  11. Lomuscio, A., Michaliszyn, J.: Decidability of model checking multi-agent systems against a class of EHS specifications. In: ECAI, pp. 543–548 (2014)

    Google Scholar 

  12. Lomuscio, A., Michaliszyn, J.: Model checking multi-agent systems against epistemic HS specifications with regular expressions. In: KR, pp. 298–308 (2016)

    Google Scholar 

  13. Marcinkowski, J., Michaliszyn, J.: The undecidability of the logic of subintervals. Fundamenta Informaticae 131(2), 217–240 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Molinari, A., Montanari, A., Murano, A., Perelli, G., Peron, A.: Checking interval properties of computations. Acta Informatica 53, 587–619 (2016)

    Article  MathSciNet  Google Scholar 

  15. Molinari, A., Montanari, A., Peron, A.: Complexity of ITL model checking: some well-behaved fragments of the interval logic HS. In: TIME, pp. 90–100 (2015)

    Google Scholar 

  16. Molinari, A., Montanari, A., Peron, A.: A model checking procedure for interval temporal logics based on track representatives. In: CSL, pp. 193–210 (2015)

    Google Scholar 

  17. Molinari, A., Montanari, A., Peron, A., Sala, P.: Model checking well-behaved fragments of HS: the (Almost) final picture. In: KR, pp. 473–483 (2016)

    Google Scholar 

  18. Montanari, A.: Interval temporal logics model checking. In: TIME, p. 2 (2016)

    Google Scholar 

  19. Moszkowski, B.: Reasoning about digital circuits. Ph.D. thesis, Stanford (1983)

    Google Scholar 

  20. Roeper, P.: Intervals and tenses. J. Philos. Log. 9, 451–469 (1980)

    MathSciNet  MATH  Google Scholar 

  21. Venema, Y.: Expressiveness and completeness of an interval tense logic. Notre Dame J. Formal Log. 31(4), 529–547 (1990)

    Article  MathSciNet  Google Scholar 

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Correspondence to Angelo Montanari .

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Bozzelli, L., Molinari, A., Montanari, A., Peron, A. (2017). An In-Depth Investigation of Interval Temporal Logic Model Checking with Regular Expressions. In: Cimatti, A., Sirjani, M. (eds) Software Engineering and Formal Methods. SEFM 2017. Lecture Notes in Computer Science(), vol 10469. Springer, Cham.

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  • Print ISBN: 978-3-319-66196-4

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