Abstract
The model checking problem has thoroughly been explored in the context of standard point-based temporal logics, such as LTL, CTL, and CTL\(^{*}\), whereas model checking for interval temporal logics has been brought to the attention only very recently.
In this paper, we prove that the model checking problem for the logic of Allen’s relations started-by and finished-by is highly intractable, as it can be proved to be \({{\mathrm{\mathbf {EXPSPACE}}}}\)-hard. Such a lower bound immediately propagates to the full Halpern and Shoham’s modal logic of time intervals (HS). In contrast, we show that other noteworthy HS fragments, namely, Propositional Neighbourhood Logic extended with modalities for the Allen relation starts (resp., finishes) and its inverse started-by (resp., finished-by), turn out to have—maybe unexpectedly—the same complexity as LTL (i.e., they are \({{\mathrm{\mathbf {PSPACE}}}}\)-complete), thus joining the group of other already studied, well-behaved albeit less expressive, HS fragments.
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Notes
- 1.
All the results we prove in the paper hold for the strict semantics as well.
- 2.
Note that such a \(\rho \)-position exists by definition of \(E(\varphi ,\rho )\).
References
Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)
Bozzelli, L., Molinari, A., Montanari, A., Peron, A., Sala, P.: Interval Temporal Logic Model Checking: the Border Between Good and Bad HS Fragments (2016). https://www.dimi.uniud.it/la-ricerca/pubblicazioni/preprints/1.2016
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)
Bresolin, D., Goranko, V., Montanari, A., Sala, P.: Tableau-based decision procedures for the logics of subinterval structures over dense orderings. J. Logic Comput. 20(1), 133–166 (2010)
Bresolin, D., Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood logics: expressiveness, decidability, and undecidable extensions. Ann. Pure Appl. Logic 161(3), 289–304 (2009)
Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (2002)
Emerson, E.A., Halpern, J.Y.: “Sometimes” and “not never” revisited: on branching versus linear time temporal logic. J. ACM 33(1), 151–178 (1986)
Giunchiglia, F., Traverso, P.: Planning as model checking. In: Biundo, S., Fox, M. (eds.) ECP 1999. LNCS, vol. 1809, pp. 1–20. Springer, Heidelberg (2000)
Gottlob, G.: NP trees and Carnap’s modal logic. J. ACM 42(2), 421–457 (1995)
Halpern, J.Y., Shoham, Y.: A propositional modal logic of time intervals. J. ACM 38(4), 935–962 (1991)
Harel, D.: Algorithmics: The Spirit of Computing. Wesley, Reading (1992)
Lodaya, K.: Sharpening the undecidability of interval temporal logic. In: Kleinberg, R.D., Sato, M. (eds.) ASIAN 2000. LNCS, vol. 1961, pp. 290–298. Springer, Heidelberg (2000)
Lomuscio, A., Michaliszyn, J.: An epistemic Halpern-Shoham logic. In: IJCAI, pp. 1010–1016 (2013)
Lomuscio, A., Michaliszyn, J.: Decidability of model checking multi-agent systems against a class of EHS specifications. In: ECAI, pp. 543–548 (2014)
Lomuscio, A., Michaliszyn, J.: Model checking epistemic Halpern-Shoham logic extended with regular expressions. CoRR abs/1509.00608 (2015)
Lomuscio, A., Raimondi, F.: mcmas: a model checker for multi-agent systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 450–454. Springer, Heidelberg (2006)
Marcinkowski, J., Michaliszyn, J.: The undecidability of the logic of subintervals. Fundamenta Informaticae 131(2), 217–240 (2014)
Molinari, A., Montanari, A., Murano, A., Perelli, G., Peron, A.: Checking interval properties of computations. Acta Informatica (2015, accepted for publication)
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)
Molinari, A., Montanari, A., Peron, A.: A model checking procedure for interval temporal logics based on track representatives. In: CSL, pp. 193–210 (2015)
Molinari, A., Montanari, A., Peron, A., Sala, P.: Model checking well-behaved fragments of HS: the (almost) final picture. In: KR (2016)
Moszkowski, B.: Reasoning about digital circuits. Ph.D. thesis, Dept. of Computer Science, Stanford University, Stanford, CA (1983)
Pnueli, A.: The temporal logic of programs. In: FOCS, pp. 46–57. IEEE (1977)
Pratt-Hartmann, I.: Temporal prepositions and their logic. Artif. Intell. 166(1–2), 1–36 (2005)
Roeper, P.: Intervals and tenses. J. Philos. Logic 9, 451–469 (1980)
Schnoebelen, P.: Oracle circuits for branching-time model checking. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 790–801. Springer, Heidelberg (2003)
Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logics. J. ACM 32(3), 733–749 (1985)
Venema, Y.: Expressiveness and completeness of an interval tense logic. Notre Dame J. Formal Logic 31(4), 529–547 (1990)
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Bozzelli, L., Molinari, A., Montanari, A., Peron, A., Sala, P. (2016). Interval Temporal Logic Model Checking: The Border Between Good and Bad HS Fragments. In: Olivetti, N., Tiwari, A. (eds) Automated Reasoning. IJCAR 2016. Lecture Notes in Computer Science(), vol 9706. Springer, Cham. https://doi.org/10.1007/978-3-319-40229-1_27
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