Abstract
Interval temporal logics are based on temporal structures where time intervals, rather than time instants, are the primitive ontological entities. They employ modal operators corresponding to various relations between intervals, known as Allen’s relations. Technically, validity in interval temporal logics translates to dyadic second-order logic, thus explaining their complex computational behavior. The full modal logic of Allen’s relations, called HS, has been proved to be undecidable by Halpern and Shoham under very weak assumptions on the class of interval structures, and this result was discouraging attempts for practical applications and further research in the field. A renewed interest has been recently stimulated by the discovery of interesting decidable fragments of HS. This paper contributes to the characterization of the boundary between decidability and undecidability of HS fragments. It summarizes known positive and negative results, it describes the main techniques applied so far in both directions, and it establishes a number of new undecidability results for relatively small fragments of HS.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allen, J.F.: Maintaining knowledge about temporal intervals. Communications of the ACM 26(11), 832–843 (1983)
Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. In: Perspectives of Mathematical Logic. Springer, Heidelberg (1997)
Bresolin, D., Goranko, V., Montanari, A., Sala, P.: Tableau-based Decision Procedure for the Logic of Proper Subinterval Structures over Dense Orderings. In: Areces, C., Demri, S. (eds.) Proceedings of the 5th International Workshop on Methods for Modalities (M4M), pp. 335–351 (2007)
Bresolin, D., Goranko, V., Montanari, A., Sala, P.: Tableau systems for logics of subinterval structures over dense orderings. In: Olivetti, N. (ed.) TABLEAUX 2007. LNCS (LNAI), vol. 4548, pp. 73–89. Springer, Heidelberg (2007)
Bresolin, D., Goranko, V., Montanari, A., Sala, P.: Tableau-based decision procedures for the logics of subinterval structures over dense orderings. In: Journal of Logic and Computation (to appear, 2008)
Bresolin, D., Goranko, V., Montanari, A., Sciavicco, G.: On decidability and expressiveness of propositional interval neighborhood logics. In: Artemov, S.N., Nerode, A. (eds.) LFCS 2007. LNCS, vol. 4514, pp. 84–99. Springer, Heidelberg (2007)
Bresolin, D., Goranko, V., Montanari, A., Sciavicco, G.: Propositional Interval Neighborhood Logics: Expressiveness, Decidability, and Undecidable Extensions. Annals of Pure and Applied Logic (to appear, 2008)
Bresolin, D., Montanari, A., Sala, P.: An optimal tableau-based decision algorithm for propositional neighborhood logic. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 549–560. Springer, Heidelberg (2007)
Bresolin, D., Montanari, A., Sala, P., Sciavicco, G.: Optimal Tableaux for Right Propositional Neighborhood Logic over Linear Orders. In: JELIA 2008. LNCS (LNAI), vol. 5293, pp. 62–75. Springer, Heidelberg (2008)
Bresolin, D., Montanari, A., Sciavicco, G.: An optimal decision procedure for Right Propositional Neighborhood Logic. Journal of Automated Reasoning 38(1-3), 173–199 (2007)
Chaochen, Z., Hoare, C.A.R., Ravn, A.P.: A calculus of durations. Information Processing Letters 40(5), 269–276 (1991)
Emerson, E.A.: Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, Formal Models and Semantics, vol. B, pp. 995–1072. MIT Press, Cambridge (1990)
Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood temporal logics. Journal of Universal Computer Science 9(9), 1137–1167 (2003)
Goranko, V., Montanari, A., Sciavicco, G.: A road map of interval temporal logics and duration calculi. Journal of Applied Non-Classical Logics 14(1–2), 9–54 (2004)
Halpern, J., Shoham, Y.: A propositional modal logic of time intervals. Journal of the ACM 38(4), 935–962 (1991)
Hodkinson, I., Montanari, A., Sciavicco, G.: Non-finite axiomatizability and undecidability of interval temporal logics with C, D, and T. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 308–322. Springer, Heidelberg (2008)
Krokhin, A.A., Jeavons, P., Jonsson, P.: Reasoning about temporal relations: The tractable subalgebras of Allen’s interval algebra. Journal of the ACM 50(5), 591–640 (2003)
Lodaya, K.: Sharpening the undecidability of interval temporal logic. In: He, J., Sato, M. (eds.) ASIAN 2000. LNCS, vol. 1961, pp. 290–298. Springer, Heidelberg (2000)
Marx, M., Reynolds, M.: Undecidability of compass logic. Journal of Logic and Computation 9(6), 897–914 (1999)
Montanari, A., Sciavicco, G., Vitacolonna, N.: Decidability of interval temporal logics over split-frames via granularity. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS, vol. 2424, pp. 259–270. Springer, Heidelberg (2002)
Moszkowski, B.: Reasoning about digital circuits. Tech. rep. stan-cs-83-970, Dept. of Computer Science, Stanford University, Stanford, CA (1983)
Otto, M.: Two variable first-order logic over ordered domains. Journal of Symbolic Logic 66(2), 685–702 (2001)
Venema, Y.: Expressiveness and completeness of an interval tense logic. Notre Dame Journal of Formal Logic 31(4), 529–547 (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bresolin, D., Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G. (2008). Decidable and Undecidable Fragments of Halpern and Shoham’s Interval Temporal Logic: Towards a Complete Classification. In: Cervesato, I., Veith, H., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2008. Lecture Notes in Computer Science(), vol 5330. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89439-1_41
Download citation
DOI: https://doi.org/10.1007/978-3-540-89439-1_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-89438-4
Online ISBN: 978-3-540-89439-1
eBook Packages: Computer ScienceComputer Science (R0)