Abstract
The primary characteristic of interval temporal logic is that intervals, rather than points, are taken as the primitive ontological entities. Their computational behaviour is generally bad, and several restrictions have been considered in order to define decidable and computationally affordable temporal logics based on intervals. In this paper we take inspiration from Golumbic and Shamir’s coarser interval algebras, which generalize the classical Allen’s Interval Algebra, in order to define two previously unknown variants of Halpern and Shoham’s logic (\(\mathrm{HS} \)). We prove that one of them (denoted here by \(\mathrm{HS_7} \)) is still generally undecidable, while the other one (\(\mathrm{HS_3} \)) becomes, perhaps surprisingly, PSpace-complete, at least in the finite case.
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References
Allen, J.F.: Maintaining knowledge about temporal intervals. Commun. ACM 26(11), 832–843 (1983)
Artale, A., Bresolin, D., Montanari, A., Ryzhikov, V., Sciavicco, G.: DL-lite and interval temporal logics: a marriage proposal. In: Proceedings of the 21st European Conference of Artificial Intelligence (ECAI), pp. 957–958 (2014)
Artale, A., Franconi, E.: A temporal description logic for reasoning about actions and plans. J. Artif. Intell. Reasoning 9, 463–506 (1998)
Bettini, C.: Time-dependent concepts: representation and reasoning using temporal description logics. Data Knowedge Engeneering 22(1), 1–38 (1997)
Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Perspectives of Mathematical Logic. Springer, Berlin (1997)
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., Della Monica, D., Montanari, A., Sala, P., Sciavicco, G.: Interval temporal logics over finite linear orders: the complete picture. In: Proceedings of the 20th European Conference on Artificial Intelligence (ECAI), pp. 199–204 (2012)
Bresolin, D., Della Monica, D., Montanari, A., Sala, P., Sciavicco, G.: Interval temporal logics over strongly discrete linear orders: the complete picture. In: Proceedings of the 4th International Symposium on Games, Automata, Logics, and Formal Verification (GANDALF). EPTCS, vol. 96, pp. 155–169 (2012)
Bresolin, D., Della Monica, D., Montanari, A., Sala, P., Sciavicco, G.: On the complexity of fragments of the modal logic of Allen’s relations over dense structures. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 511–523. Springer, Heidelberg (2015)
Bresolin, D., Della Monica, D., Montanari, A., Sciavicco, G.: The light side of interval temporal logic: the Bernays-Schönfinkel fragment of CDT. Ann. Math. Artif. Intell. 71(1–3), 11–39 (2014)
Bresolin, D., Muñoz-Velasco, E., Sciavicco, G.: Sub-propositional fragments of the interval temporal logic of Allen’s relations. In: Fermé, E., Leite, J. (eds.) JELIA 2014. LNCS, vol. 8761, pp. 122–136. Springer, Heidelberg (2014)
Chaochen, Z., Hansen, M.R.: Duration Calculus: A Formal Approach to Real-Time Systems. EATCS: Monographs in Theoretical Computer Science, Springer (2004)
Della Monica, D., Goranko, V., Montanari, A., Sciavicco, G.: Expressiveness of the interval logics of Allen’s relations on the class of all linear orders: complete classification. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI), pp. 845–850. AAAI Press (2011)
Golumbic, M., Shamir, R.: Complexity and algorithms for reasoning about time: a graph-theoretic approach. J. ACM 40(5), 1108–1133 (1993)
Halpern, J., Shoham, Y.: A propositional modal logic of time intervals. J. ACM 38(4), 935–962 (1991)
Klarman, S.: Practical querying of temporal data via OWL 2 QL and SQL:2011. In: Proceedings of the 19th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR). EPiC Series, vol. 26, pp. 52–61. Center for Artificial Inteligence Research (2014)
Marcinkowski, J., Michaliszyn, J.: The undecidability of the logic of subintervals. Fundam. Inf. 131(2), 217–240 (2014)
Montanari, A., Puppis, G., Sala, P.: Maximal decidable fragments of halpern and Shoham’s modal logic of intervals. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 345–356. Springer, Heidelberg (2010)
Montanari, A., Puppis, G., Sala, P., Sciavicco, G.: Decidability of the interval temporal logic \(AB\bar{B}\) on natural numbers. In: Proceedings of the 27th Symposium on Theoretical Aspects of Computer Science (STACS), pp. 597–608. Inria Nancy Grand Est & Loria (2010)
Moszkowski, B.: Reasoning about digital circuits. Ph.D. thesis, Department of Computer Science, Stanford University, Stanford, CA (1983)
Pratt-Hartmann, I.: Temporal prepositions and their logic. Artif. Intell. 166(1–2), 1–36 (2005)
Schmiedel, A.: Temporal terminological logic. In: Proceedings of the 8th National Conference on Artificial Intelligence (AAAI), pp. 640–645. AAAI Press (1990)
Stockmeyer, L., Meyer, A.: Word problems requiring exponential time (Preliminary Report). In: Proceedings of the 5th Annual ACM Symposium on Theory of Computing (STOC), pp. 1–9. ACM (1973)
Acknowledgments
The authors acknowledge the support from the Spanish fellowship program ‘Ramon y Cajal’ RYC-2011-07821 (G. Sciavicco), and the Spanish Project TIN12-39353-C04-01 (G. Sciavicco and E. Muñoz-Velasco).
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Muñoz-Velasco, E., Pelegrín-García, M., Sala, P., Sciavicco, G. (2015). On Coarser Interval Temporal Logics and their Satisfiability Problem. In: Puerta, J., et al. Advances in Artificial Intelligence. CAEPIA 2015. Lecture Notes in Computer Science(), vol 9422. Springer, Cham. https://doi.org/10.1007/978-3-319-24598-0_10
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