Abstract
Propositional interval temporal logics are quite expressive temporal logics that allow one to naturally express statements that refer to time intervals. Unfortunately, most such logics turn out to be (highly) undecidable. In order to get decidability, severe syntactic or semantic restrictions have been imposed to interval-based temporal logics to reduce them to point-based ones. The problem of identifying expressive enough, yet decidable, new interval logics or fragments of existing ones that are genuinely interval-based is still largely unexplored. In this paper, we focus our attention on interval logics of temporal neighborhood. We address the decision problem for the future fragment of Neighborhood Logic (Right Propositional Neighborhood Logic, RPNL for short), and we positively solve it by showing that the satisfiability problem for RPNL over natural numbers is NEXPTIME-complete. Then, we develop a sound and complete tableau-based decision procedure, and we prove its optimality.
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References
Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem, Perspectives of Mathematical Logic. Springer, Berlin Heidelberg New York (1997)
Bowman, H., Thompson, S.: A decision procedure and complete axiomatization of finite interval temporal logic with projection. J. Log. Comput. 13(2), 195–239 (2003)
Bresolin, D., Montanari, A.: A tableau-based decision procedure for a branching-time interval temporal logic. In: Schlingloff, H. (ed.): Proceedings of the 4th International Workshop on Methods for Modalities, pp. 38–53. Berlin, Germany (2005a)
Bresolin, D., Montanari, A.: A tableau-based decision procedure for Right Propositional Neighborhood Logic. In: Proceedings of TABLEAUX 2005. Lecture Notes in Artificial Intelligence, vol. 3702, pp. 63–77. Koblenz, Germany (2005b)
Bresolin, D., Montanari, A.: A tableau-based decision procedure for right propositional neighborhood logic. Technical Report 02, Dipartimento di Matematica e Informatica, Università di Udine, Italy (2005c)
Emerson, E., Halpern, J.: Decision procedures and expressiveness in the temporal logic of branching time. J. Comput. Syst. Sci. 30(1), 1–24 (1985)
Emerson, E., Sistla, A.: Deciding full branching time logic. Inf. Control 61(3), 175–201 (1984)
Goranko, V., Montanari, A., Sciavicco, G.: A general tableau method for propositional interval temporal logic. In: Proceedings of TABLEAUX 2003. Lecture Notes in Artificial Intelligence, vol. 2796, pp. 102–116. Rome, Italy (2003a)
Goranko, V., Montanari, A., Sciavicco, G.: Propositional interval neighborhood temporal logics. J. Univers. Comput. Sci. 9(9), 1137–1167 (2003b)
Goranko, V., Montanari, A., Sciavicco, G.: A road map of interval temporal logics and duration calculi. J. Appl. Non-class. Log. 14(1–2), 9–54 (2004)
Goranko, V., Montanari, A., Sciavicco, G., Sala, P.: A general tableau method for propositional interval temporal logics: theory and implementation. J. Appl. Logic 4(3), 305–330 (2006)
Halpern, J., Shoham, Y.: A propositional modal logic of time intervals. ACM 38(4), 935–962 (1991)
Kesten, Y., Manna, Z., McGuire, H., Pnueli, A.: A decision algorithm for full propositional temporal logic. In: Proceedings of the 5th International Conference on Computer Aided Verification, Elounda, Greece. Lecture Notes in Computer Science, vol. 697, vol. 697, pp. 97–109. Springer, Berlin Heidelberg New York (1993)
Lodaya, K.: Sharpening the undecidability of interval temporal logic. In: Proceedings of 6th Asian Computing Science Conference. Lecture Notes in Computer Science, vol. 1961, pp. 290–298. Springer, Berlin Heidelberg New York (2000)
Montanari, A.: Propositional interval temporal logics: some promising paths. In: Proceedings of the 12th International Symposium on Temporal Representation and Reasoning (TIME), Burlington, VT, pp. 201–203. IEEE Computer Society Press, Los Alamitos, CA (2005)
Montanari, A., Sciavicco, G.: A Decidability Proof for Propositional Neighborhood Logic (Extended Abstract). Contributed Talk, Trends in Logics III Conference, Warsaw - Ruciane Nida (Poland) (2005)
Montanari, A., Sciavicco, G., Vitacolonna, N.: Decidability of interval temporal logics over split-frames via granularity. In: Proceedings of the 8th European Conference on Logics in Artificial Intelligence, Cosenza, Italy. Lecture Notes in Artificial Intelligence, vol. 2424, pp. 259–270. Springer, Berlin Heidelberg New York (2002)
Moszkowski, B.: Reasoning about digital circuits. Technical report stan-cs-83-970, Department of Computer Science, Stanford University, Stanford, CA (1983)
Otto, M.: Two variable first-order logic over ordered domains. J. Symb. Log. 66(2), 685–702 (2001)
Schmitt, P., Goubault-Larrecq, J.: A tableau system for linear-time temporal togic. In: Brinksma, E. (ed.) 3rd Workshop on Tools and Algorithms for the Construction and Analysis of Systems, Enschede, The Netherlands. Lecture Notes in Computer Science, vol. 1217, pp. 130–144. Springer, Berlin Heidelberg New York (1997)
Venema, Y.: A modal logic for chopping intervals. J. Log. Comput. 1(4), 453–476 (1991)
Wolper, P.: The tableau method for temporal logic: an overview. Log. Anal. 28, 119–136 (1985)
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Bresolin, D., Montanari, A. & Sciavicco, G. An Optimal Decision Procedure for Right Propositional Neighborhood Logic. J Autom Reasoning 38, 173–199 (2007). https://doi.org/10.1007/s10817-006-9051-0
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DOI: https://doi.org/10.1007/s10817-006-9051-0