Abstract
This article considers the temporal logic of the lexicographic products of unbounded dense linear orders and provides via mosaics a new proof of the membership in NP of the satisfiability problem it gives rise to.
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References
Balbiani, P. 2008. Time representation and temporal reasoning from the perspective of non-standard analysis. In Eleventh International Conference on Principles of Knowledge Representation and Reasoning. AAAI ed. G. Brewka, and J. Lang, 695–704.
Balbiani, P. 2009. Axiomatization and completeness of lexicographic products of modal logics. In Frontiers of Combining Systems, ed. S. Ghilardi, and R. Sebastiani, 165–180. Berlin: Springer.
Balbiani, P. 2010. Axiomatizing the temporal logic defined over the class of all lexicographic products of dense linear orders without endpoints. In Temporal Representation and Reasoning, ed. N. Markey, and J. Wijsen, 19–26. IEEE.
Balbiani, P., and S. Mikulás. 2013. Decidability and complexity via mosaics of the temporal logic of the lexicographic products of unbounded dense linear orders. In Frontiers of Combining Systems, ed. P. Fontaine, C. Ringeissen, and R. Schmidt, 151–164. Berlin: Springer.
Balbiani, P., and I. Shapirovsky. 2019. Complete axiomatizations of lexicographic sums and products of modal logics. In preparation.
Bull, R. 1966. That all normal extensions of \(S4.3\) have the finite model property. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 12: 314–344.
Caleiro, C., L. Viganò, and M. Volpe. 2013. On the mosaic method for many-dimensional modal logics: a case study combining tense and modal operators. Logica Universalis 7: 33–69.
Euzenat, J., and A. Montanari. 2005. Time granularity. In Handbook of Temporal Reasoning in Artificial Intelligence, ed. M. Fisher, D. Gabbay, and L. Vila, 59–118. Amsterdam: Elsevier.
Fine, K. 1971. The logics containing \(S4.3\). Zeitschrift für mathematische Logik und Grundlagen der Mathematik 17: 371–376.
Gabbay, D., and V. Shehtman. 1998. Products of modal logics, part 1. Logic Journal of the IGPL 6: 73–146.
Gabbay, D., A. Pnueli, S. Shelah, and J. Stavi. 1980. On the temporal analysis of fairness. In Proceedings of the 7th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, 163–173. New York: ACM.
Gabbay, D., A. Kurucz, F. Wolter, and M. Zakharyaschev. 2003. Many-Dimensional Modal Logics: Theory and Applications. Amsterdam: Elsevier.
Gagné, J.-R., and J. Plaice. 1996. A nonstandard temporal deductive database system. Journal of Symbolic Computation 22: 649–664.
Kurucz, A. 2007. Combining modal logics. In Handbook of Modal Logic, ed. P. Blackburn, J. van Benthem, and F. Wolter, 869–924. Amsterdam: Elsevier.
Marx, M., and Y. Venema. 2007. Local variations on a loose theme: Modal logic and decidability. In Finite Model Theory and Its Applications, ed. E. Grädel, P. Kolaitis, L. Libkin, M. Marx, J. Spencer, M. Vardi, Y. Venema, and S. Weinstein, 371–429. Berlin: Springer.
Marx, M., S. Mikulás, and M. Reynolds. 2000. The mosaic method for temporal logics. In Automated Reasoning with Analytic Tableaux and Related Methods, ed. R. Dyckhoff, 324–340. Berlin: Springer.
Nakamura, K., and A. Fusaoka. 2007. Reasoning about hybrid systems based on a nonstandard model. In AI 2007: Advances in Artificial Intelligence, ed. M. Orgun, and J. Thornton, 749–754. Berlin: Springer.
Reynolds, M. 1997. A decidable temporal logic of parallelism. Notre Dame Journal of Formal Logic 38: 419–436.
Reynolds, M., and M. Zakharyaschev. 2001. On the products of linear modal logics. Journal of Logic and Compution 11: 909–931.
Van Benthem, J. 1991. The Logic of Time. Dordrecht: Kluwer.
Zakharyaschev, M., and A. Alekseev. 1995. All finitely axiomatizable normal extensions of \(K4.3\) are decidable. Mathematical Logic Quarterly 41: 15–23.
Acknowledgements
The author makes a point of thanking his colleagues of the Institut de recherche en informatique de Toulouse as well as the participants of the 8th International Workshop on Logic and Cognition who, by their comments and their suggestions, contributed to the development of the present paper. We also make a point of thanking the referees for their feedback.
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Balbiani, P. (2020). About the Temporal Logic of the Lexicographic Products of Unbounded Dense Linear Orders: A New Study of Its Computability. In: Ju, S., Palmigiano, A., Ma, M. (eds) Nonclassical Logics and Their Applications. Logic in Asia: Studia Logica Library. Springer, Singapore. https://doi.org/10.1007/978-981-15-1342-8_3
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