Abstract
We investigate linear temporal logic \(\mathscr {LT L}_{{ NT}}\) with non-transitive time and operators NEXT and UNTIL, as well as some possible interpretations of logical knowledge operators in this context. We assume time to be non-transitive, linear and discrete, the former being a major innovative part of this paper. We provide motivation for our approach along with ideas of why we might want to consider time to be non-transitive, and comment on possible interpretations of logical knowledge operators. The main result of Sect. 11.5 is a solution for decidability problem for \(\mathscr {LT L}_{{ NT}}\), for which we describe in details the decision algorithm. In Sect. 11.6 we introduce non-transitive linear temporal logic \(\mathscr {LT L}_{{ NT}}(m)\) with uniform bound (m) for non-transitivity. There we also solve the admissibility problem for \(\mathscr {LT L}_{{ NT}}(m)\), that is we provide an algorithm for verifying admissibility of inference rules in \(\mathscr {LT L}_{{ NT}}(m)\). Last section contains description of remaining interesting open problems.
I dedicate this paper to Larisa Maksimova, a bright mathematician with outstanding contributions to mathematical logic and my former PhD adviser, with whom for many years I have shared a close and friendly relationship.
This is a preview of subscription content, log in via an institution to check access.
References
Artemov, S. (2006). Justified common knowledge. Theoretical Computer Science, 357, 4–22.
Artemov, S., & Nogina, E. (2005). Introducing justification into epistemic logic. Journal of Logic and Computation, 15, 1059–1073.
Babenyshev, S., & Rybakov, V. (2011). Linear temporal logic LTL: Basis for admissible rules. Journal of Logic and Computation, 21, 157–177.
Balbiani, P., & Vakarelov, D. (2002). A modal logic for indiscernibility and complementarity in information systems. Fundamenta Informaticae, 50, 243–263.
Belardinelli, F., & Lomuscio, A. (2012). Interactions between knowledge and time in a first-order logic for multi-agent systems: Completeness results. Journal of Artificial Intelligence Research, 45, 1–45.
Fagin, R., Halpern, J., Moses, Y., & Vardi, M. (1995). Reasoning about knowledge. Cambridge: MIT Press.
Friedman, H. (1975). One hundred and two problems in mathematical logic. Journal of Symbolic Logic, 40, 113–130.
Gabbay, D. M., & Hodkinson, I. M. (1990). An axiomatization of the temporal logic with until and since over the real numbers. Journal of Logic and Computation, 1, 229–260.
Gabbay, D. M., & Hodkinson, I. M. (1995). Temporal logic in context of databases. In J. Copeland (Ed.), Logic and reality, essays on the legacy of Arthur prior (pp. 89–120). Oxford: Oxford University Press.
Gabbay, D. M., Hodkinson, I. M., & Reynolds, M. A. (1994). Temporal logic: mathematical foundations and computational aspects (Vol. 1). Oxford: Clarendon Press.
Halpern, J., Samet, D., & Segev, E. (2009). Defining knowledge in terms of belief: The modal logic perspective. The Review of Symbolic Logic, 2, 469–487.
Hintikka, J. (1962). Knowledge and belief: An Introduction to the logic of the two notions. Ithaca: Cornell University Press.
Lomuscio, A., & Michaliszyn, J. (2013). An epistemic Halpern–Shoham logic. In Proceedings of the 23rd International Joint Conference on Artificial Intelligence (IJCAI13) (pp.1010–1016). Beijing: AAAI Press.
Manna, Z., & Pnueli, A. (1992). The temporal logic of reactive and concurrent systems: Specification. New York: Springer.
Manna, Z., & Pnueli, A. (1995). Temporal verification of reactive systems: Safety. Berlin: Springer.
McLean, D., & Rybakov, V. (2013). Multi-agent temporary logic \(TS4^ U_ {K_n}\) based at non-linear time and imitating uncertainty via agents interaction, in Artificial Intelligence and Soft Computing 2013, Conference Proceedings (pp. 375–384). Heidelberg: Springer.
Rybakov, V. V. (2003). Refined common knowledge logics or logics of common information. Archive for mathematical Logic, 42, 179–200.
Rybakov, V. V. (2005). Logical consecutions in discrete linear temporal logic. Journal of Symbolic Logic, 70, 1137–1149.
Rybakov, V. V. (2008). Linear temporal logic with until and next, logical consecutions. Annals of Pure and Applied Logic, 155, 32–45.
Rybakov, V. (2009a). Logic of knowledge and discovery via interacting agents. Decision algorithm for true and satisfiable statements. Information Sciences, 179, 1608–1614.
Rybakov, V. (2009b). Linear temporal logic \(LTL_{K_n}\) extended by multi-agent logic \(K_n\) with interacting agents. Journal of logic and Computation, 19, 989–1017.
Rybakov V. V. (2011). Chance discovery and unification in linear modal logic. In Knowledge-Based and Intelligent Information and Engineering Systems 2011, Proceedings (pp. 478–485). Heidelberg: Springer.
Rybakov, V. V. (2012). Logical analysis for chance discovery in multi-agents’ environment. In Knowledge-Based and Intelligent Information and Engineering Systems 2011, Proceedings (pp. 1593–1601). Heidelberg: Springer.
Rybakov, V. V. (2014a). A note on parametrized knowledge operations in temporal logic, from arXiv:1405.0559v1.
Rybakov, V. V. (2014b). Linear non-transitive temporal logic, knowledge operations, algorithms for admissibility, from arXiv:1406.2783.
Rybakov, V. V. (2014c). Non-transitive linear temporal logic and logical knowledge operations. Submitted, December 2014.
Rybakov, V., & Babenyshev. S. (2010). Multi-agent logic with distances based on linear temporal frames. In Artificial Intelligence and Soft Computing 2010, Conference Proceedings (pp. 337–344). Heidelberg: Springer.
Vakarelov, D. (2005). A modal characterization of indiscernibility and similarity relations in Pawlak’s information systems. In D. Slezak et al (Eds.), Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing: 10th International Conference, RSFDGrC 2005, Regina, Canada, August 31–September 3, 2005, Proceedings, Part I (pp. 12–22). Heidelberg: Springer.
Vardi, M. (1995). An automata-theoretic approach to linear temporal logic. In Banff Higher Order Workshop (pp. 38–266), from http://citeseer.ist.psu.edu/vardi96automatatheoretic.html.
Vardi, M. Y. (1998). Reasoning about the past with two-way automata, In K. G. Larsen, S. Skyum, & G. Winskel (Eds.), Automata, Languages and Programming, 25th International Colloquium, ICALP’98, Aalborg, Denmark July 13–17, 1998, Proceedings (pp. 628–641). Heidelberg: Springer.
Wooldridge, M. (2003). An Automata-theoretic approach to multi-agent planning, in Proceedings of the First European Workshop on Multi-agent Systems (EUMAS 2003), Oxford: Oxford University.
Wooldridge, M., & Lomuscio, A. (2000). Multi-agent VSK logic. In M. Ojeda-Aciego, I. P. de Guzman, G. Brewka, & L. M. Pereira (Eds.), Logics in Artificial Intelligence, European Workshop, JELIA 2000 Malaga, Spain, September 29–October 2, 2000 Proceedings (pp. 300–312). Heidelberg: Springer.
Wooldridge, M., Huget, M.-P., Fisher, M., & Parsons, S. (2006). Model checking multi-agent systems: The MABLE language and its applications. International Journal on Artificial Intelligence Tools, 15, 195–225.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Rybakov, V.V. (2018). Linear Temporal Logic with Non-transitive Time, Algorithms for Decidability and Verification of Admissibility. In: Odintsov, S. (eds) Larisa Maksimova on Implication, Interpolation, and Definability. Outstanding Contributions to Logic, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-69917-2_11
Download citation
DOI: https://doi.org/10.1007/978-3-319-69917-2_11
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-69916-5
Online ISBN: 978-3-319-69917-2
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)