Abstract
Under study is the temporal multiagent logic with different intervals of lost time which are individual for each of the agents. The logic bases on the frames with principal basic sets on all naturals \( N \) as temporal states, where each agent \( j \) can have their own proper sets \( X_{j} \) of inaccessible (lost, forgotten) temporal states (\( X_{j}\subset N \) for all \( j\in J \)). The unification problem and the problem of the algorithmic recognition of admissible inference rules are the main mathematical problems of the paper. The solution of the unification problem consists in finding a finite computable set of formulas which is a complete set of unifiers. The problem is solved by the Ghilardi technique of projective formulas. We prove that every formula unifiable in this logic is projective and provide some algorithm constructing its projective unifier, which solves the unification problem. This makes it possible to solve the open problem of the algorithmic recognition of admissible rules. The article ends with some generalization of the definition of projective formulas—weakly projective formulas—and exhibits an easy example of their application.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gabbay D. M., Hodkinson I. M., and Reynolds M. A., Temporal Logic, Clarendon, Oxford (1994) (Math. Found. Comput. Aspects; Vol. 1).
Gabbay D. M. and Hodkinson I. M., “An axiomatization of the temporal logic with Until and Since over the real numbers,” J. Logic Comput., vol. 1, 229–260 (1990).
Gabbay D. M. and Hodkinson I. M., Temporal Logic in the Context of Databases, Oxford University, Oxford (1995) (Logic and Reality: Essays on the Legacy of Arthur Prior. Copeland J., Ed.).
Manna Z. and Pnueli A., The Temporal Logic of Reactive and Concurrent Systems: Specification, Springer, New York (1992).
Emerson E. A. and Halpern J. Y., “Decision procedures and expressiveness in the temporal logic of branching time,” J. Comput. Syst. Sci., vol. 30, no. 1, 1–24 (1985).
Clarke E. M., Emerson E. A., and Sistla A. P., “Automatic verification of finite-state concurrent systems using temporal logic specifications,” ACM Trans. Programming Languages and Systems, vol. 8, no. 2, 244–263 (1986).
Rybakov V., “Non-transitive linear temporal logic and logical knowledge operations,” J. Logic Comput., vol. 26, no. 3, 945–958 (2016).
Rybakov V. V., “Nontransitive temporal multiagent logic, information and knowledge, deciding algorithms,” Sib. Math. J., vol. 58, no. 5, 875–886 (2017).
Rybakov V. V., “Multi-Agent logic’s modelling non-monotonic information and reasoning,” Proc. Comput. Sci., Elsevier,, vol. 176, 670–674 (2020).
Weiss G., Multiagent Systems, a Modern Approach to Distributed Artificial Intelligence, MIT, Cambridge (1999).
Wooldridge M., An Introduction to Multiagent Systems, John Wiley and Sons, Hoboken (2002).
Fagin R., Halpern J., and Moses Y., Reasoning About Knowledge, MIT, Cambridge (1996).
Robinson A., “A machine oriented logic based on the resolution principle,” J. ACM, vol. 12, no. 1, 23–41 (1965).
Knuth D., Bendix P., and Leech J., Simple Word Problems in Universal Algebras, Pergamon, Oxford (1970).
Baader F. and Snyder W., “Unification theory,” in: Handbook of Automated Reasoning. I, Elsevier, Amsterdam (2001), 445–533.
Rybakov V. V., “Problems of substitution and admissibility in the modal system Grz and in intuitionistic propositional calculus,” Ann. Pure Appl. Log., vol. 50, no. 1, 71–106 (1990).
Rybakov V. V., “Rules of inference with parameters for intuitionistic logic,” J. Symb. Log., vol. 57, no. 3, 912–923 (1992).
Ghilardi S., “Unification through projectivity,” J. Logic Comput., vol. 7, no. 6, 733–752 (1997).
Ghilardi S., “Unification in intuitionistic logic,” J. Symb. Log., vol. 64, no. 2, 859–880 (1999).
Ghilardi S., “Best solving modal equations,” Ann. Pure Appl. Logic, vol. 102, 183–198 (2000).
Dzik W. and Wojtylak P., “Projective unification in modal logic,” Logic J. IGPL, vol. 121, 121–152 (2012).
Wrónski A., “Transparent verifiers in intermediate logics,” in: Abstracts of the 54th Conference in History of Mathematics, The Jagiellonian University, Cracow (2008), 6.
Rybakov V. V., “Projective formulas and unification in linear temporal logic LTLU,” Logic J. IGPL, vol. 22, no. 4, 665–672 (2015).
Rybakov V. V., “Linear temporal logic with until and next, logical consecutions,” Ann. Pure Appl. Logic, vol. 155, 32–45 (2008).
Babenyshev S. and Rybakov V., “Linear temporal logic LTL: Basis for admissible rules,” J. Logic Comput., vol. 21, no. 2, 157–177 (2011).
Funding
The author was supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation (Grant no. 075–02–2022–876).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 4, pp. 924–934. https://doi.org/10.33048/smzh.2022.63.417
The article is dedicated to my university friend Academician Sergey Savost’yanovich Goncharov on the occasion of his 70th Birthday.
Rights and permissions
About this article
Cite this article
Rybakov, V.V. Multiagent Temporal Logics, Unification Problems, and Admissibilities. Sib Math J 63, 769–776 (2022). https://doi.org/10.1134/S0037446622040176
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446622040176