Abstract
We show that the satisfiability problem in core fragments of modal logics T, K4, and S4 in whose languages diamond modal operators are disallowed is NL-complete. Moreover, we provide deterministic procedures for satisfiability checking. We show that the above fragments correspond to certain core fragments of linear temporal logic, hence our results imply NL-completeness of the latter.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Artale, A., Kontchakov, R., Ryzhikov, V., Zakharyaschev, M.: The complexity of clausal fragments of LTL. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 35–52. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-45221-5_3
Artale, A., Kontchakov, R., Ryzhikov, V., Zakharyaschev, M.: A cookbook for temporal conceptual data modelling with description logics. ACM Trans. Comput. Logic (TOCL) 15(3), 25 (2014)
Blackburn, P., De Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2002)
Bresolin, D., Kurucz, A., Muñoz-Velasco, E., Ryzhikov, V., Sciavicco, G., Zakharyaschev, M.: Horn fragments of the Halpern-Shoham interval temporal logic. ACM Trans. Comput. Logic (TOCL) 18(3), 22:1–22:39 (2017)
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 (LNAI), vol. 8761, pp. 122–136. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11558-0_9
Bresolin, D., Munoz-Velasco, E., Sciavicco, G.: On the complexity of fragments of Horn modal logics. In: 2016 23rd International Symposium on Temporal Representation and Reasoning (TIME), pp. 186–195. IEEE (2016)
Halpern, J.Y.: The effect of bounding the number of primitive propositions and the depth of nesting on the complexity of modal logic. Artif. Intell. 75(2), 361–372 (1995)
Halpern, J.Y., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artif. Intell. 54(3), 319–379 (1992)
Horn, A.: On sentences which are true of direct unions of algebras. J. Symbolic Logic 16(1), 14–21 (1951)
Krom, M.R.: The decision problem for formulas in prenex conjunctive normal form with binary disjunctions. J. Symbolic Logic 35(2), 210–216 (1970)
Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977)
Nguyen, L.A.: Constructing the least models for positive modal logic programs. Fundam. Inf. 42(1), 29–60 (2000)
Nguyen, L.A.: On the complexity of fragments of modal logics. Adv. Modal Logic 5, 318–330 (2004)
Papadimitriou, C.H.: Computational Complexity. Wiley, Hoboken (2003)
Sciavicco, G., Muñoz-Velasco, E., Bresolin, D.: On sub-propositional fragments of modal logic. Logical Methods. Comput. Sci. 14, 1–35 (2018)
Acknowledgments
This work is supported by the National Science Centre in Poland (NCN) grant 2016/23/N/HS1/02168 and by the Foundation for Polish Science (FNP).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Wałęga, P.A. (2019). Computational Complexity of Core Fragments of Modal Logics T, K4, and S4. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_48
Download citation
DOI: https://doi.org/10.1007/978-3-030-19570-0_48
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-19569-4
Online ISBN: 978-3-030-19570-0
eBook Packages: Computer ScienceComputer Science (R0)