Abstract
Modal logics extend classical propositional logic, and they are robustly decidable. Whereas most existing decision procedures for modal logics are based on tableau constructions, we propose a framework for obtaining decision procedures by adding instantiation rules to standard SAT and SMT solvers. Soundness, completeness, and termination of the procedures can be proved in a uniform and elementary way for the basic modal logic and some extensions.
This work was partly supported by grants ANPCyT-PICT-2008-306, ANPCyT-PICT-2010-688, ANPCyT-PICT-2012-712, the FP7-PEOPLE-2011-IRSES Project “Mobility between Europe and Argentina applying Logics to Systems” (MEALS), the project ANR-13-IS02-0001 of the Agence Nationale de la Recherche, the STIC AmSud MISMT, and the Laboratoire International Associé “INFINIS”.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andréka, H., van Benthem, J., Németi, I.: Modal languages and bounded fragments of predicate logic. Journal of Philosophical Logic 27, 217–274 (1998)
Areces, C., ten Cate, B.: Hybrid logics. In: Blackburn, P., Wolter, F., van Benthem, J. (eds.) Handbook of Modal Logics, pp. 821–868. Elsevier (2006)
Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, ch. 26, pp. 825–885. IOS Press (February 2009)
Bauer, F.L., Wirsing, M.: Elementare Aussagenlogik. Springer, Heidelberg (1991)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001)
Bolander, T., Blackburn, P.: Termination for hybrid tableaus. Journal of Logic and Computation 17(3), 517–554 (2007)
Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Springer, Berlin (1997) With an appendix by C. Allauzen, B. Durand
Chandra, A., Merlin, P.: Optimal implementation of conjunctive queries in relational databases. In: Proc. 9th ACM Symp. Theory of Computing, pp. 77–90 (1977)
Church, A.: A note on the Entscheidungsproblem. Journal of Symbolic Logic 1, 40–41 (1936)
Ganzinger, H., Hustadt, U., Meyer, C., Schmidt, R.A.: A resolution-based decision procedure for extensions of K4. In: Zakharyaschev, M., Segerberg, K., de Rijke, M., Wansing, H. (eds.) Advances in Modal Logic, pp. 225–246. CSLI Publications (1998)
Giunchiglia, F., Sebastiani, R.: Building decision procedures for modal logics from propositional decision procedures: The case study of modal K(m). Information and Computation 162(1-2), 158–178 (2000)
Goranko, V., Passy, S.: Using the universal modality: Gains and questions. Journal of Logic and Computation 2(1), 5–30 (1992)
Grädel, E.: Why are modal logics so robustly decidable? Bulletin EATCS 68, 90–103 (1999)
Grädel, E., Kolaitis, P., Vardi, M.: On the decision problem for two-variable first-order logic. Bulletin of Symbolc Logic 3, 53–69 (1997)
Grädel, E., Otto, M., Rosen, E.: Two-variable logic with counting is decidable. In: Proc. 12th Ann. IEEE Symp. Logic in Computer Science (LICS 1997), pp. 306–317. IEEE Comp. Soc. (1997)
Hustadt, U., de Nivelle, H., Schmidt, R.A.: Resolution-based methods for modal logics. Logic Journal of the IGPL 8(3), 265–292 (2000)
Löwenheim, L.: Über Möglichkeiten im Relativkalkül. Mathematische Annalen 76, 447–470 (1915)
Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT Modulo Theories: From an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). J. ACM 53(6), 937–977 (2006)
Ohlbach, H.J., Schmidt, R.A.: Functional translation and second-order frame properties of modal logics. Journal of Logic and Computation 7(5), 581–603 (1997)
Pacholsky, L., Szwast, W., Tendera, L.: Complexity of two-variable logic with counting. In: Proc. 12th Ann. IEEE Symp. Logic in Computer Science (LICS 1997), pp. 318–327 (1997)
Schmidt, R.A., Hustadt, U.: First-order resolution methods for modal logics. In: Voronkov, A., Weidenbach, C. (eds.) Programming Logics. LNCS, vol. 7797, pp. 345–391. Springer, Heidelberg (2013)
Schmidt, R.A., Tishkovsky, D.: Using tableau to decide description logics with full role negation and identity. ACM Trans. Comput. Log. 15(1) (2014)
Scott, D.: A decision method for validity of sentences in two variables. Journal of Symbolic Logic 27(377), 74 (1962)
Sebastiani, R., Tacchella, A.: SAT techniques for modal and description logics. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, pp. 781–824. IOS Press (2009)
Turing, A.: On computable numbers, with an application to the ‘Entscheidungsproblem’. Proc. London Mathematical Society 2nd. series 42, 230–265 (1937)
Vardi, M.: Why is modal logic so robustly decidable? In: DIMACS Ser. Disc. Math. Theoret. Comp. Sci., vol. 31, pp. 149–184. AMS (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Areces, C., Fontaine, P., Merz, S. (2015). Modal Satisfiability via SMT Solving. In: De Nicola, R., Hennicker, R. (eds) Software, Services, and Systems. Lecture Notes in Computer Science, vol 8950. Springer, Cham. https://doi.org/10.1007/978-3-319-15545-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-15545-6_5
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-15544-9
Online ISBN: 978-3-319-15545-6
eBook Packages: Computer ScienceComputer Science (R0)