Parameterized Modal Satisfiability

  • Antonis Achilleos
  • Michael Lampis
  • Valia Mitsou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6199)


We investigate the parameterized computational complexity of the satisfiability problem for modal logic and attempt to pinpoint relevant structural parameters which cause the problem’s combinatorial explosion, beyond the number of propositional variables v. To this end we study the modality depth, a natural measure which has appeared in the literature, and show that, even though modal satisfiability parameterized by v and the modality depth is FPT, the running time’s dependence on the parameters is a tower of exponentials (unless P=NP). To overcome this limitation we propose possible alternative parameters, namely diamond dimension and modal width. We show fixed-parameter tractability results using these measures where the exponential dependence on the parameters is much milder (doubly and singly exponential respectively) than in the case of modality depth thus leading to FPT algorithms for modal satisfiability with much more reasonable running times. We also give lower bound arguments which prove that our algorithms cannot be improved significantly unless the Exponential Time Hypothesis fails.


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  1. 1.
    Blackburn, P., van Benthem, J.F.A.K., Wolter, F.: Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3. Elsevier Science Inc., New York (2006)Google Scholar
  2. 2.
    Chagrov, A.V., Rybakov, M.N.: How Many Variables Does One Need to Prove PSPACE-hardness of Modal Logics. In: Balbiani, P., Suzuki, N.-Y., Wolter, F., Zakharyaschev, M. (eds.) Advances in Modal Logic, pp. 71–82. King’s College Publications (2002)Google Scholar
  3. 3.
    Courcelle, B.: The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs. Inf. Comput. 85(1), 12–75 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Downey, R.G., Fellows, M.R.: Parameterized complexity. Springer, Heidelberg (1999)Google Scholar
  5. 5.
    Fagin, R., Halpern, J.Y., Moses, Y., Vardi, M.Y.: Reasoning About Knowledge. The MIT Press, Cambridge (1995)zbMATHGoogle Scholar
  6. 6.
    Flum, J., Grohe, M.: Parameterized complexity theory. Springer, New York (2006)Google Scholar
  7. 7.
    Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic 130(1-3), 3–31 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Grohe, M.: Logic, graphs, and algorithms. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 14(091) (2007)Google Scholar
  9. 9.
    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)zbMATHCrossRefGoogle Scholar
  10. 10.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Nguyen, L.A.: On the complexity of fragments of modal logics. Advances in Modal Logic 5, 249–268 (2005)Google Scholar
  13. 13.
    Woeginger, G.: Exact algorithms for NP-hard problems: A survey. Combinatorial Optimization–Eureka, You Shrink!, 185–207Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Antonis Achilleos
    • 1
  • Michael Lampis
    • 1
  • Valia Mitsou
    • 1
  1. 1.Computer Science DepartmentGraduate Center, City University of New YorkNYUSA

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