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.
KeywordsModal logic Parameterized complexity Satisfiability problems
Unable to display preview. Download preview PDF.
- 1.Blackburn, P., van Benthem, J.F.A.K., Wolter, F.: Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3. Elsevier, New York (2006) Google Scholar
- 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, London (2002) Google Scholar
- 7.Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, New York (2006) Google Scholar
- 9.Grohe, M.: Logic, graphs, and algorithms. Electron. Colloq. Comput. Complex. 14, 091 (2007) Google Scholar
- 14.Nguyen, L.A.: On the complexity of fragments of modal logics. Adv. Modal Logic 5, 249–268 (2005) Google Scholar
- 15.Spaan, E.: Complexity of modal logics. PhD thesis, University of Amsterdam (1993) Google Scholar