The complexity of propositional modal theories and the complexity of consistency of propositional modal theories
This paper is concerned with the computational complexity of the following problems for various modal logics L: (1). The L-deducibility problem: given a finite set of formulas S and a formula A, determine if A is in the modal theory T HL(S) formed with all theorems of the modal logic L as logical axioms and with all members of S as proper axioms. (2). The L-consistency problem: given a finite set of formulas S, determine if the theory THL(S) is consistent.
Table 1 is a comparison of complexity results of these two problems and the corresponding provability and satisfiability problems for modal logics K, T, B, S4, KD45 and S5. The complexity results of the deducibility problem for extensions of K4 are a direct consequence of a modal deduction theorem for K4 (cf. [17, 15]). The NP-completeness of the S4-consistency problem is due to Tiomkin and Kaminski .
The main contribution of this paper is that we can show that the deducibility problem and the consistency problem for any modal logic between K and B are EXPTIME-hard; in particular, for K, T and B, both problems are EXPTIME-complete.
Keywordcomputational complexity modal logic deducibility consistency provability satisfiability
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- A.K. Chandra, D.C. Kozen and L.J. Stockmeyer, Alternation, J. of ACM 28,1981, 114–133.Google Scholar
- B.F. Chellas, Modal Logic: an Introduction, Cambridge university press, 1980.Google Scholar
- C.C. Chen, The complexity of decision problems for modal propositional logics, Ph.D. thesis, Department of Computer Science and Information Engineering, National Taiwan University. Taipei, Taiwan, 1993.Google Scholar
- E.A. Emerson and J.Y. Halpern, Decision procedures and expressiveness in the temporal logic of branching time, J. of Computer and System Science 30, 1985, 1–24.Google Scholar
- M.J. Fischer and R.E. Ladner, Propositional dynamic logic of regular programs, J. of Computer and System Science 18, 1979, 194–211.Google Scholar
- M. J. Fischer and N. Immerman, Interpreting logics of knowledge in propositional dynamic logic with converse, Information Processing Letter 25, 1987, 175–181.Google Scholar
- M. Fitting, Proof Methods for Modal and Intuitionistic Logics, D. Reidel, 1983.Google Scholar
- J.Y. Halpern and Y.O. Moses, A guide to the modal logic of knowledge, Proc. of the 9th International Joint Conference on Artificial Intelligence (IJCAI-85), 1985, 480–490.Google Scholar
- D. Harel, Dynamic logic. in: D. Gabbay and F. Guenthner eds., Handbook of Philosophical Logic, vol 2: Extensions of Classical Logic, D. Reidel, 1984, 497–604.Google Scholar
- G. E. Hughes and M. J. Cresswell, An Introduction to Modal Logic, Methuen & Co., London, 1968.Google Scholar
- R.E. Ladner, The computational complexity of provability in systems of modal propositional logic, SIAM J. Comput. 6(3), 1977, 467–480.Google Scholar
- D. McDermott, Nonmonotonic logic II: nonmonotonic modal theories, J. of ACM 29(1), 1982, 33–57.Google Scholar
- R.C. Moore, Semantical considerations on non-monotonic logic, Artificial Intelligence 25, 1985, 75–94.Google Scholar
- G.F. Shvarts, Autoepistemic modal logics, In R. Parikh, ed., Proc. of the 3rd Conference on Theoretical Aspect of Reasoning about Knowledge, San Mateo, CA., Morgan Kaufmann, 1990, 97–109.Google Scholar
- M. Tiomkin and M. Kaminski, Nonmonotonic default modal logics, J. of ACM 38, 1991, 963–984.Google Scholar
- H. Tuominen, Dynamic logic as a uniform framework for theorem proving in intensional logic, Proc. of the 10th International Conference on Automated Deduction, LNCS 449 Spring-verlag, Berlin, 1990, 514–527.Google Scholar
- E. Zarnecka-Bialy, A note on deduction theorem for Gödel's propositional calculus G4. Studia Logica 23, 1968, 35–40.Google Scholar