Abstract
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 H L(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 TH L(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 [15].
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.
This research was supported by the National Science Council of ROC under contract number NSC82-0408-E-002-428.
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References
A.K. Chandra, D.C. Kozen and L.J. Stockmeyer, Alternation, J. of ACM 28,1981, 114–133.
B.F. Chellas, Modal Logic: an Introduction, Cambridge university press, 1980.
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.
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.
M.J. Fischer and R.E. Ladner, Propositional dynamic logic of regular programs, J. of Computer and System Science 18, 1979, 194–211.
M. J. Fischer and N. Immerman, Interpreting logics of knowledge in propositional dynamic logic with converse, Information Processing Letter 25, 1987, 175–181.
M. Fitting, Proof Methods for Modal and Intuitionistic Logics, D. Reidel, 1983.
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.
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.
G. E. Hughes and M. J. Cresswell, An Introduction to Modal Logic, Methuen & Co., London, 1968.
R.E. Ladner, The computational complexity of provability in systems of modal propositional logic, SIAM J. Comput. 6(3), 1977, 467–480.
D. McDermott, Nonmonotonic logic II: nonmonotonic modal theories, J. of ACM 29(1), 1982, 33–57.
R.C. Moore, Semantical considerations on non-monotonic logic, Artificial Intelligence 25, 1985, 75–94.
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.
M. Tiomkin and M. Kaminski, Nonmonotonic default modal logics, J. of ACM 38, 1991, 963–984.
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.
E. Zarnecka-Bialy, A note on deduction theorem for Gödel's propositional calculus G4. Studia Logica 23, 1968, 35–40.
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© 1994 Springer-Verlag Berlin Heidelberg
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Chen, CC., Lin, IP. (1994). The complexity of propositional modal theories and the complexity of consistency of propositional modal theories. In: Nerode, A., Matiyasevich, Y.V. (eds) Logical Foundations of Computer Science. LFCS 1994. Lecture Notes in Computer Science, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58140-5_8
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