Abstract
We show that every proper normal extension of the bi-modal system S5 2 has the poly-size model property. In fact, to every proper normal extension L of S5 2 corresponds a natural number b(L) - the bound of L. For every L, there exists a polynomial P(·) of degree b(L) + 1 such that every L-consistent formula ϕ is satisfiable on an L-frame whose universe is bounded by P(|ϕ|), where |ϕ| denotes the number of subformulas of ϕ. It is shown that this bound is optimal.
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References
Bezhanishvili, N., ‘Varieties of two-dimensional cylindric algebras. Part I: diagonal-free case’, Algebra Universalis, 48: 11-42, 2002.
Gabbay, D. and Shehtman, V., ‘Products of modal logics, Part 1’, Logic Journal of the IGPL, 6(1):73-146, 1998.
GÖdel, K., ‘Zum Entscheidungsproblem des logischen Funktionenkalküls’, Monatshefte für Mathematik und Physik, 40:433-443, 1933.
Goldfarb, W., ‘The unsolvability of the Gödel class with identity’, Journal of Symbolic Logic, 49:1237-1252, 1984.
GrÄdel, E., Kolaitis, P., and Vardi, M., ‘On the decision problem for two—variable first order logic’, Bulletin of symbolic logic, 3(1):53-69, 1997.
Gurevich, Y., ‘The decision problem for standard classes’, Journal of Symbolic Logic, 41:460-464, 1976.
Henkin, L., Monk, J. D., and Tarski, A., Cylindric Algebras, Parts I & II, North-Holland, 1971. & 1985.
Kurucz, A., ‘S5 × S5 × S5 lacks the finite model property’, in F. Wolter, H. Wansing, M. de Rijke, and M. Zakharyaschev (eds), Advances in modal logic, volume III, CSLI, 2001. To appear.
Ladner, R., ‘The computational complexity of provability in systems of modal propositional logic’, SIAM journal of computing, 6(3):467-480, 1977.
Maddux, R., ‘The equational theory of CA3 is undecidable’, Journal of Symbolic Logic, 45:311-317, 1980.
Marx, M., ‘Complexity of products of modal logics’, Journal of Logic and Computation, 9(2):197-214, 1999.
Mortimer, M., ‘On languages with two variables’, Zeitchrift für mathematische Logik und Grundlagen der Mathematik, 21:135-140, 1975.
Nemeti, I., ‘Algebraization of quantifier logics, an introductory overview’, Studia Logica, 3/4:485-571, 1991.
Raynolds, M. and Zakharyaschev, M., ‘On the products of linear modal logics’, Journal of Logic and Computation, 11:1-24, 2001.
Scott, D., ‘A decision method for validity of sentences in two variables’, Journal of Symbolic Logic, 27:477, 1962.
Scroggs, S. G., ‘Extensions of the Lewis system S5’, Journal of Symbolic Logic, 16:111-120, 1951.
Segerberg, K., ‘Two-dimensional modal logic’, Journal of Philosophical logic, 2:77-96, 1973.
Wajsberg, M., ‘Ein erweiterter Klassenkalkül’, Monatshefte für Mathematik und Physik, 40:113-126, 1933.
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Bezhanishvili, N., Marx, M. All Proper Normal Extensions of S5-square have the Polynomial Size Model Property. Studia Logica 73, 367–382 (2003). https://doi.org/10.1023/A:1023383112908
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DOI: https://doi.org/10.1023/A:1023383112908