Abstract
We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.
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References
Andréka, H., Givant, S. and Németi, I., 1994a, “Decision problems for equational theories of relation algebras”. Manuscript, Mills College, Oakland, Jan 1992, extended abstract, Bull. Sec. of Logic (Univ. of Lodz), 23(2), 47–52.
Andréka, H., Givant, S. and Németi, J., 1994b, Undecidable equational theories of relation algebras, (submitted).
Andréka, H., Kurucz, á., Németi, I., Sain, I. and Simon, A., 1994c, “Exactly which logics touched by the dynamic trend are decidable?,” pp. 67–85 inProceedings of the 9th Amsterdam Colloquium, Amsterdam.
Andréka, H., Kurucz, á., Németi, I. and Sain, I., 1994d, “Applying algebraic logic; A general methodology”, to appear in ‘Algebraic Logic and the Methodology of Applying It’, Proc. Summer School in Budapest (H. Andréka, I. Németi and I. Sain., eds), 72 pp., A shortened version appeared as Applying Algebraic Logic to Logic in ‘Algebraic Methodology and Software Technology (AMAST'93)’, (M. Nivat, C. Rattray, T. Rus and G. Scollo, eds), pp. 7–28 in series Workshops in Computing, Springer-Verlag, 1994.
Andréka, H., Mikulás, Sz. and Németi, I., 1994e,You can decide differently: deciding relativized representable relation algebras with graded modalities, (preprint). Budapest: Math. Inst. Hung. Acad. Sci.
Blok, W. J. and Pigozzi, D. L., 1989, “Algebraizable logics,”Memoirs Amer. Math. Soc. 77/396, vi+78 p.
Burris, S. and Sankappanavar, H. P., 1981, “A course in universal algebra”, (Graduate Texts in Mathematics), New York: Springer-Verlag.
Davis, M., 1977, “Unsolvable Problems”, pp. 567–594 in Handbook of Mathematical Logic, J. Barwise, ed., Amsterdam: North Holland.
Gargov, Passy, Tinchev, 1987, “Modal environment for Boolean speculations”, pp. 253–263 inMath. Logic and Applications, New York: Plenum Press.
Gurevich, Y. and Lewis, H.R., 1984, “The word problem for cancellation semigroups with zero”,Journal of Symbolic Logic, 49(1), 184–191.
Henkin, L., Monk, J.D. and Tarski, A., 1985,Cylindric Algebras Part II, Amsterdam: North Holland.
Jipsen, P., 1992,Computer aided investigations of relation algebras Ph.D. Dissertation, Nashville, Tennessee: Vanderbilt University.
Jönsson, B. and Tarski, A., 1948, “Boolean algebras with operators”,Bulletin of the Amer. Math. Soc., 54, 79–80.
Kurucz, á., Németi, I., Sain, I. and Simon, A., 1993, “Undecidable varieties of semilattice-ordered semigroups, of Boolean Algebras with Operators, and logics extending Lambek Calculus”, Bulletin of the IGPL,1(1), 91–98.
Maddux, R., 1978, Topics in relation algebras Doctoral Dissertation, Berkeley.
Marx, M., Mikulás, Sz., Németi, I. and Simon, A., 1995, “Taming arrow logics”, Journal of Logic, Language, and Information,4 ???–??? (this volume)
Mikulás, Sz., Németi, I. and Sain, I. “Decidable logics of the dynamic trend and relativized relation algebras”, inLogic Colloquium '92, L. Csirmaz, D. M. Gabbay and M. de Rijke, eds, CSLI Publications, Stanford 1995.
Németi, I., 1985, Exactly which varieties of cylindric algebras are decidable?, (preprint), Budapest: Math. Inst.
Németi, I., 1987, “Decidability of Relation Algebras with weakened associativity”, pp. 340–344 in Proceedings of the AMS,100(2).
Németi, I., 1992, “Decidability of weakened versions of first order logic”, Proceedings of ‘Logic at Work’ Conference, Amsterdam, CCSOM of Univ. of Amsterdam
Németi, I., Sain, I. and Simon, A., 1995, “Undecidability of the equational theories of some classes of residuated Boolean algebras with operators”, Bulletin of the IGPL,3(1), 93–105.
Ohlbach, H. J., 1993, “Translation methods for non-classical logics: an overview”, Bulletin of the IGPL,1(1), 69–90.
Pigozzi, D., 1974, “The join of equational theories”, Colloq. Math.,30, 15–25.
Pixley, A.F., 1971, “The ternary discriminator function in universal algebra”, Math. Ann., 191, 167–180.
Rogers, H., 1967, Theory of recursive functions and effective computability, New York: McGraw-Hill.
Sain, I., 1984, How can the difference operator improve modal logic of programs?, Manuscript of a lecture in D. Scott's seminar, Pittsburgh.
Sain, I., 1988,Is sometimes ‘some other time’ better than ‘sometimes’ ? Math. Inst. Budapest, Preprint. No.50/1985, (A version of this appeared in Studia Logica,47(3), 279–301).
Segerberg, K., 1971,An essay in classical modal logic. Philosophical Society and Department of Philosophy, Univ. of Uppsala.
Simon, A. and Kurucz, á., 1993, Euclidean residuated monoids are hereditarily equationally undecidable, (preprint), Budapest: Math, Inst.
Van Benthem, J., 1994, “A Note on Dynamic Arrow Logic”, pp. 15–29 in Logic and Information Flow, J. van Eijck and A. Visser, eds, Cambridge, Mass.: MIT Press.
Van der Hoek, W., 1992,Modalities for reasoning about knowledge and quantities Ph.D. Dissertation, Amsterdam: Free University of Amsterdam.
Venema, Y., 1992,Many-Dimensional Modal Logic, Ph.D. Dissertation, Institute for Logic, Language and Computation, Univ. of Amsterdam.
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Research supported by the Hungarian National Foundation for Scientific Research grants no. T16448, F17452, T7255. Research of the first author is also supported by a grant of Logic Graduate School of Eötvös Loránd University Budapest
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Kurucz, á., Németi, I., Sain, I. et al. Decidable and undecidable logics with a binary modality. J Logic Lang Inf 4, 191–206 (1995). https://doi.org/10.1007/BF01049412
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DOI: https://doi.org/10.1007/BF01049412