Abstract
This work is divided in two papers (Part I and Part II). In Part I, we introduced the class of Rare-logics for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability were established by faithfully translating the Rare-logics into more standard modal logics (some of them contain the universal modal operator).
In Part II, we push forward the results from Part I. For Rare-logics with nominals (present at the level of formulae and at the level of modal expressions), we show that the constructions from Part I can be extended although it is technically more involved. We also characterize a class of standard modal logics for which the universal modal operator can be eliminated as far as satifiability is concerned. Although the previous results have a semantic flavour, we are also able to define proof systems for Rare-logics from existing proof systems for the corresponding standard modal logics. Last, but not least, decidability results for Rare-logics are established uniformly, in particular for information logics derived from rough set theory.
Since this paper is the continuation of Part I, we do not recall here the definitions of Part I although we refer to them.
Similar content being viewed by others
References
Aczel, P., ‘Schematic consequence’, in D. Gabbay, editor, What is a logical system?, pages 261–272, Clarendon Press, Oxford, 1994.
Archangelsky, D., and M. Taitslin, ‘A logic for data description’, in A. Meyer and M. Taitslin, editors, Symposium on Logic Foundations of Computer Science, Pereslavl-Zalessky, pages 2–11, Springer-Verlag, LNCS 363, 1989.
Balbiani, Ph., ‘A modal logic for data analysis’, in W. Penczek and A. Szalas, editors, MFCS'96, Kraków, pages 167–179, LNCS 1113, Springer-Verlag, 1996.
Balbiani, Ph., ‘Modal logics with relative accessibility relations’, in D. Gabbay and H. J. Ohlbach, editors, FAPR'96, Bonn, pages 29–41, LNAI, Springer-Verlag, 1996.
Balbiani, Ph., ‘Axiomatization of logics based on Kripke models with relative accessibility relations’, in [Oe97], pages 553–578, 1997
van Benthem, J., Modal Logic and Classical Logic, Bibliopolis, 1983.
Blackburn, P., Nominal Tense Logic and Other Sorted Intensional Frameworks, PhD thesis, University of Edinburgh, Edinburgh, 1990.
Blackburn, P., ‘Modal logic and attribute value structures’, in M. de Rijke, editor, Diamonds and Defaults, pages 19–66, Kluwer Academic Publishers, Series Studies in Pure and Applied Intensional Logic, Volume 229, 1993.
Blackburn, P., ‘Nominal tense logic’, Notre Dame Journal of Formal Logic 34(1) (1993), 56–83.
Blackburn, P., and J. Seligman, ‘Hybrid languages’, Journal of Logic, Language and Information 4 (1995), 251–272.
Danecki, R., ‘Nondeterministic propositional dynamic logic with intersection is decidable’, in A. Skowron, editor. Computation, Theory, 5th Symposium, Zaborów, Poland, pages 34–53, LNCS 208, Springer-Verlag, 1984.
Demri, S., ‘A class of decidable information logics’, Theoretical Computer Science 195(1) (1998), 33–60.
Demri, S., ‘A logic with relative knowledge operators’, Journal of Logic, Language and Information 8(2) (1999), 167–185.
Demri, S., and D. Gabbay, ‘On modal logics characterized by models with relative accessibility relations: Part I’, Studia Logica 65 (2000), 323–353.
Demri, S., and R. GorÉ, ‘Display calculi for logics with relative accessibility relations’, Journal of Logic, Language and Information 9 (2000), 213–236.
Demri, S., and B. Konikowska, ‘Relative similarity logics are decidable: reduction to FO2 with equality’, in JELIA'98, pages 279–293, LNAI 1489, Springer-Verlag, 1998.
FariÑas del Cerro, L., and E. Orlowska, ‘DAL--A logic for data analysis’, Theoretical Computer Science 36 (1985), 251–264.
Gargov, G., ‘Two completeness theorems in the logic for data analysis’, Technical Report 581, ICS, Polish Academy of Sciences, Warsaw, 1986.
Gargov, G., and V. Goranko, ‘Modal logic with names’, Journal of Philosophical Logic 22(6) (December 1993), 607–636.
de Giacomo, G., Decidability of Class-Based Knowledge Representation Formalisms, PhD thesis, Universita Degli Studi Di Roma ‘La Sapienza’, Dipartimento Di Informatica E Sistemistica, January 1995.
GrÄdel, E., Ph. Kolattis, and M. Vardi, ‘On the decision problem for two-variable first-order logic’, Bulletin of Symbolic Logic, 3(1) (1997), 53–69.
Goranko, V., and S. Passy, ‘Using the universal modality: gains and questions’, Journal of Logic and Computation. 2(1) (1992), 5–30.
Goldblatt, R., and S. Thomason, ‘Axiomatic classes in propositional modal logic’, in J. Crossley, editor, Algebra and Logic, pages 163–173, Springer-Verlag, Lecture Notes in Mathematics 450, 1975.
Harel, D., ‘Dynamic logic’, in D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, Volume II, pages 497–604, Reidel, Dordrecht, 1984.
Hintikka, J., Knowledge and Belief, Cornell University Press, 1962.
Halpern, J., and Y. Moses, ‘A guide to completeness and complexity for modal logics of knowledge and belief’, Artificial Intelligence, 54 (1992), 319–379.
Humberstone, L., ‘The logic of non-contingency’, Notre Dame Journal of Formal Logic 36(2) (1995), 214–229.
Konikowska, B., ‘A logic for reasoning about relative similarity’, Studia Logica, 58 (1997), 185–226.
Konikowska, B., ‘A logic for reasoning about similarity’, in [Oe97], pages 462–491, 1997.
Kozen, D., and J. Tiuryn, ‘Logics of programs’, in Handbook of Theoretical Computer Science pages 790–840, Elsevier Science Publishers, 1990.
Lemmon, E., Beginning Logic, Chapman and Hall, 1965.
Marx, M., ‘Complexity of modal logics of relations’, Technical report, ILLC, May 1997.
Mortimer, M., ‘On language with two variables’, Zeit. für Math. Logik and Grund. der Math. 21 (1975), 135–140.
Montgomery, H., and R. Routley, ‘Contingency and non-contingency bases for normal modal logics’, Logique et Analyse 9 (1966), 318–328.
NÉmeti, I., and H. AndrÉka, ‘General algebraic logic: a perspective on “what is logie”’, in D. Gabbay, editor, What is a logical system?, pages 393–443, Clarendon Press, Oxford, 1994.
OrŁowska, E., editor, Incomplete Information: Rough Set Analysis, Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, 1997.
OrŁowska, E., and Z. Pawlak, ‘Expressive power of knowledge representation systems’, Int. Journal Man-Machines Studies 20 (1984), 485–500.
OrŁowska, E., ‘Logic of indiscernibility relations’, in A. Skowron, editor, 5th Symposium on Computation Theory, Zaborów, Poland, pages 177–186, LNCS 208, Springer-Verlag, 1984.
OrŁowska, E., ‘Modal logics in the theory of information systems’, Zeitschr. für Math. Logik und Grundlagen d. Math. 30(1) (1984), 213–222.
OrŁowska, E., ‘Kripke models with relative accessibility and their applications to inferences from incomplete information’, in G. Mirkowska and H. Rasiowa, editors, Mathematical Problems in Computation Theory, pages 329–339, Banach Center Publications, Volume 21, PWN, Warsaw, 1988.
OrŁowska, E., ‘Logical aspects of learning concepts’, Journal of Approximate Reasoning 2 (1988), 349–364.
OrŁowska, E., ‘Logic for reasoning about knowledge’, Zeitschr. für Math. Logik und Grundlagen d. Math. 35 (1989), 559–568.
OrŁowska, E., ‘Reasoning with incomplete information: rough set based information logics’, in V. Alagar, S. Bergler, and F. Q. Dong, editors, Incompleteness and Uncertainty in Information Systems Workshop, pages 16–33, Springer-Verlag, October 1993.
OrŁowska, E., ‘Information algebras’, in AMAST'95, Montreal, pages 50–65, LNCS 639, Springer-Verlag, 1995.
Pawlak, Z., ‘Information systems theoretical foundations’, Information Systems, 6(3) (1981), 205–218.
Passy, S., and T. Tinchev, ‘PDL with data constants’, Information Processing Letters 20 (1985), 35–41.
Passy, S., and T. Tinchev, ‘An essay in combinatory dynamic logic’, Information and Computation 93 (1991), 263–332.
Segerberg, K., ‘A completeness theorem in the modal logic of programs’, in T. Traczyk, editor, Universal algebra and applications, pages 31–46. Banach Center Publications, Volume 9, PWN, Warsaw, 1982.
Spaan, E., Complexity of Modal Logics, PhD thesis, ILLC, Amsterdam University, March 1993.
Vakarelov, D., ‘Modal logics for knowledge representation systems’, Theoretical Computer Science 90 (1991), 433–456.
Venema, Y., Many-dimensional modal logic, PhD thesis, FWI, Amsterdam University, September 1991.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Demri, S., Gabbay, D. On Modal Logics Characterized by Models with Relative Accessibility Relations: Part II. Studia Logica 66, 349–384 (2000). https://doi.org/10.1023/A:1005260600511
Issue Date:
DOI: https://doi.org/10.1023/A:1005260600511