A System of Dynamic Modal Logic
 Maarten de Rijke
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
In many logics dealing with information one needs to make statements not only about cognitive states, but also about transitions between them. In this paper we analyze a dynamic modal logic that has been designed with this purpose in mind. On top of an abstract information ordering on states it has instructions to move forward or backward along this ordering, to states where a certain assertion holds or fails, while it also allows combinations of such instructions by means of operations from relation algebra. In addition, the logic has devices for expressing whether in a given state a certain instruction can be carried out, and whether that state can be arrived at by carrying out a certain instruction.
This paper deals mainly with technical aspects of our dynamic modal logic. It gives an exact description of the expressive power of this language; it also contains results on decidability for the language with ‘arbitrary’ structures and for the special case with a restricted class of admissible structures. In addition, a complete axiomatization is given. The paper concludes with a remark about the modal algebras appropriate for our dynamic modal logic, and some questions for further work.
The paper only contains some sketchy examples showing how the logic can be used to capture situations of dynamic interest, far more detailed applications are given in a companion to this paper (De Rijke [33]).
 Alchourrón, C., Gärdenfors, P. and Makinson, D. (1985): On the logic of theory change. Journal of Symbolic Logic 50: 510 530.
 Andréka, H., Kurucz, Á., Németi, I., Sain, I. and Simon, A. (1996): Exactly which logics touched by the dynamic trend are decidable? In L. Pólós, M. Masuch, M. Marx (eds.), Arrow Logics and MultiModal Logics, Studies in Logic, Language and Information, CSLI Publications.
 Van Benthem, J. (1983): Modal Logic and Classical Logic. Bibliopolis, Naples.
 Van Benthem, J. (1989): Modal logic as a theory of information. Technical Report LP8905, ILLC, University of Amsterdam.
 Van Benthem, J. (1991): Language in Action. NortHolland, Amsterdam.
 Van Benthem, J. (1900): Logic and the flow of information. In D. Prawitz, B. Skyrms and D. Westerståhl (eds.), Proc. 9th ILMPS, NorthHolland, Amsterdam.
 Brink, C. (1981): Boolean modules. Journal of Algebra 71: 291 313.
 Brink, C., Britz, K. and Schmidt, R. (1994): Peirce algebras. Formal Aspects of Computing 6: 339 358.
 Danecki, R. (1985): Nondeterministic propositional dynamic logic with intersection is decidable. In LNCS 208, Springer, New York, pp. 34 53.
 Van Eijck, J. and de Vries, F.J. (1995): Reasoning about update logic. Journal of Philosophical Logic 24: 19 46.
 Finger, M. and Gabbay, D. M. (1992): Adding a temporal dimension to a logic system. Journal of Logic, Language and Information 1: 203 233.
 Fuhrmann, A. (1990): On the modal logic of theory change. In A. Fuhrmann and M. Morreau (eds.), LNAI 465, pp. 259 281.
 Gabbay, D. M. and Hodkinson, I. M. (1991): An axiomatization of the temporal logic with Since and Until over the real numbers. Journal of Logic and Computation 1: 229 259.
 Gabbay, D. M., Hodkinson, I. and Reynolds, M. (1994): Temporal Logic: Mathematical Foundations and Computational Aspects. Oxford University Press, Oxford.
 Gärdenfors, P. (1988): Knowledge in Flux. The MIT Press, Cambridge, MA.
 Gargov, G. and Passy, S. (1990): A note on Boolean modal logic. In Petkov, P. P. (ed.), Mathematical Logic. Proccedings of the 1988 Heyting Summerschool, Plenum Press, New York, 311 321.
 Groenendijk, J. and Stokhof, M. (1991): Dynamic predicate logic. Linguistics and Philosophy 14: 39 100.
 Gurevich, Y. and Shelah, S. (1985): The decision problem for branching time logic. Journal of Symbolic Logic 50: 668 681.
 Harel, D. (1983): Recurring dominoes: making the highly undecidable highly understandable. In LNCS 158 (Proc. of the Conference on Foundations of Computing Theory), pp. 177 194. SpringerVerlag, Berlin.
 Harel, D. (1984): Dynamic Logic. In Gabbay, D. M. and Guenthner, F. (eds.), Handbook of Philosophical Logic, vol. 2, Reidel, Dordrecht, pp. 497 604.
 Jaspars, J. (1994): Calculi for Constructive Communicaton. Ph.D. thesis. ITK, Tilburg, and ILLC, University of Amsterdam.
 Jaspars, J. and Krahmer, E. (1996): A programme of modal unification of dynamic theories. In Proceedings of the 10th Amsterdam Colloquium, ILLC, University of Amsterdam.
 Kamp, H. (1968): Tense Logic and the Theory of Linear Order. Ph.D. thesis, UCLA.
 Katsuno, H. and Mendelzon, A. O. (1991): On the difference between updating a knowledge base and revising it. In Allen, J. A., Fikes, R. and Sandewall, E. (eds.), Princ. of Knowledge Representation and Reasoning: Proc. 2nd Intern. Conf., pp. 387 394. Morgan Kaufman.
 Katsuno, H. and Mendelzon, A. O. (1992): Propositional knowledge base revision and minimal change. Artificial Intelligence 52: 263 294.
 Kozen, D. (1981): On the duality of dynamic algebras and Kripke models. In Engeler, E. (ed.), Logic of Programs 1981, LNCS 125, pp. 1 11. SpringerVerlag, Berlin.
 Marx, M. (1995): Algebraic Relativization and Arrow Logic. Ph.D. thesis, ILLC, University of Amsterdam.
 Passy, S. and Tinchev, T. (1991): An essay in combinatory dynamic logic. Information and Computation 93: 263 332.
 Pratt, V. (1990): Action logic and pure induction. In van Eijck, J. (ed.), JELIA90, pp. 97 120, SpringerVerlag, Berlin.
 Pratt, V. (1990): Dynamic algebras as a wellbehaved fragment of relation algebras. In Bergman, C. H., Maddux, R. D. and Pigozzi, D. L. (eds.), Algebraic Logic and Universal Algebra in Computer Science, LNCS 425, pp. 77 110.
 Rabin, M. O. (1969): Decidability of second order theories and automata on infinite trees. Transactions of the American Mathematical Society 141: 1 35.
 De Rijke, M. (1992): The modal logic of inequality. Journal of Symbolic Logic 57: 566 584.
 De Rijke, M. (1994): Meeting some neighbours. In van Eijck, J. and Visser, A. (eds.), Logic and Information Flow, MIT Press, Cambridge, Mass., pp. 170 195.
 De Rijke, M. (1995): The logic of Peirce algebras. Journal of Logic, Language and Information 4: 227 250.
 De Rijke, M. (1995): Modal model theory. Report CSR9517, CWI, Amsterdam. To appear in Annals of Pure and Applied Logic.
 Veltman, F. (1996): Defaults in update semantics. Journal of Philosophical Logic 25: 221 261.
 Venema, Y. (1991): ManyDimensional Modal Logic. Ph.D. Thesis, ILLC, University of Amsterdam.
 Venema, Y. (1993): Derivation rules as antiaxioms in modal logic. Journal of Symbolic Logic 58: 1003 1034.
 Title
 A System of Dynamic Modal Logic
 Journal

Journal of Philosophical Logic
Volume 27, Issue 2 , pp 109142
 Cover Date
 19980401
 DOI
 10.1023/A:1004295308014
 Print ISSN
 00223611
 Online ISSN
 15730433
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Authors

 Maarten de Rijke ^{(1)}
 Author Affiliations

 1. Department of Computer Science, University of Warwick, Coventry, CV4 7AL, U.K. (email