Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-25T12:39:03.967Z Has data issue: false hasContentIssue false
  • English
  • Français

A quantitative analysis of modal logic

Published online by Cambridge University Press:  12 March 2014

Ronald Fagin
Ronald Fagin*
Affiliation:
IBM Almaden Research Center, San Jose, California 95120-6099,, E-mail:fagin@almaden.ibm.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We do a quantitative analysis of modal logic. For example, for each Kripke structure M, we study the least ordinal μ such that for each state of M, the beliefs up to level μ characterize the agents' beliefs (that is, there is only one way to extend these beliefs to higher levels). As another example, we show the equivalence of three conditions, that on the face of it look quite different, for what it means to say that the agents' beliefs have a countable description, or putting it another way, have a “countable amount of information”. The first condition says that the beliefs of the agents are those at a state of a countable Kripke structure. The second condition says that the beliefs of the agents can be described in an infinitary language, where conjunctions of arbitrary countable sets of formulas are allowed. The third condition says that countably many levels of belief are sufficient to capture all of the uncertainty of the agents (along with a technical condition). The fact that all of these conditions are equivalent shows the robustness of the concept of the agents' beliefs having a “countable description”.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 1 , March 1994 , pp. 209 - 252
Copyright
Copyright © Association for Symbolic Logic 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Bur80]Burgess, J. P., Decidability for branching time, Studia Logica, vol. 39 (1980), pp. 203–218.CrossRefGoogle Scholar
[Dic85]Dickmann, M. A., Larger infinitary languages, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, New York, 1985, pp. 317–363.Google Scholar
[EGS80]van Emde Boas, P., Gorenendijk, J., and Stokhof, M., The Conway paradox: its solution in an epistemic framework, Proceedings of the 3rd Amsterdam Montague symposium, Mathematics Centrum, Amsterdam, 1980, pp. 87–111; reprinted by Foris Publications, 1984.Google Scholar
[End72]Enderton, H. B., A mathematical introduction to logic, Academic Press, New York, 1972.Google Scholar
[FC83]Fattorosi-Barnaba, M. and de Caro, F., Graded modalities. I, Studia Logica, vol. 44 (1983), pp. 197–221.CrossRefGoogle Scholar
[FGHV92]Fagin, R., Geanakopolos, J., Halpern, J. Y., and Vardi, M. Y., The expressive power of the hierarchical approach to modeling knowledge and common knowledge, Theoretical aspects of reasoning about knowledge: Proceedings of the fourth conference (Moses, Y., editor), 1992, Morgan Kaufmann, San Mateo, California, pp. 229–244.Google Scholar
[FHMV94]Fagin, R., Halpern, J. Y., Moses, Y., and Vardi, M. Y., Reasoning about Knowledge, MIT Press, Cambridge, Massachusetts (to appear).CrossRefGoogle Scholar
[FHV91]Fagin, R., Halpern, J. Y., and Vardi, M. Y., A model-theoretic analysis of knowledge, Journal of the Association for Computing Machinery, vol. 91 (1991), pp. 384–428.Google Scholar
[Fin72]Fine, K., In so many possible worlds, Notre Dame Journal of Formal Logic, vol. 13 (1972), pp. 516–520.CrossRefGoogle Scholar
[Fin75]Fine, K., Normal forms in modal logic, Notre Dame Journal of Formal Logic, vol. 16 (1975), pp. Keg 229–237.CrossRefGoogle Scholar
[FV85]Fagin, R. and Vardi, M. Y., An internal semantics for modal logic, Proceedings of the 17th Association for Computing Machinery Symposium on Theory of Computing, Association for Computing Machinery, New York, 1985, pp. 305–315.Google Scholar
[Hal87]Halpern, J. Y., Using reasoning about knowledge to analyze distributed systems, Annual review of computer science (Traub, J. F., Grosz, B. J., Lampson, B. W., and Nilsson, N. J., editors), vol. 2, Annual Reviews Inc., Palo Alto, California, 1987, pp. 37–68.Google Scholar
[HC84]Hughes, G. E. and Cresswell, M. J., A companion to modal logic, Methuen, London, 1984.Google Scholar
[HM90]Halpern, J. Y. and Moses, Y., Knowledge and common knowledge in a distributed environment, Journal of the Association for Computing Machinery, vol. 37 (1990), pp. 549–587.CrossRefGoogle Scholar
[HM92]Halpern, J. Y. and Moses, Y., A guide to completeness and complexity for modal logics of knowledge and belief, Artificial Intelligence, vol. 53 (1992), pp. 319–379.CrossRefGoogle Scholar
[Hoe92]van der Hoek, W., On the semantics of graded modalities, Journal of Applied Non Classical Logics, vol. 2 (1992), pp. 81–123.Google Scholar
[Hop71]Hopcroft, J. E., An nlogn algorithm for minimizing the states in a finite-automaton, Theory of Machines-and Computations (Kohavi, Z., editor), Academic Press, New York, 1971, pp. 189 196.Google Scholar
[HS93]Heifetz, A. and Samet, D., Universal partition structures, Working paper 26/93, Israel Institute of Business Research, Tel Aviv University, Tel Aviv, 06 1993.Google Scholar
[Kri63]Kripke, S., A semantical analysis of modal logic I: normal modal propositional calculi, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 67–96.CrossRefGoogle Scholar
[Lit53]Littlewood, J. E., A mathematician's miscellany, Methuen and Company, London, 1953.Google Scholar
[Mos74]Moschovakis, Y. N., Elementary induction on abstract structures, North Holland, Amsterdam, 1974.Google Scholar
[MZ85]Mertens, J. F. and Zamir, S., Formulation of Bayesian analysis for games of incomplete information, International Journal of Game Theory, vol. 14 (1985), pp. 1–29.CrossRefGoogle Scholar
[Nad85]Nadel, M., sand admissible fragments, Model-theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, New York, 1985, pp. 271–316.Google Scholar
[Par81]Park, D., Concurrency and automata on infinite sequences, Proceedings of the 5th GI Conference on theoretical computer science (Deussen, P., editor), vol. 104, Lecture Notes in Computer Science, Springer-Verlag, Berlin and New York, 1981.Google Scholar
[Par92]Parikh, R., Finite and infinite dialogues, Logic from computer science (Moshovakis, Y. N., editor), Mathematical Sciences Research Institute Publications, no. 21, 1992, pp. 481–497.CrossRefGoogle Scholar
[Pra76]Pratt, V. R., Semantical considerations on Floyd-Hoare logic, Proceedings of the 17th IEEE symposium on foundations of computer science, 1976, pp. 109–121.Google Scholar
[Rog67]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[Seg71]Segerberg, K., An essay in classical modal logic, Philosophical Studies 13, Uppsala University, Uppsala, 1971.Google Scholar