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Common knowledge logic and game logic

Published online by Cambridge University Press:  12 March 2014

Mamoru Kaneko
Mamoru Kaneko*
Affiliation:
Institute of Policy and Planning Sciences, University of Tsukuba, Ibaraki 305, Japan E-mail: kaneko@shako.sk.tsukuba.ac.jp
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Abstract

We show the faithful embedding of common knowledge logic CKL into game logic GL, that is, CKL is embedded into GL and GL is a conservative extension of the fragment obtained by this embedding. Then many results in GL are available in CKL, and vice versa. For example, an epistemic consideration of Nash equilibrium for a game with pure strategies in GL is carried over to CKL. Another important application is to obtain a Gentzen-style sequent calculus formulation of CKL and its cut-elimination. The faithful embedding theorem is proved for the KD4–type propositional CKL and GL, but it holds for some variants of them.

Keywords

Game LogicCommon Knowledge LogicFixed-Point and Iterative Definitions of Common KnowledgeNash Equilibrium
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 2 , June 1999 , pp. 685 - 700
Copyright
Copyright © Association for Symbolic Logic 1999

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References

REFERENCES

[1] Barwise, J., Three views of common knowledge, Proceedings of the second conference on theoretical aspects of reasoning about knowledge (Los Altos) (Verdi, M.Y., editor), Morgan Kaufmann Publisher Inc., 1988, pp. 365–379.Google Scholar
[2] Halpern, J. H. and Moses, Y., A guide to completeness and complexity for modal logics of knowledge and beliefs, Artificial Intelligence, vol. 54 (1992), pp. 319–379.CrossRefGoogle Scholar
[3] Kaneko, M., Decision making in partially interactive games I, IPPS, University of Tsukuba, vol. 742 (1997), II, forthcoming.Google Scholar
[4] Kaneko, M., Depth of knowledge and the Barcan inferences in game logic, forthcoming, 1997.Google Scholar
[5] Kaneko, M., Epistemic considerations of decision making in games, to appear in Mathematical Social Sciences, 1997.Google Scholar
[6] Kaneko, M., Mere and specific knowledge of the existence of a Nash equilibrium, IPPS, University of Tsukuba, vol. 741 (1997).Google Scholar
[7] Kaneko, M. and Nagashima, T., Game logic and its applications I, Studia Logica, vol. 57 (1996), pp. 325–354.Google Scholar
[8] Kaneko, M. and Nagashima, T., Axiomatic indefinability of common knowledge infinitary logics, Epistemic logic and the theory of games and decision (Bacharach, M. et al., editors), Kluwer Academic Press, 1997, pp. 69–93.CrossRefGoogle Scholar
[9] Kaneko, M. and Nagashima, T., Game logic and its applications II, Studia Logica, vol. 58 (1997), pp. 273–303.CrossRefGoogle Scholar
[10] Karp, C., Languages with expressions of infinite length, North–Holland, 1964.Google Scholar
[11] Lismont, L. and Mongin, P., On the logic of common belief and common knowledge, Theory and Decision, vol. 37 (1994), pp. 75–106.CrossRefGoogle Scholar
[12] Lismont, L. and Mongin, P., Belief closure: A semantics of common knowledge for modal propositional logic, Mathematical Social Sciences, vol. 30 (1995), pp. 127–153.Google Scholar
[13] Nash, J. F., Noncooperative games, Annals of Mathematics, vol. 54 (1951), pp. 286–295.CrossRefGoogle Scholar
[14] Ohnishi, M. and Matsumoto, M., Gentzen method in modal calculi, Osaka Journal of Mathematics, vol. 9 (1957), pp. 112–130.Google Scholar
[15] Takano, M., Subformula property as a substitute for cut-elimination in modal propositional logics, Mathematica Japonica, vol. 37 (1992), pp. 1129–1145.Google Scholar