Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-08T09:18:38.115Z Has data issue: false hasContentIssue false

LOGICS AND ALGEBRAS FOR MULTIPLE PLAYERS

Published online by Cambridge University Press:  13 July 2010

LOES OLDE LOOHUIS*
Affiliation:
The City University of New York
YDE VENEMA*
Affiliation:
Universiteit van Amsterdam
*
*DEPARTMENT OF COMPUTER SCIENCE, THE GRADUATE CENTER, THE CITY UNIVERSITY OF NEW YORK, FIFTH AVENUE, 10016 NEW YORK, NY E-mail: [email protected]
INSTITUTE FOR LOGIC LANGUAGE AND COMPUTATION, UNIVERSITEIT VAN AMSTERDAM, SCIENCE PARK 904, 1098XH AMSTERDAM, THE NETHERLANDS E-mail: [email protected]

Abstract

We study a generalization of the standard syntax and game-theoretic semantics of logic, which is based on a duality between two players, to a multiplayer setting. We define propositional and modal languages of multiplayer formulas, and provide them with a semantics involving a multiplayer game. Our focus is on the notion of equivalence between two formulas, which is defined by saying that two formulas are equivalent if under each valuation, the set of players with a winning strategy is the same in the two respective associated games. We provide a derivation system which enumerates the pairs of equivalent formulas, both in the propositional case and in the modal case. Our approach is algebraic: We introduce multiplayer algebras as the analogue of Boolean algebras, and show, as the corresponding analog to Stone’s theorem, that these abstract multiplayer algebras can be represented as concrete ones which capture the game-theoretic semantics. For the modal case we prove a similar result. We also address the computational complexity of the problem whether two given multiplayer formulas are equivalent. In the propositional case, we show that this problem is co-NP-complete, whereas in the modal case, it is PSPACE-hard.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2010

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

BIBLIOGRAPHY

Abramsky, S. (2007). A compositional game semantics for multiagent logics of partial information. In van Benthem, J., Gabbay, D., and Löwe, B., editors. Interactive Logic. Selected Papers from the 7th Augustus de Morgan Workshop, London. Volume 1 of Texts in Games and Logic. Amsterdam: Amsterdam University Press, pp. 1147.Google Scholar
Alur, R., Henzinger, T., & Kupferman, O. (2002). Alternating-time temporal logic. Journal of the Association for Computing Machinery, 49, 672713.CrossRefGoogle Scholar
Belnap, N. (1977). A useful four-valued logic. In Dunn, J., and Epstein, G., editors. Modern Uses of Multiple-Valued Logics. Dordrecht: Reidel, pp. 837.Google Scholar
van Benthem, J. (2009). Logic in Games. Texts in Games and Logic. Amsterdam: Amsterdam University Press. To appear.Google Scholar
Bezhanishvili, G., Bezhanishvili, N., Gabelaia, D., & Kurz, A. (2010). Bitopological duality for distributive lattices and Heyting algebras. Mathematical Structures in Computer Science, 135.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Number 53 in Cambridge Tracts in Theoretical Computer Science. Cambridge: Cambridge University Press.Google Scholar
Bloniarz, P., Hunt, H., & Rosenkrantz, D. (1984). Algebraic structures with hard equivalence and minimization problems. Journal of the Association for Computing Machinery, 31(4), 879904.CrossRefGoogle Scholar
Ehrenfeucht, A. (1961). An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae, 49, 129141.CrossRefGoogle Scholar
Fraïssé, R. (1950). Sur une nouvelle classification des systèmes de relations. Comptes Rendus, 230, 10221024.Google Scholar
Grädel, E., Thomas, W., & Wilke, T., editors. (2002). Automata, Logics, and Infinite Games. Volume 2500 of Lecture Notes in Computer Science. Heidelberg: Springer.Google Scholar
Henkin, L. (1959). Some remarks on infinitely long formulas. In Infinitistic Methods. Warsaw: Pergamon Press, pp. 167183.Google Scholar
Hintikka, J. (1973). Logic, Language-Games, and Information: Kantian Themes in the Philosophy of Logic. Oxford: Clarendon Press.Google Scholar
Hodges, W. (1985). Building Models by Games. Cambridge, UK: Cambridge University Press.Google Scholar
Ladner, R. (1977). The computational complexity of provability in systems of modal logic. SIAM Journal on Computing, 6, 467480.CrossRefGoogle Scholar
Lorenzen, P. (1955). Einführung in die Operative Logik und Mathematik. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Mac Lane, S. (1998). Categories for the Working Mathematician (second edition). Number 5 in Graduate Texts in Mathematics. New York: Springer Verlag.Google Scholar
Osborne, M., & Rubinstein, A. (1994). A Course in Game Theory. Cambridge, MA: MIT Press.Google Scholar
Pauly, M. (2002). A modal logic for coalitional power in games. Journal of Logic and Computation, 12(1), 149166.CrossRefGoogle Scholar
Tulenheimo, T., & Venema, Y. (2008). Propositional logics for three. In Dégremont, C., Keiff, L., and Rückert, H., editors. Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman. London, UK: College Publications, pp. 399429.Google Scholar
Väänänen, J. (2007). Team logic. In van Benthem, J., Gabbay, D., and Löwe, B., editors. Interactive Logic. Selected Papers from the 7th Augustus de Morgan Workshop, London. Volume 1 of Texts in Games and Logic. Amsterdam: Amsterdam University Press, pp. 281302.Google Scholar
Venema, Y. (2006). Algebras and coalgebras. In Blackburn, P., van Benthem, J., and Wolter, F., editors. Handbook of Modal Logic. Elsevier, pp. 331426.Google Scholar