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LOGICS AND ALGEBRAS FOR MULTIPLE PLAYERS

Published online by Cambridge University Press:  13 July 2010

LOES OLDE LOOHUIS  and
YDE VENEMA
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: l.oldeloohuis@gmail.com
†INSTITUTE FOR LOGIC LANGUAGE AND COMPUTATION, UNIVERSITEIT VAN AMSTERDAM, SCIENCE PARK 904, 1098XH AMSTERDAM, THE NETHERLANDS E-mail: y.venema@uva.nl
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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
Information
The Review of Symbolic Logic , Volume 3 , Issue 3 , September 2010 , pp. 485 - 519
Copyright
Copyright © Association for Symbolic Logic 2010

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