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Games over Formulas in Lukasiewicz Logic
by
Tomas Kroupa
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodarenskou vezi 4, 182 08 Prague, Czech Republic
The goal of this contribution is to study a certain cooperative game [1] modelled by formulas in Lukasiewicz logic. We consider a finite set F of formulas in k propositional variables, a mapping m:F→ [0, 1], and truth valuations V over formulas in k propositional variables. The intended meaning is that a player is represented by a truth valuation V determining his degree of conformity V(j) with the statement given by a formula j ∈ F, while the number m(j) determines the total worth of "respecting" j. Identifying each formula j ∈ F with the corresponding k-variable McNaughton function fj, the functions fj can be naturally viewed as coalitions. A solution to this game is any distribution of worth among coalitions that no coalition can contest. Precisely, any state s [2] on the MV-algebra of k-variable McNaughton functions is a solution to the game, when s(fj) ≥ m(j) for each j ∈ F. The set of all solutions is compact and convex. Its nonemptiness can be tested with the coherence criterion [3] for states on MV-algebras.
Acknowledgement. The work was supported by grant No. B100300502 of GA AV CR and grant No. 1M0572 of MSMT CR.
[1] D. Butnariu and E. P. Klement, Triangular Norm Based Measures and Games with Fuzzy Coalitions, Kluwer (1993).
[2] D. Mundici, Averaging the truth value in Lukasiewicz sentential logic, Studia Logica 55 (1995) 113-127.
[3] D. Mundici, Bookmaking over infinite-valued events, Internat. J. Approx. Reason. 43 (2006) 223-240.
Date received: May 14, 2007
Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caug-52.