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MV-algebras, multiple bets and subjective states. (English) Zbl 0958.06007
The game of twenty questions with lies, first considered by Ulam and Rényi, is an important chapter of the theory of error correcting codes with feedback. See the survey paper by F. Cicalese, U. Vaccaro and the present reviewer [“Rota-Metropolis cubic logic and Ulam-Rényi games”, in: H. Crapo et al. (eds.), Algebraic combinatorics and computer sciene: a tribute to Gian Carlo Rota, Springer-Verlag, to appear]. The Ulam-Rényi game also yields a natural semantics for Łukasiewicz many-valued logic, and their algebras, Chang’s MV-algebras. See the monograph: R. Cignoli, I. M. L. D’Ottaviano, and D. Mundici, Algebraic foundations of many-valued reasoning [Trends in Logic, Studia Logica Library, Vol. 7, Dordrecht: Kluwer Academic Publishers (2000; Zbl 0937.06009)]. Pursuing the approach of her own previous paper on betting in the Ulam-Rényi game [B. Gerla, “Conditioning a state by a Łukasiewicz event: a probabilistic approach to Ulam games”, Theor. Comput. Sci. 230, 149-166 (2000; Zbl 0949.06005)], in this paper the author interprets MV-algebraic operations as acting on bets. Finitely additive states on MV-algebras (as considered by the reviewer in “Averaging the truth-value in Łukasiewicz logic” [Stud. Log. 55, No. 1, 113-127 (1995; Zbl 0836.03016)] are then interpeted in terms of De Finetti’s “fair betting systems”.

MSC:
06D35 MV-algebras
91A05 2-person games
60B99 Probability theory on algebraic and topological structures
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