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The Rényi-Ulam games and many-valued logics. (English) Zbl 1396.03042
Summary: In this paper, we introduce complete game semantics for Product, Gödel, BL and SBL logics (game semantics for Łukasiewicz logic are well known). For each of these logics we introduce a variant of the Rényi-Ulam game whose states are equipped in a natural way with an algebraic structure. Moreover we prove that each logic is complete with respect to the algebras of the states for the corresponding game.

MSC:
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
06D35 MV-algebras
91A80 Applications of game theory
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[1] Aglianò, P.; Montagna, F., Varieties of BL-algebras I: general properties, J. Pure Appl. Algebra, 181, 2-3, 105-129, (2003) · Zbl 1034.06009
[2] Berlekamp, E. R., Block coding for the binary symmetric channel with noiseless, delayless feedback, (Proceedings of a Symposium Conducted by the Mathematics Research Center, United States Army at the University of Wisconsin, Error Correcting Codes, (1968)), 61-88
[3] Blok, W. J.; Ferreirim, I. M.A., On the structure of hoops, Algebra Univers., 43, 2-3, 233-257, (2000) · Zbl 1012.06016
[4] Burris, S.; Sankappanavar, H. P., A course in universal algebra, Graduate Texts in Mathematics, vol. 78, (1981), Springer-Verlag New York · Zbl 0478.08001
[5] Busaniche, M.; Montagna, F., Hájek BL-logic and BL-algebras, (Handbook on Mathematical Fuzzy Logic, Volume I, Logic, Mathematical Logic and Foundations, vols. 37-38, (2011), College Publications London), 355-447 · Zbl 1284.03176
[6] Chang, C. C., A new proof of the completeness of the łukasiewicz axioms, Trans. Am. Math. Soc., 93, 74-90, (1959) · Zbl 0093.01104
[7] Ciabattoni, A.; Fermüller, C. G.; Metcalfe, G., Uniform rules and dialogue games for fuzzy logics, (Proceedings of Logic for Programming and Logic for Programming and Automated Reasoning, LPAR’2004, Lecture Notes in Artificial Intelligence, vol. 3452, (2005)), 496-510 · Zbl 1109.03019
[8] Cicalese, F., Fault-tolerant search algorithms, Monographs in Theoretical Computer Science - An EATCS Series, vol. 15, (2013), Springer-Verlag · Zbl 1295.68006
[9] Cicalese, F.; Mundici, D., Optimal coding with one asymmetric error: below the sphere packing bound, (Proceedings of 6th Annual International Conference on Computing and Combinatorics, COCOON’2000, Lecture Notes in Computer Science, vol. 1858, (2000)), 159-169 · Zbl 0996.94054
[10] Cicalese, F.; Mundici, D., Recent developments of feedback coding and its relations with many-valued logic, (Proof, Computation and Agency, Synthese Library, vol. 352, (2011)), 115-131, Part 3 · Zbl 1319.03040
[11] Cignoli, R.; D’Ottaviano, I. M.L.; Mundici, D., Algebraic foundations of many-valued reasoning, Trends in Logic - Studia Logica Library, vol. 7, (2000), Kluwer Academic Publishers Dordrecht · Zbl 0937.06009
[12] Cintula, P.; Hájek, P.; Noguera, C., Handbook of mathematical fuzzy logic, Logic, Mathematical Logic and Foundations, vols. 37-38, (2011), College Publications London, (in 2 volumes)
[13] Cintula, P.; Majer, O., Towards evaluation games for fuzzy logics, (Majer, O.; Pietarinen, A. V.; Tulenheimo, T., Games: Unifying Logic, Language, and Philosophy, (2009), Springer), 117-138 · Zbl 1167.03021
[14] Gottwald, S., Axiomatizations of t-norm based logics - a survey, Soft Comput., 4, 63-67, (2000)
[15] Hájek, P., Metamathematics of fuzzy logic, Trends in Logic - Studia Logica Library, vol. 4, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0937.03030
[16] Hintikka, J., Logic, language-games and information. Kantian themes in the philosophy of logic, (1973), Clarendon Press Oxford · Zbl 0253.02005
[17] Lorenzen, P., Ein dialogisches konstruktivitt äskriterium, (Proceedings of a Symposium on Foundations of Mathematics, Infinitistic Methods, Warsaw, 1959, (1961)), 193-200
[18] MacWilliams, F. J.; Sloane, N. J.A., The theory of error-correcting codes, (1977), North-Holland Mathematical Library Amsterdam · Zbl 0369.94008
[19] Marini, C.; Montagna, F.; Simi, G., Product logic and probabilistic Ulam games, Fuzzy Sets Syst., 158, 6, 639-651, (2007) · Zbl 1117.03028
[20] Mundici, D., The logic of Ulam’s game with lies, (Bicchieri, C.; Dalla Chiara, M. L., Knowledge, Belief and Strategic Interaction, Cambridge Studies in Probability, Induction, and Decision Theory, (1992), Cambridge University Press Cambridge), 275-284 · Zbl 0831.90131
[21] Mundici, D., Ulam games, łukasiewicz logic, and AF C*-algebras, Fundam. Inform., 18, 151-161, (1993) · Zbl 0780.03030
[22] Pelc, A., Searching with known error probability, Theor. Comput. Sci., 63, 2, 185-202, (1989) · Zbl 0664.68062
[23] Pelc, A., Searching games with errors - fifty years of coping with liars, Theor. Comput. Sci., 270, 1-2, 71-109, (2002) · Zbl 0984.68041
[24] Rényi, A., A diary on information theory, (1984), J. Wiley and Sons New York, Translation of Napló az információméletröl, Gondolat, Budapest, 1976 · Zbl 0539.94001
[25] Ulam, S. M., Adventures of a Mathematician, (1976), Charles Scribner’s Sons New York · Zbl 0352.01009
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