Product logic and probabilistic Ulam games. (English) Zbl 1117.03028
Summary: There is a well-known game semantics for Łukasiewicz logic, introduced by Daniele Mundici, namely the Rényi-Ulam game. Records in a Rény-Ulam game are coded by functions, which constitute an MV-algebra, and it is possible to prove a completeness theorem with respect to this semantics. In this paper we investigate some probabilistic variants of the Rényi-Ulam game, and we prove that some of them constitute a complete game semantics for product logic, whilst some other constitute a game semantics for a logic between \(\Pi\)MTL and product logic.
MSC:
| 03B50 | Many-valued logic |
References:
| [1] | Aglianó, P.; Montagna, F., Varieties of BL-algebras I: general properties, J. Pure Appl. Algebra, 181, 105-129 (2003) · Zbl 1034.06009 |
| [2] | W. Blok, D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society, Vols. 396, 77, American Mathematical Society, Providence, 1989.; W. Blok, D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society, Vols. 396, 77, American Mathematical Society, Providence, 1989. · Zbl 0664.03042 |
| [3] | Burris, S.; Sankappanavar, H. P., A course in universal algebra, Graduate Texts in Mathematics (1981), Springer: Springer Berlin · Zbl 0478.08001 |
| [4] | A. Ciabattoni, C. Fermüller, G. Metcalfe, Uniform rules and dialogue games for fuzzy logics, Proceedings of Logic for Programming and Automated Reasoning (LPAR’2004), Lecture Notes in Artificial Intelligence, Vol. 3452, 2004, pp. 496-510.; A. Ciabattoni, C. Fermüller, G. Metcalfe, Uniform rules and dialogue games for fuzzy logics, Proceedings of Logic for Programming and Automated Reasoning (LPAR’2004), Lecture Notes in Artificial Intelligence, Vol. 3452, 2004, pp. 496-510. · Zbl 1109.03019 |
| [5] | F. Cicalese, D. Mundici, Recent developments of feedback coding, and its relations with many-valued logic, in: J. van Benthem, R. Parikh, R. Ramanujam, A. Gupta (Eds.), Proceedings of the First Indian Conference on Logic and its Applications, (FICL 2005) Bombay, India, January 2005, to appear.; F. Cicalese, D. Mundici, Recent developments of feedback coding, and its relations with many-valued logic, in: J. van Benthem, R. Parikh, R. Ramanujam, A. Gupta (Eds.), Proceedings of the First Indian Conference on Logic and its Applications, (FICL 2005) Bombay, India, January 2005, to appear. · Zbl 1319.03040 |
| [6] | Cignoli, R.; Torrens, A., An algebraic analysis of product logic, Mult. Val. Logic, 5, 45-65 (2000) · Zbl 0962.03059 |
| [7] | Cox, D.; Little, J.; O’Shea, D., Ideals, Varieties and Algorithms (1996), Springer: Springer Berlin · Zbl 0861.13012 |
| [8] | I.M.A. Ferreirim, On varieties and quasi varieties of hoops and their reducts, Ph.D. Thesis, University of Illinois, Chicago, 1992.; I.M.A. Ferreirim, On varieties and quasi varieties of hoops and their reducts, Ph.D. Thesis, University of Illinois, Chicago, 1992. |
| [9] | P. Hájek, Metamathematics of Fuzzy Logic, Trends in Logic-Studia Logica Library no. 4, Kluwer, Dordercht/Boston/London, 1998.; P. Hájek, Metamathematics of Fuzzy Logic, Trends in Logic-Studia Logica Library no. 4, Kluwer, Dordercht/Boston/London, 1998. · Zbl 0937.03030 |
| [10] | Hájek, P., Observations on the monoidal \(t\)-norm logic, Fuzzy Sets and Systems, 132, 107-112 (2002) · Zbl 1012.03035 |
| [11] | Horčik, R., Standard completeness theorem for \(\Pi\) MTL, Arch. Math. Logic, 44, 4, 477-503 (2005) · Zbl 1071.03013 |
| [12] | D. Mundici, The logic of Ulam’s game with lies, Knowledge, Belief and Strategic Interaction, Cambridge Studies in Probability, Induction, and Decision Theory, 1992, pp. 275-284.; D. Mundici, The logic of Ulam’s game with lies, Knowledge, Belief and Strategic Interaction, Cambridge Studies in Probability, Induction, and Decision Theory, 1992, pp. 275-284. · Zbl 0831.90131 |
| [13] | Pelc, A., Searching with known error probability, Theoret. Comput. Sci., 63, 185-202 (1989) · Zbl 0664.68062 |
| [14] | Pelc, A., Searching games with errors—fifty years of coping with liars, Theoret. Comput. Sci., 270, 71-109 (2002) · Zbl 0984.68041 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
