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Fuzzy logics based on \([0,1)\)-continuous uninorms. (English) Zbl 1128.03015
Preceding studies of fuzzy logics were based on triangular norms. The authors turn attention to fuzzy logics where uninorms play the same role. Uninorms are common generalizations of triangular norms and conorms; the neutral element can be an arbitrary number from the interval \([0,1]\).
The only continuous residuated uninorms are triangular norms; thus the authors assume continuity only on the halfopen interval \([0,1)\) and require so-called conjunctive uninorms, i.e., attaining \(0\) at \((0,1)\). This class includes a lot of interesting examples, including the cross ratio uninorm.
Following the work of Hájek, the authors first introduce an analogue of the basic (fuzzy) logic, a logic where the conjunction (resp. implication) is evaluated by the above uninorms (resp. their residua). Then they study extensions whose semantics is based on specific uninorms. Completeness theorems are proved and the results are put into the context of previous fuzzy logics. A Gentzen-style hypersequent calculus is provided and used to establish co-NP completeness results.

MSC:
03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
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[1] Avron A. (1987). A constructive analysis of RM. J. Symb. Log. 52(4): 939–951 · Zbl 0639.03017
[2] Avron A. (1991). Hypersequents, logical consequence and intermediate logics for concurrency. Ann. Math. Artif. Intell. 4(3–4): 225–248 · Zbl 0865.03042
[3] Baaz M., Hájek P., Montagna F. and Veith H. (2001). Complexity of t-tautologies. Ann. Pure Appl. Log. 113(1): 3–11 · Zbl 1006.03022
[4] Cignoli R. and Torrens A. (2005). Standard completeness of Hájek basic logic and decompositions of BL-chains. Soft Comput. 9(12): 862–868 · Zbl 1094.03013
[5] Cintula, P.: Weakly implicative (fuzzy) logics I: basic properties. Arch. Math. Log. 45(6) (2007) · Zbl 1101.03015
[6] De Baets B. and Fodor J. (1999). Residual operators of uninorms. Soft Comput. 3: 89–100
[7] De Baets B. and Fodor J. (1999). Van Melle’s combining function in MYCIN is a representable uninorm: an alternative proof. Fuzzy Sets Syst. 104: 133–136 · Zbl 0928.03060
[8] Fodor J., Yager R.R. and Rybalov A. (1997). Structure of uni-norms. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 5: 411–427 · Zbl 1232.03015
[9] Gabbay D., Metcalfe G. and Olivetti N. (2004). Hypersequents and fuzzy logic. Revista de la Real Academia de Ciencias (RACSAM) 98(1): 113–126 · Zbl 1070.03012
[10] Girard J. (1987). Linear logic. Theoret. Comput. Sci. 50: 1–102 · Zbl 0625.03037
[11] Gottwald, S.: A treatise on many-valued logics. In: Studies in Logic and Computation, vol. 9. Research Studies Press, Baldock (2000) · Zbl 1048.03002
[12] Gurevich Y.S. and Kokorin A.I. (1963). Universal equivalence of ordered abelian groups (in Russian). Algebra i logika 2: 37–39
[13] Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (1998) · Zbl 0937.03030
[14] Hájek P. and Valdés J. (1994). An analysis of MYCIN-like expert systems. Mathw. Soft Comput. 1: 45–68
[15] Hart J., Rafter L. and Tsinakis C. (2002). The structure of commutative residuated lattices. Int. J. Algebra Comput. 12(4): 509–524 · Zbl 1011.06006
[16] Hu S. and Li Z. (2001). The structure of continuous uni-norms. Fuzzy Sets Syst. 124: 43–52 · Zbl 0989.03058
[17] Metcalfe, G., Montagna, F.: Substructural fuzzy logics. J. Symb. Log. (to appear)(2007) · Zbl 1139.03017
[18] Metcalfe G., Olivetti N. and Gabbay D. (2004). Analytic proof calculi for product logics. Arch. Math. Log. 43(7): 859–889 · Zbl 1066.03036
[19] Metcalfe G., Olivetti N. and Gabbay D. (2005). Sequent and hypersequent calculi for abelian and Łukasiewicz logics. ACM Trans. Comput. Log. 6(3): 578–613
[20] Silvert W. (1979). Symmetric summation: a class of operations on fuzzy sets. IEEE Trans. Man. Cybern. 9: 657–659 · Zbl 0424.04003
[21] Yager R.R. (2001). Uninorms in fuzzy systems modelling. Fuzzy Sets Syst. 122: 167–175 · Zbl 0978.93007
[22] Yager R.R. (2002). Defending against strategic manipulation in uninorm-based multi-agent decision making. Eur. J. Oper. Res. 141(1): 217–232 · Zbl 0998.90046
[23] Yager R.R. and Rybalov A. (1996). Uninorm aggregation operators. Fuzzy Sets Syst. 80: 111–120 · Zbl 0871.04007
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