On interval fuzzy S-implications

This paper presents an analysis of interval-valued S-implications and interval-valued automorphisms, showing a way to obtain an interval-valued S-implication from two S-implications, such that the resulting interval-valued S-implication is said to be obtainable. Some consequences of that are: (1) the resulting interval-valued S-implication satisfies the correctness property, and (2) some important properties of usual S-implications are preserved by such interval representations. A relation between S-implications and interval-valued S-implications is outlined, showing that the action of an interval-valued automorphism on an interval-valued S-implication produces another interval-valued S-implication.

[1]  Benjamín R. C. Bedregal,et al.  Interval Valued Versions of T-Conorms, Fuzzy Negations and Fuzzy Implications , 2006, 2006 IEEE International Conference on Fuzzy Systems.

[2]  Radko Mesiar,et al.  A characterization of fuzzy implications generated by generalized quantifiers , 2008, Fuzzy Sets Syst..

[3]  Jerry M. Mendel,et al.  Uncertainty measures for interval type-2 fuzzy sets , 2007, Inf. Sci..

[4]  J. Fodor.  Contrapositive symmetry of fuzzy implications , 1995 .

[5]  Chris Cornelis,et al.  ON THE PROPERTIES OF A GENERALIZED CLASS OF T-NORMS IN INTERVAL-VALUED FUZZY LOGICS , 2006 .

[6]  P. Walley.  Measures of Uncertainty in Expert Systems , 1996, Artificial Intelligence.

[7]  E. Walter,et al.  Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics , 2001 .

[8]  Patricia Melin,et al.  A hybrid approach for image recognition combining type-2 fuzzy logic, modular neural networks and the Sugeno integral , 2009, Inf. Sci..

[9]  Tien-Chin Wang,et al.  Using the fuzzy multi-criteria decision making approach for measuring the possibility of successful knowledge management , 2009, Inf. Sci..

[10]  Lotfi A. Zadeh,et al.  Fuzzy Logic , 2009, Encyclopedia of Complexity and Systems Science.

[11]  Sankar K. Pal,et al.  Fuzzy sets in pattern recognition and machine intelligence , 2005, Fuzzy Sets Syst..

[12]  Humberto Bustince,et al.  Automorphisms, negations and implication operators , 2003, Fuzzy Sets Syst..

[13]  Benjamín R. C. Bedregal,et al.  Formal Aspects of Correctness and Optimality of Interval Computations , 2006, Formal Aspects of Computing.

[14]  E. Walker,et al.  Algebraic Aspects of Fuzzy Sets and Fuzzy Logic , 1998 .

[15]  Michal Baczynski,et al.  (S, N)- and R-implications: A state-of-the-art survey , 2008, Fuzzy Sets Syst..

[16]  Mark A. Stadtherr,et al.  Reliable computation of phase stability and equilibrium using interval methods , 2007 .

[17]  Didier Dubois,et al.  Interval-valued Fuzzy Sets, Possibility Theory and Imprecise Probability , 2005, EUSFLAT Conf..

[18]  Daowu Pei,et al.  Unified full implication algorithms of fuzzy reasoning , 2008, Inf. Sci..

[19]  M. H. van Emden,et al.  Interval arithmetic: From principles to implementation , 2001, JACM.

[20]  Ivor Grattan-Guinness,et al.  Fuzzy Membership Mapped onto Intervals and Many-Valued Quantities , 1976, Math. Log. Q..

[21]  MelinPatricia,et al.  A hybrid approach for image recognition combining type-2 fuzzy logic, modular neural networks and the Sugeno integral , 2009 .

[22]  Ramon E. Moore,et al.  Interval analysis and fuzzy set theory , 2003, Fuzzy Sets Syst..

[23]  Glad Deschrijver,et al.  A representation of t-norms in interval-valued L-fuzzy set theory , 2008, Fuzzy Sets Syst..

[24]  Benjamín R. C. Bedregal,et al.  On interval fuzzy negations , 2010, Fuzzy Sets Syst..

[25]  Janusz Kacprzyk,et al.  Management decision support systems using fuzzy sets and possibility theory , 1985 .

[26]  J. Buckley,et al.  Fuzzy expert systems and fuzzy reasoning , 2004 .

[27]  Vladik Kreinovich,et al.  Knowledge Processing with Interval and Soft Computing , 2008 .

[28]  Lotfi A. Zadeh,et al.  The Concepts of a Linguistic Variable and its Application to Approximate Reasoning , 1975 .

[29]  Byung-In Choi,et al.  Interval type-2 fuzzy membership function generation methods for pattern recognition , 2009, Inf. Sci..

[30]  Ronald R. Yager,et al.  On the implication operator in fuzzy logic , 1983, Inf. Sci..

[31]  G.P. Dimuro,et al.  Towards interval analysis of the load uncertainty in power electric systems , 2004, 2004 International Conference on Probabilistic Methods Applied to Power Systems.

[32]  Lotfi A. Zadeh,et al.  The concept of a linguistic variable and its application to approximate reasoning-III , 1975, Inf. Sci..

[33]  Yun Shi,et al.  On the First Place Antitonicity in QL-implications , 2008, EUSFLAT Conf..

[34]  Ronald R. Yager,et al.  On some new classes of implication operators and their role in approximate reasoning , 2004, Inf. Sci..

[35]  R. B. Kearfott,et al.  Applications of interval computations , 1996 .

[36]  Glad Deschrijver,et al.  Arithmetic operators in interval-valued fuzzy set theory , 2007, Inf. Sci..

[37]  Ronald R. Yager,et al.  Level sets and the extension principle for interval valued fuzzy sets and its application to uncertainty measures , 2008, Inf. Sci..

[38]  Graçaliz Pereira Dimuro,et al.  Interval-Based Markov Decision Processes for Regulating Interactions Between Two Agents in Multi-agent Systems , 2004, PARA.

[39]  Didier Dubois,et al.  Random sets and fuzzy interval analysis , 1991 .

[40]  Vladik Kreinovich,et al.  Interval-Valued Degrees of Belief: Applications of Interval Computations to Expert Systems and Intelligent Control , 1997, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[41]  Renata Reiser,et al.  Analyzing Properties of Fuzzy Implications Obtained via the Interval Constructor , 2006 .

[42]  Joan Torrens,et al.  Two types of implications derived from uninorms , 2007, Fuzzy Sets Syst..

[43]  Benjamín R. C. Bedregal,et al.  A Quasi-Metric Topology Compatible with Inclusion Monotonicity on Interval Space , 1997, Reliab. Comput..

[44]  Jerry M. Mendel,et al.  Advances in type-2 fuzzy sets and systems , 2007, Inf. Sci..

[45]  Marc Roubens,et al.  Fuzzy Preference Modelling and Multicriteria Decision Support , 1994, Theory and Decision Library.

[46]  Mirko Navara,et al.  Characterization of Measures Based on Strict Triangular Norms , 1999 .

[47]  Ronald R. Yager,et al.  On global requirements for implication operators in fuzzy modus ponens , 1999, Fuzzy Sets Syst..

[48]  Yun Shi,et al.  On the characterizations of fuzzy implications satisfying I(x, y)=I(x, I(x, y)) , 2007, Inf. Sci..

[49]  E. Walker,et al.  Some comments on interval valued fuzzy sets , 1996 .

[50]  Chris Cornelis,et al.  Advances and challenges in interval-valued fuzzy logic , 2006, Fuzzy Sets Syst..

[51]  George J. Klir,et al.  Fuzzy sets and fuzzy logic , 1995 .

[52]  Michal Baczynski,et al.  On the characterizations of (S, N)-implications , 2007, Fuzzy Sets Syst..

[53]  George J. Klir,et al.  Fuzzy sets and fuzzy logic - theory and applications , 1995 .

[54]  Benjamín R. C. Bedregal,et al.  Interval Additive Generators of Interval T-Norms , 2008, WoLLIC.

[55]  Hung T. Nguyen,et al.  A First Course in Fuzzy Logic , 1996 .

[56]  Radko Mesiar,et al.  Triangular Norms , 2000, Trends in Logic.

[57]  Hao Zhang,et al.  Interval Finite Elements as a Basis for Generalized Models of Uncertainty in Engineering Mechanics , 2007, Reliab. Comput..

[58]  Benjamín R. C. Bedregal,et al.  The best interval representations of t-norms and automorphisms , 2006, Fuzzy Sets Syst..

[59]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[60]  Benjamín R. C. Bedregal,et al.  Xor-Implications and E-Implications: Classes of Fuzzy Implications Based on Fuzzy Xor , 2009, LSFA.

[61]  Chris Cornelis,et al.  Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application , 2004, Int. J. Approx. Reason..

[62]  Joan Torrens,et al.  A Survey on Fuzzy Implication Functions , 2007, IEEE Transactions on Fuzzy Systems.

[63]  Graçaliz Pereira Dimuro,et al.  ICTM: An Interval Tessellation-Based Model for Reliable Topographic Segmentation , 2004, Numerical Algorithms.

[64]  Etienne Kerre,et al.  Fuzzy implication operators and generalized fuzzy method of cases , 1993 .

[65]  Benjamín R. C. Bedregal,et al.  An Approach to Interval-Valued R-Implications and Automorphisms , 2009, IFSA/EUSFLAT Conf..

[66]  Lotfi A. Zadeh,et al.  Fuzzy logic, neural networks, and soft computing , 1993, CACM.

[67]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[68]  Michal Baczynski,et al.  Residual implications revisited. Notes on the Smets-Magrez Theorem , 2004, Fuzzy Sets Syst..

[69]  Frank Mueller,et al.  Preface , 2009, 2009 IEEE International Symposium on Parallel & Distributed Processing.

[70]  Joan Torrens,et al.  S- and R-implications from uninorms continuous in ]0, 1[2 and their distributivity over uninorms , 2009, Fuzzy Sets Syst..

[71]  Didier Dubois.  Fuzzy sets and systems , 1980 .

[72]  E. Trillas,et al.  ON IMPLICATION AND INDISTINGUISHABILITY IN THE SETTING OF FUZZY LOGIC , 1993 .

[73]  Chris Cornelis,et al.  On the representation of intuitionistic fuzzy t-norms and t-conorms , 2004, IEEE Transactions on Fuzzy Systems.

[74]  J. Balasubramaniam,et al.  Non-commercial Research and Educational Use including without Limitation Use in Instruction at Your Institution, Sending It to Specific Colleagues That You Know, and Providing a Copy to Your Institution's Administrator. All Other Uses, Reproduction and Distribution, including without Limitation Comm , 2022 .

[75]  K. Jahn.  Intervall‐wertige Mengen , 1975 .

[76]  J. Mendel.  Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions , 2001 .

[77]  Lotfi A. Zadeh,et al.  Is there a need for fuzzy logic? , 2008, NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society.

[78]  Mirko Navara,et al.  A survey on different triangular norm-based fuzzy logics , 1999, Fuzzy Sets Syst..

[79]  Benjamín R. C. Bedregal,et al.  Interval Valued QL-Implications , 2007, WoLLIC.