On (G, N)-implications derived from grouping functions

Overlap and grouping functions are special kinds of non-necessarily associative aggregation operators recently proposed for applications in classification problems involving the overlap problem and/or when the associativity property is not strongly required, as in image processing and decision making based on fuzzy preference relations, respectively. The concepts of indifference and incomparability defined in terms of overlap and grouping functions may allow their application in several different contexts. This paper introduces the concept of (G,N)-implication, for a grouping function G and a fuzzy negation N. (G,N)-implications are weaker then (S,N)-implications for positive and continuous t-conorms S, in the sense that (G,N)-implications do not necessarily satisfy certain properties, as the exchange and the left neutrality principles, which are not demanded for applications in decision making based on fuzzy preference relations. We analyze several related important properties, providing a characterization of (G,N)-implications.

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

[2]  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.

[3]  Yun Shi,et al.  The role a fuzzy implication plays in a multi-criteria decision algorithm , 2013, Int. J. Gen. Syst..

[4]  Juan Fernández-Sánchez,et al.  A copula-based family of fuzzy implication operators , 2013, Fuzzy Sets Syst..

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

[6]  Humberto Bustince,et al.  Some properties of overlap and grouping functions and their application to image thresholding , 2013, Fuzzy Sets Syst..

[7]  János C. Fodor,et al.  Nonstandard conjunctions and implications in fuzzy logic , 1995, Int. J. Approx. Reason..

[8]  R. Baker Kearfott,et al.  Introduction to Interval Analysis , 2009 .

[9]  Michal Baczynski.  On the Applications of Fuzzy Implication Functions , 2012, SOFA.

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

[11]  Petr Ekel,et al.  A dynamic consensus scheme based on a nonreciprocal fuzzy preference relation modeling , 2012, Inf. Sci..

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

[13]  Michal Baczynski,et al.  Fuzzy Implications , 2008, Studies in Fuzziness and Soft Computing.

[14]  Eyke Hüllermeier,et al.  Grouping, Overlap, and Generalized Bientropic Functions for Fuzzy Modeling of Pairwise Comparisons , 2012, IEEE Transactions on Fuzzy Systems.

[15]  Joan Torrens,et al.  On the characterization of Yager's implications , 2012, Inf. Sci..

[16]  G. P. Dimuro,et al.  Analyzing the Evolution of Social Exchange Strategies in Social Preference-Based MAS through an Evolutionary Spatial Approach of the Ultimatum Game , 2012, 2012 Third Brazilian Workshop on Social Simulation.

[17]  Benjamín R. C. Bedregal,et al.  Interval representations, Łukasiewicz implicators and Smets-Magrez axioms , 2013, Inf. Sci..

[18]  Esteban de Manuel Jerez,et al.  Modeling Role Interactions in a Social Organization for the Simulation of the Social Production and Management of Urban Ecosystems: The Case of San Jerónimo Vegetable Garden of Seville, Spain , 2012, 2012 Third Brazilian Workshop on Social Simulation.

[19]  Radko Mesiar,et al.  On special fuzzy implications , 2009, Fuzzy Sets Syst..

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

[21]  Henk J. A. M. Heijmans,et al.  Grey-Scale Morphology Based on Fuzzy Logic , 2002, Journal of Mathematical Imaging and Vision.

[22]  Benjamín R. C. Bedregal,et al.  Aggregating fuzzy implications , 2013, Inf. Sci..

[23]  Gleb Beliakov,et al.  Aggregation Functions: A Guide for Practitioners , 2007, Studies in Fuzziness and Soft Computing.

[24]  Humberto Bustince,et al.  New results on overlap and grouping functions , 2013, Inf. Sci..

[25]  Graçaliz Pereira Dimuro,et al.  Centralized Regulation of Social Exchanges Between Personality-Based Agents , 2006, COIN@AAMAS/ECAI.

[26]  Benjamín R. C. Bedregal,et al.  Archimedean overlap functions: The ordinal sum and the cancellation, idempotency and limiting properties , 2014, Fuzzy Sets Syst..

[27]  Anand S. Rao,et al.  Modeling Rational Agents within a BDI-Architecture , 1997, KR.

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

[29]  Graçaliz Pereira Dimuro,et al.  Recognizing and learning models of social exchange strategies for the regulation of social interactions in open agent societies , 2011, Journal of the Brazilian Computer Society.

[30]  Joan Torrens,et al.  Residual implications on the set of discrete fuzzy numbers , 2013, Inf. Sci..

[31]  Renata Reiser,et al.  Analyzing the Relationship between Interval-valued D-Implications and Interval-valued QL-Implications , 2010 .

[32]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[33]  Dana Hlinená,et al.  A class of implications related to Yager's f-implications , 2014, Inf. Sci..

[34]  Jaime Simão Sichman,et al.  An Overview of Social Simulation Research in Brazil , 2012, 2012 Third Brazilian Workshop on Social Simulation.

[35]  Yao Ouyang,et al.  On fuzzy implications determined by aggregation operators , 2012, Inf. Sci..

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

[37]  Radosław Łukasik.  A note on the mutual independence of the properties in the characterization of residual fuzzy implications derived from left-continuous uninorms , 2014 .

[38]  Huawen Liu,et al.  A new class of fuzzy implications derived from generalized h-generators , 2013, Fuzzy Sets Syst..

[39]  Benjamín R. C. Bedregal,et al.  Interval additive generators of interval t-norms and interval t-conorms , 2011, Inf. Sci..

[40]  Daowu Pei,et al.  Formalization of implication based fuzzy reasoning method , 2012, Int. J. Approx. Reason..

[41]  Benjamín R. C. Bedregal,et al.  On additive generators of overlap functions , 2016, Fuzzy Sets Syst..

[42]  Michal Baczynski.  On two distributivity equations for fuzzy implications and continuous, Archimedean t-norms and t-conorms , 2013, Fuzzy Sets Syst..

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

[44]  Hua-Wen Liu,et al.  Solutions to the functional equation I(x, y) = I(x, I(x, y)) for three types of fuzzy implications derived from uninorms , 2012, Inf. Sci..

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

[46]  J. Fodor.  Strict preference relations based on weak t-norms , 1991 .

[47]  Gleb Beliakov,et al.  Consensus measures constructed from aggregation functions and fuzzy implications , 2014, Knowl. Based Syst..

[48]  E. Dougherty,et al.  Fuzzification of set inclusion: theory and applications , 1993 .

[49]  Michal Baczynski,et al.  Distributivity of implication operations over t-representable t-norms in interval-valued fuzzy set theory: The case of nilpotent t-norms , 2014, Inf. Sci..

[50]  Renata Reiser,et al.  Revisiting XOR-Implications: Classes of fuzzy (Co)Implications Based on F-XOR (F-XNOR) Connectives , 2013, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[51]  Radko Mesiar,et al.  On fuzzy implications: An axiomatic approach , 2013, Int. J. Approx. Reason..

[52]  Graçaliz Pereira Dimuro,et al.  Towards the Self-regulation of Personality-Based Social Exchange Processes in Multiagent Systems , 2008, SBIA.

[53]  Graçaliz Pereira Dimuro,et al.  A Fuzzy-BDI Agent Model for Exchanges of Non-economic Services, Based on the Social Exchange Theory , 2011, 2011 Workshop-School on Theoretical Computer Science.

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

[55]  Benjamín R. C. Bedregal,et al.  On interval fuzzy S-implications , 2010, Inf. Sci..

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

[57]  Gleb Beliakov,et al.  Aggregating fuzzy implications to measure group consensus , 2013, 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS).

[58]  Etienne E. Kerre,et al.  Connections between binary, gray-scale and fuzzy mathematical morphologies , 2001, Fuzzy Sets Syst..

[59]  Humberto Bustince,et al.  An alternative to fuzzy methods in decision-making problems , 2012, Expert Syst. Appl..

[60]  Benjamín R. C. Bedregal,et al.  On the Boolean-like Law I(x, I(y, x)) = 1 , 2014, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[61]  Huawen Liu,et al.  Two classes of pseudo-triangular norms and fuzzy implications , 2011, Comput. Math. Appl..