A graded notion of functionality

Abstract We will introduce the notion of functionality as a graded property of a fuzzy relation. We will extend known results relating to functional fuzzy relations to their graded counterparts and provide new properties w.r.t. various kinds of relational compositions. As a contribution to the theory of fuzzy control, we will moreover provide an example of use of graded properties in establishing implicative models of fuzzy rule bases where the non-graded property fails.

[1]  Petr Cintula,et al.  Fuzzy class theory , 2005, Fuzzy Sets Syst..

[2]  Libor Běhounek,et al.  Set Theory and Arithmetic in Fuzzy Logic , 2015 .

[3]  Martina Daňková.  Representation Theorem for Fuzzy Functions - Graded Form , 2010, IJCCI.

[4]  Petr Hájek,et al.  Metamathematics of Fuzzy Logic , 1998, Trends in Logic.

[5]  Martina Danková,et al.  Relational compositions in Fuzzy Class Theory , 2009, Fuzzy Sets Syst..

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

[7]  Miroslav Ciric,et al.  Uniform fuzzy relations and fuzzy functions , 2009, Fuzzy Sets Syst..

[8]  Petr Cintula,et al.  Features of Mathematical Theories in Formal Fuzzy Logic , 2007, IFSA.

[9]  F. Klawonn.  Fuzzy points, fuzzy relations and fuzzy functions , 2000 .

[10]  Martina Danková,et al.  On approximate reasoning with graded rules , 2007, Fuzzy Sets Syst..

[11]  V. Novák.  Fuzzy sets and their applications , 1989 .

[12]  Irina Perfilieva,et al.  Fuzzy function as an approximate solution to a system of fuzzy relation equations , 2004, Fuzzy Sets Syst..

[13]  Mustafa Demirci,et al.  Fuzzy Functions and Their Applications , 2000 .

[14]  Siegfried Gottwald,et al.  Fuzzy uniqueness of fuzzy mappings , 1980 .

[15]  V. Novák,et al.  Mathematical Principles of Fuzzy Logic , 1999 .

[16]  C. Guido,et al.  Functional Many-Valued Relations , 2015 .

[17]  Ivan Kramosil.  Fuzzy Sets and Fuzzy Logic --Foundations of Applications from a Mathematical Point of View , 1995 .

[18]  Lluis Godo,et al.  Monoidal t-norm based logic: towards a logic for left-continuous t-norms , 2001, Fuzzy Sets Syst..

[19]  R. Belohlávek.  Fuzzy Relational Systems: Foundations and Principles , 2002 .

[20]  Martina Danková,et al.  Approximation of extensional fuzzy relations over a residuated lattice , 2010, Fuzzy Sets Syst..

[21]  Vilém Novák,et al.  Operations fitting triangular-norm-based biresiduation , 1999, Fuzzy Sets Syst..

[22]  Mustafa Demirci,et al.  Gradation of being fuzzy function , 2001, Fuzzy Sets Syst..