Fuzzy implication functions based on powers of continuous t-norms

The modification (relaxation or intensification) of the antecedent or the consequent in a fuzzy “If, Then” conditional is an important asset for an expert in order to agree with it. The usual method to modify fuzzy propositions is the use of Zadeh's quantifiers based on powers of t-norms. However, the invariance of the truth value of the fuzzy conditional would be a desirable property when both the antecedent and the consequent are modified using the same quantifier. In this paper, a novel family of fuzzy implication functions based on powers of continuous t-norms which ensure the aforementioned property is presented. Other important additional properties are analyzed and from this study, it is proved that they do not intersect the most well-known classes of fuzzy implication functions.Peer ReviewedPostprint (author's final draft

Some characterizations of T-power based implications

Recently, the so-called family of T-power based implications was introduced. These operators involve the use of Zadeh’s quantifiers based on powers of t-norms in its definition. Due to the fact that Zadeh’s quantifiers constitute the usual method to modify fuzzy propositions, this family of fuzzy implication functions satisfies an important property in approximate reasoning such as the invariance of the truth value of the fuzzy conditional when both the antecedent and the consequent are modified using the same quantifier. In this paper, an in-depth analysis of this property is performed by characterizing all binary functions satisfying it. From this general result, a fully characterization of the family of T-power based implications is presented. Furthermore, a second characterization is also proved in which surprisingly the invariance property is not explicitly used.Peer ReviewedPostprint (author's final draft

Distributivity of ordinal sum implications over overlap and grouping functions

summary:In 2015, a new class of fuzzy implications, called ordinal sum implications, was proposed by Su et al. They then discussed the distributivity of such ordinal sum implications with respect to t-norms and t-conorms. In this paper, we continue the study of distributivity of such ordinal sum implications over two newly-born classes of aggregation operators, namely overlap and grouping functions, respectively. The main results of this paper are characterizations of the overlap and/or grouping function solutions to the four usual distributive equations of ordinal sum fuzzy implications. And then sufficient and necessary conditions for ordinal sum implications distributing over overlap and grouping functions are given

O równaniach funkcyjnych związanych z rozdzielnością implikacji rozmytych

In classical logic conjunction distributes over disjunction and disjunction distributes over conjunction. Moreover, implication is left-distributive over conjunction and disjunction: p ! (q ^ r) (p ! q) ^ (p ! r); p ! (q _ r) (p ! q) _ (p ! r): At the same time it is neither right-distributive over conjunction nor over disjunction. However, the following two equalities, that are kind of right-distributivity of implications, hold: (p ^ q) ! r (p ! r) _ (q ! r); (p _ q) ! r (p ! r) ^ (q ! r): We can rewrite the above four classical tautologies in fuzzy logic and obtain the following distributivity equations: I(x;C1(y; z)) = C2(I(x; y); I(x; z)); (D1) I(x;D1(y; z)) = D2(I(x; y); I(x; z)); (D2) I(C(x; y); z) = D(I(x; z); I(y; z)); (D3) I(D(x; y); z) = C(I(x; z); I(y; z)); (D4) that are satisfied for all x; y; z 2 [0; 1], where I is some generalization of classical implication, C, C1, C2 are some generalizations of classical conjunction and D, D1, D2 are some generalizations of classical disjunction. We can define and study those equations in any lattice L = (L;6L) instead of the unit interval [0; 1] with regular order „6” on the real line, as well. From the functional equation’s point of view J. Aczél was probably the one that studied rightdistributivity first. He characterized solutions of the functional equation (D3) in the case of C = D, among functions I there are bounded below and functions C that are continuous, increasing, associative and have a neutral element. Part of the results presented in this thesis may be seen as a generalization of J. Aczél’s theorem but with fewer assumptions on the functions F and G. As a generalization of classical implication we consider here a fuzzy implication and as a generalization of classical conjunction and disjunction - t-norms and t-conorms, respectively (or more general conjunctive and disjunctive uninorms). We study the distributivity equations (D1) - (D4) for such operators defined on different lattices with special focus on various functional equations that appear. In the first two sections necessary fuzzy logic concepts are introduced. The background and history of studies on distributivity of fuzzy implications are outlined, as well. In Sections 3, 4 and 5 new results are presented and among them solutions to the following functional equations (with different assumptions): f(m1(x + y)) = m2(f(x) + f(y)); x; y 2 [0; r1]; g(u1 + v1; u2 + v2) = g(u1; u2) + g(v1; v2); (u1; u2); (v1; v2) 2 L1; h(xc(y)) = h(x) + h(xy); x; y 2 (0;1); k(min(j(y); 1)) = min(k(x) + k(xy); 1); x 2 [0; 1]; y 2 (0; 1]; where: f : [0; r1] ! [0; r2], for some constants r1; r2 that may be finite or infinite, and for functions m2 that may be injective or not; g : L1 ! [1;1], for L1 = f(u1; u2) 2 [1;1]2 j u1 u2g (function g satisfies two-dimensional Cauchy equation extended to the infinities); h; c : (0;1) ! (0;1) and function h is continuous or is a bijection; k : [0; 1] ! [0; 1], g : (0; 1] ! [1;1) and function k is continuous. Most of the results in Sections 3, 4 and 5 are new and obtained by the author in collaboration with M. Baczynski, R. Ger, M. E. Kuczma or T. Szostok. Part of them have been already published either in scientific journals (see [5]) or in refereed papers in proceedings (see [4, 1, 2, 3])