Conjugate fuzzy QL-implications obtained by the OWA-operator

Resumo

Fuzzy logic is a powerful theory to make a decision which is usually irresolute for one thing or another, making it difficult to reach a final agreement. So, fuzzy connectives have been extensively studied in computer science and widely used in practical applications such as decision-making pattern recognition also including medical diagnosis, clustering analysis and image processing [3]. Fuzzy implications and aggregation functions are mainly study in this work in order to obtain new representative members in the class of fuzzy QL-operators by analysing related mathematical properties. Thus, the aggregating fuzzy QL-subimplications are introduced. They are obtained by action of the OWA-operator performed over the family of the product triangular subnorms along with standard fuzzy negation and the probabilistic sum. As the main results, this family of QL-subimplications extend related QL-implications by preserving their corresponding properties. For that, let U be the unitary interval (U  [0, 1]). Consider the subconorm Si : U2  U , Si(x, y)  1  1i (1  x  y  xy) for i  1, the product t-norm T : U2  U , T (x, y)  xy and the standard fuzzy negation NS : U  U , NS(x)  1 x. An n-tuple of real numbers belonging to Un can be aggregate to a single real number on U by an aggregation function, which is a non-decreasing operator satisfying the following boundary conditions: A(0, 0, . . . , 0)  0 and A(1, 1, . . . , 1)  1. Let σ : Nn  Nn be a permutation function ordering the elements: xσ(1)  xσ(2)  . . .  xσ(n). Let w1, w2, . . . , wn be non negative weights (wi  0) such that their sum equals one ( n i0wi  1). For all x  Un, the n-ary aggregation function A : Un  U called OWA-operator is given as: A(x) n i0 wixσ(i). (1) According with [4, 5], a function I : U2  U is a fuzzy subimplication if it satisfies the boundary conditions I(1, 1)  I(0, 1)  I(0, 0)  1 together with the left antitonicity and right isotonicity. When a subimplication also verifies I(1, 0)  0, it is called a fuzzy implication [1]. Additionally, QL-(sub)implication is a fuzzy (sub)implication defined, for all x, y  U by the following expression: IS,N,T (x, y)  S(N(x), T (x, y)), (2) when T (S) is a t-(co)norm and N is a strong fuzzy negation. This work is supported by the Brazilian funding agencies CAPES and FAPERGS (Ed. PqG 06/2010, under the process number 11/1520-1). Proposition 1. Let ρ : U  U be an automorphism [2]. The function Ji : U2  U and its conjugate fucntion Jρi : U 2  U , given in Eq.(3) and Eq.(4) respectively, are both fuzzy QL-subimplications: Ji(x, y)  Si(NS(x), TP (x, y))  1 1 i (x x2y), (3) Jρi (x, y)  S ρ i (N ρ S(x), T ρ P (x, y))  ρ 1 ( 1 ρ(x) i (1 ρ(x)ρ(y)) ) , x, y  U. (4) Such family of fuzzy QL-subimplications is referred as J . Proposition 2. The QL-subimplication Ji, Jρi  J verify the properties: I1 : If S(N(x), x)  1 then I(x, 1)  1, for all x  U ; I2a : If x1  x2 then I(x1, 0)  I(x2, 0), for all x1, x2  U . I2b : If y1  y2 then I(1, y1)  I(1, y2), for all y1, y2  U . In [6], an k-ary function FA : Uk  U is called as (A,F)-operator and given by: FA(x1, . . . , xk)  A(F1(x1, . . . , xk), . . . , Fn(x1, . . . , xk)), x1, . . . , xk  U. (5) Proposition 3. Let TP  {Si(x, y)  1  1i (1  x  y  xy) : i  1} be a family of t-subnorms. The function TOWA : U2  U is a t-subconorm given by Eq.(6) in the following: SOWA(x, y) n i0 wiSσ(i)(x, y),x, y  U. (6) As the main result, we present the subclass of fuzzy QL-subimplication represented by a t-norm TP , the standard negation NS together with a t-subconorm SP , which is obtained by aggregating n fuzzy t-subconorms of the family SP . Theorem 1. For all x, y  U , the function JOWA : U2  U is a QL-subimplication, given by JOWA(x, y)  SOWA(NS(x), TP (x, y)). (7) Proposition 4. The QL-subimplication JOWA verify Ik for k  {1, 2a, 2b}. Concluding, by Prop. 4, the operator JOWA preserves the main properties of QL-subimplications. Further work considers the interrelations between this class of subimplications and their possible conjugate functions. Another interesting issue is related to dual constructions of QL-subimplications.