Specifying t-norms

We studied here the behavior of the t-norms at the point (1/2,1/2). We indicate why this point can be considered as significant in the specification of t-norms. Then, we suggest that the image of this point can be used to classify the t-norms. We consider some usual examples. We also study the case of parameterized tnorms. Finally using the results of this study, we propose a uniform method of computing the parameters. This method allows not only having the same parameter-scale for all the families, but also giving an intuitional sense to the parameters. Introduction A goal of Fuzzy Logic is to extend the classical binary logic into an interval valued logic. When we talk about "fuzzy", we think about something between false and truth. If we denote the value truth by one and the value of false by zero, what is more fuzzy, more central than 1/2? An important component of logic is the logical operator like negation, conjunction, disjunction and implication. In extending the binary logic to fuzzy logic an interesting and central question concerns the behavior of these logical operators at this middle point of truth-value. Here we investigate this question. In this paper we concentrate our attention on one particular logical operator: the and operator. This operator is implemented in the fuzzy logic by the class of operators called t-norms. T-norms have been well-studied and very good overviews and classifications of these operators can be found in the literature, see [1 3]. T-norms are usually defined as operators for two variables, associativity allowing the generalization of the definition to n variables. In order to study the tnorms at the "most fuzzy" point we will study the t-norms on (1/2,1/2). We show, taking into account the definitional constraints, how central this point is. We also indicate that defining a tnorm on this point can be a natural step after fulfilling the classical logic constraints. These results push us to suggest that t-norms can be classified observing their image on the (1/2,1/2) point. We consider some usual t-norms. We pursuit our study by observing what happens in the case of parameterized t-norms. We consider three different families. M. Detyniecki, R. R. Yager, and B. Bouchon-Meunier, "Specifying t-norms based on the value of T(1/2,1/2)," Mathware & Soft Computing, vol. VII (1), pp. 77-87, 2000. Finally using the results of this study and taking into account the classification aspect, we invert our reasoning and we propose a uniform method for computing the parameter of each family. This method allows not only having the same parameter-scale for all the families, but also giving an intuitional sense to the parameters. 1. T-norms The concept of a triangular norm was introduced by Menger [4] in order to generalize the triangular inequality of a metric. The current notion of a t-norm and its dual operation (t-conorm) is due to Schweizer and Sklar [5]. Both of these operations can also be used as a generalization of the Boolean logic connectives to multi-valued logic. The t-norms generalize the conjunctive 'AND' operator and the t-conorms generalize the disjunctive 'OR' operator. This situation allows them to be used to define the intersection and union operation in fuzzy logic. This possibility was first noted by Hohle [9]. Klement [10], Dubois and Prade [11] and Alsina, Trillas, and Valverde [2] very early appreciated the possibilities of this generalization. Bonissone [12] investigated the properties of these operators with the goal of using them in the development of intelligent systems. In this paper we will focus on the t-norms, but one should keep in mind that analogous observations could be made for the t-conorms based on the duality between these operators. 1.1. Definition Formally, a t-norm is a function T: [0,1]x[0,1] [0,1], having the following properties • T(a,b) = T(b,a) • T(a,b) ≤ T(c,d), if a ≤ c et b ≤ d • T(a,T(b,c)) = T(T(a,b),c) • T(a,1) = a (1) Commutivity (2) Monotonicity (3) Associativity (4) One as identity 1.2. Proprieties A natural consequence of axioms (1-4) is the following property: • T(a,0) = 0 (5) Proof: We know that T(a,0)∈[0,1], so T(a,0) ≥ 0. And using axiom (2) with b = d = 0 and a ≤ 1 because a∈[0,1], we obtain the result. Another property associated with this operator is that T(a,b) ≤ Min (a,b), a special case of this is that T(a,a) ≤ a. Viewed as a logical connective, " and " operator, the t-norm has the general tendency of making truths decrease. Using the commutivity property (1), we have the limit properties: • T(a,1) = a • T(1,a) = a • T(a,0) = 0 • T(0,a) = 0 (L1) (L2) (L3) (L4) In other words the t-norms are completely defined on the edges of the unit square as shown in Figure #1. M. Detyniecki, R. R. Yager, and B. Bouchon-Meunier, "Specifying t-norms based on the value of T(1/2,1/2)," Mathware & Soft Computing, vol. VII (1), pp. 77-87, 2000. Figure #1: Definition of t-norms on edges of the unit square We notice that the vertices of the unit square correspond to the arguments of the classical binary logic and here the t-norm emulates the classical AND operator: • T(1,1) = 1 is the value of "True AND True" = "True" • T(1,0) = 0 is the value of "True AND False" = "False" • T(0,1) = 0 is the value of "False AND True" = "False" • T(0,0) = 0 is the value of "False AND False" = "False" Figure #2: Definition of t-norms on the vertices of the unit square 1.3. The middle point As noted the t-norms are constraint not only to follow the classical behavior at vertices of the unit square, but also to satisfy the limit properties (L1-L4), on the edges of the unit square. We observe that in the middle area of the unit square we have the freedom of choice. It is in this area we distinguish between different t-norm operators. An interesting point in this middle area, because of its central position is the point (1/2, 1/2). We know that it is the gravitational center of the square, it is also the intersection of the diagonals, the intersection of the middle lines and the barycenter of the edges of the square (the classical points). It also can be shown that it is the point u T(u,v) v T(1,1) = 1 T(0,1) = 1 T(1,0) = 0 T(0,0) = 0 u T(u,v) v