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4.6.1. Nonmonotonic AND and OR operations: a generalizationYager (1987) proposed nonmonotonic operations on fuzzy sets. These operations form an important enhancement to the standard conceptual framework of fuzzy sets. Let us recall that a nonmonotonic intersection (and conjunction) of two fuzzy sets A and B defined in X is defined as where t and s are t- and s-norms, respectively. Evidently, this operation is neither commutative nor defined pointwise; the latter observation comes from the fact that the definition dwells upon the (global) level of overlap between A and B (expressed by possibility measures) whose computations involve the entire universe of discourse. In the remainder of the section, we consider that B is an onto mapping so that there are some elements of X for which B assumes 0 and 1. To gain a better insight into the key features of this operation, let us discuss two specific boundary cases:
These observations underline an important role this operator plays in nonmonotonic logics and nonmonotonic reasoning, in particular. Furthermore if B = 1 (unknown) (no default imposed) then the original definition reduces to the standard AND operation on fuzzy sets. Again, as shown in Yager (1987), the nonmonotonic union (or connective) of A and B is defined as where B assumes a role of the default argument. Again as for the nonmonotonic intersection operation, we elaborate on two specific cases:
These nonmonotonic operations can be generalized by looking into the problem in which the default (B) and a current item (A) intersect. We confine ourselves to the nonmonotonic and connective and propose the extension of the original definition as follows where the additional continuous mapping w: 1- Poss(A, B) → [0,1] defined on [0, 1] is such that
The intent of the introduced functional enhancement is to modulate the impact the default value (B) could have during the process of aggregation of information becoming less or more profound. Furthermore the boundary conditions assure that the fundamental properties of the nonmonotonic AND operation are still retained. From the operational point of view we should pay more attention to all intermediate situations when the default and the actual piece of information intersect and tend to compete as to which degree the final result should be affected by each of these two components. The following numerical example sheds more light on this modulation issue. We look into the problem of modelling an interaction between the current knowledge and the default piece of knowledge. Let us consider A and B to be two trapezoidal fuzzy sets defined in R where the list of the four numbers associated with these fuzzy sets describes the knots of the piecewise linear segments of the membership function. Consider also five examples of the modulating function (w), namely w(u) = u1/4, u1/2, u, u2, u4, Fig. 4.31.
It is noticeable that if w(u) >u then the influence of the default value becomes diminished in comparison to that obtained for the standard definition (where w(u) =u). In contrast, any contraction-like mapping w increases the impact the default fuzzy set on the result of the AND aggregation. In general, the result (D) is a subnormal fuzzy set - an evident outcome of competition between A and B. 4.6.2. Estimation problem of the default fuzzy setThe formulation of the problem is straightforward: Given A and D, or more generally, a series of Ak and Dk, determine B (default fuzzy set). One can regard this task as directly emerging in the area of human - human communication when one wants to set some standards regarding somebodys intention. The entire scheme can be portrayed as shown in Fig. 4.32.
Generally, the expressed opinion (and, in particular, its lack) represented here as a fuzzy set A (intention) and its interpretation (viewed here as D) are provided while the default value (default fuzzy set) should be determined. The series of experiments collected to carry out the estimation of the default value involves a collection of pairs of fuzzy data (Ak, Dk). Assuming that the default value remains static regardless of the semantics of the experimental data, we can formulate the optimization problem in several ways. The two possible versions of the problem can be introduced as follows. While being conceptually different, both of them involve a form of parametric optimization. To concentrate on some essential details of the learning scheme, let us make several assumptions. First, the universe of discourse is finite, meaning that fuzzy sets can be viewed as vectors of their membership values. More specifically, we get and The learning is carried out on-line so that each pair from the training set (A1, D1), 1= 1, 2, ..., N, implies changes in the membership values of the default set to be determined. In this sense, the subscript (1) can be dropped; the performance index reads now as
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