Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms

Subsethood A ⊆ B and set equality A = B are among the basic notions of set theory. For traditional (“crisp”) sets, every element a either belongs to a set A or it does not belong to A, and for every two sets A and B, either A ⊆ B or A 6⊆ B. To describe commonsense and expert reasoning, it is advantageous to use fuzzy sets in which for each element a, there is a degree μA(a) ∈ [0, 1] to which a belongs to this set. For fuzzy sets A and B, it is reasonable to define a degree of subsethood d⊆(A, B) and degree of equality (degree of similarity) d=(A, B). In practice, it is often difficult to assign a definite membership degree μA(a) to each element a; it is more realistic to expect that an expert describes an interval [μ A (a), μA(a)] of possible values of this degree. The resulting interval-valued fuzzy set can be viewed as a class of all possible fuzzy sets μA(a) ∈ [μ A (a), μA(a)]. For interval-valued fuzzy sets A and B, it is therefore reasonable to define the degree of subsethood d⊆(A,B) as the range of possible values of d⊆(A, B) for all A ∈ A and B ∈ B – and similarly, we can define the degree of similarity d=(A,B). So far, no general algorithms were known for computing these ranges. In this paper, we describe such general algorithms. The newly proposed algorithms are reasonably fast: for fuzzy subsets of an n-element universal set, these algorithms compute the ranges in time O(n · log(n)). I. FORMULATION OF THE PROBLEM Subsethood and set equality are important notions of set theory. In traditional set theory, among the basic notions are the notions of set equality and subsethood: • two sets A and B are equal if they contain exactly the same elements, and • a set A is a subset of the set B if every element of the set A also belongs to B. Because of this importance, it is desirable to generalize these notions to fuzzy sets. In fuzzy set theory, it is reasonable to talk about degrees of subsethood and equality (similarity). In traditional set theory, for every two sets A and B, either A is a subset of B, or A is not a subset of B. Similarly, either the two sets A and B are equal or these two sets are different. The main idea behind fuzzy logic is that for fuzzy, imprecise concepts, everything is a matter of degree; see, e.g., [3], [9]. Thus, for two fuzzy sets A and B, it is reasonable to define degree of subsethood and degree of similarity. How to describe degree of subsethood: main idea. In fuzzy logic and fuzzy set theory, there is no built-in notion of degree of subsethood or degree of equality (similarity) between the sets. Instead, the standard descriptions of fuzzy logic and fuzzy set theory start with the notions of union and intersection. The simplest way to describe the union of the two sets is to take the maximum of the corresponding membership functions: μA∪B(x) = max(μA(x), μB(x)). Similarly, the simplest way to describe the intersection of the two sets is to take the minimum of the corresponding membership functions: μA∩B(x) = min(μA(x), μB(x)). Thus, to describe the degrees of subsethood and equality (similarity), it is reasonable to express the notions of subsethood and set equality in terms of union and intersection. This expression is well known in set theory: it is known that • in general, A ∩B ⊆ A, and • A ⊆ B if and only if A ∩B = A. So, for crisp finite sets, to check whether A is a subset of B, we can consider the ratio |A ∩B| |A| , where |A| denotes the number of elements in a set A: • in general, this ratio is between 0 and 1, and • this ratio is equal to 1 if and only if A is a subset of B. The smaller the ratio, the more there are elements from A which are not part of the intersection A ∩ B, and thus, not part of the set B. Thus, for crisp sets, this ratio can be viewed as a reasonable measure of degree to which A is a subset of B. A similar definition can be used to define degree of subsethood of two fuzzy sets. Specifically, for finite fuzzy sets, we can use a natural fuzzy extension of the notion of cardinality: |A| def = μA(x). Let us describe the resulting formulas. Since we only consider finite fuzzy sets, we can therefore consider a finite universe of discourse. Without losing generality, we can denote the elements of the universe of discourse by their numbers 1, 2, . . . , n. The values of the membership function corresponding to the fuzzy set A can be therefore denoted by a1, . . . , an. Similarly, the values of the membership function corresponding to the fuzzy set B can be denoted by b1, . . . , bn. In these notations, • the membership function corresponding to the intersection A ∩B has the values min(a1, b1), . . . , min(an, bn), • the cardinality |A| of the fuzzy set A is equal to n ∑ i=1 ai, and • the cardinality |A∩B| of the intersection A∩B is equal to n ∑