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

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Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T University of Texas at El Paso DigitalCommons@UTEP Departmental Technical Reports (CS) Department of Computer Science 8-1-2008 Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T. Nguyen Vladik Kreinovich University of Texas at El Paso, [email protected] Follow this and additional works at: http://digitalcommons.utep.edu/cs_techrep Part of the Computer Engineering Commons Comments: Technical Report: UTEP-CS-08-27a Published in Proceedings of the 9th International Conference on Intelligent Technologies InTech'08, Samui, Thailand, October 7-9, 2008, pp. 47-55. Recommended Citation Nguyen, Hung T. and Kreinovich, Vladik, "Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms" (2008). Departmental Technical Reports (CS). Paper 94. http://digitalcommons.utep.edu/cs_techrep/94 This Article is brought to you for free and open access by the Department of Computer Science at DigitalCommons@UTEP. It has been accepted for inclusion in Departmental Technical Reports (CS) by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. Computing Degrees of Subsethood and Similarity for Interval-Valued Fuzzy Sets: Fast Algorithms Hung T. Nguyen Vladik Kreinovich Department of Mathematical Sciences Department of Computer Science New Mexico State University University of Texas at El Paso Las Cruces, NM 88003, USA El Paso, TX 79968, USA [email protected] [email protected] Abstract—Subsethood A ⊆ B and set equality A = B are Thus, for two fuzzy sets A and B, it is reasonable to define among the basic notions of set theory. For traditional (“crisp”) degree of subsethood and degree of similarity. 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 How to describe degree of subsethood: main idea. In fuzzy or A 6⊆ B. To describe commonsense and expert reasoning, it is logic and fuzzy set theory, there is no built-in notion of degree advantageous to use fuzzy sets in which for each element a, there of subsethood or degree of equality (similarity) between the is a degree ¹A(a) 2 [0; 1] to which a belongs to this set. For fuzzy sets A and B, it is reasonable to define a degree of subsethood sets. Instead, the standard descriptions of fuzzy logic and fuzzy d⊆(A; B) and degree of equality (degree of similarity) d=(A; B). set theory start with the notions of union and intersection. In practice, it is often difficult to assign a definite membership The simplest way to describe the union of the two sets degree ¹A(a) to each element a; it is more realistic to expect that is to take the maximum of the corresponding membership an expert describes an interval [¹ (a); ¹ (a)] of possible values of A A functions: ¹ (x) = max(¹ (x); ¹ (x)). Similarly, the this degree. The resulting interval-valued fuzzy set can be viewed A[B A B simplest way to describe the intersection of the two sets is as a class of all possible fuzzy sets ¹A(a) 2 [¹ (a); ¹ (a)]. A A For interval-valued fuzzy sets A and B, it is therefore to take the minimum of the corresponding membership func- reasonable to define the degree of subsethood d⊆(A; B) as the tions: ¹A\B(x) = min(¹A(x); ¹B(x)). Thus, to describe the range of possible values of d⊆(A; B) for all A 2 A and B 2 B – degrees of subsethood and equality (similarity), it is reasonable and similarly, we can define the degree of similarity d=(A; B). to express the notions of subsethood and set equality in terms So far, no general algorithms were known for computing these of union and intersection. This expression is well known in ranges. In this paper, we describe such general algorithms. The newly proposed algorithms are reasonably fast: for fuzzy subsets set theory: it is known that of an n-element universal set, these algorithms compute the ² in general, A \ B ⊆ A, and ranges in time O(n ¢ log(n)). ² A ⊆ B if and only if A \ B = A. So, for crisp finite sets, to check whether A is a subset of B, I. FORMULATION OF THE PROBLEM we can consider the ratio Subsethood and set equality are important notions of set jA \ Bj ; theory. In traditional set theory, among the basic notions are jAj the notions of set equality and subsethood: where jAj denotes the number of elements in a set A: ² two sets A and B are equal if they contain exactly the same elements, and ² in general, this ratio is between 0 and 1, and ² this ratio is equal to 1 if and only if A is a subset of B. ² a set A is a subset of the set B if every element of the set A also belongs to B. The smaller the ratio, the more there are elements from A which are not part of the intersection A \ B, and thus, not Because of this importance, it is desirable to generalize these part of the set B. Thus, for crisp sets, this ratio can be viewed notions to fuzzy sets. as a reasonable measure of degree to which A is a subset of B. In fuzzy set theory, it is reasonable to talk about degrees A similar definition can be used to define degree of subset- of subsethood and equality (similarity). In traditional set hood of two fuzzy sets. Specifically, for finite fuzzy sets, we theory, for every two sets A and B, either A is a subset of can use a natural fuzzy extension of the notion of cardinality: def P B, or A is not a subset of B. Similarly, either the two sets A jAj = ¹A(x). Let us describe the resulting formulas. and B are equal or these two sets are different. Since we only consider finite fuzzy sets, we can therefore The main idea behind fuzzy logic is that for fuzzy, imprecise consider a finite universe of discourse. Without losing gener- concepts, everything is a matter of degree; see, e.g., [3], [9]. ality, we can denote the elements of the universe of discourse by their numbers 1; 2; : : : ; n. The values of the membership Thus, as an alternative degree of subsethood, we can take a function corresponding to the fuzzy set A can be therefore ratio Pn denoted by a1; : : : ; an. Similarly, the values of the membership b jBj i function corresponding to the fuzzy set B can be denoted by = i=1 : Pn b1; : : : ; bn. In these notations, jA [ Bj max(ai; bi) ² the membership function corresponding to the intersec- i=1 tion A \ B has the values min(a1; b1);:::; min(an; bn), Pn Need for interval-valued (and more general type-2) fuzzy ² the cardinality jAj of the fuzzy set A is equal to ai, i=1 sets. In the above text, we consider the situation in which the and values ai and bi of the membership function are numbers from ² the cardinality jA \ Bj of the intersection A \ B is equal the interval [0; 1]. Is this the most adequate description? n P The main objective of fuzzy logic is to describe uncertain to min(ai; bi). i=1 (“fuzzy”) knowledge, when an expert cannot describe his or Thus, the degree of subsethood of fuzzy sets A and B can be her knowledge by an exact value or by a precise set of possible defined as the ratio values. Instead, the expert describes this knowledge by using Pn words from natural language. Fuzzy logic provides a procedure min(a ; b ) i i for formalizing these words into a computer-understandable d (A; B) = i=1 : ⊆ Pn form – as fuzzy sets. ai In the traditional approach to fuzzy logic, the expert’s i=1 degree of certainty in a statement – such as the value ¹A(x) describing that the value x satisfies the property A (e.g., Comment. An alternative (probabilistic) justification of this “small”) – is characterized by a number from the interval [0; 1]. formula is given in the Appendix. However, we are considering situations in which an expert is How to describe degree of equality (similarity). It is known unable to describe his or her knowledge in precise terms. It is that not very reasonable to expect that in this situation, the same ² in general, A \ B ⊆ A [ B, and expert will be able to meaningfully express his or her degree ² A = B if and only if A \ B = A [ B. of certainty by a precise number. It is much more reasonable So, for crisp finite sets, to check whether A is equal to B, we to assume that the expert will describe these degrees also by can consider the ratio words from natural language. Thus, for every x, a natural representation of the degree jA \ Bj : ¹(x) is not a number, but rather a new fuzzy set. Such jA [ Bj situations, in which to every value x we assign a fuzzy set ² In general, this ratio is between 0 and 1, and ¹(x), are called type-2 fuzzy sets. ² this ratio is equal to 1 if and only if A is a subset of B. Successes of type-2 fuzzy sets. Type-2 fuzzy sets are actively The smaller the ratio, the more there are elements from A[B used in practice; see, e.g., [5], [6]. Since type-2 fuzzy sets which are not part of the intersection A\B, and thus, elements provide a more adequate representation of expert knowledge, it from one of the sets A and B which do not belong to the other is not surprising that such sets lead to a higher quality control, of these two sets.
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