Patterns of Fuzzy Rule-Based Inference Valerie Cross Systems Analysis Department, Miami University, Oxford, Ohio Thomas Sudkamp Department of Computer Science, Wright State University, Dayton, Ohio ABSTRACT Processing information in fuzzy rule-based systems generally employs one of two patterns of inference: composition or compatibility modification. Composition origi- nated as a generalization of binary logical deduction to fuzzy logic, while compatibility modification was developed to facilitate the evaluation of rules by separating the evaluation of the input from the generation of the output. The first step in compatibility modification inference is to assess the degree to which the input matches the antecedent of a rule. The result of this assessment is then combined with the consequent of the rule to produce the output. This paper examines the relationships between these two patterns of inference and establishes conditions under which they produce equivalent results. The separation of the evaluation of input from the generation of output permits a flexibility in the methods used to compare the input with the antecedent of a rule with multiple clauses. In this case, the degree to which the input and the rule antecedent match is determined by the application of a compatibility measure and an aggregation operator. The order in which these operations are applied may affect the assessment of the degree of matching, which in turn may cause the production of different results. Separability properties are introduced to define conditions under which compatibility modification inference is independent of the input evaluation strategy. KEYWORDS: fuz~ inference, compatibility measures, approximate analogi- cal reasoning, fuzzy if-then rules 1. INTRODUCTION Fuzzy set theory provides a formal system suitable for the representation of the vague, imprecise, and ambiguous information that pervades many Address correspondence to Valerie Cross, Systems Analysis Department, Miami University, Oxford, Ohio 45056. E-maiL" vcross@sasrisc, aps .muohio. edu. Received June 1993; accepted April 1994. International Journal of Approximate Reasoning 1994; 11:235-255 © 1994 Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010 0888-613X/94/$7.00 235 236 Valerie Cross and Thomas Sudkamp common problem domains. Approximate reasoning using fuzzy techniques has successfully been employed in decision theory, database retrieval, expert systems, and automatic control. The fundamental representation used in many of these applications is that of a fuzzy rule. This paper examines the foundations of the two standard patterns of fuzzy rule-based inference: composition and compatibility modification. Composition origi- nated as a generalization of binary logical deduction to fuzzy logic. Compatibility modification (CM), which was specifically developed for inference in fuzzy rule-based systems, separates the evaluation of the antecedent of a rule from the generation of the output. Compositional inference integrates the analysis of the input and the creation of the output into a single inference step. In CM inference, a compatibility measure is used to determine the degree to which the input matches the antecedent of a rule. After the analysis of the input, the output is constructed from the consequent of the rule and the degree of satisfaction of the antecedent. This paper examines the functions of these two distinct patterns of inference, analyzes their efficiency, and establishes relation- ships between them. The examination of fuzzy inference begins with a brief review of the semantics and notation of fuzzy set theory. This is followed by a presenta- tion of the principles of compositional and compatibility modification inference. These inference techniques are then compared, and conditions are established under which they produce the same results. An analysis of the generation of support in CM inference produces the notion of aggrega- tion separability. A compatibility measure and an aggregation operator are said to aggregation separable if the same result is obtained regardless of the manner in which the compatibility is evaluated. It is shown that several common families of CM inference techniques satisfy the separability condition. 2. BACKGROUND A fuzzy set A over domain U is a function /z4 : U ~ [0, 1]. The function /zA is called the membershipfunction of the fuzzy set. As a generalization of binary logic, a fuzzy set A may be considered to be a predicate. Under this interpretation, tZA(U)represents the degree to which A is satisfied by u. An alternative semantics views a fuzzy set as defining a concept. In this case, the membership function expresses the degree to which u matches the criteria that define the concept. Thus, tZA(U)indicates the similarity or the compatibility of u with the concept defined by A. The value ~A(u) = 1 indicates that u is completely compatible with the concept defined by A, while IZA(U)= 0 indicates u is incompatible with A. Patterns of Fuzzy Rule-Based Inference 237 Following the interpretation of fuzzy set theory as an extension of binary logic, the basic fuzzy set operations correspond to the propositional con- nectives. T-norms provide the fuzzy generalization of conjunction. For- mally, a T-norm is a nondecreasing, commutative, and associative function T :[0, 1] × [0, 1] ~ [0, 1] that satisfies T(0, x) = 0 and T(1, x) = x. Several common T-norms that will be used in the sequel are given in Table 1. Every T-norm T assumes values bounded by T O and T 3. That is, To(x, y) < T(x, y) < T3(x, y) for all x, y ~ [0, 1]. While many of the results in this paper hold for arbitrary T-norms, we will primarily be concerned with the Lukasiewicz T-norm T~, the product, and the minimum. Associated with each T-norm is a dual T-conorm S defined by S(x,y) = 1- T(1-x, 1--y). The T-conorm S is the disjunction that corresponds to the conjunction T. An exposition of the general properties of T-norms can be found in [1-3]. A fuzzy rule specifies an approximate conditional relationship between the elements in a universe U and those in a universe W. The relationship is indicated by a statement of the form "if X is A then Z is C" where the antecedent A is a fuzzy set over U and the consequent C is a fuzzy set over W. Fuzzy rule-based inference combines the relationship with input indicating the current state of U to produce an estimate C' of the state of W. The input is given by a fuzzy set A' over U. The use of fuzzy sets to describe the input permits the representation of imprecision in the speci- fication of the current state of knowledge of U. A fuzzy set is said to be normal if there is at least one element that has the maximal membership value 1. Normality implies that some element in the universe is completely compatible with the concept defined by the fuzzy set. A fuzzy set A is said to be precise if /zA(Ui) = 1 for some ug and tzA(U j) = 0 for all j ¢ i. Throughout this paper, A, B, and C will repre- sent fuzzy sets over the domains U = {u 1.... , un}, V = {v I ..... Urn}, and W = {w 1..... wp}, respectively. Table 1. T-Norms and T-Conorms T-norm T-conorm T3(x , y) = min(x, y) S3(x , y) = max(x, y) Tz(x, y) = xy Sz(x, y) = x + y - xy Ta(x,y) = max(x + y - 1,0) Sl(x,y) = min(1, x + y) To(x, y) = (10 ifxy=l So(x, y) = (1 ifx+y>0 otherwise. 0 otherwise 238 Valerie Cross and Thomas Sudkamp 3. COMPOSITIONAL INFERENCE The generalization of inference from bivalued logic to fuzzy logic employs a generalization of the deductive law modus ponens. Following the standard pattern of logical deduction, generalized modus ponens has the form if X is A then Z is C XisA' Z is C' producing an output fuzzy set C'. Formally, the relationship specified by the rule "if X is A then Z is C" is given by an implication relation RA _~ c over the Cartesian product U × W. An element R A __, c(U, w) indicates the degree to which the truth or presence of u implies that of w. As an extension of logical deduction, tZc,(W) = 1 whenever lZA,(U) = 1 and R A _, c(U, w) = 1. A decrease in the certainty of u or of the degree of the implication of w by u is accompa- nied by a decrease in the implied support for w. An entry R A _~ c(U, w) in the implication relation may be thought of as defining a rule "if u then w" whose strength is R A _~ c(U, w). The support for w given input tt,4,(u) is obtained by combining the degree to which u is true with the strength of the implication of w from u. The implied support T(l£A(u), R A_,c(u,w)) is the conjunction of txA,(u) and R A_~c(u,w), where the T-norm T is called the evaluator of the implication. The most frequently used compositional T-norms are the minimum and the product [4]. These choices provide alternative philosophies for determining support from uncertain inferential information. The standard fuzzy-set conjunction min assigns support based on the weaker of the two conjuncts. Employing the product is similar to support generation with certainty factors [5]: the support for the consequent is affected both by the degree of support for the premise and by the strength of the implication. The interpretation of the entries RA-.c(Ui, W), i = 1,..., n, as rules may be considered to produce a element wise "rule base" If u I then w if u, then w pertaining to w.