ABSTRACT

A fuzzy restriction may be visualized as an elastic constraint on the values that may be assigned to a variable. In terms of such restrictions, the meaning of a proposition of the form “x is P,” where x is the name of an object and P is a fuzzy set, may be expressed as a relational assignment equation of the form R(A(x)) = P, where A(x) is an implied attribute of x, R is a fuzzy restriction on x, and P is the unary fuzzy relation which is assigned to R. For example, “Stella is young,” where young is a fuzzy subset of the real line, translates into R(Age(Stella))= young.The calculus of fuzzy restrictions is concerned, in the main, with (a) translation of propositions of various types into relational assignment equations, and (b) the study of transformations of fuzzy restrictions which are induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations. An important application of the calculus of fuzzy restrictions relates to what might be called approximate reasoning, that is, a type of reasoning which is neither very exact nor very inexact. The main ideas behind this application are outlined and illustrated by examples.

1 INTRODUCTION

During the past decade, the theory of fuzzy sets has developed in a variety of directions, finding applications in such diverse fields as taxonomy, topology, linguistics, automata theory, logic, control theory, game theory, information theory, psychology, pattern recognition, medicine, law, decision analysis, system theory and information retrieval.

A common thread that runs through most of the applications of the theory of fuzzy sets relates to the concept of a fuzzy restriction - that is, a fuzzy relation which acts as an elastic constraint on the values that may be assigned to a variable. Such restrictions appear to play an important role in human cognition, especially in situations involving concept formation, pattern recognition, and decision-making in fuzzy or uncertain environments.

As its name implies, the calculus of fuzzy restrictions is essentially a body of concepts and techniques for dealing with fuzzy restrictions in a systematic fashion. As such, it may be viewed as a branch of the theory of fuzzy relations, in which it plays a role somewhat analogous to that of the calculus of probabilities in probability theory. However, a more specific aim of the calculus of fuzzy restrictions is to furnish a conceptual basis for fuzzy logic and what might be called approximate reasoning [1], that is, a type of reasoning which is neither very exact nor very inexact. Such reasoning plays a basic role in human decision-making because it provides a way of dealing with problems which are too complex for precise solution. However, approximate reasoning is more than a method of last recourse for coping with insurmountable complexities. It is, also, a way of simplifying the performance of tasks in which a high degree of precision is neither needed nor required. Such tasks pervade much of what we do on both conscious and subconscious levels.

What is a fuzzy restriction? To illustrate its meaning in an informal fashion, consider the following proposition (in which italicized words represent fuzzy concepts):

young_ (1.1)

gray hair_ (1.2)

approximately equal_  in  height. (1.3)

Starting with (1.1), let Age (Tosi) denote a numerically-valued variable which ranges over the interval [0,100]. With this interval regarded as our universe of discourse U, young may be interpreted as the label of a fuzzy subset1 of U which is characterized by a compatibility function, μyoung’ of the form shown in Fig. 1.1. Thus, the degree to which a numerical age, say u = 28, is compatible with the concept of young is 0.7, while the compatibilities of 30 and 35 with young are 0.5 and 0.2, respectively. (The age at which the compatibility takes the value 0.5 is the crossover point of young.) Equivalently, the function μyoung may be viewed as the membership function of the fuzzy set young, with the value of μyoung at u representing the grade of membership of u in young.

Since young is a fuzzy set with no sharply defined boundaries, the conventional interpretation of the proposition “Tosi is young,” namely, “Tosi is a member of the class of young men,” is not meaningful if membership in a set is interpreted in its usual mathematical sense. To circumvent this difficulty, we shall view (1.1)as an assertion of a restriction on the possible values of Tosi’s age rather than as an assertion concerning the membership of Tosi in a class of individuals. Thus, on denoting the restriction on the age of Tosi by R(Age(Tosi)), (1.1)may be expressed as an assignment equation

young_ (1.4)

in which the fuzzy set young (or, equivalently, the unary fuzzy relation young) is assigned to the restriction on the variable Age (Tosi). In this instance, the restriction R(Age(Tosi)) is a fuzzy restriction by virtue of the fuzziness of the set young.

Using the same point of view, (1.2)may be expressed as

 gray_ (1.5)

Thus, in this case, the fuzzy set gray is assigned as a value to the fuzzy restriction on the variable Color (Hair(Ted)).

In the case of (1.1)and (1.2), the fuzzy restriction has the form of a fuzzy set or, equivalently, a unary fuzzy relation. In the case of (1.3), we have two variables to consider, namely, Height (Sakti) and Height (Kapali). Thus, in this instance, the assignment equation takes the form

 approximately_ equal_ (1.6)

in which approximately equal is a binary fuzzy relation characterized by a compatibility matrix μapproximately equal (u, v) such as shown in Table 1.2.

Thus, if Sakti’s height is 5′8 and Kapali’s is 5′10, then the degree to which they are approximately equal is 0.9.

The restrictions involved in (1.1), (1.2)and (1.3)are unrelated in the sense that the restriction on the age of Tosi has no bearing on the color of Ted’s hair or the height of Sakti and Kapali. More generally, however, the restrictions may be interrelated, as in the following example.

small_ (1.7)

approximately equal_ (1.8)

In terms of the fuzzy restrictions on u and v, (1.7)and (1.8)translate into the assignment equations

small_ (1.9)

approximately equal_ (1.10)

where R (u) and R (u, v) denote the restrictions on u and (u, v), respectively.

As will be shown in Section 2, from the knowledge of a fuzzy restriction on u and a fuzzy restriction on (u, v) we can deduce a fuzzy restriction on v. Thus, in the case of (1.9)and (1.10), we can assert that

∘R(u,v) = small_∘approximately equal_ (1.11)

where denotes the composition2 of fuzzy relations.

The rule by which (1.11)is inferred from (1.9)and (1.10)is called the...