Calculus of Fuzzy Restrictions

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 *This work was supported in part by the Naval Electronics Systems Command, Contract N00039-75-0034, The Army Research Office, Grant DAHC04-75-G-0056,and the National Science Foundation, Grant GK-43024X. 1 FUZZY SETS AND THEIR APPLICATIONS Edited by Lotfi A. Zadeh, King-Sun Fu, Kokichi Tanaka, Masamichi Shimura 2 FUZZY SETS AND THEIR APPLICATIONS Edited by Lotfi A. Zadeh, King-Sun Fu, Kokichi Tanaka, Masamichi Shimura L. A. ZADEH main ideas behind this application are outlined and illustrated by examples.