On Robustness of Fuzzy Logics

نویسندگان

  • Hung T. Nguyen
  • Vladik Kreinovich
  • Dana Tolbert
چکیده

In this paper we propose to investigate the concept of robustness of combining operations in fuzzy systems. We use a concept similar to modulus of continuity in approximation theory to characterize robustness of fuzzy logic connectives. Speciically, it is shown that min and max are the most robust &? and _?operations. 1. INTRODUCTION In the majority of expert systems and intelligent control systems uncertainty of expert's statements is represented by a number from the interval 0,1]: t(A) = 1 means that an expert is absolutely sure in A, t(A) = 0 means that the expert is absolutely sure that A is false, and values from 0 to 1 represent diierent degrees of the expert's uncertainty. These numbers are called certainty values, degrees of belief, truth values, etc. If we know the degrees of belief t(A) and t(B) for two statements A and B, and nothing else is known about A and B, then what is the reasonable degree of belief in A&B? in A _ B? We must compute these estimates for t(A&B) and t(A _ B) from the only information that we have: from the values t(A) and t(B). The operations f(a; b) that transform t(A) and t(B) into the estimates f(t(A); t(B)) for t(A&B) and t(A _ B) are called correspondingly an &?operation and an _?operation. The rst paper by L. Zadeh Z65] that introduced this approach to knowledge representation proposed min(a; b) and ab as &?operations, and max(a; b) and a + b ? ab as _?operations. Zadeh himself stressed that these operations \are not the only operations in terms of which the union and intersection can be deened", and \which of these ... deeni-tions are more appropriate depends on the context" Z75, pp. 225{226]. Since then several dozens diierent &? and _?operations have been proposed and successfully used. Some operations have been discovered empirically while working on real expert systems (e.g., the famous MYCIN BS84]) or while analyzing commonsense reasoning O77, Z78]); some of them were proposed on a more theoretical basis (see, e.g., DP80, KF88]). A survey of such operations is given in KQLFLKBR92]. One of the most successful applications of this type of uncertainty representation is in the area of intelligent control. This area was rst outlined by L. Zadeh CZ72] and experimentally tested by E. Mamdani M74] in the framework of fuzzy set theory Z65], therefore this area of research is also called …

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تاریخ انتشار 1992