On the use of interval mathematics in fuzzy expert systems

Abstract

Fuzzy expert systems attempt to model the cognitive processes of human experts. They currently accomplish this by capturing knowledge in the form of linguistic propositions. Real-world problems dictate the need to include mathematical knowledge as well. Pattern matching is a critical part of the inference procedure in expert systems. Matches are made between data clauses, premise clauses, and conclusion clauses, forming an inference chain. Preprocessing the clauses may generate intervals of real numbers which are compared in the fuzzy matching algorithm. These same intervals may be used in arithmetic expressions. The purpose of this article is to devise a method for incorporating arithmetic expressions into inference process of Fuzzy Expert Systems. Interval arithmetic is used to evaluate these expressions. Logical relations between intervals are analyzed using probability theory.0

Numerical methods for interval and fuzzy number comparison based on the probabilistic approach and Dempster–Shafer theory

“…The idea to use the probability interpretation of intervals is not novel. Nevertheless, now we can cite only a few works based on it [6,17,19,20,24,25,[27][28][29][30]39,44]. The main advantage of the probabilistic approach is the ability to infer a complete set of interval and fuzzy number relations as well as their comparison with real numbers using only one major assumption: an interval is a support of uniformly distributed random value.…”

“…The probabilistic approach to interval and fuzzy number comparison is a powerful practical tool for modelling and optimization in interval and fuzzy settings [28,29,33]. Nevertheless, there are some methodological problems within this approach and as a consequence, the different expressions for probabilities estimation were obtained in [17,19,20,24,25,[27][28][29][30][31][32][33]39,44]. In [39], the expressions for the probabilities P(A < B) and P(A > B) were obtained.…”

“…These were the same expressions as those obtained later on in our papers [30][31][32][33] using other assumptions. On the other hand, the probability P(A = B) in [39] has been presented (in our notation) as P(A = B) = 2 /(W(A)W(B)), where W(A), W(B) are the lengths of the compared intervals and is an arbitrary small number. The authors of [39] assumed that % 0.…”

“…On the other hand, the probability P(A = B) in [39] has been presented (in our notation) as P(A = B) = 2 /(W(A)W(B)), where W(A), W(B) are the lengths of the compared intervals and is an arbitrary small number. The authors of [39] assumed that % 0. In other words, in [39] it is implicitly assumed that in any case P(A = B) = 0.…”

“…Nevertheless, there are some methodological problems within this approach and as a consequence, the different expressions for probabilities estimation were obtained in [17,19,20,24,25,[27][28][29][30][31][32][33]39,44]. In [39], the expressions for the probabilities P(A < B) and P(A > B) were obtained. These were the same expressions as those obtained later on in our papers [30][31][32][33] using other assumptions.…”