Fuzzy interpolation by convex completion of sparse rule bases

Abstract

This paper proposes an approach to the interpolation be tween sparse fDZzy rules, founded on a unique prlnclple which supple ments the classical approximate reasoning machinery, The case of rules of the form A ➔ B, where A and B are lntervals 1s flrst dlscusse and then extended to fuzzy sets.

A new perspective on reasoning with fuzzy rules

“…It is shown that both schemes derive from standard deductive inference performed on the rule base consisting of the two rules, completed with linearly interpolated rules. 45,46 The method of Kóczy and Hirota is shown to rely on implicative rules, viewed as constraints, and the other method uses conjunctive rules, viewed as imprecise data. For the sake of simplicity, this section considers that rules have only one input (the input space has one dimension) and that the conditions and conclusions of rules are nonfuzzy closed intervals (see Ref.…”

“…On the other hand, it REASONING WITH FUZZY RULES is clear that the case when the Kóczy-Hirota method gives an empty result corresponds to an incoherent completed knowledge base of implicative rules. Hence, rather than trying to mend the method, it is advisable either to change the rules (relax the constraints) or to change the interpolation method (nonlinear interpolation or interpolation in between the rules only 45 ). In addition, note that the interpolation method by completion of a set of conjunctive rules and the inf-max composition also can lead to empty results.…”

“…For the sake of simplicity, this section considers that rules have only one input (the input space has one dimension) and that the conditions and conclusions of rules are nonfuzzy closed intervals (see Ref. 45 for the extension to fuzzy sets and to multiinput rules). Moreover, as a convenient notation the two considered sparse rules are denoted ( A 0 , B 0 ) and ( A 1 , B 1 ).…”

Completing rule bases in symbolic domains by analogy making

“…Fuzzy set theory offers a quantitative framework for approximate reasoning [21], interpolative reasoning [22,23,4,3,5], and similarity-based reasoning. Thus, [24] adapts the solution of an already solved situation to a similar new situation according to the degree of resemblance between these two situations.…”

“…The setting of default rules with exceptions which are to be applied to an incompletely described situation has especially been studied in non-monotonic reasoning (see e.g., [1,2]). However, there are other forms of reasoning under incomplete information, such as reasoning with sparse parallel (fuzzy) rules (see e.g., [3,4,5]). …”