The min-max composition rule and its superiority over the usual max-min composition rule
✍ Scribed by Sukhamay Kundu
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 789 KB
- Volume
- 93
- Category
- Article
- ISSN
- 0165-0114
No coin nor oath required. For personal study only.
✦ Synopsis
A close analysis of the Syllogism inference rule shows that if one uses Zadeh's notion of fuzzy if-then, then the proper way of combining the membership values of two fuzzy rules rl: "if A, then B" and r2: "if B, then C" is not by the usual max-min composition rule, but by the following min-max rule; z~j = min{max(l~ik,Vkj): all j }, where ~ij = mA(Xi) --, rnc(z~), #ik = rnA(xi)--+ rnB(yk), and Vkj = mB(yk)-~mC(Zj). The rain-max value gives an upper bound on Zik. The min-max rule results in a new notion of transitivity and a corresponding notion of a fuzzy equivalence relation. We demonstrate the superiority of the min-max rule in terms of the properties of this equivalence relation. In particular, we argue that the new form of transitivity is particularly suitable for studying non-logical ( ~ "~") fuzzy equivalence relationships.