Abstract
Based on the concept of incompatible factor, this
paper introduces further the principle of minimal
incompatibility, and use this principle to improve the
triple I method. In addition, by the principles of
maximal support and minimal incompatibility the
solutions of the problems of Fuzzy Modus Ponens and
Fuzzy Modus Tollens are unified. The reason why the
new method is more reasonable than the triple I
method is analyzed, and the new method is
generalized by considering two different types of
implication operators.
Keywords: Fuzzy reasoning, CRI method, Triple I
method, Incompatible factor, The least incompatibility
1. Introduction
As is well known, fuzzy reasoning now has become a
theoretical basis and an important method for the
design and analysis of fuzzy controller, and it has
found a considerable number of successful industrial
applications in some fields such as intelligent control
[1]. In 1973, the Fuzzy Modus Ponens(FMP) was
introduced for the first time by Zadeh [3], and
developed by Mamdani [4], so that it becomes the
famous CRI method (Compositional Rule of Inference)
[5]. As the fundamental method of fuzzy reasoning,
the CRI method has been widely adopted and many
researchers have generalized it in many different
ways[6]-[8].
After making many detailed researches on the
method of CRI, Wang pointed out some shortcomings
about it and proposed a kind of new method for fuzzy
reasoning, called as triple I method [9]-[10], which can
be considered as a reasonable improvement to Zadeh’s
CRI method. However, by some theoretical analyses
we discover that the triple I method also has some
shortcomings. For example, it simply considers the
minimal good set [10] as the result of FMP but has no
reasonable explanation, and the result given by it may
be not a reasonable one in some cases. In this paper,
the principle of minimal incompatibility is proposed
based on the concept of incompatible factor [11], and
the result of fuzzy reasoning is renewedly given by it.
2. Fundamental concepts
2.1. Fuzzy modus ponens(FMP)
In the fundamental research on fuzzy reasoning, the
basic model of FMP can be represented as follows [6]:
rule A → B
for given A* (1)
to determine B*
where A and A* are the fuzzy sets in domain X, and B
and B* are the fuzzy sets in domain Y.
2.2. The CRI method and the triple
I method
2.2.1. The CRI method
To get the fuzzy set B* in Eq.(1), various methods
have been carried out, but the Zadeh’s CRI method is
the most typical.
According to the CRI method, the fuzzy set B* in
Eq.(1) may be determined by
* ( B y ) = A * ( x) o RZ ( A( x), B( y )) (2)
sup[ * ( ) ( ( ), ( ))] Z
x X
A x R Ax B y
∈
= ∧
(3)
where the variable RZ: [0, 1]2 → [0, 1] is Zadeh’s
implication operator defined as
( , ) ( ) z R a b = a′ ∨ a ∧ b (4)
where a' = 1−a, a∈[0, 1]. Notice that there are two
kinds of fuzzy logic operators in Eq.(2), the compound
operator and the implication operator “RZ”. These
operators have been defined by various different
methods [12].


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