### Introduction

Fuzzy reasoning, also known as approximate reasoning, is a inference procedure that derives conclusions from a set of fuzzy if-then rules and known facts. Before introducing fuzzy reasoning, we shall discuss the compositional rule of inference, which plays a key role in fuzzy reasoning.

### Compositional Rule of Inference

The compositional rule of inference is a generalization of the following notion. Suppose a curve y = f(x) that regulates the relation between x and y. When we are given x = a, then from y = f(x) we can infer that y = b = f(a); as shown in Figure 1(a). A generalization of the aforementioned process would allow a to be an interval and f(x) to be an interval-valued function, as shown in Figure 1(b). To find the resulting interval y = b corresponding to the interval x = a, we first construct a cylindrical extension of a and then find its intersection I with the interval-valued curve. The projection of I onto the y-axis yields the interval y = b.

Going one step further in our generalization, we assume that F is a fuzzy relation on X x Y and A is a fuzzy set of X, as shown in Figure 2(a) and 2(b). To find the resulting fuzzy set B, again we construct a cylindrical extension c(A) with base A. The intersection of c(A) and F forms the analog of the region of intersection I in Figure 1(b). By projecting c(A) ∩ F onto the y-axis, we infer y as a fuzzy set B on the y-axis, as shown in Figure 2(d).

Specifically, let μA, μc(A), μB, and μF be the MFs of A, c(A), B, and F, respectively, where μc(A) is related to μA through

`μc(A)(x,y) = μA(x)`

Then,

```μc(A)∩F(x,y) = min[μc(A)(x,y), μF(x,y)]
= min[μA(x), μF(x,y)]```

By projecting c(A)∩F onto the y-axis, we have

```μB = maxxmin[μA(x), μF(x,y)]
= ∨x[μA(x) ∨ μF(x,y)]```

This formula reduces to the max-min composition of two relation matrices if both A (a unary fuzzy relation) and F (a binary fuzzy relation) have finite universes of discourse. Conventionally, B is represented as

`B = A ∘ F `

where ∘ denotes the composition operator.

It is interesting to note that the extension principle is in fact a special case of the compositional rule of inference. Specifically, if y = f(x) in Figure 3.10 is common crisp one-to-one or many-to-one function, then the derivation of the induced fuzzy set B on Y is exactly what is accomplished by the extension principle.

Using the compositional rule of inference, we can formalize an inference procedure upon a set of fuzzy if-then rules. This inference procedure, generally called approximate reasoning or fuzzy reasoning, is the topic of the next subsection.

### Fuzzy Reasoning

The basic rule of inference in traditional two-value topic is modus ponens , according to which we can infer the truth of a proposition B from the truth of A and the implication A → B. For instance, if A is identified with "the tomato is red" and B with "the tomato is ripe," then if it is true that "the tomato is red," it is also true that "the tomato is ripe". This concept is illustrated as follows:

```premise 1 (fact):      x is A
premise 2 (rule):      if x is A then y is B
---------------------------------------------
consequence (conclusion):  y is B```

However, in much of human reasoning, modus ponens is employed in an approximate manner. For example, if we have the same implication rule "if the tomato is red, then it is ripe" and we know that "the tomato is more or less red," then we may infer that "the tomato is more or less ripe". This is written as

```premise 1 (fact):      x is A'
premise 2 (rule):      if x is A then y is B
---------------------------------------------
consequence (conclusion):  y is B'```

where A' is close to A and B' is close to B. When A, B, A', and B' are fuzzy sets of approximate universes, the foregoing inference procedure is called approximate reasoning or fuzzy reasoning ; it is also called generalized modus ponens (GMP for short), since it has modus ponens as special case.

Using the composition rule of inference introduced in the previous subsection, we can formulate the inference procedure of fuzzy reasoning as the following definition.