# Protoformal deduction rules

There are many ways in which generalized constraints may be combined and propagated. The principal protoformal deduction rules are the following:

(a) *Conjunction (possibilistic)*

where X is the Cartesian product.

(b) *Projection* (*possibilistic)*

where *m _{R}* and

*m*are the membership functions of

_{S}*R*and S, respectively.

(c) *Projection* (*probabilistic)*

where *X* and *Y* are real-valued random variables, and *R* and S are probability densities of (X, Y) and *X,* respectively.

(a) *Computational rule of inference* (Zadeh, 1965)

*Figure 6.18* Compositional rule of inference

**A, B***,* and C are fuzzy sets with respective membership functions *m _{A}, В_{в}, m_{C};* a is min or t-norm (see Figure 6.18).

(b) **Intersection***/** product syllogism* (Zadeh, 1983a; 1983b)

**Q***j* and **Q *** _{2}* are fuzzy quantifiers; A, B,

*are fuzzy sets; * is product in fuzzy arithmetic (Kaufmann and Gupta, 1985).*

**C**(c) * Basic extension principle* (Zadeh, 1965)

* g* is a given function or functional;

*and*

**A***are fuzzy sets (see Figure 6.19).*

**B**(d) * Extension principle* (Zadeh, 1975b)

This is the principal deduction rule governing possibilistic constraint propagation (see Figure 6.20)

*Figure 6.19* Basic extension of principle

*Figure 6.20* Extension principle

The extension principle is an instance of the generalized extension principle

The generalized extension principle may be viewed as an answer to the following question: If *f* is a function from *U =* {X} to *V =* {Y} and I can compute the singular value of *Y* given a singular value of X, what is the granular value of *Y* given a granular value of X?

* Note:* The extension principle is a primary deduction rule in the sense that many other deduction rules are derivable from the extension principle. An example is the following rule:

(e) **Probability rule**

where * X* is a real-valued random variable;

*and*

**A, B, C,***are fuzzy sets:*

**D***is the probability density of X; and*

**r***}. To derive this rule, we note that*

**U = {u**

which are generalized constraints of the form

Applying the extension principle to these expressions, we obtain the expression for * D* which appears in the basic probability rule.

(f) **Fuzzy-graph interpolation rule**

This rule is the most widely used rule in applications of fuzzy logic (Zadeh, 1975a; 1976). We have a function, * Y = f(X* ), which is represented as a fuzzy graph (see Figure 6.21). The question is: What is the value of

*when*

**Y***is*

**X***? The*

**A***and*

**A**_{i}, B_{i}*are fuzzy sets. Symbolic part*

**A**Computational part

*Figure 6.21* Fuzzy-graph interpolation

*Figure 6.22* Mamdani interpolation

where is the degree to which *A* matches *A _{i},*

When *A* is a singleton, this rule reduces to

In this form, the fuzzy-graph interpolation rule coincides with the Mamdani rule—a rule which is widely used in control and related applications (Mamdani and Assilian, 1975b) (see Figure 6.22).

In addition to basic rules, the computation/deduction module contains a number of specialized modules and submodules. Of particular relevance to GTU are Probability module and Usuality submodule. A basic rule in Probability module is the bimodal distribution interpolation rule which is stated in the following.

(g) *Bimodal distribution interpolation rule*

With reference to Figure 6.23, the symbolic and computational parts of this rule are:

Symbolic

Computational

subject to

In this rule, *X* is a real-valued random variable; *r* is the probability density of X; and *U* is the domain of X.

*Figure 6.23* Interpolation of a bimodal distribution

What is the expected value, *E(X*), of a bimodal distribution? The answer follows through application of the extension principle:

subject to

*Note: E(X)* is a fuzzy subset of U.