6.4.3 Downward Monotonic Inference

Downward monotonic inference may be useful when we believe that the prior truth

value of B represents an upper limit on what is possible. This type of inference is

useful when modifying multivalued data, necessary when defuzzifying. Suppose

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104 INFERENCE IN AN EXPERT SYSTEM I

our consequent linguistic variable is B, with members the linguistic terms Bj .

A sample such data declaration and rule might be

declare Data output fzset (Small Medium Large);

rule IF (. . .) THEN output is Small;

where the rule is of the form

if Pi then B is Bj (6:10)

A rule has ¬red, assigning a grade of membership to Bj. A membership function has

been assigned to Bj, since Bj is a linguistic term. We now wish to modify the mem-

bership function to re¬‚ect its grade of membership. To do this, we employ down-

ward monotonic reasoning.

A formula for Bj0, the new truth value of Bj using downward monotonic

inference, is

B0j ¼ P AND Bj (6:11)

The reader will recall that there are many possible de¬nitions for the AND operator.

If we adopt the Zadehian min AND operator, we can write (6.11) as

B0j ¼ min(P, Bj ) (6:12)

If we adopt the product AND operator, we can write (6.12) as

B0j ¼ P Á Bj (6:13)

Figure 6.1 Downward monotonic inference; modi¬cation of a membership function by

¬ring a rule, prior to defuzzi¬cation. Rule is “IF A THEN size is Small”; combined

antecedent and rule con¬dence (pconf) is 0.6.

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105

6.6 TESTS OF PROCEDURES

In Figure 6.1, we illustrate the modi¬cation of a membership function using down-

ward monotonic inference and the Zadehian min AND operator.

6.5 APPROXIMATE REASONING

We have discussed the theory of approximate reasoning in Section 4.3. As we have

noted, this theory is sometimes suggested for use in fuzzy expert systems. Since the

method has already been presented, we will not repeat the procedure here. We will,

in Section 6.6, attempt to place approximate reasoning for expert systems in the

same general framework as monotonic reasoning (6.4), non-monotonic reasoning

(6.8) and downward monotonic reasoning (6.11).

6.6 TESTS OF PROCEDURES TO OBTAIN THE TRUTH

VALUE OF A CONSEQUENT FROM THE TRUTH

VALUE OF ITS ANTECEDENT

We consider a rule

if X is A then Y is B (6:14)

In this rule X and A are fuzzy sets de¬ned on the same universe, as are Y and B. X is

observed to be A0 , a fuzzy set whose truth values may be different from those of

A. We denote the fuzzy set Y that corresponds to A0 by B0 . We now desire to calcu-

late the truth values of the fuzzy set B0 . Note that since X and A, Y and B are de¬ned

on the same universes, their values are the members of those universes; we are con-

cerned here with the calculation of truth values only.

B0 ¼ P OR B (Monotonic reasoning) (6:15)

B0 ¼ P (Non-monotonic reasoning) (6:16)

B0 ¼ P AND B (Downward monotonic reasoning) (6:17)

if X ¼ A then Y ¼ B: X ¼ A0 : [Y ¼ B0 ,

B0 ¼ A0 (A ! B) (Approximate reasoning) (6:18)

where denotes a fuzzy relation, and ! denotes a fuzzy implication operator.

6.6.1 Desirable Properties

Given rule (6.14), if P then (B0 ¼ B), we have now established four methods for

inferring truth values of B0 given truth values for P and B or, in the case of

approximate reasoning, A0 , A, and B. We now ask what properties these methods

should have to be useful in expert systems. We think that the following properties

are indispensable for the three different reasoning modes we have de¬ned.

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106 INFERENCE IN AN EXPERT SYSTEM I

Monotonic reasoning:

1. B0j ! Bj

2. If A0 ¼ NULL, then B0 ¼ B (6:19)

3. If A0 and A are disjoint, then B0 ¼ B

Non-monotonic reasoning:

1. B0 ¼ P ¼ (A0 ¼ A)

2. If A0 ¼ NULL, then B0 ¼ NULL (6:20)

3. If A0 and A are disjoint, then B0 ¼ NULL

Downward monotonic reasoning:

1. B0j Bj

2. If A0 ¼ NULL, then B0 ¼ NULL (6:21)

3. If A0 and A are disjoint, then B0 ¼ NULL

6.6.2 Summary of Candidate Methods

In the following, the symbols for propositions represent their truth values.

A. Fuzzy rule “if A then B”, monotonic reasoning:

B0 ¼ (A0 ¼ A) OR B (6:22)

B. Fuzzy rule “if A then B”, non-monotonic reasoning:

B0 ¼ (A0 ¼ A) (6:23)

C. Fuzzy rule “if A then B”, downward monotonic reasoning (equivalent to

Mamdani method);

B0 ¼ (A0 ¼ A) AND B (6:24)

D. Fuzzy rule “if A0 ¼ A then B0 ¼ B”, Approximate reasoning:

B0 ¼ A0 (A ! B) (6:25)

In D, approximate reasoning, A ! B produces the fuzzy relation matrix

½Ai ! Bj ; the operator denotes composition of A0 with A ! B by

B0j ¼ max(min(A0i , Ai ! Bj ) for all i in A, A0 as de¬ned in Chapter 3.