Fuzzy Conditional Inferences and Fuzzy Inferences with Fuzzy Quantifiers

1. FUZZY CONDITIONAL INFERENCES

1"/e shall consider the following form of infer­ ence in which a fuzzy conditional proposition is contained.

where x andy are the names of objects, and A, A', B and D' are the labels of fuzzy sets in universes of discourse U, U, V and V, respectively.

For this form of fuzzy conditional inference, several methods are proposed.

Let A and D be fuzzy sets in U and v, respectively , which are written as

and let x, u, n, 7 and $be cartesian product, union, intersection, complement and bounded-sum for fuzzy sets, respectively.

Then the following fuzzy relations are obtained from a fuzzy conditional statement "If x is A then y is B" in Ant 1 of (1). The fuzzy relations Rm· Ra are proposed by Zadeh [1), Rc is by Mamdani [2), and Rs, Rg, Rsg and Rgg are new methods proposed here .

Then the consequence B' in Cons of (1) can be deduced from Ant 1 and Ant 2 using the max-min composition "o" of the fuzzy set A' in U and the fuzzy relation obtained above. Thus, we can have

and so on.

In the above form of fuzzy conditional infer­ renee, it seems according to our intuitions that the relations between A' in Ant 2 and B' in Cons in (1) ought to be satisfied as shown in the left part of Table I. Relation II-2 has the result different from that of Relation II-1, but in Ant 1 if there is not a strong casual relation between "xis A" and "y is B", the satisfaction of Rela­ tion II-2 will be permitted. Relation IV-1 asserts that when x is not A, any information about y can not be deduced from Ant 1. The satisfaction of Relation IV-2 is demanded when the fuzzy proposi­ tion "If x is A then y is B" means tacitly the proposition "If x is A then y is B else y is not B." Relation V corresponds to modus tollens in which the form of this inference is

Ant 1: If x is A then y is B.

In Table I, it is noted that very A is defined as A2, more or less A as Ao.s, not A as 7A, and unknown as v.

The right part of Table I shows the satisfac­ tion (0) or failure (X) of each Relation under the methods given in (3)-(9). It is assumed here that fuzzy sets A and B in (2) satisfy the condi­ tions in the discussion of Relations I-III:

But in the discussion of Relation V, we use the condition (i)' instead of (i) ~

2. FUZZY INFERENCES WITH FUZZY QUANTIFIERS

In this section we shall consider such infer­ ences that the antecedents are quantified by the fuzzy quantifiers such as "most", "a few", etc.

Let us consider a simple form of such infer­ ence as

In general, this form of inferences may be expressed in symbol as

where

X is a certain set (X = {Swedes} in (10)), and E represents an attribute value of the element of X (E =blond in (10)). q (£Q) is a fuzzy quanti­ fier and can be interpreted as representing a fuzzy rate. p (£P) is a linguistic probability and represents a subjective fuzzy probability (which is denoted by Pr(x' is E)) of the event "x' is E" over X. Since we can assume that p is determined by the fuzzy quantifier q alone, we introduce a function f which is a mapping from a rate space (say, the interval [0,100), with its element inter­ preted as%) into a probability space [0,1). Let a fuzzy quantifier q be represented as a fuzzy set in the rate space, then we can get a fuzzy proba­ bility f(q) as a fuzzy set in [0,1) by applying the extension principle [1).

Using this function f, we can get the following statement from (11).

Based on this discussion we shall consider the following form of a slightly complicate inference which includes fuzzy quantifier "most", and fuzzy attributes "tall" and "more or less tall".

In general, this form of inferences may be repre­ sented in symbol as

where A and A' are fuzzy attributes represented by fuzzy sets in a universe of discourse U (In the case of "tall" and "more or less tall", U will be, say, 150cm + 15lcm + ··· + 200cm). p' is a fuzzy probability which can be obtained from the conse­ quence of the inference.

For this type of inference, it may be thought that the relations in Table II ought to be satis­ fied between A' and p' under Ant 1' in (14).

Ant 1' of (14) can be read formally as

Thus we can deduce p' of (14) using the methods for fuzzy conditional inferences in Sec. 1.

If Ant 1' of (14) , i.e., (15) translates into the fuzzy relation defined in (6) , namely

where U and V are universes of discourse of A and f(q), respectively, then the consequence Pr(x' is E) of (14) is given by

Thus it follows from (17) that Relations I 0 , II 0 -l, III 0 and IV 0 in Table II are satisfied.

Similarly, if (15) translates into the fuzzy relation given by (7), then we find that Relations I 0 , II 0 -2, III 0 and IV 0 are satisfied.

Example: Let us consider the following antecedents Ant 1 and Ant 2:

where tall is a fuzzy set represented by

with the universe of discourse, U, of tall being u = 150 + 155 + 160 + 165 + ... + 185 + 190.

Let the rate space of most be

and let the probability space of f(most) be

and f(x) = x ~ 100 with x in the rate space, then when most is a fuzzy set in the rate space, that is,

f(most) will be

The antecedents Ant 1 and 2 can be rewritten from the notation of (14) and (15) as

Ant 1': If xis tall then Pr(x is well-built) is f(most).

Ant 2': Tom is very tall.

Therefore R (tall, f(most)) becomes s --- ---

Pr(Tom is well-built) ve~y tall o Rs(tall,f(~))

If this fuzzy probability can be approximated by the linguistic probability, say, very likely, then the Cons of Ant 1 and 2 of (18) will be

Cons: It is very likeiy that Tom is well-built.