Characteristics of Disjuncts

Fuzzy Multi-dimensional Analysis	Alexandr A. Savinov
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3. Characteristics of Disjunctions

In this section we define three global characteristics of fuzzy distributions. Here 'global' characteristic means that to calculate it one needs to access all elements of the domain of definition and corresponding fuzzy distribution values. All these properties are formulated for fuzzy distributions but can be also used for describing their forms of representation, e.g., for characterizing fuzzy disjunctions.

The maximal value that the fuzzy distribution takes over the domain of definition is said to be the consistency (Fig. 3).

For example, the local distribution {0, 0.1, 0.3, 0.7} has the consistency 0.7, and the consistency of the disjunction {0, 0.5}.{1, 0, 0} is equal to 1.

The minimal value that fuzzy distribution takes over the domain of definition is said to be the constant (Fig. 3).

Fig. 3. Consistency and constant of the distribution.

For example, the local distribution {0.1, 0.5, 1} and the disjunction {0, 0.5}.{1, 0, 0} have the constants 0.1 and 0 respectively.

We will need a quantity called a degree of incomparability of two distributions. Let us define at first a relative degree of incomparability. The degree of incomparability of the distribution P in relation to Q is equal to the maximal value of the distribution P which is exactly greater than the corresponding (i.e., in the same point of the universe) value of the proposition Q:

incomp_Q(P) = max(P(x)), where P(x) > Q(x)

Thus in order to compute this quantity one at first needs to select in P all the values which are exactly greater than the corresponding values in Q, and then to choose among them the maximal value (Fig. 4).

Fig. 4. Relative degree of incomparability.

In the case where the condition forall x P(x) <= Q(x)

holds, i.e., there is nothing to choose the maximal value from (the distribution P is included into Q), it is supposed by definition that incomp_Q(P)=0.

For example, degree of incomparability of the proposition P={0, 0.5, 0.7, 1} in relation to the proposition Q={0.2, 0.4, 0.6, 1} is equal to incomp_Q(P)=max(0.5,0.7)=0.7, whereas incomp_P(Q)=max(0.2) = 0.2.

The (mutual) degree of incomparability is equal to the minimal of two relative degrees of incomparability:

incomp(P,Q) = min( incompQ(P),incompP(Q))

Note that it is important that the degree of incomparability is defined not from informal interpretation of the word "incomparable" but from the formal requirements of the fuzzy resolution what will be shown below.

Consequence relation on fuzzy distributions is defined in a traditional way. Namely, the distribution Q is said to be a logical consequence of the distribution P iff the condition

forall x P(x) <= Q(x)

holds, i.e., P is included into Q.

Obviously, if Q is a logical consequence of P then Q can be removed from a set axioms or theorems. In particular, disjunction which is a consequence of another disjunction can be removed from the matrix. The process of removing such disjunctions is called absorption.

Fuzzy Multi-dimensional Analysis	Alexandr A. Savinov
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