2. Method of Sectioned Vectors and Matrices

Let  x₁,x₂,...,x_n  be elementary logical variables, each of them taking its values from the sets  X₁,X₂,...,X_n  called (elementary) domains respectively. In general, domains are supposed to be any continuos interval but in examples for the sake of simplicity we will consider only the case of domains consisting of a finite number of values  a_ij, where  i=1,2,...,n, and  j=1,2,...,n_i. The Cartesian product of all domains  X₁×X₂×...×X_n  forms the universe of discourse  O  with the power in the finite case  n₁×n₂×...×n_n. Each element  o=<x₁,x₂,...,x_n> in O  is an ordered n-tuple of the values of all variables.

Below in this section we describe a technique which we use to write fuzzy disjunctions. This technique was originally proposed by Zakrevsky for the case of multi-valued variables and two-valued distributions. Later it was generalized by Savinov onto the case of fuzzy distributions. In the method of sectioned vectors we use the following terminology:

• component  v_ij  of the vector  v  is one number which is equal to the local distribution value in one point of the domain; in the case of crisp distributions it is equal either 0 or 1; in the fuzzy case it takes values from the interval [0,1]; in other words, the component  v_ij  is equal to the local distribution value in the point  x_i=a_ij, where  a_ij  is  j-th value of  i-th  variable;
• section  v_i  is a sequence of  n_i  components for all values of one local distribution, e.g., if  v_i={0.7, 1, 0.2, 0}  then  v_i1=0.7, v_i2=1, v_i3=0.2, v_i4=0;
• vector  v  consists of  n  sections  v_i  separated by points (when the operation is implicitely implied) or explicitely by the name of operation, e.g.,  {0.7, 1, 0.2, 0}.{1, 0.4, 0}.{1, 0}, where  n₁=4, n₂=3, n₃=2.
• matrix is made up of a number of vectors each of them representing one line.

An interpretation of fuzzy vector is a rule by means of which we can compute the global distribution this vector defines over the universe of discourse proceeding from the local distributions the vector is made up. There are two interpretations of fuzzy vectors: as disjunctions and as conjuncts. If the vector  d  is interpreted as disjunction then the value of its global distribution in some point is equal to the maximum of  n  corresponding components (Fig. 2).

Fig. 2. Interpretation of the vector as disjunction by means of the maximum of n corresponding components. 

For example, the disjunction

defines the distribution which in the point  <a₁₃,a₂₂,a₃₂>  is equal to  max(0.2, 0.4, 0)=0.4.

The interpretation of sectioned vectors as conjuncts is dual, i.e., it uses the operation of minimum.

Sectioned matrices have two interpretations: as conjunctive normal form (CNF) and as disjunctive normal form (DNF). The interpretation as CNF means that the value in some point of the universe of discourse is equal to the minimum of the values which are assigned to this point by its lines. The lines of the CNF are interpreted as disjunctions.

It can be easily proved that any fuzzy distribution over the universe of discourse can be represented in the form of fuzzy CNF. Such a CNF is made up of  |O|=n₁×n₂×...×n_n  line disjunctions each of them representing a value of the fuzzy distribution in the corresponding point of the universe of discourse. In other words one line of this matrix is responsible for representing the distribution value in some point and it does not influence all other points. Each section of the disjunction satisfying this condition has to consist of all 1's except for one component which is equal to the corresponding distribution value. We say that it pricks a hole down to the necessary level in the distribution surface. Of course, it is not a procedure for building and representing fuzzy multi-dimensional distributions -- it only demonstrates that for any arbitrary fuzzy distribution there exists a sectioned matrix interpreted as CNF which represents it (i.e., with the same semantics).

© Copyright 1997, Alexandr A. Savinov