New Fuzzy Resolution Operation
Alexandr A. Savinov
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New Fuzzy Resolution Operation

Alexandr A. Savinov
Institute of Mathematics
Moldavian Academy of Sciences
str. Academiei 5
MD-2028 Kishinev
Moldavia
September 1, 1997

Abstract

In the paper a new original fuzzy resolution principle is described. This operation is applied to any two disjuncts on some variable and results in a third disjunct called resolvent. It is shown that the resolution rule proposed coincides with the classical resolution for the case of Boolean propositions, i.e., two valued crisp propositions. It is shown how this operation can be used to solve several theoretical and applied problems.

Key words: Fuzzy resolution; Fuzzy logic; Fuzzy inference.

1. Multi-dimensional Space

Let us suppose that there are n variables x1,x2,...,xn taking its values in the sets X1,X2,...,Xn called domains respectively (Fig. 1).
Fig. 1. There are n variables.
Fig. 1. There are n variables. 
In general domains are supposed to be equal to any set (including continous interval [0,1]), but when they are finite their powers are equal to n1,n2,...,nn. In particular, boolean variables have only two values 0 and 1, i.e., Xi={0,1} and ni=2 forall i.

The universe of discourse O is equal to the Cartesian product of all sets of the values (Fig. 2)

O = X1×X2×...×Xn

and consists of n-tuples

o=<x1,x2,...,xn> in O

In fininte case its power is equal to n1×n2×...×nn. In the case of boolean variables the universe of discourse is equal to the n-dimensional hypercube with the power 2n.

Fig. 2. The universe of discourse 
Fig. 2. The universe of discourse is equal to the Cartesian product of all sets of the values. 

2. Fuzzy Distributions and their Representation

Fuzzy distribution is a function from the domain of definition to the interval [0,1]. We consider fuzzy distributions over the domains which are refered to as local (or one-dimensional) distributions and over the universe of discourse which are referred to as global (or multi-dimensional) distributions.

Local distributions are supposed to be represented only extensionally, i.e., by enumerating its values in all points of the domain (in finite case). Global (multi-dimensional) distributions are supposed to be represented with the help of local distributions and operations between them. In other words, global distributions can be represented by combining local distributions with some operation.

As an operation for combining local distributions we use maximum which is written as OR. To compute the value of a global distribution in some point of the universe of discourse we have to take maximum of the values of all local distributions in the corresponding local points (Fig. 3). If u1,u2,...,un are local distributions over x1,x2,...,xn and u=u1ORu2OR...ORun is representation of some global distribution then its value in the point o=<x1,x2,...,xn> in O is equal

u(o) = max( u1(x1),u2(x2),...,un(xn) )

Such representation of global distributions by means of local distributions and the operation of maximum is called fuzzy disjunct.

Fig. 3. Representation of global distributions by disjuncts. 
Fig. 3. Representation of global distributions by disjuncts. 

3. Fuzzy resolution

An operation of fuzzy resolution is applied to any two disjuncts on some variable and results in a third disjunct called the resolvent. If u and v are two disjuncts and w is their resolvent on k-th variable, then we write:
u<xk>v = w

where <xk> denotes the resolution on k-th variable.

Each local distribution of the resolvent depends on (is constructed from) only two corresponding local distributions of the premises. k-th local distribution of the resolvent (which the resolution is applied to) is equal to the conjunction of the two corresponding distribution from the source disjuncts; every non-k-th distribution of the resolvent is equal to the disjunction of the two corresponding propositions:

Conjunction and disjunction of local distributions are equal to the pointwise minimum and maximum, respectively.

The resolution operation can be represented in the form of the following pattern (Fig. 4):

x1
max
xk
min
xn
max
u 
u1 
-/-/- 
uk 
-/-/- 
un 
v 
v1 
-/-/- 
vk 
-/-/- 
vn 
w = u<xk>v 
u1 OR v1 
-/-/- 
uk AND vk 
-/-/- 
un OR vn 
 
Fig. 4. Operation of fuzzy resolution. 
Fig. 4. Operation of fuzzy resolution. 
Here are two examples of applying the resolution:
u 
 {0, 0.1, 0.2, 1} {0, 1}
v 
 {1, 0.3, 0.5, 0} {0, 0}
w = u<x1>v 
 {0, 0.1, 0.2, 0} {0, 1}
 
u 
 {0, 1} {0, 1, 0.7} {1, 0.2, 1}
v 
 {1, 0} {1, 1, 0.2} {0, 0.1, 0}
w = u<x2>v 
 {1, 1} {0, 1, 0.2} {1, 0.2, 1}

4. Some Properties of the Resolution

The main property of the resolvent is that it is a consequence of two its premises:
If w = u <xk> v, then u AND v |= w.

It means that we can any resolvent add to the source disjuncts and the the whole global distribution will not change.

In the boolean case the behaviour of this resolution coinsides with that for the classical case of boolean propositions. Here are three examples of applying the resolution for this case.

u 
 {0, 1}=x1 {0, 1}=x2 {0, 0}=0
v 
 {0, 0}=0 {1, 0}=~ x2 {0, 1}=x3
w = u<x2>v 
 {0, 1}=x1 {0, 0}=0 {0, 1}=x3
 
u 
 {0, 1}=x1 {0, 1}=x2 {0, 0}=0
v 
 {0, 0}=0 {0, 1}=x2 {0, 1}=x3
w = u<x2>v 
 {0, 1}=x1 {0, 1}=x2 {0, 1}=x3
 
u 
 {0, 1}=x1 {0, 1}=x2 {0, 0}=0
v 
 {1, 0}=0 {1, 0}=~ x2 {0, 1}=x3
w = u<x2>v 
 {1, 1}=1 {0, 0}=0 {0, 1}=x3
 
 
New Fuzzy Resolution Operation
Alexandr A. Savinov
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© Copyright 1997, Alexandr A. Savinov