Fuzzy Multi-dimensional Analysis
Alexandr A. Savinov
[Home] [Contents] [Previous] [Next] [E-mail] 
[0] [1] [2] [3] [4] [5] [6] [7] [8]

1. Introduction

Let us consider the following problem. Given a n-dimensional space called the universe of discourse equal to the Cartesian product of  n  variables. There is some (global) distribution over this space which is supposed to be represented by means of a combination of elementary (local) distributions over individual variables. A global characteristic of the distribution is said to be some quantity which depends on the values in all (or almost all) points of the universe of discourse. The problem is how to calculate a global characteristic of the distribution without the necessity to access values in all points of the universe of discourse, i.e., taking into account only local distributions over individual variables by means of which the multi-dimensional distribution is represented..

Such a formulation is obviously too general therefore to obtain concrete results we have to reduce it to some more concrete case by specifying more exactly types of variables, their sets of values, operations used for combining distributions, global characteristics to find.

One such particular but probably the most important case has been paid a lot of attention in Logic, Algebra, Switching Theory, Cybernetics, Artificial Intelligence, and other fields where it usually has its own name and is described in special terms depending on the problem being solved. The main assumptions for this case are that

  1. all variables have only two values 0 and 1,
  2. distributions take values from the set {0,1},
  3. logical connectives AND and OR are used for combining elementary distributions, and
  4. the maximal or minimal value is usually searched as a global characteristic.
There are only 4 different two-valued distributions over two-valued variables which in propositional logic are called elementary propositions (Fig. 1):
  1. truth constant 0,
  2. truth constant 1,
  3. proposition P, and
  4. proposition ~ P.
Fig. 2. Four Bollean propositions. Both the variable and the distribution are two valued.
Fig. 1. Four Boolean propositions. Both the variable and the distribution are two-valued. 
Combining different local distributions with logical connectives AND and OR which are interpreted with the help of conventional truth tables we can represent different global distributions over n-dimensional hyper-cube. One traditional problem that many other theoretical and applied problems are reduced to, is the problem of satisfiability which is obviously equivalent to finding the maximal value of the global distribution over the n-dimensional universe of discourse. There is a lot of different methods and their modifications for solving this problem, e.g., based on the operation of consensus (resolution in the Artificial Intelligence), transformation into the dual form, covering techniques etc.

In this paper it is supposed that

  1. all variables take their values from finite sets,
  2. all distributions are fuzzy membership functions from the domain of definition (values of individual variables or their Cartesian product) to the unit interval [0,1], and
  3. logical connectives AND and OR interpreted with the help of the minimum and maximum operations are used for combining distributions.
In fact, only the supposition 3 is principal. Instead of the finite sets of values of variables we can consider the unit interval [0,1] (assumption 1) or any other continuos interval. Maximal value of the global distribution over the universe of discourse is considered as a global characteristic to find. We also consider a more general problem of finding a projection of the global distribution on some variable which allows us to solve more efficiently the problem of logical inference. There can be formulated also other useful problems, e.g., finding a global entropy of a multi-dimensional fuzzy distribution.

Currently there are not exact methods for solving this problem. However, there have been proposed a lot of inexact methods in the field of approximate reasoning (mainly for application to knowledge based systems). Probably the most well-known of them is the Zadeh's combination and projection principle [Zd75, Zd79]. This method consists in that

  1. each statement is translated into a possibility distribution,
  2. all possibility distributions are conjunctively (with the help of minimum operation) combined into an overall possibility distribution PI,
  3. the distribution PI is projected on various variables of interest (for instance, using the generalized modus ponens).
Unfortunately, it is only a principle and does not provide us a concrete procedure for finding projections. The main disadvantage of other approaches see, e.g., [KS90]) is that they do not guarantee that the conclusion (projection) obtained is correct like similar methods in the Boolean fields (Switching Theory, Boolean functions, Propositional Logic etc.). In other words, we do not know whether the projection resulted from the procedure is equal to the real projection of our distribution.

In this paper we propose a new original operation of fuzzy resolution which be can used to solve this problem. We will suppose that the global distribution is represented by means of a number of fuzzy disjunctions combined with the connective AND (minimum). Each fuzzy disjunction consist of  n  local distributions (possibly trivial) combined with the connective OR. The operation of fuzzy resolution is applied to any two disjunctions on some variable and results in a third disjunction called resolvent (consensus). The resolvent possesses several useful properties (described in below in the paper) which allows us to say that this operation is a generalization of the conventional resolution. Thus having this fuzzy resolution we can more or less easily transfer onto fuzzy case almost all resolution (consensus) based methods developed for the boolean case.

This paper originates from an original approach of Zakrevsky [Zk89, Zk94] called the logic of finite predicates where a new technique of sectioned boolean vectors for representing disjunctions and the corresponding consensus operation was proposed. On the basis of the technique of Boolean sectioned vectors and matrices a diagnostic expert system shell EDIP was implemented [L90, LS91, LS92]. An inference process in the system EDIP is based on the procedure of finding all prime vector disjunctions by means of the operation of generalized consensus.

Later [S91, S93a, LS93a, S93b] the formalism of Zakrevsky including the technique of sectioned vectors and the operation of consensus was generalized on fuzzy case where the components take their values from the unit interval [0,1]. In addition some new properties degenerated in the crisp case were studied, as well as new procedures of logical inference were developed which underlie an expert system shell EDIP for Windows 3.x [S96b]. In this generalization of the formalism of Zakrevsky instead of the term 'consensus' the term 'resolution' was used, which is conventional in the Artificial Intelligence.

This approach to fuzzy multi-dimensional analysis was reformulated in logical terms as a generalized fuzzy propositional logic [S93b, S93c] and a logic of possibility distributions [S96a]. It was also applied to such fields as diagnosis [LS94a], fuzzy control [LS94b], aggregation of information [LS94c, LS96].
 

Fuzzy Multi-dimensional Analysis
Alexandr A. Savinov
[Home] [Contents] [Previous] [Next] [E-mail] 
[0] [1] [2] [3] [4] [5] [6] [7] [8]
© Copyright 1997, Alexandr A. Savinov