Fuzzy Multi-dimensional Analysis

Abstract. In this paper a new original approach to the analysis of fuzzy multi-dimensional distributions is described. A uniform method for representing the fuzzy multi-dimensional distributions by means of sectioned vectors and matrices is proposed. The sectioned matrix is interpreted as fuzzy conjunctive normal form, while its line vectors are interpreted as fuzzy disjunctions. Several useful characteristics of fuzzy distributions and disjunctions are defined and studied. The main operation for manipulating by fuzzy multi-dimensional distributions is a new original fuzzy resolution which is applied to any two disjunctions on some variable and results in a third disjunction called resolvent. The property of adjacency of two disjunctions is defined and the criterion of adjacency is formulated. It is shown that the resolution operation proposed is a generalization of the conventional resolution and the whole approach can be viewed as a generalization of propositional logic. Methods for finding prime disjunctions, projection on a variable (and thus solving the satisfiability problem) and transforming into the dual form are proposed.

Key words: Fuzzy multi-dimensional analysis; Fuzzy distributions; Fuzzy resolution; Fuzzy prime disjunction; Fuzzy logic; Finding projections.

1. Introduction
2. Method of Sectioned Vectors and Matrices
3. Characteristics of Disjunctions
4. Reduced Forms of Disjunctions
5. Resolution Principle
5.1. General Definition
5.2. Resolution on Reduced Disjunctions
5.3. Adjacency of Disjunctions
6. Equivalent Transformations of Fuzzy Sectioned Matrices
6.1. Finding Prime Disjunctions
6.2. Transforming Matrix into the Dual Form
7. Finding Projections on Variables
8. Conclusion and References