#### Publication Date

7-1-2023

#### Abstract

To adequately represent human reasoning in a computer-based systems, it is desirable to select fuzzy operations that are as close to human reasoning as possible. In general, every real-valued function can be approximated, with any desired accuracy, by polynomials; it is therefore reasonable to use polynomial fuzzy operations as the appropriate approximations. We thus need to select, among all polynomial operations that satisfy corresponding properties -- like associativity -- the ones that best fit the empirical data. The challenge here is that properties like associativity mean satisfying infinitely many constraints (corresponding to infinitely many possible triples of values), while most effective optimization techniques assume that the number of equality or inequality constraints is finite. Thus, it is desirable to find, for each corresponding family of infinitely many constraints, an equivalent finite set of constraints. Such sets have been found for many fuzzy operations -- e.g., for implication operations represented by polynomials of degree 4. In this paper, we show that such equivalent finite sets always exist, and we describe an algorithm for generating these sets.

## Comments

Technical Report: UTEP-CS-23-30

To appear in

Proceedings of the 20th World Congress of the International Fuzzy Systems Association IFSA'2023, Daegu, South Korea, August 20-24, 2023.