Comparison of solutions resulted from direct problems formulated as FRE

TL;DR

This paper examines the direct solutions of fuzzy relation equations (FRE), compares different t-norms and t-conorms, and identifies conditions for solutions aligning with human reasoning and real-world data.

Contribution

It introduces conditions ensuring FRE solutions match human intuition and proposes a new t-conorm for better real-world similarity.

Findings

FRE with maximum t-conorm and minimum t-norm can differ from human reasoning.

Necessary and sufficient conditions for solution alignment are provided.

A new t-conorm is proposed to improve solution similarity to real data.

Abstract

In this paper, we investigate direct solution of FRE and compare their results with expected real consequences. We give an applied example formulated by a FRE problem and show that FRE defined by maximum t-conorm and an arbitrary t-norm can yield different interpretations for our example. A necessary condition and a sufficient condition are presented that guarantee FRE defined by maximum t-conorm and minimum t-norm attain the same solutions as human mind does. Also, we present a t-conorm and use it instead of maximum t-conorm in FRE to obtain solutions with highest similarity with real ones. Finally, we show that under some conditions, FRE defined by any t-conorm and any t-norm may find a solution which is not reasonable.