Symmetric I* Restriction Method of Fuzzy Inference

Yiming Tang,Guangqing Bao

doi:10.11648/j.mcs.20190406.14

Abstract

As one of important parts of fuzzy logic, fuzzy inference plays a vital role in the fields of fuzzy control, artificial intelligence, affective computing, image processing and so forth. Two key problems of fuzzy inference are FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens). How to get the ideal solution for FMP and FMT is a difficult problem in the area of fuzzy logic. Aiming at such problem, from the idea of symmetric implicational reasoning, triple I* method and restriction theory, we put forward and investigate the α-symmetric I* restriction method, and then generalize it to the α(x,y)-symmetric I* restriction method. To begin with, the α-symmetric I* restriction principle and the α(x,y)-symmetric I* restriction principle are established. Furthermore, the equivalent condition to let a basic restriction solution exist is given. Then the unified solutions of the α-symmetric I* restriction method and the α(x,y)-symmetric I* restriction method are achieved for R-implications and (S, N)-implications. Besides, some special cases of optimal solutions are shown. Finally, the corresponding conclusions are provided when the two methods degenerate into the α-triple I* restriction method and α(x,y)-triple I* restriction method. These research results would be an important improvement for the fields of fuzzy inference, fuzzy logic and related applications.

Highlights

Nowadays fuzzy inference plays a vital role in the fields of fuzzy control, artificial intelligence, affective computing, image processing and so on [1,2,3,4,5]
We verify that C is an α(x,y)-symmetric I* restriction solution, i.e., the following formula holds for any x∈ X, y∈Y : ( A(x) →1 B( y)) →2 (B* ( y) →1 C(x)) < α (x, y)
We consider the idea of symmetric implicational reasoning, triple I* method and restriction theory, we put forward the α-symmetric I* restriction method and the α(x,y)-symmetric I* restriction method

Summary

Introduction

Nowadays fuzzy inference plays a vital role in the fields of fuzzy control, artificial intelligence, affective computing, image processing and so on [1,2,3,4,5]. The triple I method shows several good properties, which includes strict logical basis, continuity, reversibility, robustness, and so on Regarding this topic, Pei proposed the triple I* method of FMT [24] from the perspective of another kind of reversibility, which focused on ( A(x) → B( y)) → (B* ( y) → A* (x)). The first and third fuzzy implications in (3) correspond to the implication connective in a logic system; and the second fuzzy implication in (3) reflects the “if-” relation of fuzzy inference model Based upon this idea, we extend (3) as follows:. We think about all of these formulas including (4), (5) and (6), a new fuzzy inference method called the α-symmetric I* restriction method is proposed, which focuses on ( α ∈ (0,1] ). The aim of this study is to research the α-symmetric I* restriction method and the α(x,y)-symmetric I* restriction method

Preliminaries

Basic Structure

Optimal Solutions

Conclusions