Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

When is the convolution a t-norm on normal, convex and upper semicontinuous fuzzy truth values?

Jie Sun

TL;DR

This paper characterizes when the convolution $\ast_{\vartriangle}$ yields a t-norm on the lattice of normal convex and upper semicontinuous fuzzy truth values $(L_u,\sqsubseteq)$. It establishes a necessary and sufficient condition: $\ast_{\vartriangle}$ is a t-norm on $(L_u,\sqsubseteq)$ if and only if the base t-norm $\ast$ on $[0,1]$ is continuous and the t-conjunction $\vartriangle$ on $[0,1]$ is right-continuous. The sufficiency and necessity are proven in separate sections, leveraging $t_r$-norm concepts and the convolution framework, with the result requiring $\ast$ to be continuous and $\vartriangle$ to be right-continuous. The finding extends a prior result on $(L,\sqsubseteq)$ to the broader $(L_u,\sqsubseteq)$ setting, providing a practical blueprint for constructing t-norms to support Type-2 rule-based fuzzy systems.

Abstract

In Type-2 rule-based fuzzy systems (T2 RFSs), triangular norms on complete lattice $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$ can be used to model the compositional rule of inference, where $\textbf{L}$ is the set of all convex normal fuzzy truth values, $\mathbf{L_u}$ is the set of all convex normal and upper semicontinuous fuzzy truth values, and $\sqsubseteq$ is the so-called convolution order. Hence, the choice of t-norms on $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$ may influence the performance of T2 RFSs, and thus, it is significant to broad the set of t-norms among which domain experts can choose most suitable one. To construct t-norms on $(\mathbf{L},\sqsubseteq)$ or $(\mathbf{L_u},\sqsubseteq)$, the mainstream method is based on convolution $\ast_\vartriangle$ induced by two operators $\ast$ and $\vartriangle$ on the unit interval $[0,1]$. Recently, we have complete solve the question when convolution $\ast_\vartriangle$ is a t-norm on $(\mathbf{L},\sqsubseteq)$. This paper aim to provide the necessary and sufficient conditions under which convolution $\ast_\vartriangle$ is a t-norm on $(\mathbf{L_u}, \sqsubseteq)$.

When is the convolution a t-norm on normal, convex and upper semicontinuous fuzzy truth values?

TL;DR

This paper characterizes when the convolution yields a t-norm on the lattice of normal convex and upper semicontinuous fuzzy truth values . It establishes a necessary and sufficient condition: is a t-norm on if and only if the base t-norm on is continuous and the t-conjunction on is right-continuous. The sufficiency and necessity are proven in separate sections, leveraging -norm concepts and the convolution framework, with the result requiring to be continuous and to be right-continuous. The finding extends a prior result on to the broader setting, providing a practical blueprint for constructing t-norms to support Type-2 rule-based fuzzy systems.

Abstract

In Type-2 rule-based fuzzy systems (T2 RFSs), triangular norms on complete lattice or can be used to model the compositional rule of inference, where is the set of all convex normal fuzzy truth values, is the set of all convex normal and upper semicontinuous fuzzy truth values, and is the so-called convolution order. Hence, the choice of t-norms on or may influence the performance of T2 RFSs, and thus, it is significant to broad the set of t-norms among which domain experts can choose most suitable one. To construct t-norms on or , the mainstream method is based on convolution induced by two operators and on the unit interval . Recently, we have complete solve the question when convolution is a t-norm on . This paper aim to provide the necessary and sufficient conditions under which convolution is a t-norm on .
Paper Structure (10 sections, 24 theorems, 70 equations)

This paper contains 10 sections, 24 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Triangular norms
  4. Fuzzy truth values
  5. Convolutions
  6. $t_r$-norms
  7. Main Results
  8. Sufficiency
  9. Necessity
  10. Conclusion

Key Result

Proposition 2.2

A t-norm $\ast:[a,b]^2\to[a,b]$ is continuous (left continuous, right continuous) if and only if for each $x\in[a,b]$, the section $x\ast(-):[a,b]\to [a,b]$ is continuous (left continuous, right continuous).

Theorems & Definitions (42)

  • Example 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • ...and 32 more