Some novel fuzzy logic operators with applications in fuzzy neural networks

Abstract

T-norms, t-conorms, uninorms, grouping functions, overlap functions, etc., are important fuzzy logic operators, they have been widely used in fuzzy reasoning, fuzzy control, information fusion, intelligent decision-making and fuzzy neural network. Recently, as a unified form of 1-grouping functions and 0-overlap functions, the new concept of $\Theta -\Xi$ function has been proposed. It is a new class of fuzzy logic operators with strong expressive power. However, we find that the parameter k in $\Theta -\Xi$ functions only belongs to {0,1} rather than [0,1], which limits their application scope. This article first delves into the characteristics of $\Theta -\Xi$ functions and provides several new construction theorems for $\Theta -\Xi$ functions. Then, more extensive OG-functions are proposed, proving that OG-functions are joint extension of the general grouping functions and general overlap functions. Multiple methods for constructing OG-functions are provided, and the structural theorem of OG-functions is proved (i.e., the necessary and sufficient conditions for generating OG-functions from “continuous symmetric nondecreasing function pairs”). Thirdly, OG-functions are extended to ($a,b$)-OG functions, and a novel neuron model based on ($a,b$)-OG functions (OG-neuron) is proposed for the first time. We also demonstrate OG-neurons have stronger approximation ability than traditional MP neurons (a single OG-neuron can achieve XOR operation). Finally, we establish novel artificial neural network OG-ANN and convolutional neural network OG-CNN. Comparative experimental results show that the introduction of ($a,b$)-OG functions improves the classification accuracy of neural networks by 5.23%, 6.02%, 7.77% in mnist, cifar10 and fashion datasets, respectively.