Logical Divergence, Logical Entropy, and Logical Mutual Information in Product MV-Algebras.

Dagmar Markechová,Abolfazl Ebrahimzadeh,Batool Mosapour

doi:10.3390/e20020129

Abstract

In the paper we propose, using the logical entropy function, a new kind of entropy in product MV-algebras, namely the logical entropy and its conditional version. Fundamental characteristics of these quantities have been shown and subsequently, the results regarding the logical entropy have been used to define the logical mutual information of experiments in the studied case. In addition, we define the logical cross entropy and logical divergence for the examined situation and prove basic properties of the suggested quantities. To illustrate the results, we provide several numerical examples.

Highlights

In all areas of empirical research, it is very important to know how much information we gain by the realization of experiments
For some recent works related to the concept of logical entropy on algebraic structures based on fuzzy set theory, we refer the reader to [58,59,60,61,62,63,64,65]
Using the notion of logical conditional mutual information, we present chain rules for logical mutual information in product MV-algebras

Summary

Introduction

In all areas of empirical research, it is very important to know how much information we gain by the realization of experiments. We note that in the recently published paper [48], the results regarding the Shannon entropy of partitions in product MV-algebras were exploited to define the notions of Kullback-Leibler divergence and mutual information of partitions in product MV-algebras. For some recent works related to the concept of logical entropy on algebraic structures based on fuzzy set theory, we refer the reader to (for example) [58,59,60,61,62,63,64,65]. The purpose of this article is to extend the study of logical entropy provided in [50] to the case of product MV-algebras. It is shown that by replacing the Shannon entropy function (Equation (1)) by the logical entropy function (Equation (2)) we obtain the results analogous to the results given in [48]

Preliminaries

Logical Entropy of Partitions in Product MV-Algebras

Logical Mutual Information in Product MV-Algebras

Logical Cross Entropy and Logical Divergence in Product MV-Algebras

Findings

Conclusions