Abstract
Recently the logical entropy was suggested by D. Ellerman (2013) as a new information measure. The present paper deals with studying logical entropy and logical mutual information and their properties in a fuzzy probability space. In particular, chain rules for logical entropy and for logical mutual information of fuzzy partitions are established. Using the concept of logical entropy of fuzzy partition we define the logical entropy of fuzzy dynamical systems. Finally, it is proved that the logical entropy of fuzzy dynamical systems is invariant under isomorphism of fuzzy dynamical systems.
Highlights
The classical approach in information theory [1] is based on Shannon’s entropy [2]
The aim of this paper is to study the logical entropy in fuzzy probability spaces and fuzzy dynamical systems
We introduced the notion of logical entropy of fuzzy partition of a given fuzzy probability space
Summary
The classical approach in information theory [1] is based on Shannon’s entropy [2]. Using. Shannon entropy Kolmogorov and Sinai [3,4] defined the entropy hpTq of dynamical systems. In the paper by Markechová [5] the Shannon entropy of fuzzy partitions has been defined This concept was exploited to define the Kolmogorov-Sinai entropy hm of fuzzy dynamical systems [6]. The aim of this paper is to study the logical entropy in fuzzy probability spaces and fuzzy dynamical systems. Entropy 2016, 18, 157 invariant under isomorphism of fuzzy dynamical systems (Theorem 12) In this way, we obtained a new tool for distinction of non-isomorphic fuzzy dynamical systems; this result is demonstrated by Example 4.
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