Logical Entropy of Fuzzy Dynamical Systems

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.

Logical Entropy of Dynamical Systems—A General Model

“…In the paper by Markechová and Riečan [23], the notions of logical entropy and logical mutual information of fuzzy partitions in the given fuzzy probability space were introduced and studied. By means of the suggested concept of entropy of fuzzy partitions the authors have defined the logical entropy of fuzzy dynamical systems.…”

“…Thereby they obtained a new tool for distinction of non-isomorphic fuzzy dynamical systems, since, as it is proved in Theorem 12 of the cited paper, the logical entropy of fuzzy dynamical systems coincides on isomorphic fuzzy dynamical systems. It is noted that the definition of fuzzy dynamical systems (Ω, F, m, τ) considered in [23] is based on Zadeh's connectives.…”

“…The recently published paper [23] deals with studying logical entropy of fuzzy partitions in a fuzzy probability space and logical entropy of fuzzy dynamical systems.…”

“…In [21] Ebrahimzadeh defined and studied the notions of logical entropy of partitions and logical entropy of dynamical systems in quantum logics and in [22] the author introduced the notion of conditional logical entropy of dynamical systems on quantum logics. In the recently published paper [23], Markechová and Riečan deal with the study of logical entropy of fuzzy partitions and fuzzy dynamical systems.…”

“…We define the logical entropy and logical mutual information of finite partitions on an appropriate algebraic structure and prove basic properties of these measures. As a special case we obtain the logical entropy of fuzzy partitions introduced in [23]. It is noted that some investigations concerning entropy of dynamical systems and related notions in the above setup were conducted in [24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”