Abstract
In this contribution, we introduce the concepts of logical entropy and logical mutual information of experiments in the intuitionistic fuzzy case, and study the basic properties of the suggested measures. Subsequently, by means of the suggested notion of logical entropy of an IF-partition, we define the logical entropy of an IF-dynamical system. It is shown that the logical entropy of IF-dynamical systems is invariant under isomorphism. Finally, an analogy of the Kolmogorov-Sinai theorem on generators for IF-dynamical systems is proved.
Highlights
The notions of entropy and mutual information are basic notions in information theory [1] and, as is known, the customary approach is based on Shannon’s entropy [2]
It is shown that the logical entropy of IF-dynamical systems is invariant under isomorphism
If ξ = { A1, . . . , A I }, η = B1, . . . , B J are two IF-partitions of F, the conditional logical entropy of ξ assuming a realization of the IF-experiment η is defined by the formula: I
Summary
The notions of entropy and mutual information are basic notions in information theory [1] and, as is known, the customary approach is based on Shannon’s entropy [2]. In [6], we generalized the Kolmogorov–Sinai entropy concept to the case of a fuzzy probability space [7]. Since in the fuzzy case the inequality μ A ≤ μ B implies νA = 1Ω − μ A ≥ 1Ω − μ B = νB , in the family F it is natural to define the relation ≤ as follows: if A = (μ A , νA ), and B = (μ B , νB ) are two IF-events, A ≤ B if and only if μ A ≤ μ B , and νA ≥ νB.
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