Relative model of the logical entropy of sub-$\sigma_\Theta$-algebras

Document Type : Research Paper


Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran


‎In the context of observers‎, ‎any mathematical model according to the viewpoint of an observer $ \Theta $ is called a relative model‎. ‎The purpose of the present paper is to study the relative model of logical entropy‎. ‎Given an observer $ \Theta $‎, ‎we define the relative logical entropy and relative conditional logical entropy of a sub-$ \sigma_\Theta $-algebra having finitely many atoms on the relative probability $ \Theta‎- ‎$measure space and prove the ergodic properties of these measures‎. ‎Finally‎, ‎it is shown that the relative logical entropy is invariant under‎
‎the relation of equivalence modulo zero.


Main Subjects

[1] Xi. Dong, The gravity dual of Renyi entropy, Nature Communications, (2016), 1{6.
[2] D. Ellerman, An introduction to logical entropy and its relation to Shannon entropy, Int. J. Semant. Comput, 7 (2013), 121{145.
[3] A. Ebrahimzadeh, J. Jamalzadeh, Conditional logical entropy of fuzzy -algebras, J. Intell. Fuzzy Syst, 33 (2017), 1019{1026.
[4] U. Mohammadi, Observers and relative entropy of G-sets, Eur. Phys. J. Plus, (2020), 135:492.
[5] G. Manfredi, Logical Entropy { Special Issue, 4Open, 5 (2022).
[6] U. Mohammadi, Observational modeling of the Kolmogorov-Sinai entropy, Sahand Communications in Mathematical Analysis, 13 (2019), 101-114.
[7] D. Markechova, B. Riecan Logical entropy and logical mutual information of experiments in the intuitionistic fuzzy case, Entropy, 19 (2017), 1-19.
[8] M. R. Molaei, Mathematical modeling of observer in physical systems, Journal of Dynamical Systems and Geometric Theories, (2006) 183-186.
[9] M. R. Molaei and B. Ghazanfari, Relative entropy of relative measure preserving maps with constant observers, Journal of Dynamical Systems and Geometric Theories, (2007) 179-191.
[10] M. R. Molaei, Observational modeling of topological spaces, Choas, Solitons and Fractals, 42 (2009), 615-619.
[11] C.-R. Rao, Diversity and dissimilarity coecients, Auni ed approach, Theoretical Population Biology 21 (1982), 24{43.
[12] T.Waezizadeh, A. Mehrpooya, M. Rezaeizadeh, Sh. Yarahmadian, Mathematical models for the e ects of hypertension and stress on kidney and their uncertainty, Mathematical Biosciences, 305 (2018), 77-95.