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The Non-Archimedean Theory of Discrete Systems

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Abstract

In the paper, we study behaviour of discrete dynamical systems (automata) w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be behaviour of the system w.r.t. variety of word transformations performed by the system: we call a system completely transitive if, given arbitrary pair a, b of finite words that have equal lengths, the system \({\mathfrak{A}}\) , while evolution during (discrete) time, at a certain moment transforms a into b. To every system \({\mathfrak{A}}\) , we put into a correspondence a family \({\mathcal{F}_{\mathfrak{A}}}\) of continuous mappings of a suitable non-Archimedean metric space and show that the system is completely transitive if and only if the family \({\mathcal{F}_{\mathfrak{A}}}\) is ergodic w.r.t. the Haar measure; then we find easy-to-verify conditions the system must satisfy to be completely transitive. The theory can be applied to analyse behaviour of straight-line computer programs (in particular, pseudo-random number generators that are used in cryptography and simulations) since basic CPU instructions (both numerical and logical) can be considered as continuous mappings of a (non-Archimedean) metric space \({\mathbb{Z}_{2}}\) of 2-adic integers.

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Correspondence to Vladimir Anashin.

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Supported in parts by Russian Foundation for Basic Research Grant No. 12-01-00680-a and by Chinese Academy of Sciences visiting professorship for senior international scientists Grant No. 2009G2-11.

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Anashin, V. The Non-Archimedean Theory of Discrete Systems. Math.Comput.Sci. 6, 375–393 (2012). https://doi.org/10.1007/s11786-012-0132-7

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