The Non-Archimedean Theory of Discrete Systems

In the paper, we study behavior of discrete dynamical systems (automata) w.r.t. transitivity; that is, speaking loosely, we consider how diverse may be behavior 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 maps 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 analyze behavior 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 maps of a (non-Archimedean) metric space $\mathbb Z_2$ of 2-adic integers.Comment: The extended version of the talk given at MACIS-201

Digital algebra and circuits

Abstract. Digital numbers D are the world’s most popular data representation: nearly all texts, sounds and images are coded somewhere in time and space by binary sequences. The mathematical construction of the fixed-point D � Z2 and floating-point D ′ � Q2 digital numbers is a dual to the classical constructions of the real numbers R. The domain D ′ contains the binary integers N and Z, as well as Q. The arithmetic operations in D ′ are the usual ones when restricted to integers or rational numbers. Similarly, the polynomial operations in D ′ are the usual ones when applied to finite binary polynomials F2[z] or their quotients F2(z). Finally, the set operations in D ′ are the usual ones over finite or infinite subsets of N. The resulting algebraic structure is rich, and we identify over a dozen rings, fields and Boolean algebras in D ′. Each structure is well-known in its own right. The unique nature of D ′ is to combine all into a single algebraic structure, where operations of different nature happily mix