3,467 research outputs found

    Automata and Differentiable Words

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    We exhibit the construction of a deterministic automaton that, given k > 0, recognizes the (regular) language of k-differentiable words. Our approach follows a scheme of Crochemore et al. based on minimal forbidden words. We extend this construction to the case of C\infinity-words, i.e., words differentiable arbitrary many times. We thus obtain an infinite automaton for representing the set of C\infinity-words. We derive a classification of C\infinity-words induced by the structure of the automaton. Then, we introduce a new framework for dealing with \infinity-words, based on a three letter alphabet. This allows us to define a compacted version of the automaton, that we use to prove that every C\infinity-word admits a repetition in C\infinity whose length is polynomially bounded.Comment: Accepted for publicatio

    The Non-Archimedean Theory of Discrete Systems

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    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,ba,b of finite words that have equal lengths, the system A\mathfrak A, while evolution during (discrete) time, at a certain moment transforms aa into bb. To every system A\mathfrak A, we put into a correspondence a family FA\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 FA\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 Z2\mathbb Z_2 of 2-adic integers.Comment: The extended version of the talk given at MACIS-201

    A Garden of Eden theorem for Anosov diffeomorphisms on tori

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    Let ff be an Anosov diffeomorphism of the nn-dimensional torus Tn{\mathbb{T}}^n and Ď„\tau a continuous self-mapping of Tn{\mathbb{T}}^n commuting with ff. We prove that Ď„\tau is surjective if and only if the restriction of Ď„\tau to each homoclinicity class of ff is injective.Comment: 9 page
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