3,467 research outputs found
Automata and Differentiable Words
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
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
of finite words that have equal lengths, the system , while
evolution during (discrete) time, at a certain moment transforms into .
To every system , we put into a correspondence a family of continuous maps of a suitable non-Archimedean metric space
and show that the system is completely transitive if and only if the family
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 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
Let be an Anosov diffeomorphism of the -dimensional torus
and a continuous self-mapping of
commuting with . We prove that is surjective if and only if the
restriction of to each homoclinicity class of is injective.Comment: 9 page
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