2,281 research outputs found
Bounds for the discrete correlation of infinite sequences on k symbols and generalized Rudin-Shapiro sequences
Motivated by the known autocorrelation properties of the Rudin-Shapiro
sequence, we study the discrete correlation among infinite sequences over a
finite alphabet, where we just take into account whether two symbols are
identical. We show by combinatorial means that sequences cannot be "too"
different, and by an explicit construction generalizing the Rudin-Shapiro
sequence, we show that we can achieve the maximum possible difference.Comment: Improved Introduction and new Section 6 (Lovasz local lemma
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
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