607 research outputs found
On the reversibility and the closed image property of linear cellular automata
When is an arbitrary group and is a finite-dimensional vector space,
it is known that every bijective linear cellular automaton is reversible and that the image of every linear cellular automaton is closed in for the prodiscrete topology. In this
paper, we present a new proof of these two results which is based on the
Mittag-Leffler lemma for projective sequences of sets. We also show that if
is a non-periodic group and is an infinite-dimensional vector space, then
there exist a linear cellular automaton which is
bijective but not reversible and a linear cellular automaton whose image is not closed in for the prodiscrete topology
On injective endomorphisms of symbolic schemes
Building on the seminal work of Gromov on endomorphisms of symbolic algebraic
varieties [10], we introduce a notion of cellular automata over schemes which
generalize affine algebraic cellular automata in [7]. We extend known results
to this more general setting. We also establish several new ones regarding the
closed image property, surjunctivity, reversibility, and invertibility for
cellular automata over algebraic varieties with coefficients in an
algebraically closed field. As a byproduct, we obtain a negative answer to a
question raised in [7] on the existence of a bijective complex affine algebraic
cellular automaton whose inverse
is not algebraic
On algebraic cellular automata
We investigate some general properties of algebraic cellular automata, i.e.,
cellular automata over groups whose alphabets are affine algebraic sets and
which are locally defined by regular maps. When the ground field is assumed to
be uncountable and algebraically closed, we prove that such cellular automata
always have a closed image with respect to the prodiscrete topology on the
space of configurations and that they are reversible as soon as they are
bijective
Post-surjectivity and balancedness of cellular automata over groups
We discuss cellular automata over arbitrary finitely generated groups. We
call a cellular automaton post-surjective if for any pair of asymptotic
configurations, every pre-image of one is asymptotic to a pre-image of the
other. The well known dual concept is pre-injectivity: a cellular automaton is
pre-injective if distinct asymptotic configurations have distinct images. We
prove that pre-injective, post-surjective cellular automata are reversible.
Moreover, on sofic groups, post-surjectivity alone implies reversibility. We
also prove that reversible cellular automata over arbitrary groups are
balanced, that is, they preserve the uniform measure on the configuration
space.Comment: 16 pages, 3 figures, LaTeX "dmtcs-episciences" document class. Final
version for Discrete Mathematics and Theoretical Computer Science. Prepared
according to the editor's request
Reversibility of Symmetric Linear Cellular Automata with Radius r = 3
The aim of this work is to completely solve the reversibility problem for symmetric linear
cellular automata with radius r = 3 and null boundary conditions. The main result obtained is the
explicit computation of the local transition functions of the inverse cellular automata. This allows
introduction of possible and interesting applications in digital image encryption.This research was funded by Ministerio de Ciencia, Innovación y Universidades (MCIU, Spain), Agencia Estatal de Investigación (AEI, Spain), and Fondo Europeo de Desarrollo Regional (FEDER, UE) under project TIN2017-84844-C2-2-R (MAGERAN) and project SA054G18 supported by Consejería de Educación (Junta de Castilla y León, Spain)
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