59 research outputs found
Multidimensional cellular automata and generalization of Fekete's lemma
Fekete's lemma is a well known combinatorial result on number sequences: we
extend it to functions defined on -tuples of integers. As an application of
the new variant, we show that nonsurjective -dimensional cellular automata
are characterized by loss of arbitrarily much information on finite supports,
at a growth rate greater than that of the support's boundary determined by the
automaton's neighbourhood index.Comment: 6 pages, no figures, LaTeX. Improved some explanations; revised
structure; added examples; renamed "hypercubes" into "right polytopes"; added
references to Arratia's paper on EJC, Calude's book, Cook's proof of Rule 110
universality, and arXiv paper 0709.117
Fekete's lemma for componentwise subadditive functions of two or more real variables
We prove an analogue of Fekete's subadditivity lemma for functions of several
real variables which are subadditive in each variable taken singularly. This
extends both the classical case for subadditive functions of one real variable,
and a result in a previous paper by the author. While doing so, we prove that
the functions with the property mentioned above are bounded in every closed and
bounded subset of their domain. The arguments follows those of Chapter 6 in E.
Hille's 1948 textbook.Comment: 22 pages. Revised and expanded. Longer introduction, more detailed
background, statement of main theorem extende
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
When--and how--can a cellular automaton be rewritten as a lattice gas?
Both cellular automata (CA) and lattice-gas automata (LG) provide finite
algorithmic presentations for certain classes of infinite dynamical systems
studied by symbolic dynamics; it is customary to use the term `cellular
automaton' or `lattice gas' for the dynamic system itself as well as for its
presentation. The two kinds of presentation share many traits but also display
profound differences on issues ranging from decidability to modeling
convenience and physical implementability.
Following a conjecture by Toffoli and Margolus, it had been proved by Kari
(and by Durand--Lose for more than two dimensions) that any invertible CA can
be rewritten as an LG (with a possibly much more complex ``unit cell''). But
until now it was not known whether this is possible in general for
noninvertible CA--which comprise ``almost all'' CA and represent the bulk of
examples in theory and applications. Even circumstantial evidence--whether in
favor or against--was lacking.
Here, for noninvertible CA, (a) we prove that an LG presentation is out of
the question for the vanishingly small class of surjective ones. We then turn
our attention to all the rest--noninvertible and nonsurjective--which comprise
all the typical ones, including Conway's `Game of Life'. For these (b) we prove
by explicit construction that all the one-dimensional ones are representable as
LG, and (c) we present and motivate the conjecture that this result extends to
any number of dimensions.
The tradeoff between dissipation rate and structural complexity implied by
the above results have compelling implications for the thermodynamics of
computation at a microscopic scale.Comment: 16 page
On the Induction Operation for Shift Subspaces and Cellular Automata as Presentations of Dynamical Systems
We consider continuous, translation-commuting transformations of compact,
translation-invariant families of mappingsfrom finitely generated groups into
finite alphabets. It is well-known that such transformations and spaces can be
described "locally" via families of patterns and finitary functions; such
descriptions can be re-used on groups larger than the original, usually
defining non-isomorphic structures. We show how some of the properties of the
"induced" entities can be deduced from those of the original ones, and vice
versa; then, we show how to "simulate" the smaller structure into the larger
one, and obtain a characterization in terms of group actions for the dynamical
systems admitting of presentations via structures as such. Special attention is
given to the class of sofic shifts.Comment: 20 pages, no figures. Presented at LATA 2008. Extended version,
submitted to Information and Computatio
Multidimensional cellular automata and generalization of Fekete's lemma
Automata, Logic and Semantic
Surjective cellular automata far from the Garden of Eden
Automata, Logic and SemanticsInternational audienceOne of the first and most famous results of cellular automata theory, Moore's Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure
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