30 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
STATE-OF-ART Algorithms for Injectivity and Bounded Surjectivity of One-dimensional Cellular Automata
Surjectivity and injectivity are the most fundamental problems in cellular
automata (CA). We simplify and modify Amoroso's algorithm into optimum and make
it compatible with fixed, periodic and reflective boundaries. A new algorithm
(injectivity tree algorithm) for injectivity is also proposed. After our
theoretic analysis and experiments, our algorithm for injectivity can save much
space and 90\% or even more time compared with Amoroso's algorithm for
injectivity so that it can support the decision of CA with larger neighborhood
sizes. At last, we prove that the reversibility with the periodic boundary and
global injectivity of one-dimensional CA is equivalent
Cellular automata on regular rooted trees
We study cellular automata on regular rooted trees. This includes the
characterization of sofic tree shifts in terms of unrestricted Rabin automata
and the decidability of the surjectivity problem for cellular automata between
sofic tree shifts
Revisiting the Rice Theorem of Cellular Automata
A cellular automaton is a parallel synchronous computing model, which
consists in a juxtaposition of finite automata whose state evolves according to
that of their neighbors. It induces a dynamical system on the set of
configurations, i.e. the infinite sequences of cell states. The limit set of
the cellular automaton is the set of configurations which can be reached
arbitrarily late in the evolution.
In this paper, we prove that all properties of limit sets of cellular
automata with binary-state cells are undecidable, except surjectivity. This is
a refinement of the classical "Rice Theorem" that Kari proved on cellular
automata with arbitrary state sets.Comment: 12 pages conference STACS'1
Computation and construction universality of reversible cellular automata
An arbitrary d-dimensional cellular automaton can be constructively embedded in areversible one having d+1 dimensions. In particular, there exist computation- and construction-universal reversible cellular automata. Thus, we explicitly show a way of implementing nontrivial irreversible processes in a reversible medium. Finally, we derive new results for the bounding problem for configurations, both in general and for reversible cellular automata