114 research outputs found
Undecidable Properties of Limit Set Dynamics of Cellular Automata
Cellular Automata (CA) are discrete dynamical systems and an abstract model
of parallel computation. The limit set of a cellular automaton is its maximal
topological attractor. A well know result, due to Kari, says that all
nontrivial properties of limit sets are undecidable. In this paper we consider
properties of limit set dynamics, i.e. properties of the dynamics of Cellular
Automata restricted to their limit sets. There can be no equivalent of Kari's
Theorem for limit set dynamics. Anyway we show that there is a large class of
undecidable properties of limit set dynamics, namely all properties of limit
set dynamics which imply stability or the existence of a unique subshift
attractor. As a consequence we have that it is undecidable whether the cellular
automaton map restricted to the limit set is the identity, closing, injective,
expansive, positively expansive, transitive
On Factor Universality in Symbolic Spaces
The study of factoring relations between subshifts or cellular automata is
central in symbolic dynamics. Besides, a notion of intrinsic universality for
cellular automata based on an operation of rescaling is receiving more and more
attention in the literature. In this paper, we propose to study the factoring
relation up to rescalings, and ask for the existence of universal objects for
that simulation relation. In classical simulations of a system S by a system T,
the simulation takes place on a specific subset of configurations of T
depending on S (this is the case for intrinsic universality). Our setting,
however, asks for every configurations of T to have a meaningful interpretation
in S. Despite this strong requirement, we show that there exists a cellular
automaton able to simulate any other in a large class containing arbitrarily
complex ones. We also consider the case of subshifts and, using arguments from
recursion theory, we give negative results about the existence of universal
objects in some classes
Ultimate Traces of Cellular Automata
A cellular automaton (CA) is a parallel synchronous computing model, which
consists in a juxtaposition of finite automata (cells) whose state evolves
according to that of their neighbors. Its trace is the set of infinite words
representing the sequence of states taken by some particular cell. In this
paper we study the ultimate trace of CA and partial CA (a CA restricted to a
particular subshift). The ultimate trace is the trace observed after a long
time run of the CA. We give sufficient conditions for a set of infinite words
to be the trace of some CA and prove the undecidability of all properties over
traces that are stable by ultimate coincidence.Comment: 12 pages + 5 of appendix conference STACS'1
Sofic Trace of a Cellular Automaton
The trace subshift of a cellular automaton is the subshift of all possible
columns that may appear in a space-time diagram, ie the infinite sequence of
states of a particular cell of a configuration; in the language of symbolic
dynamics one says that it is a factor system. In this paper we study conditions
for a sofic subshift to be the trace of a cellular automaton.Comment: 10 pages + 6 for included proof
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
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
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