99 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 the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures
We study the notion of limit sets of cellular automata associated with
probability measures (mu-limit sets). This notion was introduced by P. Kurka
and A. Maass. It is a refinement of the classical notion of omega-limit sets
dealing with the typical long term behavior of cellular automata. It focuses on
the words whose probability of appearance does not tend to 0 as time tends to
infinity (the persistent words). In this paper, we give a characterisation of
the persistent language for non sensible cellular automata associated with
Bernouilli measures. We also study the computational complexity of these
languages. We show that the persistent language can be non-recursive. But our
main result is that the set of quasi-nilpotent cellular automata (those with a
single configuration in their mu-limit set) is neither recursively enumerable
nor co-recursively enumerable
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
Conjugacy of one-dimensional one-sided cellular automata is undecidable
Two cellular automata are strongly conjugate if there exists a
shift-commuting conjugacy between them. We prove that the following two sets of
pairs of one-dimensional one-sided cellular automata over a full shift
are recursively inseparable: (i) pairs where has strictly larger
topological entropy than , and (ii) pairs that are strongly conjugate and
have zero topological entropy.
Because there is no factor map from a lower entropy system to a higher
entropy one, and there is no embedding of a higher entropy system into a lower
entropy system, we also get as corollaries that the following decision problems
are undecidable: Given two one-dimensional one-sided cellular automata and
over a full shift: Are and conjugate? Is a factor of ? Is
a subsystem of ? All of these are undecidable in both strong and weak
variants (whether the homomorphism is required to commute with the shift or
not, respectively). It also immediately follows that these results hold for
one-dimensional two-sided cellular automata.Comment: 12 pages, 2 figures, accepted for SOFSEM 201
-Limit Sets of Cellular Automata from a Computational Complexity Perspective
This paper concerns -limit sets of cellular automata: sets of
configurations made of words whose probability to appear does not vanish with
time, starting from an initial -random configuration. More precisely, we
investigate the computational complexity of these sets and of related decision
problems. Main results: first, -limit sets can have a -hard
language, second, they can contain only -complex configurations, third,
any non-trivial property concerning them is at least -hard. We prove
complexity upper bounds, study restrictions of these questions to particular
classes of CA, and different types of (non-)convergence of the measure of a
word during the evolution.Comment: 41 page
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
Probabilistic cellular automata, invariant measures, and perfect sampling
A probabilistic cellular automaton (PCA) can be viewed as a Markov chain. The
cells are updated synchronously and independently, according to a distribution
depending on a finite neighborhood. We investigate the ergodicity of this
Markov chain. A classical cellular automaton is a particular case of PCA. For a
1-dimensional cellular automaton, we prove that ergodicity is equivalent to
nilpotency, and is therefore undecidable. We then propose an efficient perfect
sampling algorithm for the invariant measure of an ergodic PCA. Our algorithm
does not assume any monotonicity property of the local rule. It is based on a
bounding process which is shown to be also a PCA. Last, we focus on the PCA
Majority, whose asymptotic behavior is unknown, and perform numerical
experiments using the perfect sampling procedure
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