291 research outputs found
Topology Inspired Problems for Cellular Automata, and a Counterexample in Topology
We consider two relatively natural topologizations of the set of all cellular
automata on a fixed alphabet. The first turns out to be rather pathological, in
that the countable space becomes neither first-countable nor sequential. Also,
reversible automata form a closed set, while surjective ones are dense. The
second topology, which is induced by a metric, is studied in more detail.
Continuity of composition (under certain restrictions) and inversion, as well
as closedness of the set of surjective automata, are proved, and some
counterexamples are given. We then generalize this space, in the sense that
every shift-invariant measure on the configuration space induces a pseudometric
on cellular automata, and study the properties of these spaces. We also
characterize the pseudometric spaces using the Besicovitch distance, and show a
connection to the first (pathological) space.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Bulking II: Classifications of Cellular Automata
This paper is the second part of a series of two papers dealing with bulking:
a way to define quasi-order on cellular automata by comparing space-time
diagrams up to rescaling. In the present paper, we introduce three notions of
simulation between cellular automata and study the quasi-order structures
induced by these simulation relations on the whole set of cellular automata.
Various aspects of these quasi-orders are considered (induced equivalence
relations, maximum elements, induced orders, etc) providing several formal
tools allowing to classify cellular automata
-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
Expressing the entropy of lattice systems as sums of conditional entropies
Whether a system is to be considered complex or not depends on how one
searches for correlations. We propose a general scheme for calculation of
entropies in lattice systems that has high flexibility in how correlations are
successively taken into account. Compared to the traditional approach for
estimating the entropy density, in which successive approximations builds on
step-wise extensions of blocks of symbols, we show that one can take larger
steps when collecting the statistics necessary to calculate the entropy density
of the system. In one dimension this means that, instead of a single sweep over
the system in which states are read sequentially, one take several sweeps with
larger steps so that eventually the whole lattice is covered. This means that
the information in correlations is captured in a different way, and in some
situations this will lead to a considerably much faster convergence of the
entropy density estimate as a function of the size of the configurations used
in the estimate. The formalism is exemplified with both an example of a free
energy minimisation scheme for the two-dimensional Ising model, and an example
of increasingly complex spatial correlations generated by the time evolution of
elementary cellular automaton rule 60
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