35 research outputs found
Restricted density classification in one dimension
The density classification task is to determine which of the symbols
appearing in an array has the majority. A cellular automaton solving this task
is required to converge to a uniform configuration with the majority symbol at
each site. It is not known whether a one-dimensional cellular automaton with
binary alphabet can classify all Bernoulli random configurations almost surely
according to their densities. We show that any cellular automaton that washes
out finite islands in linear time classifies all Bernoulli random
configurations with parameters close to 0 or 1 almost surely correctly. The
proof is a direct application of a "percolation" argument which goes back to
Gacs (1986).Comment: 13 pages, 5 figure
Statistical Mechanics of Surjective Cellular Automata
Reversible cellular automata are seen as microscopic physical models, and
their states of macroscopic equilibrium are described using invariant
probability measures. We establish a connection between the invariance of Gibbs
measures and the conservation of additive quantities in surjective cellular
automata. Namely, we show that the simplex of shift-invariant Gibbs measures
associated to a Hamiltonian is invariant under a surjective cellular automaton
if and only if the cellular automaton conserves the Hamiltonian. A special case
is the (well-known) invariance of the uniform Bernoulli measure under
surjective cellular automata, which corresponds to the conservation of the
trivial Hamiltonian. As an application, we obtain results indicating the lack
of (non-trivial) Gibbs or Markov invariant measures for "sufficiently chaotic"
cellular automata. We discuss the relevance of the randomization property of
algebraic cellular automata to the problem of approach to macroscopic
equilibrium, and pose several open questions.
As an aside, a shift-invariant pre-image of a Gibbs measure under a
pre-injective factor map between shifts of finite type turns out to be always a
Gibbs measure. We provide a sufficient condition under which the image of a
Gibbs measure under a pre-injective factor map is not a Gibbs measure. We point
out a potential application of pre-injective factor maps as a tool in the study
of phase transitions in statistical mechanical models.Comment: 50 pages, 7 figure
Conservation Laws in Cellular Automata
Conservation laws in physics are numerical invariants of the dynamics of a system. In cellular automata (CA), a similar concept has already been defined and studied. To each local pattern of cell states a real value is associated, interpreted as the “energy” (or “mass”, or . . . ) of that pattern.The overall “energy” of a configuration is simply the sum of the energy of the local patterns appearing on different positions in the configuration. We have a conservation law for that energy, if the total energy of each configuration remains constant during the evolution of the CA. For a given conservation law, it is desirable to find microscopic explanations for the dynamics of the conserved energy in terms of flows of energy from one region toward another. Often, it happens that the energy values are from non-negative integers, and are interpreted as the number of “particles” distributed on a configuration. In such cases, it is conjectured that one can always provide a microscopic explanation for the conservation laws by prescribing rules for the local movement of the particles. The onedimensional case has already been solved by Fuk´s and Pivato. We extend this to two-dimensional cellular automata with radius-0,5 neighborhood on the square lattice.
We then consider conservation laws in which the energy values are chosen from a commutative group or semigroup. In this case, the class of all conservation laws for a CA form a partially ordered hierarchy. We study the structure of this hierarchy and prove some basic facts about it. Although the local properties of this hierarchy (at least in the group-valued case) are tractable, its global properties turn out to be algorithmically inaccessible. In particular, we prove that it is undecidable whether this hierarchy is trivial (i.e., if the CA has any non-trivial conservation law at all) or unbounded. We point out some interconnections between the structure of this hierarchy and the dynamical properties of the CA. We show that positively expansive CA do not have non-trivial conservation laws. We also investigate a curious relationship between conservation laws and invariant Gibbs measures in reversible and surjective CA. Gibbs measures are known to coincide with the equilibrium states of a lattice system defined in terms of a Hamiltonian. For reversible cellular automata, each conserved quantity may play the role of a Hamiltonian, and provides a Gibbs measure (or a set of Gibbs measures, in case of phase multiplicity) that is invariant. Conversely, every invariant Gibbs measure provides a conservation law for the CA. For surjective CA, the former statement also follows (in a slightly different form) from the variational characterization of the Gibbs measures. For one-dimensional surjective CA, we show that each invariant Gibbs measure provides a conservation law. We also prove that surjective CA almost surely preserve the average information content per cell with respect to any probability measure.Siirretty Doriast
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
Equivalence of relative Gibbs and relative equilibrium measures for actions of countable amenable groups
We formulate and prove a very general relative version of the
Dobrushin-Lanford-Ruelle theorem which gives conditions on constraints of
configuration spaces over a finite alphabet such that for every absolutely
summable relative interaction, every translation-invariant relative Gibbs
measure is a relative equilibrium measure and vice versa. Neither implication
is true without some assumption on the space of configurations. We note that
the usual finite type condition can be relaxed to a much more general class of
constraints. By "relative" we mean that both the interaction and the set of
allowed configurations are determined by a random environment. The result
includes many special cases that are well known. We give several applications
including (1) Gibbsian properties of measures that maximize pressure among all
those that project to a given measure via a topological factor map from one
symbolic system to another; (2) Gibbsian properties of equilibrium measures for
group shifts defined on arbitrary countable amenable groups; (3) A Gibbsian
characterization of equilibrium measures in terms of equilibrium condition on
lattice slices rather than on finite sets; (4) A relative extension of a
theorem of Meyerovitch, who proved a version of the Lanford--Ruelle theorem
which shows that every equilibrium measure on an arbitrary subshift satisfies a
Gibbsian property on interchangeable patterns.Comment: 37 pages and 3 beautiful figure
First-order transition in Potts models with "invisible' states: Rigorous proofs
In some recent papers by Tamura, Tanaka and Kawashima [arXiv:1102.5475,
arXiv:1012.4254], a class of Potts models with "invisible" states was
introduced, for which the authors argued by numerical arguments and by a
mean-field analysis that a first-order transition occurs. Here we show that the
existence of this first-order transition can be proven rigorously, by
relatively minor adaptations of existing proofs for ordinary Potts models. In
our argument we present a random-cluster representation for the model, which
might be of independent interest
Characterisation of the Set of Ground States of Uniformly Chaotic Finite-Range Lattice Models
Chaotic dependence on temperature refers to the phenomenon of divergence of
Gibbs measures as the temperature approaches a certain value. Models with
chaotic behaviour near zero temperature have multiple ground states, none of
which are stable. We study the class of uniformly chaotic models, that is,
those in which, as the temperature goes to zero, every choice of Gibbs measures
accumulates on the entire set of ground states. We characterise the possible
sets of ground states of uniformly chaotic finite-range models up to computable
homeomorphisms.
Namely, we show that the set of ground states of every model with
finite-range and rational-valued interactions is topologically closed and
connected, and belongs to the class of the arithmetical hierarchy.
Conversely, every -computable, topologically closed and connected set of
probability measures can be encoded (via a computable homeomorphism) as the set
of ground states of a uniformly chaotic two-dimensional model with finite-range
rational-valued interactions.Comment: 46 pages, 12 figure