3,758 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
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