9,846 research outputs found
Majority-Vote Cellular Automata, Ising Dynamics, and P-Completeness
We study cellular automata where the state at each site is decided by a
majority vote of the sites in its neighborhood. These are equivalent, for a
restricted set of initial conditions, to non-zero probability transitions in
single spin-flip dynamics of the Ising model at zero temperature.
We show that in three or more dimensions these systems can simulate Boolean
circuits of AND and OR gates, and are therefore P-complete. That is, predicting
their state t time-steps in the future is at least as hard as any other problem
that takes polynomial time on a serial computer.
Therefore, unless a widely believed conjecture in computer science is false,
it is impossible even with parallel computation to predict majority-vote
cellular automata, or zero-temperature single spin-flip Ising dynamics,
qualitatively faster than by explicit simulation.Comment: 10 pages with figure
Communications in cellular automata
The goal of this paper is to show why the framework of communication
complexity seems suitable for the study of cellular automata. Researchers have
tackled different algorithmic problems ranging from the complexity of
predicting to the decidability of different dynamical properties of cellular
automata. But the difference here is that we look for communication protocols
arising in the dynamics itself. Our work is guided by the following idea: if we
are able to give a protocol describing a cellular automaton, then we can
understand its behavior
Predicting Non-linear Cellular Automata Quickly by Decomposing Them into Linear Ones
We show that a wide variety of non-linear cellular automata (CAs) can be
decomposed into a quasidirect product of linear ones. These CAs can be
predicted by parallel circuits of depth O(log^2 t) using gates with binary
inputs, or O(log t) depth if ``sum mod p'' gates with an unbounded number of
inputs are allowed. Thus these CAs can be predicted by (idealized) parallel
computers much faster than by explicit simulation, even though they are
non-linear.
This class includes any CA whose rule, when written as an algebra, is a
solvable group. We also show that CAs based on nilpotent groups can be
predicted in depth O(log t) or O(1) by circuits with binary or ``sum mod p''
gates respectively.
We use these techniques to give an efficient algorithm for a CA rule which,
like elementary CA rule 18, has diffusing defects that annihilate in pairs.
This can be used to predict the motion of defects in rule 18 in O(log^2 t)
parallel time
Two-dimensional cellular automata and the analysis of correlated time series
Correlated time series are time series that, by virtue of the underlying
process to which they refer, are expected to influence each other strongly. We
introduce a novel approach to handle such time series, one that models their
interaction as a two-dimensional cellular automaton and therefore allows them
to be treated as a single entity. We apply our approach to the problems of
filling gaps and predicting values in rainfall time series. Computational
results show that the new approach compares favorably to Kalman smoothing and
filtering
A review of wildland fire spread modelling, 1990-present 3: Mathematical analogues and simulation models
In recent years, advances in computational power and spatial data analysis
(GIS, remote sensing, etc) have led to an increase in attempts to model the
spread and behvaiour of wildland fires across the landscape. This series of
review papers endeavours to critically and comprehensively review all types of
surface fire spread models developed since 1990. This paper reviews models of a
simulation or mathematical analogue nature. Most simulation models are
implementations of existing empirical or quasi-empirical models and their
primary function is to convert these generally one dimensional models to two
dimensions and then propagate a fire perimeter across a modelled landscape.
Mathematical analogue models are those that are based on some mathematical
conceit (rather than a physical representation of fire spread) that
coincidentally simulates the spread of fire. Other papers in the series review
models of an physical or quasi-physical nature and empirical or quasi-empirical
nature. Many models are extensions or refinements of models developed before
1990. Where this is the case, these models are also discussed but much less
comprehensively.Comment: 20 pages + 9 pages references + 1 page figures. Submitted to the
International Journal of Wildland Fir
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