22,409 research outputs found
Freezing, Bounded-Change and Convergent Cellular Automata *
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension
Freezing, Bounded-Change and Convergent Cellular Automata *
This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension
Computing Aggregate Properties of Preimages for 2D Cellular Automata
Computing properties of the set of precursors of a given configuration is a
common problem underlying many important questions about cellular automata.
Unfortunately, such computations quickly become intractable in dimension
greater than one. This paper presents an algorithm --- incremental aggregation
--- that can compute aggregate properties of the set of precursors
exponentially faster than na{\"i}ve approaches. The incremental aggregation
algorithm is demonstrated on two problems from the two-dimensional binary Game
of Life cellular automaton: precursor count distributions and higher-order mean
field theory coefficients. In both cases, incremental aggregation allows us to
obtain new results that were previously beyond reach
Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems
Most current methods for identifying coherent structures in
spatially-extended systems rely on prior information about the form which those
structures take. Here we present two new approaches to automatically filter the
changing configurations of spatial dynamical systems and extract coherent
structures. One, local sensitivity filtering, is a modification of the local
Lyapunov exponent approach suitable to cellular automata and other discrete
spatial systems. The other, local statistical complexity filtering, calculates
the amount of information needed for optimal prediction of the system's
behavior in the vicinity of a given point. By examining the changing
spatiotemporal distributions of these quantities, we can find the coherent
structures in a variety of pattern-forming cellular automata, without needing
to guess or postulate the form of that structure. We apply both filters to
elementary and cyclical cellular automata (ECA and CCA) and find that they
readily identify particles, domains and other more complicated structures. We
compare the results from ECA with earlier ones based upon the theory of formal
languages, and the results from CCA with a more traditional approach based on
an order parameter and free energy. While sensitivity and statistical
complexity are equally adept at uncovering structure, they are based on
different system properties (dynamical and probabilistic, respectively), and
provide complementary information.Comment: 16 pages, 21 figures. Figures considerably compressed to fit arxiv
requirements; write first author for higher-resolution version
Mean-Field Theory of Meta-Learning
We discuss here the mean-field theory for a cellular automata model of
meta-learning. The meta-learning is the process of combining outcomes of
individual learning procedures in order to determine the final decision with
higher accuracy than any single learning method. Our method is constructed from
an ensemble of interacting, learning agents, that acquire and process incoming
information using various types, or different versions of machine learning
algorithms. The abstract learning space, where all agents are located, is
constructed here using a fully connected model that couples all agents with
random strength values. The cellular automata network simulates the higher
level integration of information acquired from the independent learning trials.
The final classification of incoming input data is therefore defined as the
stationary state of the meta-learning system using simple majority rule, yet
the minority clusters that share opposite classification outcome can be
observed in the system. Therefore, the probability of selecting proper class
for a given input data, can be estimated even without the prior knowledge of
its affiliation. The fuzzy logic can be easily introduced into the system, even
if learning agents are build from simple binary classification machine learning
algorithms by calculating the percentage of agreeing agents.Comment: 23 page
Scaling and Inverse Scaling in Anisotropic Bootstrap percolation
In bootstrap percolation it is known that the critical percolation threshold
tends to converge slowly to zero with increasing system size, or, inversely,
the critical size diverges fast when the percolation probability goes to zero.
To obtain higher-order terms (that is, sharp and sharper thresholds) for the
percolation threshold in general is a hard question. In the case of
two-dimensional anisotropic models, sometimes correction terms can be obtained
from inversion in a relatively simple manner.Comment: Contribution to the proceedings of the 2013 EURANDOM workshop
Probabilistic Cellular Automata: Theory, Applications and Future
Perspectives, equation typo corrected, constant of generalisation correcte
Multiband linear cellular automata and endomorphisms of algebraic vector groups
We propose a correspondence between certain multiband linear cellular
automata - models of computation widely used in the description of physical
phenomena - and endomorphisms of certain algebraic unipotent groups over finite
fields. The correspondence is based on the construction of a universal element
specialising to a normal generator for any finite field. We use this
correspondence to deduce new results concerning the temporal dynamics of such
automata, using our prior, purely algebraic, study of the endomorphism ring of
vector groups. These produce 'for free' a formula for the number of fixed
points of the -iterate in terms of the -adic valuation of , a
dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic
formula for the number of periodic orbits. Since multiband linear cellular
automata simulate higher order linear automata (in which states depend on
finitely many prior temporal states, not just the direct predecessor), the
results apply equally well to that class.Comment: 11 page
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