2,752 research outputs found
A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications
Cellular automata (CAs) are dynamical systems which exhibit complex global
behavior from simple local interaction and computation. Since the inception of
cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention
of several researchers over various backgrounds and fields for modelling
different physical, natural as well as real-life phenomena. Classically, CAs
are uniform. However, non-uniformity has also been introduced in update
pattern, lattice structure, neighborhood dependency and local rule. In this
survey, we tour to the various types of CAs introduced till date, the different
characterization tools, the global behaviors of CAs, like universality,
reversibility, dynamics etc. Special attention is given to non-uniformity in
CAs and especially to non-uniform elementary CAs, which have been very useful
in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin
Recommended from our members
Pseudorandom number generation with self programmable cellular automata
In this paper, we propose a new class of cellular automata – self programming cellular automata (SPCA) with specific application to pseudorandom number generation. By changing a cell's state transition rules in relation to factors such as its neighboring cell's states, behavioral complexity can be increased and utilized. Interplay between the state transition neighborhood and rule selection neighborhood leads to a new composite neighborhood and state transition rule that is the linear combination of two different mappings with different temporal dependencies. It is proved that when the transitional matrices for both the state transition and rule selection neighborhood are non-singular, SPCA will not exhibit non-group behavior. Good performance can be obtained using simple neighborhoods with certain CA length, transition rules etc. Certain configurations of SPCA pass all DIEHARD and ENT tests with an implementation cost lower than current reported work. Output sampling methods are also suggested to improve output efficiency by sampling the outputs of the new rule selection neighborhoods
Communication Complexity and Intrinsic Universality in Cellular Automata
The notions of universality and completeness are central in the theories of
computation and computational complexity. However, proving lower bounds and
necessary conditions remains hard in most of the cases. In this article, we
introduce necessary conditions for a cellular automaton to be "universal",
according to a precise notion of simulation, related both to the dynamics of
cellular automata and to their computational power. This notion of simulation
relies on simple operations of space-time rescaling and it is intrinsic to the
model of cellular automata. Intrinsinc universality, the derived notion, is
stronger than Turing universality, but more uniform, and easier to define and
study. Our approach builds upon the notion of communication complexity, which
was primarily designed to study parallel programs, and thus is, as we show in
this article, particulary well suited to the study of cellular automata: it
allowed to show, by studying natural problems on the dynamics of cellular
automata, that several classes of cellular automata, as well as many natural
(elementary) examples, could not be intrinsically universal
Convolution equations on lattices: periodic solutions with values in a prime characteristic field
These notes are inspired by the theory of cellular automata. A linear
cellular automaton on a lattice of finite rank or on a toric grid is a discrete
dinamical system generated by a convolution operator with kernel concentrated
in the nearest neighborhood of the origin. In the present paper we deal with
general convolution operators. We propose an approach via harmonic analysis
which works over a field of positive characteristic. It occurs that a standard
spectral problem for a convolution operator is equivalent to counting points on
an associate algebraic hypersurface in a torus according to the torsion orders
of their coordinates.Comment: 30 pages, a new editio
Complexity Measures from Interaction Structures
We evaluate new complexity measures on the symbolic dynamics of coupled tent
maps and cellular automata. These measures quantify complexity in terms of
-th order statistical dependencies that cannot be reduced to interactions
between units. We demonstrate that these measures are able to identify
complex dynamical regimes.Comment: 11 pages, figures improved, minor changes to the tex
Periodic harmonic functions on lattices and points count in positive characteristic
This survey addresses pluri-periodic harmonic functions on lattices with
values in a positive characteristic field. We mention, as a motivation, the
game "Lights Out" following the work of Sutner, Goldwasser-Klostermeyer-Ware,
Barua-Ramakrishnan-Sarkar, Hunzikel-Machiavello-Park e.a.; see also 2 previous
author's preprints for a more detailed account. Our approach explores harmonic
analysis and algebraic geometry over a positive characteristic field. The
Fourier transform allows us to interpret pluri-periods of harmonic functions on
lattices as torsion multi-orders of points on the corresponding affine
algebraic variety.Comment: These are notes on 13p. based on a talk presented during the meeting
"Analysis on Graphs and Fractals", the Cardiff University, 29 May-2 June 2007
(a sattelite meeting of the programme "Analysis on Graphs and its
Applications" at the Isaac Newton Institute from 8 January to 29 June 2007
Recommended from our members
Layered cellular automata for pseudorandom number generation
The proposed Layered Cellular Automata (L-LCA), which comprises of a main CA with L additional layers of memory registers, has simple local interconnections and high operating speed. The time-varying L-LCA transformation at each clock can be reduced to a single transformation in the set formed by the transformation matrix of a maximum length Cellular Automata (CA), and the entire transformation sequence for a single period can be obtained. The analysis for the period characteristics of state sequences is simplified by analyzing representative transformation sequences determined by the phase difference between the initial states for each layer. The L-LCA model can be extended by adding more layers of memory or through the use of a larger main CA based on widely available maximum length CA. Several L-LCA (L=1,2,3,4) with 10- to 48-bit main CA are subjected to the DIEHARD test suite and better results are obtained over other CA designs reported in the literature. The experiments are repeated using the well-known nonlinear functions and in place of the linear function used in the L-LCA. Linear complexity is significantly increased when or is used
Upper Bound on the Products of Particle Interactions in Cellular Automata
Particle-like objects are observed to propagate and interact in many
spatially extended dynamical systems. For one of the simplest classes of such
systems, one-dimensional cellular automata, we establish a rigorous upper bound
on the number of distinct products that these interactions can generate. The
upper bound is controlled by the structural complexity of the interacting
particles---a quantity which is defined here and which measures the amount of
spatio-temporal information that a particle stores. Along the way we establish
a number of properties of domains and particles that follow from the
computational mechanics analysis of cellular automata; thereby elucidating why
that approach is of general utility. The upper bound is tested against several
relatively complex domain-particle cellular automata and found to be tight.Comment: 17 pages, 12 figures, 3 tables,
http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and
accompanying text modified, to comply with legal demands arising from
on-going intellectual property litigation among third parties. V3: Accepted
for publication in Physica D. References added and other small changes made
per referee suggestion
- …