1,387 research outputs found
Intrinsically Universal Cellular Automata
This talk advocates intrinsic universality as a notion to identify simple
cellular automata with complex computational behavior. After an historical
introduction and proper definitions of intrinsic universality, which is
discussed with respect to Turing and circuit universality, we discuss
construction methods for small intrinsically universal cellular automata before
discussing techniques for proving non universality
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
Impartial games emulating one-dimensional cellular automata and undecidability
We study two-player \emph{take-away} games whose outcomes emulate two-state
one-dimensional cellular automata, such as Wolfram's rules 60 and 110. Given an
initial string consisting of a central data pattern and periodic left and right
patterns, the rule 110 cellular automaton was recently proved Turing-complete
by Matthew Cook. Hence, many questions regarding its behavior are
algorithmically undecidable. We show that similar questions are undecidable for
our \emph{rule 110} game.Comment: 22 pages, 11 figure
Computational Processes and Incompleteness
We introduce a formal definition of Wolfram's notion of computational process
based on cellular automata, a physics-like model of computation. There is a
natural classification of these processes into decidable, intermediate and
complete. It is shown that in the context of standard finite injury priority
arguments one cannot establish the existence of an intermediate computational
process
Sensitivity to noise and ergodicity of an assembly line of cellular automata that classifies density
We investigate the sensitivity of the composite cellular automaton of H.
Fuk\'{s} [Phys. Rev. E 55, R2081 (1997)] to noise and assess the density
classification performance of the resulting probabilistic cellular automaton
(PCA) numerically. We conclude that the composite PCA performs the density
classification task reliably only up to very small levels of noise. In
particular, it cannot outperform the noisy Gacs-Kurdyumov-Levin automaton, an
imperfect classifier, for any level of noise. While the original composite CA
is nonergodic, analyses of relaxation times indicate that its noisy version is
an ergodic automaton, with the relaxation times decaying algebraically over an
extended range of parameters with an exponent very close (possibly equal) to
the mean-field value.Comment: Typeset in REVTeX 4.1, 5 pages, 5 figures, 2 tables, 1 appendix.
Version v2 corresponds to the published version of the manuscrip
Cellular automaton supercolliders
Gliders in one-dimensional cellular automata are compact groups of
non-quiescent and non-ether patterns (ether represents a periodic background)
translating along automaton lattice. They are cellular-automaton analogous of
localizations or quasi-local collective excitations travelling in a spatially
extended non-linear medium. They can be considered as binary strings or symbols
travelling along a one-dimensional ring, interacting with each other and
changing their states, or symbolic values, as a result of interactions. We
analyse what types of interaction occur between gliders travelling on a
cellular automaton `cyclotron' and build a catalog of the most common
reactions. We demonstrate that collisions between gliders emulate the basic
types of interaction that occur between localizations in non-linear media:
fusion, elastic collision, and soliton-like collision. Computational outcomes
of a swarm of gliders circling on a one-dimensional torus are analysed via
implementation of cyclic tag systems
A Computation in a Cellular Automaton Collider Rule 110
A cellular automaton collider is a finite state machine build of rings of
one-dimensional cellular automata. We show how a computation can be performed
on the collider by exploiting interactions between gliders (particles,
localisations). The constructions proposed are based on universality of
elementary cellular automaton rule 110, cyclic tag systems, supercolliders, and
computing on rings.Comment: 39 pages, 32 figures, 3 table
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