11,106 research outputs found
Causal graph dynamics
We extend the theory of Cellular Automata to arbitrary, time-varying graphs.
In other words we formalize, and prove theorems about, the intuitive idea of a
labelled graph which evolves in time - but under the natural constraint that
information can only ever be transmitted at a bounded speed, with respect to
the distance given by the graph. The notion of translation-invariance is also
generalized. The definition we provide for these "causal graph dynamics" is
simple and axiomatic. The theorems we provide also show that it is robust. For
instance, causal graph dynamics are stable under composition and under
restriction to radius one. In the finite case some fundamental facts of
Cellular Automata theory carry through: causal graph dynamics admit a
characterization as continuous functions, and they are stable under inversion.
The provided examples suggest a wide range of applications of this mathematical
object, from complex systems science to theoretical physics. KEYWORDS:
Dynamical networks, Boolean networks, Generative networks automata, Cayley
cellular automata, Graph Automata, Graph rewriting automata, Parallel graph
transformations, Amalgamated graph transformations, Time-varying graphs, Regge
calculus, Local, No-signalling.Comment: 25 pages, 9 figures, LaTeX, v2: Minor presentation improvements, v3:
Typos corrected, figure adde
Intrinsic Simulations between Stochastic Cellular Automata
The paper proposes a simple formalism for dealing with deterministic,
non-deterministic and stochastic cellular automata in a unifying and composable
manner. Armed with this formalism, we extend the notion of intrinsic simulation
between deterministic cellular automata, to the non-deterministic and
stochastic settings. We then provide explicit tools to prove or disprove the
existence of such a simulation between two stochastic cellular automata, even
though the intrinsic simulation relation is shown to be undecidable in
dimension two and higher. The key result behind this is the caracterization of
equality of stochastic global maps by the existence of a coupling between the
random sources. We then prove that there is a universal non-deterministic
cellular automaton, but no universal stochastic cellular automaton. Yet we
provide stochastic cellular automata achieving optimal partial universality.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Quantum Cellular Automata
Quantum cellular automata (QCA) are reviewed, including early and more recent
proposals. QCA are a generalization of (classical) cellular automata (CA) and
in particular of reversible CA. The latter are reviewed shortly. An overview is
given over early attempts by various authors to define one-dimensional QCA.
These turned out to have serious shortcomings which are discussed as well.
Various proposals subsequently put forward by a number of authors for a general
definition of one- and higher-dimensional QCA are reviewed and their properties
such as universality and reversibility are discussed.Comment: 12 pages, 3 figures. To appear in the Springer Encyclopedia of
Complexity and Systems Scienc
When--and how--can a cellular automaton be rewritten as a lattice gas?
Both cellular automata (CA) and lattice-gas automata (LG) provide finite
algorithmic presentations for certain classes of infinite dynamical systems
studied by symbolic dynamics; it is customary to use the term `cellular
automaton' or `lattice gas' for the dynamic system itself as well as for its
presentation. The two kinds of presentation share many traits but also display
profound differences on issues ranging from decidability to modeling
convenience and physical implementability.
Following a conjecture by Toffoli and Margolus, it had been proved by Kari
(and by Durand--Lose for more than two dimensions) that any invertible CA can
be rewritten as an LG (with a possibly much more complex ``unit cell''). But
until now it was not known whether this is possible in general for
noninvertible CA--which comprise ``almost all'' CA and represent the bulk of
examples in theory and applications. Even circumstantial evidence--whether in
favor or against--was lacking.
Here, for noninvertible CA, (a) we prove that an LG presentation is out of
the question for the vanishingly small class of surjective ones. We then turn
our attention to all the rest--noninvertible and nonsurjective--which comprise
all the typical ones, including Conway's `Game of Life'. For these (b) we prove
by explicit construction that all the one-dimensional ones are representable as
LG, and (c) we present and motivate the conjecture that this result extends to
any number of dimensions.
The tradeoff between dissipation rate and structural complexity implied by
the above results have compelling implications for the thermodynamics of
computation at a microscopic scale.Comment: 16 page
Complex dynamics of elementary cellular automata emerging from chaotic rules
We show techniques of analyzing complex dynamics of cellular automata (CA)
with chaotic behaviour. CA are well known computational substrates for studying
emergent collective behaviour, complexity, randomness and interaction between
order and chaotic systems. A number of attempts have been made to classify CA
functions on their space-time dynamics and to predict behaviour of any given
function. Examples include mechanical computation, \lambda{} and Z-parameters,
mean field theory, differential equations and number conserving features. We
aim to classify CA based on their behaviour when they act in a historical mode,
i.e. as CA with memory. We demonstrate that cell-state transition rules
enriched with memory quickly transform a chaotic system converging to a complex
global behaviour from almost any initial condition. Thus just in few steps we
can select chaotic rules without exhaustive computational experiments or
recurring to additional parameters. We provide analysis of well-known chaotic
functions in one-dimensional CA, and decompose dynamics of the automata using
majority memory exploring glider dynamics and reactions
Complex dynamics emerging in Rule 30 with majority memory
In cellular automata with memory, the unchanged maps of the conventional
cellular automata are applied to cells endowed with memory of their past states
in some specified interval. We implement Rule 30 automata with a majority
memory and show that using the memory function we can transform quasi-chaotic
dynamics of classical Rule 30 into domains of travelling structures with
predictable behaviour. We analyse morphological complexity of the automata and
classify dynamics of gliders (particles, self-localizations) in memory-enriched
Rule 30. We provide formal ways of encoding and classifying glider dynamics
using de Bruijn diagrams, soliton reactions and quasi-chemical representations
On Factor Universality in Symbolic Spaces
The study of factoring relations between subshifts or cellular automata is
central in symbolic dynamics. Besides, a notion of intrinsic universality for
cellular automata based on an operation of rescaling is receiving more and more
attention in the literature. In this paper, we propose to study the factoring
relation up to rescalings, and ask for the existence of universal objects for
that simulation relation. In classical simulations of a system S by a system T,
the simulation takes place on a specific subset of configurations of T
depending on S (this is the case for intrinsic universality). Our setting,
however, asks for every configurations of T to have a meaningful interpretation
in S. Despite this strong requirement, we show that there exists a cellular
automaton able to simulate any other in a large class containing arbitrarily
complex ones. We also consider the case of subshifts and, using arguments from
recursion theory, we give negative results about the existence of universal
objects in some classes
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