5,635 research outputs found
Right-Permutative Cellular Automata on Topological Markov Chains
In this paper we consider cellular automata with
algebraic local rules and such that is a topological Markov
chain which has a structure compatible to this local rule. We characterize such
cellular automata and study the convergence of the Ces\`aro mean distribution
of the iterates of any probability measure with complete connections and
summable decay.Comment: 16 pages, 2 figure. A new version with improved redaction of Theorem
6.3(i)) to clearify its consequence
Exploring the concept of interaction computing through the discrete algebraic analysis of the Belousov–Zhabotinsky reaction
Interaction computing (IC) aims to map the properties of integrable low-dimensional non-linear dynamical systems to the discrete domain of finite-state automata in an attempt to reproduce in software the self-organizing and dynamically stable properties of sub-cellular biochemical systems. As the work reported in this paper is still at the early stages of theory development it focuses on the analysis of a particularly simple chemical oscillator, the Belousov-Zhabotinsky (BZ) reaction. After retracing the rationale for IC developed over the past several years from the physical, biological, mathematical, and computer science points of view, the paper presents an elementary discussion of the Krohn-Rhodes decomposition of finite-state automata, including the holonomy decomposition of a simple automaton, and of its interpretation as an abstract positional number system. The method is then applied to the analysis of the algebraic properties of discrete finite-state automata derived from a simplified Petri net model of the BZ reaction. In the simplest possible and symmetrical case the corresponding automaton is, not surprisingly, found to contain exclusively cyclic groups. In a second, asymmetrical case, the decomposition is much more complex and includes five different simple non-abelian groups whose potential relevance arises from their ability to encode functionally complete algebras. The possible computational relevance of these findings is discussed and possible conclusions are drawn
Statistical Mechanics of Surjective Cellular Automata
Reversible cellular automata are seen as microscopic physical models, and
their states of macroscopic equilibrium are described using invariant
probability measures. We establish a connection between the invariance of Gibbs
measures and the conservation of additive quantities in surjective cellular
automata. Namely, we show that the simplex of shift-invariant Gibbs measures
associated to a Hamiltonian is invariant under a surjective cellular automaton
if and only if the cellular automaton conserves the Hamiltonian. A special case
is the (well-known) invariance of the uniform Bernoulli measure under
surjective cellular automata, which corresponds to the conservation of the
trivial Hamiltonian. As an application, we obtain results indicating the lack
of (non-trivial) Gibbs or Markov invariant measures for "sufficiently chaotic"
cellular automata. We discuss the relevance of the randomization property of
algebraic cellular automata to the problem of approach to macroscopic
equilibrium, and pose several open questions.
As an aside, a shift-invariant pre-image of a Gibbs measure under a
pre-injective factor map between shifts of finite type turns out to be always a
Gibbs measure. We provide a sufficient condition under which the image of a
Gibbs measure under a pre-injective factor map is not a Gibbs measure. We point
out a potential application of pre-injective factor maps as a tool in the study
of phase transitions in statistical mechanical models.Comment: 50 pages, 7 figure
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
Attractiveness of the Haar measure for linear cellular automata on Markov subgroups
For the action of an algebraic cellular automaton on a Markov subgroup, we
show that the Ces\`{a}ro mean of the iterates of a Markov measure converges to
the Haar measure. This is proven by using the combinatorics of the binomial
coefficients on the regenerative construction of the Markov measure.Comment: Published at http://dx.doi.org/10.1214/074921706000000130 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
On the structure of Clifford quantum cellular automata
We study reversible quantum cellular automata with the restriction that these
are also Clifford operations. This means that tensor products of Pauli
operators (or discrete Weyl operators) are mapped to tensor products of Pauli
operators. Therefore Clifford quantum cellular automata are induced by
symplectic cellular automata in phase space. We characterize these symplectic
cellular automata and find that all possible local rules must be, up to some
global shift, reflection invariant with respect to the origin. In the one
dimensional case we also find that every uniquely determined and
translationally invariant stabilizer state can be prepared from a product state
by a single Clifford cellular automaton timestep, thereby characterizing these
class of stabilizer states, and we show that all 1D Clifford quantum cellular
automata are generated by a few elementary operations. We also show that the
correspondence between translationally invariant stabilizer states and
translationally invariant Clifford operations holds for periodic boundary
conditions.Comment: 28 pages, 2 figures, LaTe
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