A full computation-relevant topological dynamics classification of elementary cellular automata (original) (raw)

Topological chaos for elementary cellular automata

Lecture Notes in Computer Science, 1997

We apply the definition of chaos given by Devaney for discrete time dynamical systems to the case of elementary cellular automata, i.e., 1-dimensional binary cellular automata with radius 1. A discrete time dynamical system is chaotic according to the Devaney's definition of chaos if it is topologically transitive, is sensitive to initial conditions, and has dense periodic orbits. We enucleate an easy-to-check property of the local rule on which a cellular automaton is based which is a necessary condition for chaotic behavior. We prove that this property is also sufficient for a large class of elementary cellular automata. The main contribution of this paper is the formal proof of chaoticity for many non additive elementary cellular automata. Finally, we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata.

Investigating topological chaos by elementary cellular automata dynamics

Theoretical Computer Science, 2000

We apply the two di erent deÿnitions of chaos given by Devaney and by Knudsen for general discrete time dynamical systems (DTDS) to the case of elementary cellular automata, i.e., 1-dimensional binary cellular automata with radius 1. A DTDS is chaotic according to the Devaney's deÿnition of chaos i it is topologically transitive, has dense periodic orbits, and it is sensitive to initial conditions. A DTDS is chaotic according to the Knudsen's deÿnition of chaos i it has a dense orbit and it is sensitive to initial conditions. We enucleate an easy-to-check property (left or rightmost permutivity) of the local rule associated with a cellular automaton which is a su cient condition for D-chaotic behavior. It turns out that this property is also necessary for the class of elementary cellular automata. Finally, we prove that the above mentioned property does not remain a necessary condition for chaoticity in the case of non elementary cellular automata.

Complex Dynamics of Elementary Cellular Automata Emerging from Chaotic Rules

International Journal of Bifurcation and Chaos, 2012

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, λ and Zparameters, 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 onedimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions.

Cellular automata and dynamical systems

1989

In this thesis we investigate the theoretical nature of the mathematical structures termed cellular automata. Chapter 1: Reviews the origin and history of cellular automata in order to place the current work into context. Chapter 2: Develops a cellular automata framework which contains the main aspects of cellular automata structure which have appeared in the literature. We present a scheme for specifying the cellular automata rules for this general model and present six examples of cellular automata within the model. Chapter 3: Here we develop a statistical mechanical model of cellular automata behaviour. We consider the relationship between variations within the model and their relationship to dynamical systems. We obtain results on the variance of the state changes, scaling of the cellular automata lattice, the equivalence of noise, spatial mixing of the lattice states and entropy, synchronous and asynchronous cellular automata and the equivalence of the rule probability and the ...

Complex dynamics of elementary cellular automata emerging in chaotic rules

2010

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, λ and Zparameters, 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 onedimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions.

Some results about the chaotic behavior of cellular automata

Theoretical Computer Science, 2005

We study the behavior of cellular automata (CA for short) in the Cantor, Besicovitch and Weyl topologies. We solve an open problem about the existence of transitive CA in the Besicovitch topology. The proof of this result has some interest of its own since it is obtained by using Kolmogorov complexity. To our knowledge it is the first result about discrete dynamical systems obtained using Kolmogorov complexity. We also prove that in the Besicovitch topology every CA has either a unique periodic point (thus a fixed point) or an uncountable set of periodic points. This result underlines the fact that CA have a great degree of stability; it may be considered a further step towards the understanding of CA periodic behavior.

On the Dynamical Behavior of Chaotic Cellular Automata

Theoretical Computer Science, 1999

The shift (bi-infinite) cellular automaton is a chaotic dynamical system according to all the definitions of deterministic chaos given for discrete time dynamical systems (e.g., those given by Devaney [6] and by Knudsen [10]). The main motivation to this fact is that the temporal evolution of the shift cellular automaton under finite description of the initial state is unpredictable. Even

A geometrical interpretation of the chaotic state of inhomogeneous deterministic cellular automata

Physica A: Statistical Mechanics and its Applications, 1989

We propose a geometrical interpretation of the chaotic state of inhomogeneous cellular automata. From the ru:!es of the cellular automaton we construct a network. The percolating phase of this network ,:oincides with the chaotic phase of the cellular automaton. We also report numerical tests < f these ideas on several cellular automata.

Cellular Automata - The Stanford Encyclopedia of Philosophy

Cellular automata (henceforth: CA) are discrete, abstract computational systems that have proved useful both as general models of complexity and as more specific representations of non-linear dynamics in a variety of scientific fields. Firstly, CA are (typically) spatially and temporally discrete: they are composed of a finite or denumerable set of homogenous, simple units, the atoms or cells. At each time unit, the cells instantiate one of a finite set of states. They evolve in parallel at discrete time steps, following state update functions or dynamical transition rules: the update of a cell state obtains by taking into account the states of cells in its local neighborhood (there are, therefore, no actions at a distance). Secondly, CA are abstract: they can be specified in purely mathematical terms and physical structures can implement them. Thirdly, CA are computational systems: they can compute functions and solve algorithmic problems. Despite functioning in a different way from traditional, Turing machine- like devices, CA with suitable rules can emulate a universal Turing machine (see entry), and therefore compute, given Turing’s thesis (see entry on Church-Turing thesis), anything computable...

Global Dynamics of Finite Cellular Automata

Lecture Notes in Computer Science

A novel algebraic and dynamic systems approach to finite elementary cellular automata is presented. In particular, simple algebraic expressions for the local rules of elementary cellular automata are deduced and the cellular automata configurations are represented via Fourier analysis. This allows for a further analysis of the global dynamics of cellular automata by the use of tools derived from functional analysis and dynamical system theory.