On dynamical properties of neural networks

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.

Applied Mathematics Letters, 2006

The main goal of this work is to show an extension of well known Hedlund's theorem which states that in the Cantor topology the cellular automata are the continuous shift preserving maps. This extension maintains the topological structure and uses the concept of barriers of Set Theory to generalize the notion of local rule in the definition of cellular automata.

Complex Systems, 1989

Neurocomputing, 2011

Infinite countable or uncountable systems of nonlinear ordinary and partial differential equations, which are discrete and continuous models, respectively, of real-world phenomena and processes were analysed in physics, biology, neuroscience, in studying of neural systems or in pure mathematics. Discrete models and corresponding infinite countable systems have been investigated by numerous authors in Banach sequence spaces ' 1. But continuous model and corresponding infinite uncountable systems have been investigated by a few authors only. It is interesting that we can attack the above problems by using the same topological fixed-point theory tools as in the countable case. It is the study of infinite uncountable systems of parabolic reaction-diffusion-convection equations that is the subject matter of this paper. Motivation for considering such systems and examples thereof are presented. The paper is an introduction to the study of infinite uncountable systems of parabolic equations.

Proceedings of the 2002 7th IEEE International Workshop on Cellular Neural Networks and Their Applications

The stability and dynamics of a class of Cellular Neural Networks (CNN's) in the central linear pan is investigated using the decoupling lechnique based on discrete spatial transforms, Nyquist and root locus techniques.

Encyclopedia of Complexity and Systems Science, 2009

Almost equicontinuous CA: a CA which has at least one equicontinuous configuration.

We study discrete dynamical systems through the topological concepts of limit set, which consists of all points that can be reached arbitrarily late, and asymptotic set, which consists of all adhering values of orbits. In particular, we deal with the case when each of these are a singleton, or when the restriction of the system is periodic on them, and show that this is equivalent to some simple dynamics in the case of subshifts or cellular automata. Moreover, we deal with the stability of these properties with respect to some simulation notions.

Complex Systems, 1990

Abstract. Determining just what tasks are computable by neural networks is of fundamental importance in neural computing. The configuration space of several models of parallel computation is essentially the Cantor middle-third set of real numbers. The Hedlund-Richardson theorem states that a transformation from the Cantor set to itself can be realized as the global dynamics of a cellular automaton if and only if it takes the quiescent configuration to itself, commutes with shifts, and is continuous in the product topology. An ...

2013

In this paper, we study the quantitative behavior of one-dimensional linear cellular automata T f [−r,r] , defined by local rule f (x −r , . . . , xr) = r i=−r λ i x i (mod m), acting on the space of all doubly infinite sequences with values in a finite ring Zm, m ≥ 2. Once generalize the formulas given by Ban et al. [J. Cellular Automata 6 (2011) 385-397] for measure-theoretic entropy and topological pressure of one-dimensional cellular automata, we calculate the measure entropy and the topological pressure of the linear cellular automata with respect to the Bernoulli measure on the set Z Z m . Also, it is shown that the uniform Bernoulli measure is the unique equilibrium measure for linear cellular automata. We compare values of topological entropy and topological directional entropy by using the formula obtained by Akın [J. Computation and Appl. Math. 225 (2) (2009) 459-466]. The topological directional entropy is interpreted by means of figures. As an application, we demonstrate that the Hausdorff of the limit set of a linear cellular automaton is the unique root of Bowen's equation. Some open problems remain to be of interest.

This paper investigates the ergodicity and the power rule of the topological pressure of a cellular automaton. If a cellular automaton is either leftmost or rightmost permutive (due to the terminology given by Hedlund [Math. Syst. Theor. 3, 320-375, 1969]), then it is ergodic with respect to the uniform Bernoulli measure. More than that, the relation of topological pressure between the original cellular automaton and its power rule is expressed in a closed form. As an application, the topological pressure of a linear cellular automaton can be computed explicitly. 2010 MSC: 37B10

J. Cell. Autom., 2019

In this article, we consider a topological dynamical system. The generic limit set is the smallest closed subset which has a comeager realm of attraction. We study some of its topological properties, and the links with equicontinuity and sensitivity. We emphasize the case of cellular automata, for which the generic limit set is included in all subshift attractors, and discuss directional dynamics, as well as the link with measure-theoretical similar notions.

2009

Lecture Notes in Computer Science, 2008

A one-dimensional cellular automaton is a dynamical system which consisting in a juxtaposition of cells whose state changes over discrete time according to that of their neighbors. One of its simplest behaviors is nilpotency: all configurations of cells are mapped after a finite time into a given "null" configuration. Our main result is that nilpotency is equivalent to the condition that all configurations converge towards the null configuration for the Cantor topology, or, equivalently, that all cells of all configurations asymptotically reach a given state.

Lecture Notes in Computer Science, 2003

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

Entropy, 2009

This elucidation studies ergodicity and equilibrium measures for additive cellular automata with prime states. Additive cellular automata are ergodic with respect to Bernoulli measure unless it is either an identity map or constant. The formulae of measure-theoretic and topological entropies can be expressed in closed forms and the topological pressure is demonstrated explicitly for potential functions that depend on finitely many coordinates. According to these results, Parry measure is inferred to be an equilibrium measure.