Related papers: Level Set Structure of an Integrable Cellular Auto…
We show that a large number of elementary cellular automata are computationally simple. This work is the first systematic classification of elementary cellular automata based on a formal notion of computational complexity. Thanks to the…
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…
Let L:=Z^D be a D-dimensional lattice. Let A^L be the Cantor space of L-indexed configurations in a finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F:A^L-->A^L. An…
In this paper, we explore relationships between two models of systems which are governed by only the local interactions of large collections of simple components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM). While…
A multi-component semi-discrete nonlinear integrable system associated with the relevant third-order auxiliary linear problem is claimed to be the prototype system for several reduced integrable systems formulated in terms of true dynamical…
We develop a rather elaborate computer program to investigate the jointly periodic points of one-dimensional cellular automata. The experimental results and mathematical context lead to questions, conjectures and a contextual theorem.
A novel, information-based classification of elementary cellular automata is proposed that circumvents the problems associated with isolating whether complexity is in fact intrinsic to a dynamical rule, or if it arises merely as a product…
Particle-like objects are observed to propagate and interact in many spatially extended dynamical systems. For one of the simplest classes of such systems, one-dimensional cellular automata, we establish a rigorous upper bound on the number…
We consider random boolean cellular automata on the integer lattice, i.e., the cells are identified with the integers from 1 to $N$. The behaviour of the automaton is mainly determined by the support of the random variable that selects one…
Hierarchical structure is an essential part of complexity, important notion relevant for a wide range of applications ranging from biological population dynamics through robotics to social sciences. In this paper we propose a simple…
State-of-the-art review of cellular automata, cellular automata for partial differential equations, differential equations for cellular automata and pattern formation in biology and engineering.
The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter…
Gauge-invariance is a fundamental concept in physics---known to provide the mathematical justification for all four fundamental forces. In this paper, we provide discrete counterparts to the main gauge theoretical concepts, directly in…
A deautonomized version of the two-dimensional Toda lattice equation is presented. Its ultra-discrete analogue and soliton solutions are also discussed.
In a probabilistic cellular automaton in which all local transitions have positive probability, the problem of keeping a bit of information indefinitely is nontrivial, even in an infinite automaton. Still, there is a solution in 2…
In this chapter, we consider a class of discrete dynamical systems defined on the homogeneous space associated with a regular tiling of $\R^N$, whose most familiar example is provided by the $N-$dimensional torus $\T ^N$. It is proved that…
A cellular automaton model is used to describe the dynamics of the catalytic oxidation of $CO$ on a $Pt(100)$ surface. The cellular automaton rules account for the structural phase transformations of the $Pt$ substrate, the reaction…
We investigate a soliton cellular automaton (Box-Ball system) with periodic boundary conditions. Since the cellular automaton is a deterministic dynamical system that takes only a finite number of states, it will exhibit periodic motion. We…
Deterministic many-body systems governed by simple interactions can self-organize into macroscopic patterns, and the determinants of long-time behavior are assumed to be encoded in the initial configuration. Here we show that predictability…
In order to develop systems capable of artificial evolution, we need to identify which systems can produce complex behavior. We present a novel classification method applicable to any class of deterministic discrete space and time dynamical…