Defect Particle Kinematics in One-Dimensional Cellular Automata

Let A^Z be the Cantor space of bi-infinite sequences in a finite alphabet A, and let sigma be the shift map on A^Z. A `cellular automaton' is a continuous, sigma-commuting self-map Phi of A^Z, and a `Phi-invariant subshift' is a closed, (Phi,sigma)-invariant subset X of A^Z. Suppose x is a sequence in A^Z which is X-admissible everywhere except for some small region we call a `defect'. It has been empirically observed that such defects persist under iteration of Phi, and often propagate like `particles'. We characterize the motion of these particles, and show that it falls into several regimes, ranging from simple deterministic motion, to generalized random walks, to complex motion emulating Turing machines or pushdown automata. One consequence is that some questions about defect behaviour are formally undecidable.Comment: 37 pages, 9 figures, 3 table

Upper Bound on the Products of Particle Interactions in Cellular Automata

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 of distinct products that these interactions can generate. The upper bound is controlled by the structural complexity of the interacting particles---a quantity which is defined here and which measures the amount of spatio-temporal information that a particle stores. Along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata; thereby elucidating why that approach is of general utility. The upper bound is tested against several relatively complex domain-particle cellular automata and found to be tight.Comment: 17 pages, 12 figures, 3 tables, http://www.santafe.edu/projects/CompMech/papers/ub.html V2: References and accompanying text modified, to comply with legal demands arising from on-going intellectual property litigation among third parties. V3: Accepted for publication in Physica D. References added and other small changes made per referee suggestion

Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems

Most current methods for identifying coherent structures in spatially-extended systems rely on prior information about the form which those structures take. Here we present two new approaches to automatically filter the changing configurations of spatial dynamical systems and extract coherent structures. One, local sensitivity filtering, is a modification of the local Lyapunov exponent approach suitable to cellular automata and other discrete spatial systems. The other, local statistical complexity filtering, calculates the amount of information needed for optimal prediction of the system's behavior in the vicinity of a given point. By examining the changing spatiotemporal distributions of these quantities, we can find the coherent structures in a variety of pattern-forming cellular automata, without needing to guess or postulate the form of that structure. We apply both filters to elementary and cyclical cellular automata (ECA and CCA) and find that they readily identify particles, domains and other more complicated structures. We compare the results from ECA with earlier ones based upon the theory of formal languages, and the results from CCA with a more traditional approach based on an order parameter and free energy. While sensitivity and statistical complexity are equally adept at uncovering structure, they are based on different system properties (dynamical and probabilistic, respectively), and provide complementary information.Comment: 16 pages, 21 figures. Figures considerably compressed to fit arxiv requirements; write first author for higher-resolution version

Local Causal States and Discrete Coherent Structures

Coherent structures form spontaneously in nonlinear spatiotemporal systems and are found at all spatial scales in natural phenomena from laboratory hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary climate dynamics. Phenomenologically, they appear as key components that organize the macroscopic behaviors in such systems. Despite a century of effort, they have eluded rigorous analysis and empirical prediction, with progress being made only recently. As a step in this, we present a formal theory of coherent structures in fully-discrete dynamical field theories. It builds on the notion of structure introduced by computational mechanics, generalizing it to a local spatiotemporal setting. The analysis' main tool employs the \localstates, which are used to uncover a system's hidden spatiotemporal symmetries and which identify coherent structures as spatially-localized deviations from those symmetries. The approach is behavior-driven in the sense that it does not rely on directly analyzing spatiotemporal equations of motion, rather it considers only the spatiotemporal fields a system generates. As such, it offers an unsupervised approach to discover and describe coherent structures. We illustrate the approach by analyzing coherent structures generated by elementary cellular automata, comparing the results with an earlier, dynamic-invariant-set approach that decomposes fields into domains, particles, and particle interactions.Comment: 27 pages, 10 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.ht

A Spatial Agent-Based Model of N-Person Prisoner's Dilemma Cooperation in a Socio-Geographic Community

The purpose of this paper is to present a spatial agent-based model of N-person prisoner's dilemma that is designed to simulate the collective communication and cooperation within a socio-geographic community. Based on a tight coupling of REPAST and a vector Geographic Information System, the model simulates the emergence of cooperation from the mobility behaviors and interaction strategies of citizen agents. To approximate human behavior, the agents are set as stochastic learning automata with Pavlovian personalities and attitudes. A review of the theory of the standard prisoner's dilemma, the iterated prisoner's dilemma, and the N-person prisoner's dilemma is given as well as an overview of the generic architecture of the agent-based model. The capabilities of the spatial N-person prisoner's dilemma component are demonstrated with several scenario simulation runs for varied initial cooperation percentages and mobility dynamics. Experimental results revealed that agent mobility and context preservation bring qualitatively different effects to the evolution of cooperative behavior in an analyzed spatial environment.Agent Based Modeling, Cooperation, Prisoners Dilemma, Spatial Interaction Model, Spatially Structured Social Dilemma, Geographic Information Systems

Cellular automaton supercolliders

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular-automaton analogous of localizations or quasi-local collective excitations travelling in a spatially extended non-linear medium. They can be considered as binary strings or symbols travelling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyse what types of interaction occur between gliders travelling on a cellular automaton `cyclotron' and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in non-linear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analysed via implementation of cyclic tag systems

A Genetic Algorithm to Study a P3 Non-trivial Collective Task

Here we report new results of a genetic algorithm (GA) used to evolve one dimensional Cellular Automata (CA) to perform a P3 non-trivial collective behavior task. For this task the goal is to find a CA rule that reaches one final configuration in which the concentration of active cells oscillates among three different values. Though the majority of the best evolved rules belong to the II Wolfram’s class, the GA also finds rules of the III and IV classes. The different computational mechanisms used by each rule to synchronize the entire lattice are analyzed by means of the spatio-temporal patterns generated