Complexity Measures from Interaction Structures

We evaluate new complexity measures on the symbolic dynamics of coupled tent maps and cellular automata. These measures quantify complexity in terms of $k$-th order statistical dependencies that cannot be reduced to interactions between $k-1$ units. We demonstrate that these measures are able to identify complex dynamical regimes.Comment: 11 pages, figures improved, minor changes to the tex

A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin

Celulární automat a CML systémy

The main aim of this thesis is the study of cellular automata and discrete dynamical systems on a lattice. Both tools, cellular automata as well as dynamical systems on a lattice are introduced and elementary properties described. The relation between cellular automata and dynamical system on lattice is derived. The main goal of the thesis is also the use of the cellular automata as that mathematical tool of evolution visualization of discrete dynamical systems. The theory of cellular automata is applied to the discrete dynamical systems on a lattice Laplacian type and implemented in Java language.Hlavním cílem práce je studium vztahu celulárních automatů a diskrétních dynamických systémů na mřížce. Oba nástroje, jak celulární automat tak dynamický systém na mřížce, jsou zavedeny a jejich základní vlastnosti popsány. Vztah mezi celulárními automaty a dynamickými systémy na mřížce je podrobně popsán. Hlavním cílem práce je dále použití nástroje celulárního automatu jako matematického vizualizačního prostředku evoluce diskrétních dynamických systémů. Teorie celulárních automatů je použita na dynamické systémy na mřížce Lamplaceova typu a implementována v prostředí Java.470 - Katedra aplikované matematikyvelmi dobř

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

Detecting synchronization in spatially extended discrete systems by complexity measurements

The synchronization of two stochastically coupled one-dimensional cellular automata (CA) is analyzed. It is shown that the transition to synchronization is characterized by a dramatic increase of the statistical complexity of the patterns generated by the difference automaton. This singular behavior is verified to be present in several CA rules displaying complex behavior.Comment: 4 pages, 2 figures; you can also visit http://add.unizar.es/public/100_16613/index.htm

Complex dynamics of elementary cellular automata emerging from chaotic rules

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, \lambda{} and Z-parameters, 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 one-dimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions

Spatial updating, spatial transients, and regularities of a complex automaton with nonperiodic architecture

We study the dynamics of patterns exhibited by rule 52, a totalistic cellular automaton displaying intricate behaviors and wide regions of active/inactive synchronization patches. Systematic computer simulations involving 230 initial configurations reveal that all complexity in this automaton originates from random juxtaposition of a very small number of interfaces delimiting active/inactive patches. Such interfaces are studied with a sidewise spatial updating algorithm. This novel tool allows us to prove that the interfaces found empirically are the only interfaces possible for these periods, independently of the size of the automata. The spatial updating algorithm provides an alternative way to determine the dynamics of automata of arbitrary size, a way of taking into account the complexity of the connections in the lattice