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

Complex dynamics emerging in Rule 30 with majority memory

In cellular automata with memory, the unchanged maps of the conventional cellular automata are applied to cells endowed with memory of their past states in some specified interval. We implement Rule 30 automata with a majority memory and show that using the memory function we can transform quasi-chaotic dynamics of classical Rule 30 into domains of travelling structures with predictable behaviour. We analyse morphological complexity of the automata and classify dynamics of gliders (particles, self-localizations) in memory-enriched Rule 30. We provide formal ways of encoding and classifying glider dynamics using de Bruijn diagrams, soliton reactions and quasi-chemical representations

Topology regulates pattern formation capacity of binary cellular automata on graphs

We study the effect of topology variation on the dynamic behavior of a system with local update rules. We implement one-dimensional binary cellular automata on graphs with various topologies by formulating two sets of degree-dependent rules, each containing a single parameter. We observe that changes in graph topology induce transitions between different dynamic domains (Wolfram classes) without a formal change in the update rule. Along with topological variations, we study the pattern formation capacities of regular, random, small-world and scale-free graphs. Pattern formation capacity is quantified in terms of two entropy measures, which for standard cellular automata allow a qualitative distinction between the four Wolfram classes. A mean-field model explains the dynamic behavior of random graphs. Implications for our understanding of information transport through complex, network-based systems are discussed.Comment: 16 text pages, 13 figures. To be published in Physica

Complexity and Information: Measuring Emergence, Self-organization, and Homeostasis at Multiple Scales

Concepts used in the scientific study of complex systems have become so widespread that their use and abuse has led to ambiguity and confusion in their meaning. In this paper we use information theory to provide abstract and concise measures of complexity, emergence, self-organization, and homeostasis. The purpose is to clarify the meaning of these concepts with the aid of the proposed formal measures. In a simplified version of the measures (focusing on the information produced by a system), emergence becomes the opposite of self-organization, while complexity represents their balance. Homeostasis can be seen as a measure of the stability of the system. We use computational experiments on random Boolean networks and elementary cellular automata to illustrate our measures at multiple scales.Comment: 42 pages, 11 figures, 2 table

Outer-totalistic cellular automata on graphs

We present an intuitive formalism for implementing cellular automata on arbitrary topologies. By that means, we identify a symmetry operation in the class of elementary cellular automata. Moreover, we determine the subset of topologically sensitive elementary cellular automata and find that the overall number of complex patterns decreases under increasing neighborhood size in regular graphs. As exemplary applications, we apply the formalism to complex networks and compare the potential of scale-free graphs and metabolic networks to generate complex dynamics.Comment: 5 pages, 4 figures, 1 table. To appear in Physics Letters

Designing complex dynamics in cellular automata with memory

Since their inception at Macy conferences in later 1940s, complex systems have remained the most controversial topic of interdisciplinary sciences. The term "complex system" is the most vague and liberally used scientific term. Using elementary cellular automata (ECA), and exploiting the CA classification, we demonstrate elusiveness of "complexity" by shifting space-time dynamics of the automata from simple to complex by enriching cells with memory. This way, we can transform any ECA class to another ECA class - without changing skeleton of cell-state transition function - and vice versa by just selecting a right kind of memory. A systematic analysis displays that memory helps "discover" hidden information and behavior on trivial - uniform, periodic, and nontrivial - chaotic, complex - dynamical systems. © World Scientific Publishing Company