Volume 22, Number 1 (2013)

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Exploring the Space of Substitution Systems
Richard Southwell and Chris Cannings

Substitution systems, where strings are rewritten according to local rules, have many applications. They are used to model the development of plants, as well as to generate music and architectural designs. Many substitution systems can generate highly complex patterns using only simple rules. This feature can make substitution systems difficult to analyze mathematically. A different approach, pioneered by Stephen Wolfram, is to use computer searches to reveal simple systems with interesting properties. This approach is used to explore a class of systems we call symmetric sequential substitution systems within which a string is repeatedly updated by applying rewrite rules in a non-overlapping way. In this paper several simple examples of these systems are exhibited that produce complex behavior. The dynamics of several of these systems are studied and a system is exhibited that is computationally universal. Applications of symmetric sequential substitution systems are discussed, such as compression and the evaluation of numerical functions.

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Emergence of Frontiers in Networked Schelling Segregationist Models
Philippe Collard, Salma Mesmoudi, Teodor Ghetiu, and Fiona Polack

The relation between individuality and aggregation is an important topic in complex systems sciences, both aspects being facets of emergence. This topic has frequently been addressed by adopting a classical, individual versus population level perspective. Here, however, the frontiers that emerge in segregated communities are the focus; segregation is synonymous with the existence of frontiers that delineate and interface aggregates. A generic agent-based model is defined, with which we simulate communities located on grid and scale-free networked environments. Emerging frontiers are analyzed in terms of their relative occupancy, porosity, and permeability. Results emphasize that the frontier is highly sensitive to the topology of the environment, not only to the agent tolerance. These relations are clarified while addressing the topics of frontier robustness and the trade-off between its capacity to separate and allow exchange.

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Schizophrenic Representative Investors
Philip Z. Maymin

Representative investors whose behavior is modeled by a deterministic finite automaton generate complexity both in the time series of each asset and in the cross-sectional correlation when the rule governing their behavior is schizophrenic, meaning the investor holds multiple seemingly contradictory beliefs simultaneously, either by switching between two different rules at each time step, or computing different responses to different assets.

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Continuum versus Discrete: A Physically Interpretable General Rule for Cellular Automata by Means of Modular Arithmetic
Luan C. de S. M. Ozelim, André L. B. Cavalcante, and Lucas P. de F. Borges

Describing complex phenomena by means of cellular automata (CAs) has shown to be a very effective approach in pure and applied sciences. In fact, the number of published papers concerning this topic has tremendously increased over the last 20 years. Most of the applications use CAs to qualitatively describe the phenomena, which is surely a consequence of the way the automata rules are commonly defined. In this paper, a physical application of a general rule that describes each of Stephen Wolfram's CAs is discussed. The new representation is given in terms of the so-called iota-delta function. The latter function is further generalized in order to provide a general rule for not only Wolfram's but also to every CA rule that depends on the sum and products of the values of cells in the automaton mesh. By means of a parallel between the finite difference method and the iota-delta function, a straightforward physical interpretation of CAs is derived. Such an application regards advective-diffusive phenomena without a constant source. Finally, the relation between CAs and anomalous diffusion is briefly discussed.