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Computation at the edge of chaos: phase transitions and emergent computation

Computations are dynamical systems. The formal study of dynamical systems has revealed a spectrum of behaviors ranging from fixed-point dynamics to fully developed chaos. How does computation--especially universal computation--fit into this spectrum of dynamical behaviors?
This thesis addresses the question via the study of Cellular Automata (CA). CA are discrete space/time dynamical systems which (a) exhibit the full spectrum of dynamical behaviors, and (b) are known to support universal computation. Therefore, if we can locate the CA which support computation in the full spectrum of CA dynamics, we may learn a great deal about the general relationship between computation and dynamics.
We find that, with an appropriate parameterization of the space of CA, we can order CA dynamical behaviors from fixed-point, through periodic, to chaotic behavior. We also find that there exists a phase-transition in this sequence, located between the periodic and the chaotic rules. The dynamics in the vicinity of this phase-transition are the most complex exhibited anywhere in the spectrum, both qualitatively and quantitatively. These "complex" rules exhibit many of the properties deemed necessary to support computation, and therefore we are able to identify computation with a phase-transition in the space of CA. We are also able to explain the existence of--and relationships between--previously suggested classes of CA behavior (such as the four Wolfram classes) in terms of this phase-transition in the space of CA.
The results reported here suggest a general relationship between computation and phase-transitions, especially second-order phase-transitions. When viewed from this perspective, one is able to explain a great deal of the phenomenology of computation, such as the existence of complexity classes, computability classes, and the Halting problem. This view also leads to some new suggestions about the nature and origin of such complex natural phenomena as Life and Intelligence.

Computation at the edge of Chaos: phase transitions and emergent computation

Transition phenomena in cellular automata rule space

Some of the major outstanding problems in biology are related to issues of emergence and evolution. These include: (a) how populations of organisms traverse their adaptive landscapes; (b) what the relation between adaptedness and fitness is; and (c) the formation of multicellular organisms from basic units or cells. In this article we study these issues using a model that is both general and simple. The system, derived from the CA (cellular automata) model, consists of a two-dimensional grid of interacting organisms that may evolve over time. We first present designed multicellular organisms that display several interesting behaviors, including reproduction, growth, and mobility. We then turn our attention to evolution in various environments, including an environment in which competition for space occurs, an IPD (Iterated Prisoner's Dilemma) environment, an environment of spatial niches, and an environment of temporal niches. One of the advantages of artificial life (AL) models is the opportunities they offer in performing in-depth studies of the evolutionary process. This is accomplished in our case by observing not only phenotypic effects but also such measures as fitness, operability, energy and the genescape. Our work sheds light on the problems raised above, and offers a possible path toward the long-term, two-fold goal of ALife research: (a) increasing our understanding of biology, and (b) enhancing our understanding of artificial models, thereby providing us with the ability to improve their performance.

Studying artificial life using a simple, general cellular model

Dynamical systems with nonlocal connections have potential applications to economic and biological systems. This paper studies the dynamics of nonlocal cellular automata. In particular, all two-state, three-input nonlocal cellular automata are classified according to the dynamical behavior starting from random initial configurations and random wirings, although it is observed that sometimes a rule can have different dynamical behaviors with different wirings. The nonlocal cellular automata rule space is studied using a mean-field parametrization which is ideal for the situation of random wiring. Nonlocal cellular automata can be considered as computers carrying out computation at the level of each component. Their computational abilities are studied from the point of view of whether they contain many basic logical gates. In particular, I ask the question of whether a three-input cellular automaton rule contains the three fundamental logical gates: two-input rules AND and OR, and one-input rule NOT. A particularly interesting “edge-of-chaos” nonlocal cellular automaton, the rule 184, is studied in detail. It is a system of coupled “selectors” or “multiplexers.” It is also part of the Fredkin's gate—a proposed fundamental gate for conservative computations. This rule exhibits irregular fluctuations of density, large coherent structures, and long transient times.

Phenomenology of nonlocal cellular automata

A probabilistic one-dimensional cellular automaton model by Domany and Kinzel is mapped into an inhomogeneous cellular automaton with the Boolean functions XOR and AND as transition rules. Wolfram's classification is recovered by varying the frequency of these two simple rules and by quenching or annealing the inhomogeneity. In particular, “class 4” is related to critical behavior in directed percolation. Also, the critical slowing down of second-order phase transitions is related to a stochastic version of the classical “halting problem” of computation theory.

Annealed and quenched inhomogeneous cellular automata (INCA)

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Annealed and quenched inhomogeneous cellular automata (INCA)