Christopher G. Langton

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

Abstract In order for computation to emerge spontaneously and become an important factor in the dynamics of a system, the material substrate must support the primitive functions required for computation: the transmission, storage, and modification of information. Under what conditions might we expect physical systems to support such computational primitives? This paper presents research on cellular automata which suggests that the optimal conditions for the support of information transmission, storage, and modification, are achieved in the vicinity of a phase transition. We observe surprising similarities between the behaviors of computations and systems near phase transitions, finding analogs of computational complexity classes and the halting problem within the phenomenology of phase transitions. We conclude that there is a fundamental connection between computation and phase transitions, especially second-order or “critical” transitions, and discuss some of the implications for our understanding of nature if such a connection is borne out.

Studying artificial life with cellular automata

Abstract Biochemistry studies the way in which life emerges from the interaction of inanimate molecules. In this paper we look into the possibility that life could emerge from the interaction of inanimate artificial molecules. Cellular automata provide us with the logical universes within which we can embed artificial molecules in the form of propagating, virtual automata. We suggest that since virtual automata have the computational capacity to fill many of the functional roles played by the primary biomolecules, there is a strong possibility that the ‘molecular logic’ of life can be embedded within cellular automata and that, therefore, artificial life is a distinct possibility within these highly parallel computer structures.

Self-reproduction in cellular automata

Abstract Self-reproduction in cellular automata is discussed with reference to the models of von Neumann and Codd. The conclusion is drawn that although the capacity for universal construction is a sufficient condition for self-reproduction, it is not a necessary condition. Slightly more “liberal” criteria for what constitutes genuine self-reproduction are introduced, and a simple self-reproducing structure is exhibited which satisfies these new criteria. This structure achieves its simplicity by storing its description in a dynamic “loop”, rather than on a static “tape”.

Transition phenomena in cellular automata rule space

Abstract We define several qualitative classes of cellular automata (CA) behavior, based on various statistical measures, and describe how the space of all cellular automata is organized. As a cellular automaton is changed by varying entries in its rule table, abrupt changes in qualitative behavior may occur. These abrupt changes have the character of bifurcations in smooth dynamical systems, or of phase transitions in statistical mechanical systems. The most complex CA rules exhibit many of the characteristics of second-order transitions, suggesting an association between computation, complexity, and critical phenomena.

Mean Field Theory of the Edge of Chaos

Is there an Edge of Chaos, and if so, can evolution take us to it? Many issues have to be settled before any definitive answer can be given. For quantitative work, we need a good measure of complexity. We suggest that convergence time is an appropriate and useful measure. In the case of cellular automata, one of the advantages of the convergence-time measure is that it can be analytically approximated using a generalized mean field theory.

Is there a sharp phase transition for deterministic cellular automata

Abstract Previous work has suggested that there is a kind of phase transition between deterministic automata exhibiting periodic behavior and those exhibiting chaotic behavior. However, unlike the usual phase transitions of physics, this transition takes place over a range of values of the parameter rather than at a specific value. The present paper asks whether the transition can be made sharp, either by taking the limit of an infinitely large rule table, or by changing the parameter in terms of which the space of automata is explored. We find strong evidence that, for the class of automata we consider, the transition does become sharp in the limit of an infinite number of symbols, the size of the neighborhood being held fixed. Our work also suggests an alternative parameter in terms of which it is likely that the transition will become fairly sharp even if one does not increase the number of symbols. In the course of our analysis, we find that mean field theory, which is our main tool, gives surprisingly good predictions of the statistical properties of the class of automata we consider.