Computation at the Onset of Chaos

Computation at levels beyond storage and transmission of information appears in physical systems at phase transitions. We investigate this phenomenon using minimal computational models of dynamical systems that undergo a transition to chaos as a function of a nonlinearity parameter. For period-doubling and band-merging cascades, we derive expressions for the entropy, the interdependence of -machine complexity and entropy, and the latent complexity of the transition to chaos. At the transition deterministic finite automaton models diverge in size. Although there is no regular or context-free Chomsky grammar in this case, we give finite descriptions at the higher computational level of context-free Lindenmayer systems. We construct a restricted indexed context-free grammar and its associated one-way nondeterministic nested stack automaton for the cascade limit language. This analysis of a family of dynamical systems suggests a complexity theoretic description of phase transitions based on the informational diversity and computational complexity of observed data that is independent of particular system control parameters. The approach gives a much more refined picture of the architecture of critical states than is available via correlation functions, mutual information, and statistical mechanics generally. The analytic methods establish quantitatively the longstanding observation that significant computation is associated with the critical states found at the border between order and chaos. Appearing in Complexity, Entropy, and Physics of Information, W. Zurek, editor, Addison-Wesley, Reading, Massachusetts (1989). *JPC’s Internet address is [email protected]. KY’s permanent address: Physics Board of Studies, University of California, Santa Cruz, CA 95064. His Internet address is [email protected]