A nonlinear dynamics perspective of wolfram's new kind of science. Part I: threshold of complexity

摘要

This tutorial provides a nonlinear dynamics perspective to Wolfram's monumental work on A New Kind of Science. By mapping a Boolean local Rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram's empirical observations can be justified on firm ground. The advantage of this new approach for studying Cellular Automata phenomena is that it is based on concepts from nonlinear dynamics and attractors where many fuzzy concepts introduced by Wolfram via brute force observations can be defined and justified via mathematical analysis. The main result of Part I is the introduction of a fundamental concept called linear separability and a complexity index k for each local Rule which characterizes the intrinsic geometrical structure of an induced "Boolean cube" in three-dimensional Euclidean space. In particular, Wolfram's seductive idea of a "threshold of complexity" is identified with the class of local Rules having a complexity index equal to 2.