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A nonlinear dynamics perspective of Wolfram’s new kind of science. III. Predicting the unpredictable. (English) Zbl 1091.37500
Summary: We prove rigorously the four cellular automata local rules 110, 124, 137 and 193 have identical dynamic behaviors capable of universal computations. We exploit Felix Klein’s remarkable Vierergruppe to partition the 256 local rules studied empirically by Wolfram into 89 global equivalence classes of which only 50 may exhibit complex dynamics. We define a 24-element rotation group which induces 30 local equivalence classes of nonlinear difference equations whose parameters can be mapped into each other among members of the same class. For part II see ibid. 13, 2377--2491 (2003; Zbl 1046.37004) and part IV see ibid. 15, 1045--1183 (2005; Zbl 1084.37011).

37B10Symbolic dynamics
37N20Dynamical systems in other branches of physics
68Q80Cellular automata (theory of computing)
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