Chapter 10: Processes of Perception and Analysis

Section 11: Traditional Mathematics and Mathematical Formulas

Cellular automaton [Boolean] formulas

See page 869. The maximum length DNF for elementary rules after 1 step is 4, and this is achieved by rules 105, 107, 109, 121, 150, 151, 158, 182, 214 and 233. These rules have behavior of quite varying complexity. Rules 150 and 105 are additive, and correspond to Xor and its negation. After t steps the maximum conceivable DNF would be of length 22t. In practice, after 2 steps, the maximum length is 9, achieved by rules 107, 121 and 182; after 3 steps, it is 33 achieved by rule 182; after 4 steps, 78 achieved by rule 129; after 5 steps 256 achieved by rules 105 and 150. The distributions of lengths for all elementary rules are shown below.

Note that the length of a minimal DNF representation cannot be considered a reliable measure of the complexity of a function, since among other things, just exchanging the role of black and white can substantially change this length (as in the case of rule 126 versus rule 129).

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From Stephen Wolfram: A New Kind of Science [citation]