Nonlinear Dynamics Perspective of Wolfram's New Kind of Science, a (Volume V)

This penultimate volume contains numerous original, elegant, and surprising results in 1-dimensional cellular automata. Perhaps the most exciting, if not shocking, new result is the discovery that only 82 local rules, out of 256, suffice to predict the time evolution of any of the remaining 174 local rules from an "arbitrary" initial bit-string configuration. This is contrary to the well-known folklore that 256 local rules are necessary, leading to the new concept of "quasi-global...
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This penultimate volume contains numerous original, elegant, and surprising results in 1-dimensional cellular automata. Perhaps the most exciting, if not shocking, new result is the discovery that only 82 local rules, out of 256, suffice to predict the time evolution of any of the remaining 174 local rules from an "arbitrary" initial bit-string configuration. This is contrary to the well-known folklore that 256 local rules are necessary, leading to the new concept of "quasi-global equivalence."

Another surprising result is the introduction of a simple, yet explicit, infinite bit string called the super string "S," which contains all random bit strings of finite length as sub-strings. As an illustration of the mathematical subtlety of this amazing discrete testing signal, the super string "S" is used to prove mathematically, in a trivial and transparent way, that rule 170 is as "chaotic" as a "coin toss."

Yet another unexpected new result, among many others, is the derivation of an explicit "basin tree generation formula" which provides an analytical relationship between the basin trees of globally-equivalent local rules. This formula allows the "symbolic," rather than numerical, generation of the time evolution of any local rule corresponding to any initial bit-string configuration, from one of the 88 globally-equivalent local rules.

But perhaps the most provocative idea is the proposal for adopting rule 137, over its three "globally-equivalent" siblings, including the heretofore more well-known "rule 110," as the prototypical "universal Turing machine."
Contents: "Period-2 Rules: "Recap of Period-2 RulesBasin Tree DiagramsRobust ω-Limit Orbits of Local Rules Belonging to Group 2Quasi Global-EquivalenceSuper String "S" and Super Decimal x"S"Concluding Remarks"Period-3, Period-6, and Permutive Rules: "List of the 88 Minimal Equivalence RulesBasin Tree Diagrams, Omega-Limit Orbits and Time-τ Characteristic Function of Rules from Group 3Robust ω-Limit Orbits of Rules from Group 3Permutive RulesConcluding Remarks
Readership: Graduate students, researchers and academics interested in nonlinear dynamics, computer science and complexity theory.