This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. In particular, the period 'T' of all attractors of rules, and, as well as of all isles of Eden of rules and, increases linearly with unit slope, in logarithmic scale, with the length L. In particular, local rules and are globally quasi-equivalent where corresponding space-time patterns can be derived from each other by simply complementing every other row.Īnother important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules, and. Such formulas led to the rigorous proof of several surprising periodicity constraints for rule, and to the discovery of a new global, quasi-equivalence class, defined via an alternating transformation. Two such properties form the main results of this paper namely, Įxplicit global state transition formulas are given for local rules, and. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞.īasin tree diagrams of all ten complex Bernoulli σ τ -shift rules are exhibited for lengths L = 3, 4, …, 8. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulliσ τ -shift rules and eight hyper Bernoulliσ τ -shift rules, the latter including such famous rules and. This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. In particular, the period 'T ' of all attractors of rules 90, 105 and 150, as well as of all isles of Eden of rules 105 and 150, increases linearly with unit slope, in logarithmic scale, with the length L. Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules 90, 105 and 150. In particular, local rules 105 and 150 are globally quasi-equivalent where corresponding space-time patterns can be derived from each other by simply complementing every other row. Such formulas led to the rigorous proof of several surprising periodicity constraints for rule 90, and to the discovery of a new global, quasi-equivalence class, defined via an alternating transformation. Explicit global state transition formulas are given for local rules 90, 105 and 150. Rules 105 and 150 are composed of nothing but Isles of Eden for all string lengths L not divisible by 3.Two such properties form the main results of this paper namely, Basin tree diagrams of all ten complex Bernoulli σ τ -shift rules are exhibited for lengths L = 3, 4. All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulli σ τ -shift rules and eight hyper Bernoulli σ τ -shift rules, the latter including such famous rules 30 and 110.

Bernoulli subshift