科研之友
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摘要
The 11th part of our tour through one-dimensional binary Cellular Automata concerns period-2 rules, which form the second group in our classification of the 88 globally-independent CA rules according to the properties of their periodic orbits. In this article, we display the basin tree diagrams of all period-2 rules along with their time-2 characteristic functions, and then we prove that all rules belonging to group 2 have robust period-2 omega-limit orbits for any finite, and infinite, bit string length. This rigorous result, which pairs with the one about period-1 rules given in the tenth installment of our chronicle, confirms what we stated about period-2 rules on the basis of empirical evidence. In the second part of this tutorial, we introduce the notion of quasi global-equivalence and prove that there are only 82 quasi globally-independent CA rules. For the first time, we show that the space-time patterns of globally-independent local rules can depend on each other, and we present an example of quasi-global transformation. We also define the super string, and its unique decimal representation x, dubbed the super decimal, which provides a completely transparent yet rigorous proof that rule 170 is chaotic when L -> 8. Moreover, we present the basin tree generation formulas, which uncover the analytical relationships between basin trees of globally-equivalent rules. Last but not least, for pedagogical and epistemological reasons, we conclude this paper with the selection of rule 137, instead of rule 110, as the prototypic universal Turing machine for our future discourse.
- 出版日期2009-6
