[Continuous 110 paper] - A New Kind of Science: The NKS Forum

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Continuous 110 paper

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Posted by: Jason Cawley

Fuzzy Rule 110 and the golden number

A. Mingarelli

WSEAS Transactions on Computers
2003 Volume: 2 Issue: 4 Pages: 1102-1107

We continue the investigation into the dynamics and evolution
of fuzzy rules, obtained by the fuzzification of the disjunctive normal form, and initiated for Rule 90 in Flocchini-Geurts-Mingarelli- Santoro (2000). Interest in the boolean form of Rule 110 has been tremendous since the discovery that it supports Turing-complete computation.

Inspired by this, we present new results regarding the dynamics of fuzzy Rule 110 whose boolean evolution is also known to be chaotic [Wolfram (2002), p.871]. In particular, we show that in the fuzzy case with a finite support configuration all spatiotemporal sequences are aperiodic, and their convergence is strongly dependent upon their positions along key diagonals. It follows that fuzzy Rule 110 is neither chaotic nor random. It turns out that the evolution and dynamics in this case differ radically from those in fuzzy Rule 90. We will show that the dynamics of
fuzzy Rule 110 make it such that every key sequence converges to interesting distinct mathematical constants that, themselves, converge to a key mathematical constant.


Posted by: Jason Cawley

You can readily impliment these as continuous CAs and examine their characteristics experimentally, rather than just proving theorems about their infinite limit behavior. Ben Koo of MIT worked on similar stuff at the NKS 2004 summer program at Brown. He was interested in Bayesian nets laid out in CA form. The convergence to the golden ratio in 110 - and to 1/2 in rules like 30 and 90 - is readily seen. You can put in any level of uncertainty about specific points or entire initial conditions and see how the uncertainty propagates. Typically you get a greyed out region in the lower middle of the pattern (how far down depends on how close to 0 or 1 you set initial values), while e.g. the periodic upper left boundary area in rule 30 remains nearly certain - and the like.


Posted by: Emmanuel Garces

I'm actually working on a continuous CA with the ability to reproduce the same behaviour that a simple one does.

I used boolean rules in order to work with values between 0 and 1. The only difference is the negation operator, that should look like a sigmoid function. And I used a larger size neighborhood.

Here are some of the pictures that my CA was able to generate. Those correspond to rule 30 and 110 respectively. It can also reproduce another patterns for another rules. But I'm still working on improving it.




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