From: Andrew Wuensche (100020.2727@CompuServe.COM)
Date: Sat Mar 27 1993 - 14:04:42 UTC
...recent mail on "better parameters than lambda"
the Z parameter
---------------
...is the probability that the "next cell" in a partial pre-image has a
unique value. It predicts quantifiable properties of the basin of
attraction fields of CA such as the degree of pre-imaging and garden of
Eden density, which reflect the convergence of state space and relate
strongly with behaviour.
low Z --> simple dynamics, high convergence,
short transients - attractors, high g of E density.
high Z --> chaotic dynamics, low convergence
long transients - attractors, low g of E density
intermediate Z --> phase transition, possibly "complex" dynamics
ie. emergence of interacting gliders etc.
Z is derived from the rule table - the first approximation is the density
of "deterministic neighbourhoods" (which come in pairs) such that,
for binary CA with n'hood size = n..
if the n'hood a1,a2,..an-1,1 --> T (its output)
then a1,a2,..an-1,0 --> not T
Z may need to be corrected to account for additional deterministic
n'hoods (which come in 4-tuples) such that,
if the n'hood a1,a2,..an-2,1,* --> T (its output)
then a1,a2,..an-2,0,* --> not T
where "*" is a wildcard value, 0 or 1 with equal validity.
this correction is repeated if nessesary to take account of additional
deterministic n'hoods consisting of 8-tuples, 16-tuples etc,..up to a
2^n-tuple.
The proceedure gives Z-left, a converse proceedure gives Z-right, and the
greater of these values gives Z.
The lambda parameter and Z are related because lambda gives an indication
of the probability of a particular value of Z. Exceptions to lambda's
predictive powers (and correspondence with Z) are due to the fact that
lambda does not take into account "symmetries" or bunching in the
distribution of values in the rule table. (it only looks at the
proportions of the different values)
Details on the Z parameter are in chapter 3.6 of my book..
"The Global Dynamics of Cellular Automata; An Atlas of Basin of
Attraction Fields of One-Dimensional Cellular Automata"
Andrew Wuensche and Mike Lesser
(diskette included, IBSN 0-201-55740-1),
Santa Fe Institute Studies in the Sciences of Complexity, Reference
Vol.I, Addison-Wesley, Reading, M.A, 1992.
Andy Wuensche
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