mean field software?
Johnicholas
Are there any open-source tools for doing mean-field analysis on CAs?
I've looked, and while there are a LOT of simulators and a significant
number of search programs of various sorts, mean field theory seems to
be mentioned only in academic papers (without source code).
Johnicholas
Kent Paul Dolan
> Johnicholas
Well, not having a clue what mean field theory _is_, I'm helpless to
read
the hits from this search intelligently to see if they'll help you, but
the
fourth one down does say something like "for source code, contact",
which seems at least hope-provoking.
http://scholar.google.com/scholar?q=mean.field.theory+cellular.automata+source.code
HTH
xanthian.
Johnicholas
I don't know much about mean field theory in physics. Howard Gutowitz
wrote some about applying mean field theory to CAs.
(http://www.santafe.edu/~hag/ecal95/ecal95.html)
If I understand it correctly, the idea is to work with n-gram
statistical models to say things like "this CA tends to become mostly
white space" or "this CA tends to become filled with this pattern".
I can give a simple example, applying mean field theory to the 110
rule.
1. Start with a 2-gram model, which specifies for each pair ("00",
"01", "10", "11") a probability.
2. Construct a 3-gram model from the 2-gram model. Set the probability
of a 3-gram "abc" to the probability of "ab" in the previous model
times the probability of "bc".
3. Construct a 4-gram model from the 3-gram model. Set the probability
of a 4-gram "abcd" to the probability of "abc" in the previous
4. Apply rule 110 to each 4-gram, shrinking it to a 2-gram. This
transforms the 4-gram model into a 2-gram model. (The probability of a
2-gram "ab" is the sum of the probabilities of all the 4-grams that 110
transformed to "ab".) In general, this 2-gram model will be different
from the first 2-gram model.
Of course, you can do it with n-grams instead of 2-grams, other rules,
more colors, different neighborhood, etc. The idea is that the
n-gram-to-n-gram transformation approximates the cellular automaton
rule; at least closely enough to predict things like "If I start with a
random mix of 60% white cells and 40% black cells, and run this rule
for 100 generations, I will have approximately 35% white cells and 65%
black cells".
Johnicholas
McIntosh Harold V.
news:1167251750.0...@79g2000cws.googlegroups.com
> Are there any open-source tools for doing mean-field analysis on CAs?
You could try the LCAU series of programs (later NXLCAU), but if you're
just interested in mean field theory you can do the algebra by hand.
Trouble is, mean field theory isn't very accurate except in special
situations, such as single generation evolution.
> I've looked, and while there are a LOT of simulators and a significant
> number of search programs of various sorts, mean field theory seems to
> be mentioned only in academic papers (without source code).
It isn't always called by that neme:
M. Dresden and D. Wong,
"Life Games and Statistical Models,"
Proceedings of the National Academy of Sciences
(U.S.A.) 72 956-960 (1975)
L. S. Schulman and P. E. Seiden,
"Statistical mechanics of a dynamical
system based on Conway's game of Life,"
Journal of Statistical Physics 19 293-314 (1978)
With varying degrees of applicability, one can think about
1) Langton's Lambda parameter. Crude but informative.
2) Wuensche's Z parameter More informative. Related to variances.
3) Gutowitz's block probabilities. Works well in one dimension.
Related to Bayes estimates and Kolmogoroff's interpretation.
Wuensche has a suite of programs; the LCAU programs are also applicable.
Trying to do a statistical analysis on Rule 110 is probably not likely
to uncover the structure which has made it famous. Nevertheless note
that is an environment describable in terms of macrocells and membranes.
The distinction between ether and vacuum is also important; why should
an ether exist and how can it be predicted? Rule 110 abhors a vacuum.
- hvm
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