From: Harold V. McIntosh (MCINTOSH@unamvm1.dgsca.unam.mx)
Date: Sat May 15 1993 - 02:11:56 UTC
Howard Andrew GUTOWITZ <gutowitz(at)AMOCO.SACLAY.CEA.FR> comments: > > In reply to: > > >From: Harold V. McIntosh <MCINTOSH@unamvm1.dgsca.unam.mx> > > Message-Id: <9305140414.AA19868(a)Early-Bird.Think.COM> > >Here is a bit of discrepancy: Dewdney, in an 'Armchair Universe' addendum, > >implies that Piers Anthony's 6777 grows without limit, but the mean field > > curve is subtangent, implying the contrary - fading away to nothing. Maybe > >there is a bug in my program. > > or maybe mean field theory has to be improved :-) > > (difficult, of course, in 3-D) > Well, it's a new day, and another afternoon of work puts evolutions (in cubes up to 20x20x20) into the program (give me another day and there'll be monte carlo mean fields, and maybe even cross sections). However, the actual return map (and the actual density curve) allow some further conclusions. - 1) Walsh's (John Walsh <zed(at)MATHS.TCD.IE>) observations are confirmed: 2233 levels off somewhere around the observed fixed point (but hasn't been run through all initial probabilities - that's for tomorrow). A word of caution - if you set the initial density too high, say 50%, everything dies off right away for many of these rules. That's consistent with their mean field curves at 50%. - 2) Anthony's 6777 dies out in six generations, contrary to how I read Dewdney. This does not exclude that there are some very interesting figures down there at low density; after all, Conway Life also sinks down to a fairly low final density. Time to turn to Bays' Scientific American supplement and articles (when all else fails, read the manual...). While we wonder whether I understand Dewdney and whether Dewdney and Bays understood each other, we might speculate as to where Anthony got his rule. As a novellist, he has inserted a lot of mathematical games into his stories, but that does not tell us whether he actually played a 3-D Life, or just extrapolated from Martin Gardner's articles. - 3) The Chat\'e-Mannevill automata have, for the most part, used von Neumann neighborhoods. The Swiss Cheese model says that voids fill, trying to encase themselves in regular surfaces. They don't quite make it, erode, collapse, and the cycle starts anew. The model is more poetic than real, because you get a lot of froth. But if it were true, Moore neighborhoods have 9-cell surfaces, and one might try out an 8/10/0/0 rule. So much for that hypothesis. - However, if one looks for a mean field which resembles the one for a 7-cell von Neumann neighborhood, but scaled up to 27 cells, 16/18/0/0 performs like a charm. - 4) Howard seems to be asking us, how can such a miserable theory give such good results? Mainly by bending it all out of shape. - Recall Wolfram's four Classes - static, periodic, chaotic, 'islands of chaos in a sea of order?' Taking Class IV (Life and similar) as a transition regime between order and chaos (there are some nice papers on this), mean field theory seems to work moderately well for the order and the chaos, leaving the transition (the interesting region) up for 'interpretation.' ----- An aside here - some inquiries have arrived asking, 'what is mean field theory?' It is a fancy word for just using naive probability to figure out what the automaton is likely to do. Simplest thing is to judge the number of live cells by the number of rules giving them - all the different ways a live cell can survive or be born. But not all neighborhoods are equally likely, and live and dead aren't equally likely. So you assign them probabilities, complement, multiply and add probabilities they way you learn in textbooks, and look for a self-consistent probability (fixed point). Trouble is, assumptions of independence aren't always true, and so the calculations aren't always right. Totalistic rules are the worst. ----- A year ago I went to considerable trouble trying to prove that the Chat\'e- Manneville phenomonon (that ubiquitous period-3) was due to an unstable fixed point of period 3 in the mean field curve, which would be approximated more and more exactly the higher the dimension and (which was really the supposition) the more voluminous the neighborhood (27-cell neighborhoods were beyond my aspirations at the time). Well, it didn't work out that way. - First, the windows for period 3 in the return map are too, too tiny (think of the relative size of the complex period-3 'ears' on the Mandelbrot set (not to mention its infinitesimal real period-3 cardioid). Second, the (rigorously accurate) mean field curves for subsequent generations are NOT iterates of the first one, and don't even have obliging fixed points. But there is some hope that some members of the family may be related by iteration, and some understanding may procede from there. - Getting a good theory still has a long way to go, but the bad theory seems to still be capable of pointing us in interesting directions. - Harold V. McIntosh |Depto. de Aplicaci'on de Microcomputadoras MCINTOSH@UNAMVM1.BITNET |Instituto de Ciencias/UAP mcintosh@unamvm1.dgsca.unam.mx |Apdo. Postal 461 (+52+22)43-6330 |72000 Puebla, Pue., MEXICO
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