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From: khorsell@ee.latrobe.edu.au (Kym Horsell)
Subject: Re: Reversing Life/C.A.s
Sender: news@lugb.latrobe.edu.au (News System)
Message-ID: <3qm60k$18b@feynman.ee.latrobe.edu.au>
Date: Fri, 2 Jun 1995 05:02:12 GMT
Lines: 68
References: <3qfrnv$q4h@ns.cityscape.co.uk> <3qgqf2$3k3@fourier.ee.latrobe.edu.au> <3qibrv$6fe@ns.cityscape.co.uk>
Organization: Department of Electronic Engineering, La Trobe University

In article <3qibrv$6fe@ns.cityscape.co.uk>,
Kasprzyk  <dr37@cityscape.co.uk> wrote:
>khorsell@ee.latrobe.edu.au (Kym Horsell) wrote:
>
>>I believe there is a theorem to the effect that if a CA is reversible then
>>it is not interesting. Conway's life is not reversible -- the mapping
>>from t to t+1 is a function but NOT vice-versa (i.e. there may be more than 
>>one predecessor state for a given state).
>
>I am almost certain that when life/C.A.'s were first suggested that they can "mirror" 
>patterns in real life they were rejected because they couldn't be reversed (and the theory 
>went that if you can't reverse C.A.'s then they can't be interesting because you can 
>"reverse" real life), and that it was eventually discovered that Conway's life could be 
>reversed - C.A.'s were accepted.

But I don't think life IS reversible.  In "real life" there are thermodynamic 
considerations that indicate, basically, "things run down -- they don't run 
up". This is believed to be responsible for "the arrow of time". While in 
Physics it can be shown that (some) quantum systems can be viewed in forward 
or reverse directions (e.g. Feynman showed that electroncs might be viewed as 
positrons travelling in the "opposite direction" and vice versa) this is not 
true in general.

From what I remember of it a book by Margolis about CA's gives a basic 
explanation of why reversible CA's are  "not interesting". My
probably recollection is as follows.

The number of behaviour of a system that is also reversible is severely limited.
Consider a sequence of N different numbers that must look the same backwards 
as forwards.  Normally there would be N! possible sequences. How many 
palendromic sequences are there? Around (N/2)! For N getting large the ratio
is asymptotic to sqrt(1/2) (1/2)^(N/2) / (N/e)^(N/2)
which falls off "very rapidly" with N.

N  O(factor)

10  10^-6
20  10^-18
30  10^-32
40  10^-47
50  10^-64
60  10^-81
70  10^-100
80  10^-118
90  10^-138
100  10^-157
200  10^-374
300  10^-614
400  10^-868
500  10^-1134
600  10^-1408
700  10^-1689
800  10^-1976
900  10^-2269
1000  10^-2563

So roughly if we're talking about a system with only 10 distinct states 
then only about 1 in 1 million are also "reversible", etc. A corolary
is that reversible solutions of a given complexity will therefore
tend to be much larger than ones that are not -- and by a very
very vary large margin. It might therefore be argued that evolution would 
have to be quite a trick to find solutions to problems with (e.g.) 100 states 
with its hands tied like that (so to speak ;-).  It would seem to
require a large part of the universe to build it.

-- 
R. Kym Horsell
khorsell@EE.Latrobe.EDU.AU              kym@CS.Binghamton.EDU 
