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From: flake@scr.siemens.com (Gary William Flake)
Subject: Re: Chaos in Turing Machines
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Date: Fri, 15 Mar 1996 15:34:02 GMT
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In article <4i7m9p$q9a@newsbf02.news.aol.com>,

Storgodar <storgodar@aol.com> argued that discrete systems are
essentially linear.  I've deleted the first paragraph and will only
quote the second which is the one that contains, IMHO, the error.

> First, one must remember that subsequent operations will be *linear* if
> the now-discrete values are only manipulated in discrete form, by discrete
> functions. Thus, stopping at one step introduces a non-linearity only to
> cause the systems total non-linearity to dwindle. But this all raises an
> intriguing idea: is there possibly great value in constant digitization
> and DEdigitization: a computer that uses a combination of interacting
> digital AND analog components? 

I think you are confusing the terms "linear" and "discrete".  A linear
system is one that obeys the two rules:

  f(x + y) = f(x) + f(y) and f(x * y) = x * f(y), for all x and y.

Period.  How discrete the system is irrelevant, except to note that it
is "hard" to make a linear discrete system but "easy" to make a
nonlinear discrete function.  This is because all logical operations
can be expressed in terms of step functions, which are clearly
nonlinear.  Also note, that if you take any program and look at it as
a function from the natural numbers to the natural numbers, you are
almost guaranteed that it will fail this test for linearity.

Re. your argument that Turing machines can't express "real" chaos, the
shadowing lemma implies that simulated chaos is for all practical
purposes as "real" as natural chaos.  No one even knows if the world
is temporally and spatially continuous.  In fact, the current thought
in physics is that both are discrete.

Furthermore, I would claim that the "realness" of a chaotic system is
interesting, but not as interesting as the type of uncertainty
introduced by incomputability.  Specifically, measurement error
prevents you from predicting the long-term future of a chaotic
system.  But with a perfect description of a computing system we have
the same problem.  One could also look at things from the opposite
direction and consider chaotic system that compute.

But I digress.

Regards,
Gary
-- 
Gary W. Flake, PhD, flake@scr.siemens.com, Phone:609-734-3676, Fax:609-734-6565
Project Manager and MTS,  Adaptive Information and Signal Processing Department
Siemens Corporate Research,  755 College Road East,  Princeton, NJ  08540,  USA
