Newsgroups: comp.ai.philosophy,sci.cognitive,sci.nonlinear
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!news.mathworks.com!gatech!newsxfer.itd.umich.edu!agate!news.Stanford.EDU!rpal.rockwell.com!news.cs.indiana.edu!jwmills@cs.indiana.edu
From: "Jonathan W. Mills" <jwmills@cs.indiana.edu>
Subject: Re: Chaos and Computation
Message-ID: <1995May6.030340.28974@news.cs.indiana.edu>
Organization: Computer Science, Indiana University
References: <3o8o3a$9a5@portal.gmu.edu> <3odifr$6o3@mp.cs.niu.edu>
Date: Sat, 6 May 1995 03:03:36 -0500
Lines: 71
Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:27665 sci.cognitive:7516 sci.nonlinear:3074


In article <3odifr$6o3@mp.cs.niu.edu>, Neil Rickert <rickert@cs.niu.edu> wrote:
>In <3o8o3a$9a5@portal.gmu.edu> herwin@osf1.gmu.edu (HARRY R. ERWIN) writes:
>
>>Ref: Siegelmann, H. T., "Computation beyond the Turing limit," Science, 
>>268:545-548, 28 April 1995.
>
>>The author uses a highly chaotic dynamic system to gain computational 
>>power beyond the Turing limit. The implementation the author describes 
>>appears (at first look) to have similar dynamics to the neural systems 
>>discussed in Skarda and Freeman.
>
>It appears to depend on infinitely precise analog computation.  As
>such I am skeptical as to whether it has any relevance to problems of
>human cognition or artificial intelligence.
>

siegelmann & sontag's first works depended on the ability to load at
least one weight of an artificial neural net with an
infinitely-precise irrational, and do so in finite time.  in numerous
discussions with siegelmann, and a few with sontag, their position
was that 'reals will arise in such a physical system.'

most physicists i talked to -- and virtually every computer scientist
-- would argue against this, primarily from the position that while
infinitely-precise numbers are useful abstractions, they do not have
physical analogs.  even pour-el and richard's demonstration that under
certain conditions a physical system will lead to an uncomputable real
is NOT an example of such a number:  the uncomputable real is the
measure of a set of values describing the system, and not a physical
quantity.

with this in mind, and having just obtained the paper (but having
discussed these issues with siegelmann for over a year), i interpret
the concept of linear precision as a way for a single non-algorithmic
(ie, analog) system to generate increasingly longer approximations
to a real.  if the real is kolmogorov complex, then it cannot be
represented by any description less complex than the number itself.
but siegelmann's point is that the analog system, with inherent
chaos (or randomness) is capable of generating that number.

a digital computer programmed with a single algorithm can generate
some approximation to the real, but because it is kolmogorov complex,
the approximation will deviate from the value generated by the analog
system...at which time an improved algorithm must be constructed...
but that algorithm will eventually fail, too...etc, etc.

another way to look at this is that the analog system is described
over time by some increasingly complex algorithm, or a growing
ensemble of algorithms, each suited to describe the system up to
some point in time.

or, think of it as an infinite game, with the analog system and the
digital system as players.  for some finite time an algorithm will
describe the system, but eventually the minimal description must be the
number itself...which means that for finite algorithms and time
intervals, there will always come a point at which the analog system
'wins' the game because it correctly generates the next sequence in
the kolmogorov real, while the algorithm cannot.

this is my interpretation of 'super-turing,' and probably -- almost
certainly -- is not siegelmann's, and no doubt is open to question.
i'd certainly appreciate comment, as i have struggled to understand
the physical meaning of 'superturing' computation for over a year:
as a vlsi designer of some (theoretically) real-valued analog
computers, the question of physically-implementing these machines
is of interest.  of course, i also know that no digital computer is
a physical implementation of a turing machine either...

sincerely,
jonathan
