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From: "Jonathan W. Mills" <jwmills@cs.indiana.edu>
Subject: Re: Chaos and Computation
Message-ID: <1995May13.214733.18164@news.cs.indiana.edu>
Organization: Computer Science, Indiana University
References: <3o8o3a$9a5@portal.gmu.edu> <3odifr$6o3@mp.cs.niu.edu> <3og3jo$ftu@uuneo.neosoft.com> <hubey.799797500@pegasus.montclair.edu>
Date: Sat, 13 May 1995 21:47:28 -0500
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Xref: glinda.oz.cs.cmu.edu comp.ai.philosophy:28050 sci.cognitive:7594 sci.nonlinear:3135

In article <hubey.799797500@pegasus.montclair.edu>,
H. M. Hubey <hubey@amiga.montclair.edu> wrote:
>
>It might be possible to solve some of these problems using analog
>computers. I don't know about the TS but there are parallel problems
>that electrical circuits solve almost instantaneously. It depends on
>whether some problems can be reduced to be special cases of analog
>problems.
>

a consequence of lee rubel's work (the extended analog computer) and
siegelmann & sontag's work (neural nets with rational weights) is that
these devices are known to be turing-equivalent.

thus, an analog computer exists that can, theoretically, with finite
resources solve np-complete problems.  however, this illustrates the
difficulty we have realizing such machines:  because the analog
computer represents quantities as distinct physical analogs, not
binary encoded values, we must be able to measure these values to
distinguish between them and so recognize a solution to the problem.

problems that are especially difficult are those that depend on an
encoding, such as partition and knapsack, where very large and very
small values can be mixed in the problem instance.  analog computers
do not represent this kind of data very well at all.  a good
discussion of the issues is found in a paper by vergis, steiglitz and
dickinson, 'the complexity of analog computation.'  in a personal
communication, steiglitz has recently commented that even the graph
connectivity problem he used in that paper to illustrate the utility
on analog computers for polynomial computations might be too
optimistic.

having said that, i must also say that the work i am doing with vlsi
implementations of rubel's model (kirchhoff machines) may be able to
overcome some limitations by hiding high-precision computation within
the operation of the analog computer, leaving the inputs and outputs
measurable.  for an np-complete problem, consider that it can be
reduced to a decision problem whose output can be expressed using 0 or
1.  there is a long way to go on this work, but it is much less
controversial than siegelmann's proposals that depend on either
absolute or linearly precise reals internally.

sincerely,
jonathan
