X-Message-Number: 8633
From: Thomas Donaldson <>
Subject: Re: CryoNet #8631
Date: Sun, 28 Sep 1997 16:47:44 -0700 (PDT)

Hi once more!

Perhaps this is way out of order, but so be it.

Some time ago, when I mentioned that some people in computer science had
discussed variations of Turing machines which used real numbers (I DON'T
mean floating point numbers, I mean real numbers ie. sequences of integers 
potentially going off to infinity to the right but not to the left) I was
queried on this. I said that I had not invented the idea, I read it
somewhere. Peter Merel asked me to provide a reference, and I said that I
did not recall it offhand but would look. I also said that I didn't publish
the article in PERIASTRON because it did not obviously pertain.

I did spend some time trying to find the reference, without success. And 
just now, looking through PERIASTRON for experiments and names of those
who did them on LTP and brain slices, I found a short article describing 
just the result I had remembered. 

The reference is by HT Siegelmann, ED Sontag, SCIENCE (268(1995) 545-548).
In the particular article given, they discussed neural nets and how their
results with such real-number computers led to conclusions about neural
nets also (their article naturally refers to previous work on such 
computers). It turns out that a neural net with real-number weights on 
each connection cannot be imitated by a Turing machine. FULL STOP. This,
in a way, is a version of an analog Turing machine, but we have to be
careful what we mean by "analog". Their general results apply not just to
Turing machines but to other computing devices also. The two authors have
basically shown that computing with real numbers can do MORE than a Turing
machine can do. In each case the infinite real number isn't used directly;
instead, as part of the computation it is calculated out to the required
number of decimal places for the computation the machine is performing.

So I guess it did go into PERIASTRON. After all, neural nets do provide the
closest version of neural circuits that computer science has yet come to. The
implication that our brain's neural nets might do some things impossible for
a Turing machine bears a lot on the various controversies that have been
going on in Cryonet.

And in case some of this audience believe I came out of biology and 
biotechnology, that's not true at all. I have a PhD in mathematics (1969)
and only started studying biological issues when they began to look highly
pertinent to cryonics. In 1985 I moved into computing. That was going 
fine until I got a brain tumor. I remain interested in computing, and 
remain a member of IEEE.

			Best and long long life,

				Thomas Donaldson

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