X-Message-Number: 8614
From:  (Thomas Donaldson)
Subject: Re: CryoNet #8596 - #8599
Date: Fri, 19 Sep 1997 23:55:07 -0700 (PDT)

Hi again!

You see, I have a little time now. It's the evening of 19 September,
11:10 PM. And I'm going through the different Cryonets which have been posted'
up to now.

To John Pietrzak:

You're strictly correct that there is no such thing as a PRACTICAL Turing
machine (Turing machines are hardly practical, being intellectual constructions
only). I do hope, though, that you understood the point I was making when
I made that statement.

I will say the same about your ideas for using a Turing machine to simulate
a brain. Sure. But they tell us nothing about simulating a brain on a
computer in the real world. 

The issue of what is and isn't practical, or will someday become practical,
is critical to me. I think of intellectual constructions such as Turing 
machines more as ways to exclude some things from possibility than as ways
of proving that something can really exist. The first is useful to keep 
me from trying to do the mathematically impossible, but I also find it
very useful to look at what kinds of limits there may be in the real world.
One reason I asked my question about neural nets made of real neurons was
that in some ways the behavior of such a system emulated the behavior of
a generalized Turing machine able to write real numbers: real numbers
are infinite sequences of digits, rather than finite. That means that 
important information may be stored arbitrarily far out on the sequence.

When we use intellectual constructions we can make all kinds of things. Did
you know (another cute idea) that one guy actually produced a consistent,
axiomatic version of Newton's infinitesimals? An infinitesmal is another
kind of number, you see. They lie between any two real numbers, and 
if we consider numbers from (say) 0 to .00000000000000000001, the infinites-
mal is smaller than any given real number but still not equal to 0. And we
can add them together and even discuss calculus in terms of them. Limits?
Who needs limits? We could use infinitesmals instead.

This idea didn't catch on, but it remains cute.

Whether or not anyone has defined one, I believe it would be useful to 
consider a device which is like a Turing machine but suffers from limits
of various kinds: length of tape, speed with which it can write or read
from its tape, and so on. Such a machine might allow us to find better
bounds for the behavior of real machines at any given time, for instance.
I will call such machines "practical Turing machines". They are yet another
intellectual construction, and in the sense of existence that an idea has
existence, I guess they must exist.

And such machines raise other issues too: what about an abstract parallel
machine consisting of N communicating practical Turing machines? 

Someone else on this forum has raised the question of whether I really meant
von Neumann machines. I did not, but the ABSTRACT behavior of a von Neumann
machine might be that of a practical Turing machine. At some point, analog
or digital or whatever, we will come up with an abstract device which can
do the processing that a human brain can do. The fact that we exist shows
me that someday we will be able to make a device capable of that. But you'll
note that I'm playing here with intellectual constructions of various kinds,
ideas derived from the idea of a Turing machine but by no means the same.
And it is in that sense that the device we find may not qualify as a 
computing device, though all our current computers do.

Incidentally, in one sense an analog machine may be thought of as one
which uses real numbers rather than single digits or finite strings of 
digits. (See computer definition of "real number" below). The idea of a Turing
machine which wrote real numbers or even FP numbers didn't arise out of
nowhere. As for what is practical, you might think of such a machine as 
a Turinglike machine operating not on a tape but on an N-dimensional
flat surface, moving from point to point (of course with infinite speed)
to write both horizontally and vertically. The horizontal writes are the 
original writes on the tape. The vertical writes are the real numbers.
The limits, for a practical machine, would then be the lengths of the
surface in each dimension and the speed with which the machine could do
its writing.

And for that matter, what about an infinite dimensional surface....

To Peter Merel:

OK. I thought a "floating point Turing machine" or "real number Turing 
machine" were interesting ideas, and I'll try to find them. It was weeks
ago that I read about these, though, and I'll have to go back through my
piles of magazines... or when I'm next in a math and computer library,
I'll look through their database. (As for a distinction between real
numbers and floating point numbers --- remembering that we are discussing
intellectual constructions --- a real number is an infinite sequence
of digits to the left, with a decimal point at some finite point from
its beginning ie. it isn't infinite in both directions. A floating point
number is a finite sequence of digits followed by another one giving
its "exponent". In the small way enforced by practical issues, the 
language Forth allowed real numbers AND floating point numbers: in both
cases, the number of digits had an absolute upper bound controlled by
the computer running the Forth code. In Forth they were called "fixed point
numbers". I've never seen them used elsewhere in any form, but in the
proper setting they actually were faster than FP numbers for computation).

Well, what about sequences of digits infinite in BOTH directions. Another
intellectual construction, of course. NO more strange than the idea of 
a function on the whole real line...

			Long long life,

				Thomas Donaldson

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