Re: [EGN] Re: turing completeness

From: Gerry Quinn (gerryq_at_indigo.ie)
Date: 02/18/04 

Date: Wed, 18 Feb 2004 11:51:28 GMT

In article <rqoYb.4089$d34.694963@news20.bellglobal.com>, "Michael N. Christoff" <mchristoff@sympatico.caREMOVETHIS> wrote:
>"Gerry Quinn" <gerryq@indigo.ie> wrote in message
>> >>>
>> >>>An artifical neural network can distuinguish scents better than any
>> >>>human. And artificial neural networks are purely mathematical things.
>> >>
>> >> I do not believe you - it seems to me that physical components such as
>> >> wires and chips would be needed to make an artifical neural network.
>> >>
>> >Fair enough. But the process can be described perfectly by mathematics.
>>
>> I don't think so, unless you invented lots and lots of axioms describing
>> the system, the stimuli given, and the response.
>
>That doesn't seem right. Inputs are digitized and the neural nets I've
>dealt with are simulated in software, and hence must be formalizable. Do
>you mean that the output of a neural net may not be calculable without
>actually running it (ie: not analytically solvable)? Why would a
>description of a neural net require information about its responses? Its
>seems like saying that the description of a function is not only a rule for
>calculating the mapping of inputs to outputs, but must also include
>information about its responses to certain inputs.

What I mean is that to train a NN to do anything useful, you have to
provide lots of stimuli and responses according to whether it reacts
correctly. To express this mathematically would require many terms.
You can describe a useless untrained NN in terms of simple mathematics,
but to describe either a trained one, or a process sufficient to go from
an untrained to a trained one, you require lots and lots of data which
in mathematics correspond to a complicated system of axioms.

A 100x100 array of neurons linked orthogonally, and a description of the
properties of each, might be a simple mathematical system. But if you
describe the weighting of each neuron, that is instantly 10000 extra
axioms. Describing how it was trained to get this weighting would be
just as difficult.

We could, I suppose, say "The 100x100 array that is best [by some
criterion] at recognising a chess knight", and that is a simple-ish
description that in principle defines a particular array. If there
are finitely many possible weightings, linkages, and inputs, the problem
is certainly decidable. But we are going far away from simulating a
single network in mathematics here.

- Gerry Quinn

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