Re: [EGN] Re: turing completeness
From: Gerry Quinn (gerryq_at_indigo.ie)
Date: 02/19/04
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Date: Thu, 19 Feb 2004 10:58:26 GMT
In article <_PVYb.11315$d34.1196716@news20.bellglobal.com>, "Michael N. Christoff" <mchristoff@sympatico.caREMOVETHIS> wrote:
>"Gerry Quinn" <gerryq@indigo.ie> wrote in message
>> 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 simpleish
>> 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.
>>
>A mathematical description may contain raw data, like lists of weights. I
>agree that describing a single NN is not the same as providing a general
>mathematical system for creating NNs that can perform specific tasks.
>However, if I remember correctly, the point was that one could give a
>mathematical description of something (in this case an NN) that could smell
>better than humans. To do this, one need only provide a single NN that does
>this. A sort of 'proof by construction'. Also I don't see why each weight
>would require an axiom. A set of input data and an algorithm for processing
>the data constitutes a mathematical description. A closedform solution is
But it will be a complicated description, requiring a very long string.
>not required. In fact, I'd be surprised if there was a closed form
>mathematical description of any but the most rudimentary NNs. Although an
>algorithm and list of weights may not really help us understand "how" or
>"why" the NN is successful, its still a mathematical description. But maybe
>thats not what you are getting at.
No, I'm not interested in whether the description is 'closed form'. the
point I'm making is that the long description is not 'mathematics'  it
is more like a blueprint for the thing itself.
>In terms of having to describe how its weights were attained: Couldn't one
>also use that argument for arbitrary programs? ie: One may say that in
>order to fully describe Dijkstra's shortest path algorithm that one must
>include a description of the mental process he employed to discover it. In
Not at all, the shortest path algorithm can be described simply, as can
the principle of a neural network. But a neural network that does
something useful is unlikely to have a simple description.
>the case of NNs, if empirical evidence agrees that an NN, with weights etc.
>applied, is able to distinguish certain scents better than a human, then it
>should be enough to present that description without mention of the process
>by which it was developed.
I agree  I'm just pointing out that the description would be a
blueprint for the neural network, not any 'mathematical' alternative.
A cube is of mathematical interest, it is one of a set of Platonic
solids, for example. The blueprint for a house contains cubes and other
shapes, indeed we can use geometric theorems when discussing how to
build the house. Nevertheless, the blueprint is not "part of
mathematics".
 Gerry Quinn
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