Re: [EGN] Re: turing completeness

From: Gerry Quinn (gerryq_at_indigo.ie)
Date: 02/19/04


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 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.
>>
>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 closed-form 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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