# 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
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*>> properties of each, might be a simple mathematical system. But if you
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*>> describe the weighting of each neuron, that is instantly 10000 extra
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*>> axioms. Describing how it was trained to get this weighting would be
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*>> just as difficult.
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*>>
*

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

*>A mathematical description may contain raw data, like lists of weights. I
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*>agree that describing a single NN is not the same as providing a general
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*>mathematical system for creating NNs that can perform specific tasks.
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*>However, if I remember correctly, the point was that one could give a
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*>mathematical description of something (in this case an NN) that could smell
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*>better than humans. To do this, one need only provide a single NN that does
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*>this. A sort of 'proof by construction'. Also I don't see why each weight
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*>would require an axiom. A set of input data and an algorithm for processing
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*>the data constitutes a mathematical description. A closed-form solution is
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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
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*>mathematical description of any but the most rudimentary NNs. Although an
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*>algorithm and list of weights may not really help us understand "how" or
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*>"why" the NN is successful, its still a mathematical description. But maybe
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*>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
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*>also use that argument for arbitrary programs? ie: One may say that in
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*>order to fully describe Dijkstra's shortest path algorithm that one must
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*>include a description of the mental process he employed to discover it. In
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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.
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*>applied, is able to distinguish certain scents better than a human, then it
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*>should be enough to present that description without mention of the process
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*>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

**Next message:**Richard Harter: "Re: RFCAS - The San Data Flow and Messaging Engine"**Previous message:**Alf P. Steinbach: "Re: Fast pancake flipping"**In reply to:**Michael N. Christoff: "Re: [EGN] Re: turing completeness"**Next in thread:**Michael N. Christoff: "Re: [EGN] Re: turing completeness"**Reply:**Michael N. Christoff: "Re: [EGN] Re: turing completeness"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]