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From: sdo@linus.UUCP (Sean D. O'Neil)
Newsgroups: comp.ai.neural-nets
Subject: Re: Prove non-existence with neural net?
Keywords: Cray, finite projective plane, confusion
Message-ID: <45171@linus.UUCP>
Date: 22 Feb 89 16:08:20 GMT
References: <1637@cps3xx.UUCP> <1233@usfvax2.EDU> <476@madnix.UUCP> <44911@linus.UUCP> <9905@nsc.nsc.com>
Reply-To: sdo@faron.UUCP (Sean D. O'Neil)
Organization: The MITRE Corporation, Bedford MA
Lines: 54

In article <9905@nsc.nsc.com> andrew@nsc.nsc.com (andrew) writes:
>Hmmmm. Re-phrase your title: "prove non-existence with neural net?" to
>"prove existence after termination with neural net" and all becomes clear.
>
>Perhaps Sean is being unduly pessimistic. Perhaps Sean realises that if
         ^^^^                                      ^^^^
     ,
Touche.  I apologize if my previous note sounded patronizing---that was
not my intention.


>an FPP can be found - e.g. by a neural net - that is of an order
>formally excluded by conjecture - then that conjecture is disproved.


I agree totally that the contribution the neural network can make is to
find an FPP.  Where my original concern arose was with the idea of
comparing the neural network with the Cray (i.e. "do battle") on this 
order 10 example since the contribution of the Cray was to help prove
the non-existence of an FPP of order 10.  I think we both agree that
the neural network setup you are talking about cannot help us prove
non-existence (correct me if I am wrong on assuming this, as it's an
important point).  Therefore, we cannot interpret the results of the
Cray and the results of the neural network in the same manner (i.e.
a positive result for either proves existence, while a negative
result for the Cray proves non-existence, but for the neural network
proves nothing).  Thus, it would be impossible to state whether the
neural network or the Cray is `better' at this particular task, as
the tasks they are carrying out are not identical (i.e. the Cray
is performing a task that comes out with a stronger result).


>The nice thing thing about the net idea is how it scales with order.
>Order **6 is much nicer than (NcK)**N, but of course this is, as he
>point out, apples with non-exhaustive pears. "Exactly" is quite easy -
>just look at the NN settled state, and count 1's for a few seconds.
>This is eyeball and brain territory, and no big deal. For lack of


I agree that checking whether the answer matches the constraints is
trivial.


>decent simulation x-ware, I'm unable to say what happens with order 6.  


I think the contribution that could be made here is to find an FPP
of order 12, if such a thing exists (read my previous note for details).
Depending on the amount of effort required to set up the proposed
neural network search, I might want to have some reasonable assurances
that it would find an FPP if it existed.  What such assurances should
be are, of course, a matter of opinion.

Sean