On 1/22/2014 2:12 AM, Bruno Marchal wrote:
On 22 Jan 2014, at 01:41, Russell Standish wrote:
On Tue, Jan 21, 2014 at 12:53:33PM +0100, Bruno Marchal wrote:
With some competence, I guess you mean.
Without competence, and giving time to the creature, any universal
machine do have an open-ended creativity. Well, certainly in the
sense of Post (I can explain this, but it is a bit technical).
I'm interested to hear your explanation, but if its what I suspect it
will be, I'll be disappointed :).
A set (of natural numbers) is creative if
1) it is RE (and thus is some w_k)
2) its complement (N - w_k) is productive, and this means that for all w_y included in,
we can recursively (mechanically) find an element in it, not in W_y.
It means that the set is RE and his complement is constructively NOT RE. Each attempt
to recursively enumerate he complement can be mechanically refuted by showing
explicitlky a counterexample in it, and this gives the ability to such a creative set to
approximate its complement in a transfinite progressions of approximation. this gives an
ability to jump to a bigger picture out of the cuurent conception of the big picture. I
find it a reasonable definition of creativity.
So what would be an example of a creative set of natural numbers? Are there sets of
natural numbers such that both the set and its complement are not RE?
Brent
The John Myhill proved that a set is creative iff it is Turing complete, i.e. Turing
universal.
So that RE set
Basically stating that the universal dovetailer emulates creative
conscious being does not demonstrate a creative program, which needs
to be creative relative to us (as observers).
I agree. The UD is not creative. But it generates all creative programs or sets.
Note that the UD can be considered as creative though, if you conceive it as the set of
all initial segment of UD*. In particular the set define by the diophantine polynomial
that I send today to Brent, *is* probably creative itself.
But if your idea is something different, I'm all ears!
I haven't had a chance to study and understand Post's definition (sure
I've looked at it, but didn't grok it), but if you say it is
equivalent to universality, then its not really going to contribute to
the solution.
I am not sure. Open ended creativity seems to me well captured by
Post. It makes the machine able to defeat all effective complete
theories about itself. It gives what I often called the comp vaccine
against reductionism.
Well - maybe if you explain more?
I hope that what is above is not too much concise.
Bruno
Cheers
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Prof Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Professor of Mathematics hpco...@hpcoders.com.au
University of New South Wales http://www.hpcoders.com.au
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