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From: Bruno Marchal <marchal.domain.name.hidden>

Date: Thu, 4 Jun 2009 16:17:04 +0200

Hi Jesse,

On 01 May 2009, at 19:36, Jesse Mazer wrote:

*>
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*> I found a paper on the Mandelbrot set and computability, I
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*> understand very little but maybe Bruno would be able to follow it:
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*>
*

*> http://arxiv.org/abs/cs.CC/0604003
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*>
*

*> The same author has a shorter outline or slides for a presentation
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*> on this subject at http://www.cs.swan.ac.uk/cie06/files/d37/PHP_MandelbrotCiE2006Swansea_Jul2006.pdf
*

*> and at the end he asks the question "If M (Mandelbrot set) not Q-
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*> computable, can the Halting Problem be reduced to determining
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*> membership of (intersection of M and Q^2), i.e. how powerful a
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*> 'hypercomputer' is the Mandelbrot set?" I believe Q^2 here just
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*> refers to the set of all possible pairs of rational numbers. Maybe
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*> by "reducing" the Halting Problem he means that for any Turing
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*> machine + input, there might be some rule that would translate it
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*> into a pair of rational numbers such that the computation will halt
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*> iff the pair is included in the Mandelbrot set? Whatever he means,
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*> it sounds like he's saying it's an open question...
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*>
*

*> Jesse
*

*>
*

*> >
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*> >
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*> > On Thu, Apr 30, 2009 at 10:35 AM, Bruno Marchal
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*> <marchal.domain.name.hidden> wrote:
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*> >>
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*> >>
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*> >> The mathematical Universal Dovetailer, the splashed universal
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*> Turing
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*> >> Machine, the rational Mandelbrot set, or any creative sets in the
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*> >> sense of Emil Post, does all computations. Really all, with Church
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*> >> thesis. This is a theorem in math. The rock? Show me just the 30
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*> first
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*> >> steps of a computation of square-root(2). ...
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*> >
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*> > Bruno,
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*> >
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*> > I am interested about your statement regarding the Mandelbrot set
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*> > implementing all computations, could you elaborate on this?
*

So, indeed the conjecture I made on the Mandelbrot Set concerns the

decidability-on-the-rationals of the set M intersected with QXQ. And

it is indeed still an open problem. Actually my question is the

"creativity" (in the sense of Post) of M, and this would mean that you

can reduce the halting problem of any Turing machine into a problem of

membership of a rational complex number a+bi (a, b, in Q) to M. There

would be one fixed algorithm transforming any computable problem on N

into such a membership problem. If the solution is positive, then the

Mandelbrot Set would be a compact representation of a Universal

Dovetailing. Also, this would entail the existence of interesting

relationship between classical computability theory and the theory of

Chaos on the reals. The universality in chaos phenomenon (Feigenbaum)

would be related to the Turing Universality. Also, each of us would

be, in a sense, distributed densely on the boundary of M, and each

little Mandelbrot would represent the third person projection view of

each of our "first person plenitude". That would be cute, mainly for

the pedagogy of the UD, but also, it would made it possible to borrow

mathematical tools from chaos theory theory for the pursue of deriving

physics from numbers.

Not everything is clear for me in Potgieter paper, probably a result

of my incomptence, but it is very interesting. Thanks for the link.

Did I give you the link of the last, impressive M-zoom by phaumann?

Look at it with the high quality option + full screen, if you are

patient enough. Love it!

http://www.youtube.com/watch?v=x6DD1k4BAUg&feature=channel_page

Bruno

http://iridia.ulb.ac.be/~marchal/

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Received on Thu Jun 04 2009 - 16:17:04 PDT

Date: Thu, 4 Jun 2009 16:17:04 +0200

Hi Jesse,

On 01 May 2009, at 19:36, Jesse Mazer wrote:

So, indeed the conjecture I made on the Mandelbrot Set concerns the

decidability-on-the-rationals of the set M intersected with QXQ. And

it is indeed still an open problem. Actually my question is the

"creativity" (in the sense of Post) of M, and this would mean that you

can reduce the halting problem of any Turing machine into a problem of

membership of a rational complex number a+bi (a, b, in Q) to M. There

would be one fixed algorithm transforming any computable problem on N

into such a membership problem. If the solution is positive, then the

Mandelbrot Set would be a compact representation of a Universal

Dovetailing. Also, this would entail the existence of interesting

relationship between classical computability theory and the theory of

Chaos on the reals. The universality in chaos phenomenon (Feigenbaum)

would be related to the Turing Universality. Also, each of us would

be, in a sense, distributed densely on the boundary of M, and each

little Mandelbrot would represent the third person projection view of

each of our "first person plenitude". That would be cute, mainly for

the pedagogy of the UD, but also, it would made it possible to borrow

mathematical tools from chaos theory theory for the pursue of deriving

physics from numbers.

Not everything is clear for me in Potgieter paper, probably a result

of my incomptence, but it is very interesting. Thanks for the link.

Did I give you the link of the last, impressive M-zoom by phaumann?

Look at it with the high quality option + full screen, if you are

patient enough. Love it!

http://www.youtube.com/watch?v=x6DD1k4BAUg&feature=channel_page

Bruno

http://iridia.ulb.ac.be/~marchal/

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Received on Thu Jun 04 2009 - 16:17:04 PDT

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