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

> I found a paper on the Mandelbrot set and computability, I  
> understand very little but maybe Bruno would be able to follow it:
> http://arxiv.org/abs/cs.CC/0604003
> The same author has a shorter outline or slides for a presentation  
> 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- 
> computable, can the Halting Problem be reduced to determining  
> membership of (intersection of M and Q^2), i.e. how powerful a  
> 'hypercomputer' is the Mandelbrot set?" I believe Q^2 here just  
> refers to the set of all possible pairs of rational numbers. Maybe  
> by "reducing" the Halting Problem he means that for any Turing  
> machine + input, there might be some rule that would translate it  
> into a pair of rational numbers such that the computation will halt  
> iff the pair is included in the Mandelbrot set? Whatever he means,  
> it sounds like he's saying it's an open question...

Thanks! Very interesting. It confirms my feeling that the result Blum,  
Smale and Shub cannot really help to figure out if the "digital  
Mandelbrot Set" is a compact form of a universal dovetaling ... or the  
exponential complex would already be one .... Hmm....

Another way to digitalize the M set would be to consider its digital,  
ste by step enlargement on the Gaussian Integers (n + mi, n, m in Z).

I will study those papers, soon or later. I really love the Mandelbrot  
set. Look at this beautiful musical zoom by Ubermari0:


Look at this new very impressive zoom by Phaumann, with a 10^333   
enlargement, in an hard to compute part of the M-set!


You can see that the computations is deep in Bennett sense, like most  
object in "nature" plausibly: it is both very involved and  
sophisticated yet incredibly redundant, and it is itself the product  
of a very tiny algorithm. It can be used in practice to compress data.


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


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