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...

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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: http://www.youtube.com/watch?v=KgM3XJmH768&feature=channel_page Look at this new very impressive zoom by Phaumann, with a 10^333 enlargement, in an hard to compute part of the M-set! http://www.youtube.com/watch?v=x6DD1k4BAUg&feature=channel_page 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. Bruno > > >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 > > > > > > http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---