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 > 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... > > Jesse > > > > > > > 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? 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/ --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---