Stephen P. King (firstname.lastname@example.org)
Fri, 23 Jul 1999 09:51:39 -0400
This is a very good intro to the computation question as I think of it!
Date: Fri, 2 Jan 1998 00:57:05 -0800 (PST) From: Mitchell Porter <email@example.com> To: firstname.lastname@example.org Subject: Noncomputable dynamics for quantum theory Part of Roger Penrose's hypothesis regarding the form of a final theory is that its dynamics should be noncomputable. Recall that this simply means that no Turing machine could reproduce this dynamics in the output of a calculation. Mathematics already offers examples of noncomputable sequences whose first few elements we know, but only because we have hit upon methods that suffice to identify those particular elements. When looking for noncomputability in physics, Penrose suggests that quantum gravity with topology change might be noncomputable, since four-manifolds are not classifiable, and four-manifolds would interpolate between the spacelike hypersurfaces at either end of a sum over histories. This would give us noncomputable amplitudes, and so noncomputable transition probabilities. This might be suitable for a noncomputable stochastic theory, but I wonder if we could go further and have a deterministic noncomputable theory. In this regard I find Chaitin's number interesting. Chaitin's number is the halting probability for a Turing machine, given certain weightings on initial conditions. Not only is Chaitin's number noncomputable, it is a random real, which means that it is statistically indistinguishable from a random series. Could the apparent randomness of quantum behavior, rather than resulting from real (albeit structured) randomness, be the result of a pseudorandom, deterministic noncomputable dynamic? -mitch http://www.thehub.com.au/~mitch on Chaitin's number http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer2.html
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