Infinite computations with random oracles

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Merlin Carl and Philipp Schlicht have posted an article to the arXiv entitled infinite computations with random oracles. A basic result of classical computability theory states that if a real x is Turing-computable relative to all oracles in a set of positive measure, then this real number is Turing-computable in the empty oracle. In different words,  the use of random generators does not enrich the set of computable functions. The authors examine whether this is true for models of infinite computation. In particular, for Infinite Time Turing Machine, Infinite Time Register machines, α-Turing machines, and ordinal Turing/register machines, the authors ask whether there are noncomputable reals (in the sense of the machine type) that are reducible to all oracles in a set of positive measure. The authors show that the answer is negative for most models of infinite computation.

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A comment on a comment

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Recently, "someone" posted a review of my book on hypercomputation on the Brains site. The reviewer argues that This area is full of problems, some of which are physico/mathematical and some of which are conceptual. Unfortunately, from a quick sample, Syropolous's book does not avoid common mistakes and confusions—some of which I've been trying to correct in my own work. First of all, I have to admit that back in 2004 I read a paper by this reviewer, but at that time I did not considered it interesting. However, after I read this review of my book, I read The Physical Church-Turing Thesis: Modest or Bold? , the reviewer's latest manuscript, in order to see what he meant by common mistakes and confusions and how he was trying to solve them. In this manuscript, the reviewer puts forth a number of criteria that every machine has to satisfy in order to be ``useful for physical computing.'' These criteria are: Readable Inputs and Outputs Process-Independ...
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Is the Universe a Quantum Computer?

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 The idea that the Universe (whatever we may mean by this) is a some sort of a computer is a recurring theme. Recently, Nature published an article by David L. Chandler that is entitled Could the Universe be a giant quantum computer? This article is actually a tribute to Edward Fredkin, who passed away last June. The main contribution of Fredkin to the philosophy of science is his "digital philosophy" that advocates the idea that the universe is a computer and everything in our universe is discrete. I would not like to argue against the validity of these ideas (I have done so in Hypercomputation: Computing Beyond the Church-Turing Barrier and in Demystifying Computation: A Hands-On Introduction ) . However, I would like to point out that if the Universe is not a computing device, in general, then it cannot be a quantum computer, in particular. Also, what I did not like with Chandler's article is that it takes it for granted that the Universe is a computer. He did not e...
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