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Jürgen Schmidhuber pointed out that a simple explanation of the universe would be a Turing machine analogy programmed to execute all possible programs computing all possible histories for all types of computable physical laws. His work was based on Zuse's thesis.

This hypothesis would be inside the domain of Digital Physics, a group of hypothetical models that propose that the universe is analogous to a computer or an automata.

(https://en.wikipedia.org/wiki/Digital_physics)

In 2000, he expanded this work by combining Ray Solomonoff's theory of inductive inference with the assumption that quickly computable universes are more likely than others. This work on digital physics also led to limit-computable generalizations of algorithmic information or Kolmogorov complexity and the concept of Super Omegas, which are limit-computable numbers that are even more random (in a certain sense) than Gregory Chaitin's number of wisdom Omega.

So since Chaitin's constant (https://en.wikipedia.org/wiki/Chaitin%27s_constant) would allow hypercomputation if it existed in the pysics of the universe, would then Schmidhuber's hypothesis be able to perform hypercomputation? Also, in the hytpercomputation wikipedia entry (https://en.wikipedia.org/wiki/Hypercomputation) it says (In "eventually correct" systems):

A symbol sequence is computable in the limit if there is a finite, possibly non-halting program on a universal Turing machine that incrementally outputs every symbol of the sequence. This includes the dyadic expansion of π and of every other computable real, but still excludes all noncomputable reals. Traditional Turing machines cannot edit their previous outputs; generalized Turing machines, as defined by Jürgen Schmidhuber, can. He defines the constructively describable symbol sequences as those that have a finite, non-halting program running on a generalized Turing machine, such that any output symbol eventually converges; that is, it does not change any more after some finite initial time interval. Due to limitations first exhibited by Kurt Gödel (1931), it may be impossible to predict the convergence time itself by a halting program, otherwise the halting problem could be solved. Schmidhuber uses this approach to define the set of formally describable or constructively computable universes or constructive theories of everything. Generalized Turing machines can eventually converge to a correct solution of the halting problem by evaluating a Specker sequence.

And also, Super-recursive algorithms are closely related to hypercomputation, and super recursive algorithm is just one way of defining hypercomputation. In super-recursive algorithms wikipedia entry (https://en.wikipedia.org/wiki/Super-recursive_algorithm), Schmidhuber's model is included as a super-recursive algorithm-based model

So, knowing all of this, then, would Schmidhuber's hypothetical universes be based on hypercomputation?

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  • $\begingroup$ "Chaitin's constant would allow hypercomputation if it existed in the pysics of the universe" - that doesn't sound right. It's not clear what what it would mean for a constant to "exist" in the physics of the universe; I doubt that that is a well-defined notion. $\endgroup$ – D.W. Feb 17 at 20:57
  • $\begingroup$ "A real computer (a sort of idealized analog computer) can perform hypercomputation if physics admits general real variables (not just computable reals), and these are in some way "harnessable" for useful (rather than random) computation. This might require quite bizarre laws of physics (for example, a measurable physical constant with an oracular value, such as Chaitin's constant), and would require the ability to measure the real-valued physical value to arbitrary precision" (from en.m.wikipedia.org/wiki/Hypercomputation) @D.W. $\endgroup$ – sztorwi Feb 17 at 21:06
  • $\begingroup$ Yes, I can understand the statement from Wikipedia. That's a different statement, though. That requires more than that it "exists"; but that it be measurable. It is also necessary that it be measurable to arbitrarily high precision. In any case, it's not clear to me what any of this has to do with Schmidhuber's hypothesis. $\endgroup$ – D.W. Feb 17 at 21:10
  • $\begingroup$ "Jürgen Schmidhuber (2000) constructed a limit-computable "Super Ω" which in a sense is much more random than the original limit-computable Ω, as one cannot significantly compress the Super Ω by any enumerating non-halting algorithm" "(...) This work on digital physics also led to limit-computable generalizations of algorithmic information or Kolmogorov complexity and the concept of Super Omegas, which are limit-computable numbers that are even more random (in a certain sense) than Gregory Chaitin's number of wisdom Omega." (Omega basically means Chaitin's constant) @D.W. $\endgroup$ – sztorwi Feb 17 at 21:17
  • $\begingroup$ Sorry, I don't know what connection you are seeing. $\endgroup$ – D.W. Feb 17 at 21:20
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If the hypothesis is that we live in a universe whose physics are computed by a Turing machine, then hypercomputation is trivially impossible in our universe.

The constants you're taking about are limits of computable numbers. If all you have is a computer you can't produce such a number in finite time.

It sounds like you're taking two different things named after the same mathematician and assuming they're somehow connected when they're not.

(Disclaimer: I'm going off the information in the question alone.)

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