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If we had some sort of black box that allows us to run an infinite amount of steps of a Turing machine in a finite amount of time (no limitation on the length of the tape), and be able to output something based on everything that happened during those infinite steps, what would the consequences be?

I can see a few interesting consequences from the get go, but I'm sure there'd be even deeper consequences for that:

  • We would be able to solve the Halting Problem by just having the black box machine tell us whether the Turing machine inside stopped.
  • We would be able to find all propositions that can be proven true in a finite amount of deductions from a given (recursively enumerable?) set of axioms, and the propositions which cannot, by enumerating all finite strings of symbols and checking whether they are a proof for a given proposition.
  • We might be able to find the "best" algorithms for a given task as long as we can come up with a way to formalize it and a relevent fitness function that can take any algorithm and tell how bad it is at accomplishing the task. The black box would just test every algorithm and keep a list of the ones with the highest fitness score.
  • Being able to do computations with any computable real number with no loss of precision

What about more abstract results? Are there more fundamental things we would be able to tell about mathematics? For instance, while it's evident all fields would be shaken by this, there would be interesting developments in logic regarding infinitary logics, possibly reaching another road-block down the road that brings us to a "higher halting problem" regarding the black boxes themselves, and other related to infinitary logic that can't be proven with the black boxes alone. Regarding first order logic, could it allow us to "find" a non-recursively enumerable consistent and complete theory that implies ZFC?

I'm sure this has been thought about and studied by passionate people that have actual knowledge about this, that's I'm still asking even though the question is a mess. If this is better suited for another stack please refer me to it, but I doubt the Science-Fiction & Fantasy stack exchange is a place I could hope to get an answer to this question.

Mitchell Spector
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Uro
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    ''run an infinite amount of steps of a Turing machine in a finite amount of time'' is a contradiction in terms. Every step counts 1 tick. – Wuestenfux Dec 31 '21 at 15:02
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    @Wuestenfux The question is apparently about "overclocking" your TM such that the $n$th tick takes only $2^{-n}$ years, so the inifinitely many steps are "completed" after one year. Apart from computational problems, this might also solve interstellar superluminal information exchange (unless we are guaranteed that the $n$th tape field is shorter than $2^{-n}$ lightyears; but then the fields would sooner or later be shorter than the Planck length, sigh). – Hagen von Eitzen Dec 31 '21 at 15:04
  • By "finite amount of time" here I mean in a physical sense, as in: Being able to run an infinite amount of steps of the Turing machine in the blackbox in 3 seconds and output something as a result. – Uro Dec 31 '21 at 15:04
  • @HagenvonEitzen Yes something like that. I'm aware for the superliminal information exchange, but that element itself is to be ignored, that's why I refer to it as a "black box", just assume it works no matter how, ignoring physical implications. – Uro Dec 31 '21 at 15:10
  • @Uro So you want to use "finite amount of time" in a physical sense, but ignore physical implications? – Hagen von Eitzen Dec 31 '21 at 15:27
  • @HagenvonEitzen Yes, I'm not saying there aren't, just that it's another topic, another question for another stack. – Uro Dec 31 '21 at 15:30
  • Because the machine attains a discrete state after each step, you arrive effectively at (a slightly generalized version of) Thomson's lamp. You can find further answers on this site by using this search term. (The Wikipedia article isn't of much use, IMHO.) – John Bentin Dec 31 '21 at 17:01
  • @Hagen Interesting concept. – Wuestenfux Dec 31 '21 at 17:36
  • One possible resolution of this is to replace the natural numbers with a nonstandard version. (looking like: https://victoriagitman.github.io/images/PAmodel.jpg) This adds a bunch of new numbers that we know to be infinite, but the model does not. Let $n$ be such a number. As such, we can run Turing machines for $n$ steps and use tape up to and beyond position $n$. If we're talking about nonstandard models of an incomplete theory (Peano Arithmetic, for example), the behavior of these machines depends on our choice of model. – TomKern Dec 31 '21 at 23:10
  • It seems oracle TMs do exactly what you're asking. IF $A$ is, say, $\Pi_1$ complete, then in a sense it takes $\omega$ many steps to compute whether $n\in A$ , but a TM equipped with an oracle for $A$ computes that in one atomic step. – BrianO Jan 01 '22 at 01:07
  • The possibility of hypercomputers is actually not completely off the table. If the Malament–Hogarth spacetime is realizable, then we could perform something like what you're describing. Of course, it is likely that this kind of spacetime is simply physically unrealizable just like many other spacetimes of GR. – Maximal Ideal Feb 28 '22 at 01:54

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I don’t have time to write up a detailed answer right now, but the sets accepted by these generalized Turing machines are the hyperarithmetic (or $\Delta^1_1)$ sets. This has been studied extensively; look up Addison and the $\aleph_0$-mind in Hartley Rogers Jr., The Theory of Recursive Functions and Effective Computability.

Note that these generalized Turing machines are not physically realizable; this is an abstract model of computation.

The OP didn’t say, but I’ve assumed here that these “infinite Turing machines” can be used as blocks in other more complicated infinite Turing machines. Without that, it’s much less interesting.

Mitchell Spector
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    The assumption is right. I'm aware those machines are not physically realizable, it is more of a "what if" question wondering about the practical consequences it would have on maths and what could be achieved with it. I guess it means we could decide arithmetic sets but I currently lack the knowledge to understand the implications. Thanks for the reference, I'll look into that. – Uro Jan 01 '22 at 10:29