[Sharlin]

It doesn't matter whether we can physically do it, as long as we can mathematically do it. In math running a TM for an arbitrary, finite number of steps is not a problem. The actual answer to OP's question was given in one of the other subthreads, namely: the result tells us that BB(754) is not computable in ZFC. Some 754-state TMs halt, others run forever, but there is no way to figure out (without an oracle) which of them runs the longest while still eventually halting.

[smaudet]

>"It doesn't matter whether we can physically do it,"

Hmm. I'm not sure that I agree - philosophically if you could prove that something is computable mathematically and yet at the same time know that physically it cannot be done due to the laws of physics, I would say that imposes a "second order" halting problem.

By which I mean if you had some mathematical algorithm that could be proven to tell you some property of a system like halting, and you knew the number of steps required by that algorithm to produce that answer, if that number of steps is beyond the capacity of the physical world to provide, then there is a set of algorithms that can be proven to never be decidable.

I'm sure you are thinking, oh maybe we get more efficient, but to solve the bounding problem itself you must provide a constant time solution within the bounds of the universe, or else the "mathematical" solution is itself undecidable.

And even if you found a fast, low constant time solution (the holy grail), it still must be shown that the categories of problems are themselves infinite variations of finite categories, so that your low constant time solution could conceivably be applied within the limits of computation capacity, or else there will always be for some large enough set of problems some that are not decidable.

[Sharlin]

If you insist that only tractable problems are decidable (or computable), you don't really get a nice or useful theory of computation. You need a more fine-grained hierarchy.

Almost all real numbers are uncomputable (computable reals have measure zero; in other words, the probability of an arbitrary real number being computable is exactly 0). BB(754) is uncomputable in this sense even though its definition is extremely concise. Still, its value could be computed by a machine stronger than a TM (but it would again be unable to compute its own BB problem).

Of the countable subset of reals that are computable, almost all are like 𝜋, in that computability does not mean being able to actually calculate or represent its exact numeric value in the physical world. Only that it's in principle possible to compute it to any precision desired – but of course this isn't possible in reality either. Only in math.

Then there are numbers that are unbelievably large, like 3^^^^3 or (the immensely larger) Graham's number G, which nevertheless admit an incredibly compact definition, and we can prove various properties of them. Yet their numeric representation would not fit in our future light cone (either in space or in time). Where do you draw the line? After all, G is an entirely ordinary natural number, and indeed almost all natural numbers are greater than G.

Complexity theory draws a line, somewhat arbitrarily, between polynomial-time and superpolynomial-time problems, the former being tractable and the latter intractable. But a number might be easily defined in terms of a tractable function and still be entirely unfeasible to ever calculate or represent in the physical world. (Such a number wouldn't even have to be particularly large if the function that defines it grows sufficiently slowly.)

Nowhere is there a distinction made between "computable in reality" and "only computable in math", because it's impossible to formally define such a distinction. Some numbers are obviously within our reach, almost all others obviously aren't. In between there's a huge gray area.

[smaudet]

Hmm, I think its the same problem, you are somewhat arbitrarily drawing the line between "math" and the universe (or maybe I'm agreeing with you and don't realize it).

A halting/decide-able problem is one that is precisely about tractability - does it end, and can you decide that. The maths is about whether there is a tractable solution, in theory ignoring any energy/mass requirements.

The maths of the maths of the tractability issue is also one of tractability - if you provide an infinite solution to an infinite problem then is it a solution or an admission of defeat? You can't even decide if you can decide that you can decide.

If there is no maths for the maths for the decidability question of is it tractable (not even energy computeable), then in my mind there is no difference between the two, i.e. you have no higher order reduction with which to express the lower order problem. Without a constant time solution, there may not be enough math to express the solution to the problem, i.e. this may just be another set of infinities, as you mentioned the integers, irrational numbers, precision, etc. (which are lower order entities for which we have tractable mathematical solutions).

And the lower energy bound of the universe to the set of problems to me remains interesting, in that maths itself is an invented set of relationships, in the sense that the set is invented and less in that the relationships are invented (they exist but we value certain relationships over others, so the set is invented), so any maths that cannot survive and exist within the information space that is the universe is to some degree irrelevant - if there is no way to relate to it, does it exist? I think so, we simply lack the ability to process or observe it.

Consider, for instance, a constant time solution to the decideability problem, however the maths to prove it require a set of equations and theorems larger than the processing time of the universe. Then a solution exists, perhaps, but we will never discover it. It doesn't mean it doesn't exist, but it does not exist for us, and we can neither prove nor disprove that one exists.

[irchans]

I think that it would be fairly easy (meaning that I think I could do it in 8 hours) to build a Turing Machine that prints 10^1000 1's on its tape and then halts. We would not be able to construct and run physical Turing machine that could do every move physically (by moving electrons) within 100 billion years, but we can still write the proof that it halts, probably with less than 3 pages. (In fact, we could probably tell you which state it is in for any i in the set {1, ...., halting step} without ever constructing and running a physical machine.)

On the other hand, if the "shortest" proof of halting is more than 10^20 pages long, then we would have major problems "writing" it.

[adrianN]

If you restrict yourself to problems that fit in the physical universe, everything can be done using finite automata. That’s a bit boring.