A proof longer than the size of the universe is also pretty useless and probably not something we need to worry about.
Like i guess you are saying we couldn't really use such a machine to determine whether a certain conjecture just has a very long proof/disproof or is actually undecidable. Which sure, but umm i think that is the sort of problem most mathematicians would love to have.
The real reason non-deterministic turing machines can't help us is that they dont actually exist.
Of course it would, you would enumerate lengths too. If the lengths need to be larger than polynomially bounded then we can be sure it would never be found by a human anyway.
> If the lengths need to be larger than polynomially bounded then we can be sure it would never be found by a human anyway.
We cannot, as a human might be able to intuitively devise proof that might be quite large in the target language but easily constructible in a separate one.
Then in that target language, found by a clever human, you could do the same type of enumeration...
My whole point is that humans simply cannot process/create by themselves any truly long proof (we can obviously create a process for that). Therefore enumeration puts everything achievable by humans in the NP complexity. Feel free to disagree, this is not so much a theorem as more of a thesis.
> Therefore enumeration puts everything achievable by humans in the NP complexity.
This is a severe misunderstanding of NP. Hindley Milner type inference is worst case doubly exponential, for example, and we solve that all the time. We just happen to rarely hit the hard instances. I think the statement you're trying to make is something more along the lines of https://en.wikipedia.org/wiki/Skolem%27s_paradox, which is definitely fascinating but doesn't have much to do with P vs NP. Notably, this paradox does not apply to higher order logics like CiC with more than one universe, which lack countable models (which is why your intuitions about provability may not apply there).
No misunderstanding about NP here for sure. As I said, this is about as much of a thesis as Church Turing is about what can be computed.
I have no clue about CiC, lean and whatnot. It was never my field and I don't doubt there can be some very weird things you can do with some fancy logic models that are rarely discussed by non logicians.
What I'm claiming is that anything a human could prove without a computer could be done by a non deterministic machine in poly time under a very reasonable assumption of proof length. I'm baking in the assumption of proof length, so I can claim something about NP...
Let me put it this way: if you can write a paper with your proof and I can check it with only my brain without a computer helping me, then a poly time algorithm could also check that proof. Unless you believe something exotic about how the brain computes, how would it not be the case?
> Let me put it this way: if you can write a paper with your proof and I can check it with only my brain without a computer helping me, then a poly time algorithm could also check that proof. Unless you believe something exotic about how the brain computes, how would it not be the case?
What does "without a computer" have to do with anything, and what do you mean by saying "in polynomial time" when restricted to a specific instance of the problem? To talk about solutions being checkable in polynomial time in the size of the input (which is what NP actually means, more or less, since every problem in NP can be encoded as a 3SAT formula with at most polynomial blowup), you have to actually specify some (infinite) class of problems first, otherwise "in polynomial time" doesn't really mean anything. Then, you have to specify a polynomial time verification algorithm for every instance in the class.
T he problem is that "proofs of length N" in CiC doesn't have a polynomial time verification algorithm (for checking proof correctness) in the size of the proof (and as I mentioned elsewhere, attempts to translate to an encoding that does can result in exponential or much worse blowup in the proof size, which does not allow you to put them in the same complexity class in this context). Moreover, it's very unclear how you'd even begin to carve out the complexity class of "proofs that can be verified quickly" because that pretty much requires you to write an algorithm to determine the complexity of the proof, which includes being able to determine the complexity of arbitrary functions in CiC (which as you can imagine, is not easy in such an expressive system! AFAIK, the totality proof requires axioms beyond ZFC, which means that the proof is not constructive and cannot provide any specific upper bound on the number of steps taken by an expression).
This means that there is no polynomial function f(n) such that a nondeterministic Turing Machine could always terminate after f(n) steps and (correctly) report "yes" when there is a proof with n or fewer steps and "no" otherwise, and it is rather unlikely that there is a decision procedure to restrict the proofs to "proofs that run in polynomial time in the size of the proof input" so you could practically apply your claim just to that subset of proofs (though I don't know whether the absence of such a decision procedure has actually been proven, but generally speaking problems like this are very rarely decidable for interesting programming languages like CiC, even if they are total). It does not mean that a human, or human with a computer, couldn't come up with a clever short proof in O(n) tokens that takes longer than f(n) steps to check on some particular instance of the problem for some particular f... because why would it?
I think what you're trying to say (to make your argument hopefully what you had in mind) is that if humans can verify a proof in "a reasonable amount of time", then a nondeterministic Turing Machine could produce the proof in "a reasonable amount of time." Which might well be true, but the point is that in CiC, how long the proof takes to verify has very little to do with the size of the proof, so this claim has nothing to do with proof verification being in P (and therefore, has nothing to do with proof search being in NP, which it isn't for CiC). More to the point, this is pretty much true of any problem of interest--if it takes more than a reasonable amount of time in practice to verify whether a solution is correct (where verification can include stuff like "I ran polynomial-time UNSAT on this instance of the problem" in a universe where P=NP), it isn't practically solvable, regardless of what fancy methods we used to arrive at the solution and regardless of the length of the solution. So I don't find this particularly insightful and don't think it says anything deep about humanity, mathematics, or the universe.
Anyway, I'm mostly just pointing out that even if P=NP, you can't really leverage that to discover whether a short proof exists in CiC in polynomial time. More concretely, there isn't an algorithm to convert an arbitrary CiC formula to a 3SAT formula without greater than polynomial size blowup (we know this because we can run non-elementary algorithms in CiC!). So even if we had a polynomial time 3SAT algorithm, which would solve a great many things, it wouldn't solve proof synthesis, which is another way to phrase "proof synthesis in CiC isn't in NP."
There’s always another encoding, all you prove is that there doesn’t exist a proof in the encoding you selected less than the length you specified, it does nothing at all to disprove the existence of a short proof in general.
Right, and this is also the current status of handmade mathematics. All we know is that we did not find a proof yet with everything that has been tried. This typically means that a problem is harder the more stuff has been thrown at it and failed.
A non deterministic Turing machine would absolutely crunch through math theorems, I really don't know why there is so much push back against what I am stating. Basically, if you had such a machine, there essentially would no longer be any room left for proving stuff the manual way, you would get completely outclassed by them.
The real consequence for me though is that this is a strong evidence (call it faith) against P=NP.
> A non deterministic Turing machine would absolutely crunch through math theorems
Can you formalize that statement in any useful way? It seems you believe proof validation to be in NP, when it most certainly is not. Accordingly an N-TM would be of relatively little power against the problem.
Like i guess you are saying we couldn't really use such a machine to determine whether a certain conjecture just has a very long proof/disproof or is actually undecidable. Which sure, but umm i think that is the sort of problem most mathematicians would love to have.
The real reason non-deterministic turing machines can't help us is that they dont actually exist.