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I am looking for an example of decidable problems the decision procedures of which are unknown. I believe someone mentioned one to me once, and I also have read somewhere, but my memory is corrupted. I suppose the decidability of these problems are proved non-constructively, so that they have this kind of counter-intuitive property.

I think that in graph theory, there are a number of problems in this trait. I am wondering if you are aware of any of these.

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  • $\begingroup$ I doubt there is any such kind of decidable problem, because of the definition of decidable. There are lots of decidable problems for which people do not care enough to figure out or specify a decision algorithm in its full detail. However, I would not be willing to call those situations as without known concrete decision procedure. It is more like no explicit implementation in a programming language has been done. Or is that your real intention? $\endgroup$ – Apass.Jack Apr 2 at 17:26
  • $\begingroup$ Very similar questions have been considered. With a lot of confusion again! Like here: cs.stackexchange.com/questions/32325/… and here: cs.stackexchange.com/questions/367/… $\endgroup$ – Hendrik Jan Apr 2 at 22:41
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Here is a trivial example of such a problem: consider any yes-no question with an unknown answer, e.g. $P=NP$. For any $n \in \mathbb{N}$, let $A(n)$ be true iff $P=NP$. Then $A$ is decidable by one of two decision procedures:

  1. Always return "true".

  2. Always return "false".

But you need to solve $P=NP$ to know which one.

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  • $\begingroup$ ah I really like this sort of construction. $\endgroup$ – Jason Hu Apr 2 at 18:19
  • $\begingroup$ In fact, some people believe $P=NP$ is undecidable. Can you prove that there is a Turing machine that will either prove or disprove $P=NP$? So the meaning of "let $A(n)$ be true iff $P=NP$" is somewhat ambiguous. What I am saying is that I would like to require that a valid or, you say, interesting example should not contain a part that is not known to be decidable from the ground, for example, ZFC system. $\endgroup$ – Apass.Jack Apr 2 at 19:17
  • $\begingroup$ Has $P=NP$ been proved to be falsifiable in ZFC system? It looks like I am determined to confuse myself. $\endgroup$ – Apass.Jack Apr 2 at 19:21
  • $\begingroup$ @Apass.Jack This construction doesn't need $P=NP$ to be decidable. Taking an undecidable (in some system, e.g. in ZFC) statement would actually work a bit better: then no explicit decision procedure can ever be proven to work (in that system), as opposed to just not having one now. I am not sure exactly what you mean by "falsifiable" in the second comment. $\endgroup$ – Alexey Romanov Apr 2 at 20:29
  • $\begingroup$ @Apass.Jack This might be a bit confusing because there are two different senses of "(un)decidable" involved: for logical formulas and for decision problems. $\endgroup$ – Alexey Romanov Apr 2 at 20:39
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For the record, I found the one in graph theory, which is called graph minor theorem, or Robertson–Seymour theorem.

https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem

Indeed, this theorem is proved non-constructively.

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    $\begingroup$ The truth for formidden minors is much more complicated and refined than this answer. Check is there an algorithm that finds the forbidden minors?. Note that Robertson–Seymour theorem is a theorem, not a decidable problem. It might mention a decidable problem, but it is not a decidable problem by itself. I recommend that you state the problem, the more formally the better. Explain why the problem is decidable, which might not be obvious at all and why there is no algorithm found for it, which might not be obvious at all again. $\endgroup$ – Apass.Jack Apr 2 at 17:13
  • $\begingroup$ In short, I am saying this answer cannot be counted as correct. $\endgroup$ – Apass.Jack Apr 2 at 17:30
  • $\begingroup$ @Apass.Jack I don't understand the theorem in its full details. I am search for an example of such kind as a concrete example, and it stops there. from wiki, it says that "As a result, the theorem proves that the problem can be solved in polynomial time, but does not provide a concrete polynomial-time algorithm for solving it. Such proofs of polynomiality are non-constructive: they prove polynomiality of problems without providing an explicit polynomial-time algorithm." I suppose it says that the poly algorithm is just unknown. what's your concern? $\endgroup$ – Jason Hu Apr 2 at 17:34
  • $\begingroup$ You are talking about polynomial-time instead of decidable. These two concepts are very different. They are plenty of problems that are decidable but without polynomial-time algorithm. Did you mean to ask "decision problems that are solvable in polynomial-time for which no concrete polynomial-time algorithm is known"? That sounds like an interesting question (which might have been asked before) that makes much more sense. $\endgroup$ – Apass.Jack Apr 2 at 17:41
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    $\begingroup$ @JasonHu Compare statements: 1. There's no known explicit algorithm for solving problem $X$. 2. The proof of R-S theorem doesn't provide an explicit polynomial-time algorithm for solving problem $X$. The Wikipedia only says something like 2 ("like" because as Apass.Jack says, you haven't specified the problem yet). Does this imply 1? $\endgroup$ – Alexey Romanov Apr 2 at 18:40

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