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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.

Jason Hu
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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.

Alexey Romanov
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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.

Jason Hu
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