This problem is not decidable (reducible to halting problem) but is semi-decidable and therefor verifiable (as those two definitions are equivalent: How to prove semi-decidable = verifiable?).

However, is this problem poly-time verifiable? A decision problem π is in poly time verifiable iff

there is an algorithm π called verifier such that V runs in $ O(x^{c})$ for some constant c for all inputs π₯,

if π(π₯)=ππΈπ then there is a string π¦ such that π(π₯,π¦)=ππΈπ, if π(π₯)=ππ then for all strings π¦, π(π₯,π¦)=ππ.

Example: for an enumeration of P (such as this): How does an enumerator for machines for languages work? for each string $ p$ in the enumeration, does there exist some other string (certificate) $ c$ that allows you to verify $ p$ is a member of the enumeration in poly time?