Decide for a TM M whether L(M) is finite when you already know thet L(M) is regular. Consider the following problem: For finite automata it is of course decidable to check if the recognized language is finite, but this obviously not the case for TMs but I wonder if it is possible to decide whether $L(M)$ is finite if you already know (e.g. by some oracle) that $L(M)$ is regular.
I am thinking of an answer where one might construct an FA $\mathcal{A}$ from a TM $M$ with $L(\mathcal{A}) = L(M)$. Is there any way?
Best,
Niklas
 A: I think it depends on what information the oracle tells you. 
Imagine the oracle takes in $\langle M\rangle$, a description of a TM, and returns $\langle D\rangle$, a description of a DFA.  $D$ is guaranteed to have the same language as $M$ if $L(M)$ is regular. $D$ may be an arbitrary DFA otherwise.  Given that you trust this oracle, you can simply test whether $L(D)$ is finite instead of whether $L(M)$ is finite, a decideable problem.
Alternatively, imagine the oracle takes in $\langle M\rangle$ and returns an integer $n$, such that there's an $n$-state DFA with the same language as $M$ (if $L(M)$ is regular, otherwise return an arbitrary integer). If you trust the oracle, I believe you can find a DFA $D$ that corresponds to $M$ by simply iterating over all $n$-state DFAs and comparing their results to $M$ on all strings up to length $2n$ or something.  Then test whether $L(D)$ is finite, as before.
Finally, imagine the oracle takes in $\langle M\rangle$ and returns a single bit indicating REGULAR or NOT-REGULAR. That information actually doesn't help you— when you restrict to machines whose languages actually are regular, the oracle's output is just constant. It doesn't give you any increased ability to decide finiteness.
In summary, I would say that the answer depends on how we formalize the oracle. I think that interestingly, if the oracle tells you anything useful, then the problem becomes relatively straightforward to decide. If the oracle only asserts that the language is regular, then the finiteness problem is no more solvable than it was originally. 
