$$E_{\text{TM}}=\{\langle M\rangle|M\text{ is a TM and $L(M)=\emptyset$}\}.$$ $E_{\text{TM}}$ is undecidable $$PCP=\{\langle P\rangle|P\text{ is an instance of the Post Correspondence Problem with a match}\}.$$ $PCP$: The Post Correspondence Problem is to determine whether a collection of dominos has a match. PCP is undecidable
Show that there is an oracle Turing machine having an oracle for $E_{\text{TM}}$ which can be used to decide $PCP$. Show that there is an oracle Turing machine having an oracle for $PCP$ which can be used to decide $E_{\text{TM}}$.
$PCP$ is decidable relative to $E_{\text{TM}}$ Can we contrustruct reduction of $PCP$ to $E_{\text{TM}}$ like that?:
T(Q) =
if Q is an instance of the Post Correspondence Problem with a match then P else reject
then we know that language of $T$ is empty exactly when $P$ is not an instance of the Post Correspondence Problem with a match
and then the decider for $PCP$ with access to oracle for $E_{\text{TM}}$ can look like that:
Write T on the tape.
Query oracle with contents of tape.
Output the opposite of what the oracle outputs.
Am I completely wrong? Can somebody help me please? And also I have no idea how to do the opposite direction.