On Bi-infinite and Conjugate Post Correspondence Problems

We study two modifications of the Post Correspondence Problem (PCP), namely 1) the bi-infinite version, where it is asked whether there exists a bi-infinite word such that two given morphisms agree on it, and 2) the conjugate version, where we require the images of a solution for two given morphisms are conjugates of each other. For the bi-infinite PCP we show that it is in the class $\Sigma_2^0$ of the arithmetical hierarchy and for the conjugate PCP we give an undecidability proof by reducing it to the word problem for a special type of semi-Thue systems.