In 1971 C.F. Miller associated to every finitely presented group $G$ a free-by-free group $M(G)$ known as the Miller Machine, whose conjugacy problem is closely related to the conjugacy and word problems of $G$. In this talk, I will discuss a complete description of this relationship; namely, that the conjugacy problem in $M(G)$ is computationally equivalent to a strong form of list conjugacy in $G$, which we term iso-computational list conjugacy (ICLC for short). I will also discuss the geometric tools, such as annular diagrams, and algebraic tools, such as HNN extensions, that are used to prove this equivalence. If time allows I will use these tools to prove an important special case.