\begin{frame}{Modified Post Correspondence Problem}
  \begin{goal}{}
    We will show that the PCP is undecidable.
  \end{goal}
  \pause\bigskip\bigskip

  We first prove that the \emph{modified PCP (MPCP)} is undecidable.
  \pause\medskip

  \begin{block}{Modified PCP (MPCP)}
    Given $n$ pairs of words:
    \begin{talign}
      (w_1,v_1),\;\ldots,\;(w_n,v_n)
    \end{talign}
    Are there indices $i_1,i_2\ldots,i_k$ ($k \ge 1$) such that
    \begin{talign}
      \alert{w_1} w_{i_1} w_{i_2} \cdots w_{i_k} ~~ = ~~ \alert{v_1} v_{i_1} v_{i_2} \cdots v_{i_k} ~~ ?
    \end{talign}
  \end{block}
\end{frame}