\begin{frame}{Paths in DFAs}
  Let $M = (Q,\Sigma,\delta,q_0,F)$ be a DFA. 
  \begin{block}{}
    For a word $w = a_1 \cdots a_n$, $n \ge 0$, we write 
    \begin{talign}
      \alert{q_0 \apath{w} q_n}
    \end{talign}
    if there are states $q_1,\ldots,q_{n-1}$ such that
    \begin{talign}
      q_0 \stackrel{a_1}{\to} q_1 \quad\quad
      q_1 \stackrel{a_2}{\to} q_2 \quad\quad
      \ldots \quad\quad
      q_{n-1} \stackrel{a_n}{\to} q_n
    \end{talign}
  \end{block}
  \pause
  \begin{exampleblock}{}
    \centerline{
    \begin{tikzpicture}[default,node distance=20mm,->]
      \node (q0) [fstate] {$z_0$}; \draw ($(q0) + (-10mm,0mm)$) -- (q0); 
      \node (q2) [state,right of=q0,yshift=-10mm] {$z_2$};
      \node (q1) [fstate,right of=q2,yshift=10mm] {$z_1$};
      \draw (q0) to[bend left=15] node [label,above left] {$a$} (q1);
      \begin{scope}[label/.style={inner sep=0.5mm}]
        \draw (q0) to[bend left=10] node [label,above right] {$b$} (q2);
        \draw (q1) to[bend left=10] node [label,below right,inner sep=0mm] {\alert<1>{$a,b$}} (q2);
        \draw (q2) to[bend left=10] node [label,above left] {$a$} (q1);
        \draw (q2) to[bend left=10] node [label,below left] {$b$} (q0);
      \end{scope}
      \begin{scope}[nodes={rectangle}]
        \node at (6.5cm,-0.3cm) [align=left] {
          $z_1 \apath{\lambda} z_1$\\[1mm]
          $z_0 \apath{ab} z_2$\\[1mm]
          $z_0 \apath{abba} z_1$
        };
      \end{scope}
    \end{tikzpicture}\quad
    }
  \end{exampleblock}
  \pause\bigskip
  
  \begin{goal}{}
    Theorem: \;\;$q \apath{w} q' \;\iff\; (q,w) \vdash^* (q',\lambda)$.
  \end{goal}
\end{frame}