\begin{frame}{Regular Languages}
  \begin{goal}{}
    A DFA defines (accepts) a language! 
  \end{goal}  
  \pause
  
  \begin{block}{}
    The \emph{language accepted by} DFA $M = (Q,\Sigma,\delta,q_0,F)$ is
    \begin{talign}
      \alert{L(M)} 
      &= \{\, w \in \Sigma^* \mid (q_0,w) \vdash^* (q,\lambda) \text{ with } q \in F \,\}\\
      &= \{\, w \in \Sigma^* \mid q_0 \apath{w} q \text{ with } q \in F \,\}
    \end{talign}
  \end{block}
  \pause
  \begin{exampleblock}{}   
    \begin{center}
      \input{tikz/dfa_even_bs.tex}
      \vspace{-2ex}
    \end{center}
    We have
    \begin{talign}
      (q_0,abba) \vdash 
      (q_0,bba) \vdash 
      (q_1,ba) \vdash
      (q_0,a) \vdash 
      (q_0,\lambda)
    \end{talign}
    \pause
    The word $abba$ is accepted by $M$, that is, $abba \in L(M)$. 
  \end{exampleblock}
  \pause
  
  \begin{block}{}
  A language $L$ is \emph{regular} if there exists a DFA $M$ with $L(M) = L$.
  \end{block}
  \bigskip
\end{frame}