\begin{frame}{Minimal DFAs (2)}
  \begin{goal}{}
    States $q_1,q_2 \in Q$ are \emph{distinguishable} if there exists $w \in \Sigma^*$ s.t.
    \begin{talign}
      q_1 \apath{w} q_1' \in F &&
      q_2 \apath{w} q_2' \not\in F\;,
    \end{talign}
    or vice versa.
  \end{goal}
  \pause
  
  \begin{block}{} 
    \emph{Step 2:} Partition $Q$ in indistinguishable states.
    \medskip
  
    We construct the partitioning stepwise:
    \begin{itemize}
    \pause\smallskip
    \item Initial partitioning is $\{\, Q\setminus F,\,F\,\}$.
    \pause
    \item If there are partitions $R$ and $S$ such that 
      \begin{talign}
        \delta(q,a)\in S &&\text{and}&& \delta(q',a)\not\in S,
      \end{talign}
      for some $a \in \Sigma$ and $q,q' \in R$, then we split $R$ in
      \begin{talign}
        \{\, q \in R \mid \delta(q,a)\in S \,\} && \{\,q \in R \mid \delta(q,a) \not\in S\,\}
      \end{talign}
    \end{itemize}
    \pause
    We keep splitting until no more split is possible.
  \end{block}
\end{frame}