\begin{frame}{Regular Expressions}
  \begin{block}{}
  We define the \emph{regular expressions} over an alphabet $\Sigma$:

  \begin{itemize}
    \item \alert{$\emptyset$} is a regular expression
    \item \alert{$\lambda$} is a regular expression
    \item \alert{$a$} is a regular expression for every $a \in \Sigma$
    \item \alert{$r_1+r_2$} is a regular expression for all regular expr.\ $r_1$ and $r_2$
    \item \alert{$r_1\cdot r_2$} is a regular expression for all regular expr.\ $r_1$ and $r_2$
    \item \alert{$r^*$} is a regular expression for all regular expressions $r$
  \end{itemize}
  \end{block}
  \pause\medskip
  
  A regular expression is syntax, describing a language.
  \begin{goal}{}
    Every \emph{regular expression} $r$ defines a \emph{language} $L(r)$:
    \begin{talign}
      L(\emptyset) &= \emptyset &
      L(r_1 + r_2) &= L(r_1) \cup L(r_2) \\      
      L(\lambda) &= \{\, \lambda \,\} &
      L(r_1 \cdot r_2) &= L(r_1)  L(r_2) \\
      L(a) &= \{\, a \,\} \text{\;\; for $a\in\Sigma$} &
      L(r^*) &= L(r)^*
    \end{talign}
  \end{goal}
\end{frame}