\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}