lec14.tex

% context free grammar

\subsection{Push Down Automata}

\begin{example}[Bracket Matching]
    The push down automaton described as following
    \centgraph[7cm]{mp/push_down-0}
    recognizes the language 
    \[
        \set{ 0^n 1^n \mid n \ge 0 }.
    \]
\end{example}

\begin{example}[Same number of 0's and 1's]
    A push down automaton can be easily constructed to recognize the language,
    \centgraph[7cm]{mp/push_down-1}
%     \[
%         S \mapsto SS \mid 0S1 \mid 1S0 \mid \varepsilon
%     \]
\end{example}

\section{PDA}

\begin{definition}[PDA]
    A PDA is 
    \[
        M = \lst{Q, \Sigma, \Gamma, \delta, s, F}
    \]
    where
    \begin{compactitem}
    \item
        $\Sigma, \Gamma$ are finite sets of alphabets,
    \item
        $Q$ is a finite set of states,
    \item
        $s \in Q$ is the starting state,
    \item
        $F \subseteq Q$ is a finite set of the final states, and
    \item
        $\delta \colon Q \times \Sigma_\varepsilon \times \Gamma_\varepsilon
        \mapsto
        \gamma(Q \times \Gamma_\varepsilon)$
    \end{compactitem}
\end{definition}

\begin{definition}[Configurations]
    \begin{align*}
        \Conf_M &= Q \times \Sigma^* \times \Gamma^*    \\
        I_M(w)  &= (s,w,\varepsilon)                    \\
        H_M     &= \set{ (q,w,z) \mid w = \varepsilon } \\
        O_M\( (q,\varepsilon,z)\) &=
                    \begin{cases}
                        \text{accept} & q \in F  \\
                        \text{reject} & \text{else}
                    \end{cases}                         \\
        R(q',w,cz)
    \end{align*}
\end{definition}