lec13.tex

\[
    G = \lst{\Sigma, V, S, R}
\]
where
\begin{compactitem}
\item
    $\Sigma, V$ are finite sets for input alphabet and variable symbols,
\item
    $S \in V$ is the start variable,
\item
    $R \subseteq V \times (V \cup \Sigma)^*$ is a finites set of rules.
\end{compactitem}

\begin{example}[A simple grammar]
    \label{exa:a_simple_grammer}
    A grammar 
    \[
        G = \lst{\Sigma, V, S, R}
    \]
    \begin{align*}
        \Sigma = \set{0,1,+,*,(,)}  \\
        V = \set{E,T,F}             \\
        S = E                       \\
        \abs R = 7
    \end{align*}
    with rules
    \begin{align*}
        E &\mapsto E + T \mid T      \\
        T &\mapsto T * F \mid F      \\
        F &\mapsto 0 \mid 1 \mid (E)
    \end{align*}
\end{example}

\begin{example}[Is $(0+1)*0$ in $L(G)$?]
    With the grammar defined in \autoref{exa:a_simple_grammer}, 
    consider the expression $(0+1)*0 \in L(G)$?

    \[
        E \mapsto T \mapsto T*F \mapsto F*F \mapsto (E)*F \mapsto (E+T)*F
        \mapsto (T+F)*F \mapsto (F+F)*F \mapsto (0+1)*0 \in \Sigma^*
    \]
\end{example}

\begin{definition}[Leftmost Derivatioonn]
    leftmost derivation
    is a sequence 
    $w_1, w_2, \cdots, w_n \in (V \cup \Sigma)^*$ 
    such that 
    \begin{compactitem}
    \item $w_1 = S$
    \item $w_n \in S^*$
    \item $\forall i = 1, \cdots, n-1$:
        \begin{compactitem}
        \item
            $w_i = \alpha X \beta \quad \text{ where }
            \alpha \in \Sigma^*, X \in V, \beta \in (\Sigma \cup V)^*,$
        \item
            $w_{i+1} = \alpha \gamma \beta \quad \text{ where }
            (X, \gamma) \in R.$
        \end{compactitem}
    \end{compactitem}
\end{definition}

\begin{definition}[Language of a Grammar]
    \begin{align*}
    L(G) &= \set{ w \in \Sigma^* \mid S \mapsto_G^* w }   \\
         &= \set{ w \mid \text{ there is a leftmost derivation of $w$ in $G$} }
    \end{align*}
\end{definition}

\begin{example}[Palindromes]
    Write a grammar for the following language,
    \[
        P = \set{ w \in {0,1} \mid w^R = w }.
    \]

    \[
        E \mapsto 0E0 \mid 1E1 \mid 0 \mid 1 \mid \varepsilon
    \]
\end{example}

push what has been read onto the stack and check for reversion instantly.
Push a special symbol in the very first for the ``beginning of string.''