lec15.tex

\begin{align*}
    S &\mapsto aSc \mid T \\
    T &\mapsto bT  \mid \varepsilon
\end{align*}

Clearly, the grammar above describes a language
\[
    L(G) = \set{ a^m b^n c^m },
\]
e.g. abc, which is generated through
\[
    S \mapsto aSc \mapsto aTc \mapsto abTc \mapsto ab\varepsilon c \mapsto abc.
\]

Claim:
\begin{align*}
    \forall \text{\upshape CFG} G = \lst{V,\Sigma,S,R}, \\
    \exists \text{\upshape PDA} P = \lst{Q,\Sigma,\Gamma,\delta,s,F} \text{ s.t.\ }  \\
    L(G) = L(P)
\end{align*}

\begin{proof}
    \[
        \abs Q = 4 + \sum_{(x \to \alpha) \in R} \max( \abs \alpha -1, 0 )
    \]
\end{proof}

Claim: $P \mapsto G$
\begin{align*}
    V &= \set{ A_{pq} \mid p, q \in Q }  \\
    S &= A_{sf}  \\
    A_{pp} \mapsto \varepsilon
    A_{pq} \mapsto A_{pr} A_{rq}
    A_{pq} \mapsto A_{pr} A_{rq}
\end{align*}