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\begin{document}



\prof{Peter Cholak}
\class{Math 441}
\quizdate{Due Wednesday, November 6}
\examname{Homework}

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\heading

\begin{problem} 3.3.2, 3.3.3, 3.3.4. \end{problem}


\begin{problem}
  Consider $L = \{ x^ny^n : n \in \mathbb{N}, x \in \{ab, ba\}, y
  \in \{cd,dc\}\}$.  Construct a grammar $G$ such that $L(G)=L$.
  Using Lemma 3.4.3 find a PDA $M$ such that $L(M)=L$.
\end{problem}
\vspace{.5in}

\begin{problem}
  Let $M = ( \{s,f \}, \{a,b,c,d\},\{a\}, s, \{f\},\Delta)$,
  where $\Delta = \{((g,aa,e),(g,aaa))$, $((g,b,e),(g,aa))$,
  $((g,e,e),(f,e))$, $((f,d,a),(f,e))$, $((f,d,aaa),(f,e))$, $
  ((f,c,aa),(f,e))$.  Using Lemma 3.4.4 find a context-free
  grammar $G$ such that $L(G)=L(M)$.  Describe $L(M)$.
\end{problem} 


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