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\begin{document}
  \klausur{GdI-GTI-B Machines and Languages}
        {Prof. M. Mendler, Ph.D.}
        {Wintersemester 16/17}
        {90}
        {Wörterbuch (Englisch-Deutsch/Deutsch-Englisch)}

  \begin{enumerate}
  \item{
      Consider the following non-deterministic automaton $M$ over input alphabet
      $\Sigma = \{a,b\}$:
      \image{0.6}{WS1617-GTI_Diagram_1.png}{NDA $M$}{NDA $M$}
      \begin{enumerate}
      \item Construct a \textbf{deterministic} automaton $M'$ that accepts the
        same language, i.e., such that $L(M') = L(M)$.
      \item Find a regular expression $r$ that represents the language of the
        automaton $M$ (and $M'$), i.e., so that $L(r) = L(M) = L(M')$.
      \end{enumerate}

      Make sure you \textbf{explain} how you obtained your answer in each of
      the sub-questions!
    }

  \item{
      Consider the following pushdown automaton (PDA) $M$ over input alphabet
      $\Sigma = \{0,1\}$ and stack alphabet $\Gamma = \{0, \#\}$:
      \image{0.8}{WS1617-GTI_Diagram_2.png}{PDA $M$}{PDA $M$}

      Give a formal (or an informal but precise) description of the language
      $L(M) \subseteq \{0,1\}^*$ that is accepted by $M$ and \textbf{justify
        your answer} by referring to the operational behavior of $M$ as
      specified by the transition diagram given for $M$ above.
    }

  \item{
      Consider the context-free grammar $G = (\{S, B\}, \{a,b\}, S, P)$ with
      production rules $P$ given as:
      \begin{align*}
        S \quad &\rightarrow \quad aSaB \; | \; aaB \\
        B \quad &\rightarrow \quad bB \; | \; \Delta
      \end{align*}

      \begin{enumerate}
      \item The word $w = aaaabaabb$ is a word of the language $L_G = L(G)$
        generated b grammar $G$. Draw a derivation tree (parse tree) for $w$
        with respect to $G$.
      \item Construct a pushdown automaton $M_G$ that accepts the same language,
        ,i.e., for which $L(M_G) = L(G)$.
      \end{enumerate}
    }
  \item What is \textit{Cook's Theorem} and what is its significance in
    complexity theory?
  \end{enumerate}
\end{document}

      %\image{1}{Capture3.PNG}{DNS-Anfrage}{DNS-Anfrage}