forked from klausuren/klausuren-allgemein
86 lines
3.2 KiB
TeX
86 lines
3.2 KiB
TeX
\input{../settings/settings}
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\usepackage{amsmath,amssymb}
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\usepackage{tikz}
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\usetikzlibrary{shapes,arrows}
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\begin{document}
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\klausur{GdI-GTI-B}
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{Prof. Dr. M. Mendler}
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{Sommersemester 2017}
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{90}
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{Wörterbuch Englisch-Deutsch\,/\,Deutsch-Englisch}
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\begin{enumerate}
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\item
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Consider the context-free grammar $G = (\{S, A, B\}, \{a, b\}, S, P)$ with production rules $P$ given as
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\begin{align*}
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S &\rightarrow AB \\
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A &\rightarrow aAa~|~\Lambda \\
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B &\rightarrow bbbB~|~\Lambda
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\end{align*}
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\begin{enumerate}
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\item Give a formal (or an informal but precise) definition of $L(G)$ that describes the general structure of the words which are produced by $G$.
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\item Show that $L(G)$ is a \emph{regular} language by constructing an adequate automaton $M$ such that $L(M) = L(G)$.
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\end{enumerate}
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(15 marks)
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\item
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Consider the language $L = \{0^i1^j2^k ~|~ i, j, k \geq 0 ~\textrm{and}~ k \geq 2i+j\}$.
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\begin{enumerate}
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\item Construct a \emph{pushdown automaton} (PDA) $M$ which accepts exactly the words of $L$. (15~marks)
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\item Use the \emph{Pumping Lemma for regular languages} to prove that $L$ is \emph{not} a \emph{regular} language. (15~marks)
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\end{enumerate}
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\item
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Remember the definition of $\mathcal{O}$:
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$$\mathcal{O}(f) =_{df} \{g \in \Omega ~|~ \exists c.\: \exists n_0.\: \forall n > n_0.\: g(n) \leq cf(n)\}$$
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for $\Omega$ being the set of all functions with type $\mathbb{N} \rightarrow \mathbb{N}$, and $f \in \Omega$.
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Consider the function $f^\prime(n) =_{df} 12n^3 + 7n^2$.
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Use the definitions from above to show formally that $f^\prime\in\mathcal{O}(n^3)$. (10~marks)
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\item
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Answer both of the following questions:
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\begin{enumerate}
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\item Consider arbitrary languages $L_1$, $L_2$, and $L = L_1 \cap L_2$. Assume that $L_1$ and $L_2$ are both \emph{Turing-decidable.}
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Explain briefly why $L$ is then also \emph{Turing-decidable.} You do \emph{not} need to construct machines explicitly.
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\item Consider an arbitrary language $L$ and assume that $L$ is \emph{polytime acceptable.} Explain briefly why $L$ is then also \emph{polytime decidable.} You do \emph{not} need to construct machines explicitly.
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\end{enumerate}
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(15 marks)
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\newpage\item
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Let $M$ be the following single tape Turing machine with input alphabet $\Sigma = \{x, y\}$ and tape alphabet $\Gamma = \{x, y, \#, \Delta\}$:
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\begin{center}
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\input{SS17_GTI_Figure.pdf_tex}
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\end{center}
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$M$ uses the building block Turing machines introduced in the lectures and tutorials (where $a \in \Gamma$):
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\begin{itemize}
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\item left move $L$, right move $R$, halt $stop$, and print $a$;
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\item the searching machines $R_a$, $R_{\neg a}$, $L_a$, and $L_{\neg a}$; and
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\item the shifting machines $S_R$ and $S_L$.
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\end{itemize}
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Describe the behaviour of $M$ starting from an arbitrary initial tape configuration $\underline{\Delta}w\Delta\Delta\dots$, where $w \in \Sigma^*$. What does the final tape look like when $M$ halts and how does the result depend on $w$? In other words, what computation does $M$ perform? (20 marks)
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% \item{
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% Hier könnte dein Bild stehen:
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% %\image{1}{Capture3.PNG}{DNS-Anfrage}{DNS-Anfrage}
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% }
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\end{enumerate}
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\end{document}
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