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z3JAar2jXFZ61%Wjusv7viHLb1$>60}6IhCBUkv0cg3&*tgT2?|Q@}fyvM=A-`E=9* zBg0eKk+j``+_7NdL(41KYeXu-!UVh@$3`Erf;AHDfaS2hJX8uGkTwIrzYIEssMix5 z{mW7AA-b4L*d8U4mm`59xN5L5M__-Mf!ywfd9>Btt<(PGO%Vw!GhCxTCvq#|N|C(C zsXNYSz&y^g6`$p8V120Z=h(!?BS61zLo${IFmB?yz$)}1AV_fR_IzoN{e_!{ym)&+kY5a})5qwE`YC>%9`TV*HI(Nfr3}@M1twVB)`9us?v< z{7qc_gc3`v_@Y9PWoq;ht}o-SS*OkZZ zz03|?tIqKRQj!+T1}RGU!X%R z>&gu] + + % Knoten + \node (Zug3) at (-2,0) {Zug 3: MIN}; + \node (A) at (1,0) [circle, minimum size=1cm, draw] {5}; + \node (B) at (3,0) [circle, minimum size=1cm, draw, fill] { }; + \node (Zug2) at (-2,2) {Zug 2: MAX}; + \node (C) at (0,2) [circle, minimum size=1cm, draw] {-2}; + \node (D) at (2,2) [circle, minimum size=1cm, draw] { }; + \node (Zug1) at (-2,4) {Zug 1: MIN}; + \node (E) at (1,4) [circle, minimum size=1cm, draw] { }; + + % Kanten + \draw[-{>[scale=4, + length=2, + width=3]},line width=0.4pt] (A) to (D); + \draw[-{>[scale=4, + length=2, + width=3]},line width=0.4pt] (B) edge (D); + \draw[-{>[scale=4, + length=2, + width=3]},line width=0.4pt] (C) edge (E); + \draw[-{>[scale=4, + length=2, + width=3]},line width=0.4pt] (D) edge (E); + + \end{tikzpicture} +\end{center} + +\section*{Problem 3} + +\begin{enumerate}[label=\alph*)] + \item The crossword shown below has to be filled with the words \textbf{IS}, \textbf{IN}, \textbf{SO} and \textbf{NOT}. Each word may be used once only. Represent the task as a \textit{constraint satisfaction problem}. Describe the variables, their domains, and the constraints. Draw the constraint graph. (8 P) + + \begin{figure}[H] + \centering + \includegraphics[scale=0.3]{ws1516-SemInf-problem3a} + \label{fig:ws1516-seminf-problem3a} + \end{figure} + + \item Aplly the \textit{arc consistency algorithm} to the constraint satisfaction problem. Describe the intermediate steps of the computation in form of a table (variables and domains). Does the algorithm solve the puzzle? (7 P) +\end{enumerate} + +\section*{Problem 4} + +\begin{enumerate}[label=\alph*)] + \item Formalize sentences (1), (2), and (3) in \textit{predicate logic}. Use a predicate $K(x, y)$, which expresses that person $x$ knows person $y$ and a predicate $B(x, z)$, which expresses that person $x$ attends course $z$. Avoid using further predicates. (3 P) + + \begin{enumerate}[label=(\arabic*)] + \item If x and y attend the same course, then person x knows person y. + \item Alice attends the course \glqq cryptography\grqq\ and Bob attends the course \glqq cryptography\grqq . + \item Alice knows Bob and Bob knows Alice. + \end{enumerate} + + \item Specify the \textit{clause forms}, which are needed to show by resolution that formula (3) can be proved from formulas (1) and (2). (3 P) + + \item Provide the \textit{resolution proof}. Specifiy all resolution steps and the term substitutions they involve. (9 P) + +\end{enumerate} + +\section*{Problem 5} + + +\begin{enumerate}[label=\alph*)] + \item Explain the difference between \textit{terminological} and \textit{assertional} knowledge. Give an example for each. (4P) + + \item Model the following definitions using a \textit{description logic} (e.g. SHIQ): (4 P) + + \begin{enumerate}[label=(\arabic*)] + \item A Student is a person that is enrolled in a university. + \item An interested student is a student, who is interested in all lectures. + \end{enumerate} + + \item Consider the following \textit{Bayesian network} with nodes \textit{A, B, C, D, E,} and probabilites as specified in the tables. What is the probability of E, if it is known that $\neg B$ and $C$. (4 P) + + \vspace{1cm} + \begin{multicols}{2} + + \begin{center} + + \begin{tikzpicture}[thick, scale=2, every node/.style={scale=2}] + % Knoten + \node (A) at (0,2) [circle, draw] {A}; + \node (B) at (2,2) [circle, draw] {B}; + \node (C) at (1,1) [circle, draw] {C}; + \node (D) at (0,0) [circle, draw] {D}; + \node (E) at (2,0) [circle, draw] {E}; + + % Kanten + \draw[-{>[scale=4, + length=2, + width=3]},line width=0.4pt] (A) to (C); + \draw[-{>[scale=4, + length=2, + width=3]},line width=0.4pt] (B) edge (C); + \draw[-{>[scale=4, + length=2, + width=3]},line width=0.4pt] (C) edge (E); + \draw[-{>[scale=4, + length=2, + width=3]},line width=0.4pt] (C) edge (D); + + \end{tikzpicture} + \end{center} + + \begin{multicols}{2} + \begin{tabular}{|l|} + \hline + P(A) \\ + \hline + 0,4 \\ + \hline + \end{tabular} + + \begin{tabular}{|l|} + \hline + P(B) \\ + \hline + 0,5 \\ + \hline + \end{tabular} + \end{multicols} + + + \begin{tabular}{|l|l|l|} + \hline + A & B & P(C) \\ + \hline + 1 & 1 & 0,9 \\ + \hline + 1 & 0 & 0,3 \\ + \hline + 0 & 1 & 0,6 \\ + \hline + 0 & 0 & 0,2 \\ + \hline + \end{tabular} + + \begin{multicols}{2} + \begin{tabular}{|l|l|} + \hline + C & P(D) \\ + \hline + 1 & 0,8 \\ + \hline + 0 & 0,1 \\ + \hline + \end{tabular} + + + \begin{tabular}{|l|l|} + \hline + C & P(D) \\ + \hline + 1 & 0,8 \\ + \hline + 0 & 0,1 \\ + \hline + \end{tabular} + \end{multicols} + + \end{multicols} + + \item Explain what it means that the event $X$ is \textit{conditionally independent} of event $Y$ given event $Z$. Use your definition to check in the Bayesian network above, whether or not $D$ is conditionally independent of $E$ given $C$. (3 P) + +\end{enumerate} + +\section*{Problem 6} +Users of an online dating platform are shown profile pictures from other users. Based on the picture, the user decides on whether or not to signal interest. To improve the service, prior classification results of the user are evaluated with the goal of identifying pictures that are likely to raise interest. + +For user Alice the interaction history is listed below: + +\begin{center} + \begin{tabular}{|l|l|l|c|} + \hline + \textbf{Alter} & \textbf{Figur} & \textbf{Haarfarbe} & \\ + \hline + alt & sportlich & blond & $\heartsuit$ \\ + \hline + jung & unsportlich & brünett & $\times$ \\ + \hline + alt & sportlich & brünett & $\heartsuit$ \\ + \hline + jung & sportlich & brünett & $\times$ \\ + \hline + jung & unsportlich & blond & $\times$ \\ + \hline + alt & sportlich & brünett & $\heartsuit$ \\ + \hline + jung & sportlich & blond & $\times$ \\ + \hline + jung & unsportlich & brünett & $\times$ \\ + \hline + alt & unsportlich & brünett & $\heartsuit$ \\ + \hline + alt & unsportlich & blond & $\heartsuit$ \\ + \hline + \end{tabular} +\end{center} + +\begin{enumerate}[label=\alph*)] + \item Apply the \textit{version space algorithm} to the first \textbf{four} examples and determine the set of most general consistent hypothesis and the set of most specific consistent hypothesis. (8 P) + \item What is \textit{information gain} used for in decision tree learning? Make an educated guess abour which of the three attributes produces the highest information gain and which the lowest. Underpin your argument with a formula -- You do \textbf{not} need to numerically evaluate the formula. (5 P) + \item Contrast \textit{feed-forward} and \textit{recurrent} neural networks. (2 P) +\end{enumerate} + +\end{document} \ No newline at end of file