release symbenc

compile

@@ -102,7 +102,7 @@ compile_wrapper() {
 
 compile_chapter() {
   #for chapter in smtzthree proplogic satsolver ndpred predlogic ndpred smt bdd eqchecking symbenc temporal
-  for chapter in  smt
+  for chapter in  symbenc
   do
     set_chapter "\\chapter${chapter}true"
     compile_wrapper "$chapter" "$@"

symbolic_encoding/0001.tex

@@ -0,0 +1,6 @@
+\item \lect Draw the graph for the \emph{transition system} $\mathcal{T}$ with:
+\begin{itemize}[itemsep=-0.9em, leftmargin=0.8em]
+\item $S = \{s_1, s_2, s_3, s_4\}, $\\
+\item $S_0 = \{s_2\}, $\\
+\item $R = \{\{s_1, s_2\}, \{s_1, s_1\}, \{s_2, s_4\}, \{s_2, s_3\}, \{s_3, s_1\}, \{s_4, s_2\}, \{s_4, s_3\}\}, $
+\end{itemize}

symbolic_encoding/0001_sol.tex

@@ -0,0 +1,29 @@
+\begin{center}
+\begin{tikzpicture}[scale=0.2]
+\tikzstyle{every node}+=[inner sep=0pt]
+\draw [black] (11,-26.9) circle (3);
+\draw (11,-26.9) node {$s_2$};
+\draw [black] (23.6,-15) circle (3);
+\draw (23.6,-15) node {$s_1$};
+\draw [black] (23.6,-38) circle (3);
+\draw (23.6,-38) node {$s_4$};
+\draw [black] (36.7,-26.9) circle (3);
+\draw (36.7,-26.9) node {$s_3$};
+\draw [black] (21.42,-17.06) -- (13.18,-24.84);
+\fill [black] (13.18,-24.84) -- (14.11,-24.65) -- (13.42,-23.93);
+\draw [black] (3.7,-26.9) -- (8,-26.9);
+\fill [black] (8,-26.9) -- (7.2,-26.4) -- (7.2,-27.4);
+\draw [black] (25.89,-36.06) -- (34.41,-28.84);
+\fill [black] (34.41,-28.84) -- (33.48,-28.98) -- (34.12,-29.74);
+\draw [black] (34.48,-24.88) -- (25.82,-17.02);
+\fill [black] (25.82,-17.02) -- (26.08,-17.93) -- (26.75,-17.18);
+\draw [black] (14,-26.9) -- (33.7,-26.9);
+\fill [black] (33.7,-26.9) -- (32.9,-26.4) -- (32.9,-27.4);
+\draw [black] (21.35,-36.02) -- (13.25,-28.88);
+\fill [black] (13.25,-28.88) -- (13.52,-29.79) -- (14.18,-29.04);
+\draw [black] (13.25,-28.88) -- (21.35,-36.02);
+\fill [black] (21.35,-36.02) -- (21.08,-35.11) -- (20.42,-35.86);
+\draw [black] (25.456,-12.658) arc (169.34618:-118.65382:2.25);
+\fill [black] (26.59,-15.05) -- (27.28,-15.69) -- (27.47,-14.7);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/0002.tex

@@ -0,0 +1,38 @@
+
+\item \lect 
+Consider the example of an elevator.
+Initially, the elevator is in the ground floor. 
+From the ground floor, it can either go basement, stay there for a while,
+and then go back to the ground floor, or it can go from the ground floor 
+to the second floor, stay there for a while, and go back to the ground floor.
+While traveling between ground floor to second floor, the elevator passes the first floor, but it cannot stop there.
+
+Model this elevator as \emph{transition system}.
+
+
+
+
+\iffalse
+\item \lect 
+  Given the following graph representation of a \textit{transition system}, state the contents of the sets $S, S_0 and R$
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=20pt}]
+    \node[node] (s0) {$s0$};
+    \node (s0s2) [below= 1cm of s0] {};
+    \node[node] (s1) [right of=s0s2] {$s1$};
+    \node[node] (s2) [below of=s0]   {$s2$};
+    \node (start1) [left=1cm of s0] {};
+    
+    \path[->] (start1) edge (s0);
+    \path[->] (s1) edge (s0);
+    \path[->] (s0) edge [bend left] (s2);
+    \path[->] (s2) edge (s1);
+    \path[->] (s2) edge [bend left] (s0);
+    \path[->] (s0.30) edge[bend right=90, looseness=15, out=240, in=300] (s0.60);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+\end{tikzpicture}
+\end{center}
+\fi

symbolic_encoding/0002_sol.tex

@@ -0,0 +1,44 @@
+We use the following states:
+%ground floor, basement,second floor, passing ground floor
+\begin{itemize}
+\item $s_g$ indicates that the elevator is on the ground floor.
+\item $s_b$ indicates that the elevator is in the basement.
+\item $s_s$ indicates that the elevator is on the second floor.
+\item $s_f$ indicates that the elevator is passing the first floor.
+\end{itemize}
+The transition system is then given by: $\mathcal{T}$  = ($S$, $S_0$, $R$) with $S$ = $\{s_g, s_b, s_s, s_f\}$, $S_0$ = $\{s_g\}$, $R$ = $\{(s_g, s_g), (s_g, s_b), (s_b, s_b), (s_b, s_g), (s_g, s_f), (s_f, s_s), (s_s, s_s), (s_s, s_f), (s_f, s_g)\}$\\
+
+
+\begin{center}
+\begin{tikzpicture}[scale=0.2]
+\tikzstyle{every node}+=[inner sep=0pt]
+\draw [black] (13.9,-26) circle (3);
+\draw (13.9,-26) node {$s_g$};
+\draw [black] (27.8,-35.6) circle (3);
+\draw (27.8,-35.6) node {$s_b$};
+\draw [black] (27.8,-26) circle (3);
+\draw (27.8,-26) node {$s_f$};
+\draw [black] (27.8,-15.8) circle (3);
+\draw (27.8,-15.8) node {$s_s$};
+\draw [black] (4.8,-26) -- (10.9,-26);
+\fill [black] (10.9,-26) -- (10.1,-25.5) -- (10.1,-26.5);
+\draw [black] (11.501,-24.218) arc (261.13116:-26.86884:2.25);
+\fill [black] (13.85,-23.01) -- (14.47,-22.3) -- (13.48,-22.14);
+\draw [black] (16.37,-27.7) -- (25.33,-33.9);
+\fill [black] (25.33,-33.9) -- (24.96,-33.03) -- (24.39,-33.85);
+\draw [black] (26.957,-38.467) arc (11.34364:-276.65636:2.25);
+\fill [black] (25.01,-36.67) -- (24.13,-36.34) -- (24.33,-37.32);
+\draw [black] (25.33,-33.9) -- (16.37,-27.7);
+\fill [black] (16.37,-27.7) -- (16.74,-28.57) -- (17.31,-27.75);
+\draw [black] (16.9,-26) -- (24.8,-26);
+\fill [black] (24.8,-26) -- (24,-25.5) -- (24,-26.5);
+\draw [black] (27.8,-23) -- (27.8,-18.8);
+\fill [black] (27.8,-18.8) -- (27.3,-19.6) -- (28.3,-19.6);
+\draw [black] (28.483,-12.891) arc (194.52895:-93.47105:2.25);
+\fill [black] (30.52,-14.57) -- (31.42,-14.86) -- (31.17,-13.89);
+\draw [black] (27.8,-18.8) -- (27.8,-23);
+\fill [black] (27.8,-23) -- (28.3,-22.2) -- (27.3,-22.2);
+\draw [black] (24.8,-26) -- (16.9,-26);
+\fill [black] (16.9,-26) -- (17.7,-26.5) -- (17.7,-25.5);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/0003.tex

@@ -0,0 +1 @@
+\item \lect Given a state space of size $|S| = 2^4 = 16$, give the symbolic encoding for the following states: (a) $s_{7}$, (b) $s_{15}$, and (c) $s_{10}$.

symbolic_encoding/0003_sol.tex

@@ -0,0 +1,6 @@
+For the symbolic encoding we need 4 Boolean variables, \{$v_3$, . . . , $v_0$\}.
+Let $v_3$ be the most significant bit, and $v_0$ the least significant bit.\\
+
+(a) $s_{7}$ = $\lnot v_3 \land v_2 \land v_1 \land v_0$\\
+(b) $s_{15}$ = $v_3 \land v_2 \land v_1 \land v_0$\\
+(c) $s_{10}$ = $v_3 \land \lnot v_2 \land v_1 \land \lnot v_0$

symbolic_encoding/0004.tex

@@ -0,0 +1,2 @@
+\item \lect Given is the  set of states $S=\{s_0,\dots,s_7\}$.
+Find formulas in propositional logic that symbolically represent the sets $A = \{s_7, s_6,s_3, s_2 \}$, $B=\{s_1, s_3,s_5, s_7\}$, and $C=\{s_7, s_6,s_0, s_1\}$.

symbolic_encoding/0004_sol.tex

@@ -0,0 +1,6 @@
+$A = \{s_7, s_6,s_3, s_2 \}$ = $(v_2 \land v_1 \land v_0) \lor (v_2 \land v_1 \land \lnot v_0) \lor (\lnot v_2 \land v_1 \land v_0) \lor (\lnot v_2 \land v_1 \land \lnot v_0)$\\ 
+\hspace*{0,34cm} = $v_1$\\\\
+$B=\{s_1, s_3,s_5, s_7\}$ = $(\lnot v_2 \land \lnot v_1 \land v_0) \lor (\lnot v_2 \land v_1 \land v_0) \lor (v_2 \land \lnot v_1 \land v_0) \lor (v_2 \land v_1 \land v_0)$\\ 
+\hspace*{0,34cm} = $v_0$\\\\
+$C=\{s_7, s_6,s_0, s_1\}$ = $(v_2 \land v_1 \land v_0) \lor (v_2 \land v_1 \land \lnot v_0) \lor (\lnot v_2 \land \lnot v_1 \land \lnot v_0) \lor (\lnot v_2 \land \lnot v_1 \land v_0)$ \\ 
+\hspace*{0,34cm} = $(v_2 \land v_1) \lor (\lnot v_2 \land \lnot v_1)$

symbolic_encoding/0005.tex

@@ -0,0 +1,24 @@
+\item \lect Find a \textit{symbolic encoding} for the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v_1 \land \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    %\path[<->] (s0) edge (s1);
+    \path[->] (s0) edge (s2);
+    \path[<-] (s0) edge (s3);
+    %\path[<->] (s1) edge (s2);
+    \path[<->] (s1) edge (s3);
+    \path[<-] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    %\path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    %\path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+   % \path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/0005_sol.tex

@@ -0,0 +1,13 @@
+Using the variables $v_1$ and $v_0$, we can define the transition relation using the following formula:\\
+\begin{center}
+    $\lnot v_1 \land \lnot v_0 \land (\lnot v'_1 \land  \lnot v'_0 \lor v'_1 \land \lnot v'_0) \ \lor$\\
+    $\lnot v_1 \land v_0 \land v'_1 \land v'_0 \ \lor$\\
+    $v_1 \land v_0 \land (\lnot v'_1 \land v'_0  \lor \lnot v'_1 \land \lnot v'_0 \lor v'_1 \land \lnot v'_0)$\\
+\end{center}
+
+We can further simplify the formula to:
+\begin{center}
+    $\lnot v_1 \land \lnot v_0 \land \lnot v'_0 \ \lor$\\
+    $\lnot v_1 \land v_0 \land v'_1 \land v'_0 \ \lor$\\
+    $v_1 \land v_0 \land (\lnot v'_1 \land v'_0  \lor  \lnot v'_0)$\\
+\end{center}

symbolic_encoding/0006.tex

@@ -0,0 +1,24 @@
+\item \lect Find a \textit{symbolic encoding} for the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v_1 \land \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    \path[->] (s0) edge (s1);
+    \path[<->] (s0) edge (s2);
+    \path[<->] (s0) edge (s3);
+    \path[<->] (s1) edge (s2);
+    \path[<->] (s1) edge (s3);
+    \path[<->] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    %\path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+    \path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/0006_sol.tex

@@ -0,0 +1,15 @@
+Using the variables $v_1$ and $v_0$, we can define the transition relation using the following formula:\\
+\begin{center}
+    $\lnot v_1 \land \lnot v_0 \land (\lnot v'_1 \land \lnot v'_0 \lor \lnot v'_1 \land v'_0 \lor v'_1 \land \lnot v'_0 \lor v'_1 \land v'_0) \ \lor$\\
+    $\lnot v_1 \land v_0 \land (\lnot v'_1 \land v'_0 \lor v'_1 \land \lnot v'_0 \lor v'_1 \land v'_0) \ \lor$\\
+    $v_1 \land \lnot v_0 \land (\lnot v'_1 \land \lnot v'_0 \lor \lnot v'_1 \land v'_0 \lor v'_1 \land v'_0) \ \lor$\\
+    $v_1 \land v_0 \land (\lnot v'_1 \land \lnot v'_0 \lor \lnot v'_1 \land v'_0 \lor v'_1 \land \lnot v'_0 \lor v'_1 \land v'_0)$
+\end{center}
+
+We can further simplify the formula to:
+\begin{center}
+    $\lnot v_1 \land \lnot v_0 \lor$\\ 
+    $\lnot v_1 \land v_0 \land (v'_0 \lor v'_1 \land \lnot v'_0) \ \lor$\\
+    $v_1 \land \lnot v_0 \land (\lnot v'_1 \lor v'_1 \land v'_0) \ \lor$\\
+    $v_1 \land v_0$
+\end{center}

symbolic_encoding/0007.tex

@@ -0,0 +1,39 @@
+
+\item \lect
+Consider the domain $A=\{Spain, France, Italy, Germany\}$ and
+    the two different symbolic encodings for $A$ given below.
+    Which one gives a shorter symbolic representation for the set
+    $B=\{France, Germany\}$? Illustrate your answer by giving the
+    representing formulas for $B$ in both encodings.
+
+\vspace{.5cm}
+
+%  \begin{minipage}
+    \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 1}} \\
+    \hline
+    Element & $v_1$ & $v_0$ \\
+    \hline
+    Spain & $0$ & $0$ \\
+    France & $1$ & $0$ \\
+    Italy & $0$ & $1$ \\
+    Germany & $1$ & $1$
+   \end{tabular}
+%\end{minipage}
+%\begin{minipage}
+\hspace{3cm}
+   \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 2}} \\
+    \hline
+    Element & $v_1$ & $v_0$ \\
+    \hline
+    Spain & $0$ & $0$ \\
+    France & $1$ & $0$ \\
+    Italy & $1$ & $1$ \\
+    Germany & $0$ & $1$
+   \end{tabular}
+%\end{minipage}
+  
+  

symbolic_encoding/0007_sol.tex

@@ -0,0 +1,7 @@
+Using encoding 1, we end up in the following formula:\\
+$b = v_1$\\\\
+Using encoding 2, we end up in the following formula:\\
+$b = (v_1 \land \lnot v_0) \lor (\lnot v_1 \land v_0)$\\
+
+Encoding 1 gives a shorter symbolic representation
+for the set $B=\{France, Germany\}$.

symbolic_encoding/0008.tex

@@ -0,0 +1,9 @@
+\item  \lect 
+Find a symbolic binary encoding for $X = \{ 0,1, \ldots, 31 \}$.
+  Use it to find formulas that symbolically represent the sets $A$ and $B$ and simplify the formulas:
+  \begin{itemize}
+    \item $A =\{ 12, 13, 14, 15, 28, 29, 30, 31 \}$
+    \item $B =\{x \in X \mid 0 \leq x \leq 15\}$
+  \end{itemize}
+  Furthermore, give the formulas representing the sets
+  $C=A\cap B$ and $D = A \cup B$.

symbolic_encoding/0008_sol.tex

@@ -0,0 +1,4 @@
+We use 5 Boolean variables, \{$v_4$, . . . , $v_0$\}, for the encoding.\\
+
+$A = (v_2 \land v_3)$\\
+$B = \lnot v_4$

symbolic_encoding/1000.tex

@@ -0,0 +1,17 @@
+\item \self 
+Listed are the participants of a seminar as well as their choice of snacks. Find a symbolic encodings for the participants.
+For for this encoding, give the symbolic representation of the set $B$ of all participants that ordered \emph{bananas}, and the set $C$ of all participants that ordered cake.
+
+  \begin{tabular}{l|l}
+    Name  & Snack    \\
+    \hline
+    Alice & banana     \\
+    Bob   & cake     \\
+    Carl  & banana     \\
+    David & banana     \\
+    Eve   & cake     \\
+    Frank & cake     \\
+    Greg  & orange     \\
+    Hank  & cake     \\
+  \end{tabular}
+

symbolic_encoding/1001.tex

@@ -0,0 +1,8 @@
+\item \self
+
+Consider the example of a controller for a lamp.
+
+Initially the light is off. Pressing the button once turns on the light and the light glows white. From this state, any short-lasting pressure of the button causes the light to switch its color randomly between white, red, green, blue, and yellow.
+At any state, pressing the button for a longer time turns the light off.
+
+Model the lamp controller as \emph{transition system}.

symbolic_encoding/1002.tex

@@ -0,0 +1 @@
+\item \self Given a state space of size $|S| = 2^4 = 16$. Give the symbolic encoding for the following states: (a) $s_{4}$, (b) $s_{9}$, and (c) $s_{13}$.

symbolic_encoding/1003.tex

@@ -0,0 +1,2 @@
+\item \self Given is the  set of states $S=\{s_0,\dots,s_7\}$.
+Find formulas in propositional logic that symbolically represent the sets $A = \{s_0, s_2,s_4, s_6 \}$, $B=\{s_0, s_1,s_2, s_3\}$, and $C=\{s_7, s_1\}$.

symbolic_encoding/1004.tex

@@ -0,0 +1,25 @@
+\item \self
+  Find a \textit{symbolic encoding} for the set of initial states and the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v1 \wedge \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    %\path[->] (s0) edge (s1);
+    \path[->] (s0) edge (s2);
+    \path[->] (s0) edge (s3);
+    \path[->] (s1) edge (s2);
+    \path[->] (s1) edge (s3);
+    %\path[->] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    %\path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    %\path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+    %\path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/1005.tex

@@ -0,0 +1,25 @@
+\item \self
+  Find a \textit{symbolic encoding} for the set of initial states and the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v1 \wedge \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    \path[<->] (s0) edge (s1);
+    \path[<->] (s0) edge (s2);
+    \path[<->] (s0) edge (s3);
+    \path[<->] (s1) edge (s2);
+    \path[<->] (s1) edge (s3);
+    %\path[<->] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    \path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+    \path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/1006.tex

@@ -0,0 +1,25 @@
+\item \self
+Find a \textit{symbolic encoding} for the set of initial states and the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v1 \wedge \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    \path[<->] (s0) edge (s2);
+    %\path[<->] (s0) edge (s2);
+    %\path[<->] (s0) edge (s3);
+    %\path[<->] (s1) edge (s2);
+    \path[<->] (s1) edge (s3);
+    %\path[<->] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    \path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+    \path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/1007.tex

@@ -0,0 +1,7 @@
+\item \self Define the \textit{transition system} from the following symbolically encoded transition relations and draw the
+  corresponding graph:
+  \begin{align*}
+    (v_1 \land v_0 \land \lnot v'_1 \land \neg v'_0)  & \enspace \lor  \\
+    (\neg v_1 \land v_0 \land \lnot v'_1 \land  v'_0) & \enspace \lor  \\
+     (v_1 \land v_0 \land  v'_1 \land v'_0)    &
+  \end{align*}

symbolic_encoding/1008.tex

@@ -0,0 +1,7 @@
+\item \self Define the \textit{transition system} from the following symbolically encoded transition relations and draw the
+  corresponding graph:
+  \begin{align*}
+    (\lnot v_1 \land \lnot v_0 \land  v'_1 \land v'_0)  & \enspace \lor  \\
+    (\neg v_1 \land v_0 \land \lnot v'_1 \land  \lnot v'_0) & \enspace \lor  \\
+     (\lnot v_1 \land \lnot v_0 \land  \lnot v'_1 \land \lnot v'_0)    &
+  \end{align*}

symbolic_encoding/1009.tex

@@ -0,0 +1,3 @@
+\item \self What is the main advantage of symbolic set operations over
+    non-symbolic operations that enumerate all set elements
+    explicitly?

symbolic_encoding/1010.tex

@@ -0,0 +1,17 @@
+\item \self 
+Listed are the participants of a seminar as well as their choice of snacks. Find a symbolic encodings for the participants.
+For for this encoding, give the symbolic representation of the set $B$ of all participants that ordered \emph{bananas}, and the set $C$ of all participants that ordered cake.
+
+  \begin{tabular}{l|l}
+    Name  & Snack    \\
+    \hline
+    Alice & banana     \\
+    Bob   & cake     \\
+    Carl  & banana     \\
+    David & banana     \\
+    Eve   & cake     \\
+    Frank & cake     \\
+    Greg  & orange     \\
+    Hank  & cake     \\
+  \end{tabular}
+

symbolic_encoding/1011.tex

@@ -0,0 +1 @@
+\item \self Given a state space of size $|S| = 2048$, find a symbolic binary encoding for this state space and compute the characteristic function for the sets of states $$B = \{s_0, s_1, s_2, ..., s_{1023}\} \enspace \text{and} \enspace C = \{s_{512}, s_{513},, s_{514}, ..., s_{1535}.\}$$ Then compute the characteristic function for the sets $D = B \cup C$ and $E = B \setminus C$. If possible, simplify the formulas.

symbolic_encoding/1012.tex

@@ -0,0 +1,15 @@
+\item \self The following table shows eight students and their means of transportation. Find a symbolic encodings representing the list of students.
+For this encoding, give the symbolic representation of the set $B$ of all students that go by \emph{bike}, and the set $C$ of all students that go by \emph{car}.
+
+  \begin{tabular}{l|l}
+    Name  & Transportation    \\
+    \hline
+    Alice & Car     \\
+    Bob   & Bike     \\
+    Carl  & Tram     \\
+    David & Bike     \\
+    Eve   & Tram     \\
+    Frank & Bike     \\
+    Greg  & Tram     \\
+    Hank  & Bike     \\
+  \end{tabular}

symbolic_encoding/1013.tex

@@ -0,0 +1,36 @@
+\item \self 
+Consider the domain $A=\{Spain, France, Italy, Germany\}$ and
+    the two different symbolic encodings for $A$ given below.
+    Which one gives a shorter symbolic representation for the set
+    $B=\{France, Italy\}$? Illustrate your answer by giving the
+    representing formulas for $B$ in both encodings.
+
+\vspace{.5cm}
+
+%  \begin{minipage}
+    \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 1}} \\
+    \hline
+    Element & $v_1$ & $v_0$ \\
+    \hline
+    Spain & $0$ & $0$ \\
+    France & $1$ & $0$ \\
+    Italy & $0$ & $1$ \\
+    Germany & $1$ & $1$
+   \end{tabular}
+%\end{minipage}
+%\begin{minipage}
+\hspace{3cm}
+   \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 2}} \\
+    \hline
+    Element & $v_1$ & $v_0$ \\
+    \hline
+    Spain & $0$ & $0$ \\
+    France & $1$ & $0$ \\
+    Italy & $1$ & $1$ \\
+    Germany & $0$ & $1$
+   \end{tabular}
+%\end{minipage}

symbolic_encoding/1014.tex

@@ -0,0 +1,41 @@
+\item \self Consider the following set operations and relations between
+    two sets $X$ and $Y$, and an element $a$:
+\begin{enumerate}
+\item Union: $X \cup Y$
+\item Intersection: $X \cap Y$
+\item Set Difference: $X \setminus Y$
+\item Containment: $a \in X$?
+\item Subset: $X \subseteq Y$?
+\item Strict Subset: $X \subset Y$?
+\item Emptiness: $X=\emptyset$?
+\item Equality: $X=Y$?
+\end{enumerate}
+Let $x$ and $y$ be the symbolic representations of $X$ and $Y$
+respectively, and let $\alpha$ be the symbolic encoding of element
+$a$. For each of the following items, state which of the above
+operations is performed, or which of the above questions is answered.
+Write the letters of the corresponding operation/question into the
+boxes of the items below. Note that some of the items below do not
+perform any of the above operations or answer any of the above
+questions. Put a ``--'' in the box of these items. Also note that
+some of the items below might do the same computation or answer the
+same question.
+\begin{itemize}
+\item[\Huge{$\square$}] $\neg x \vee y$
+\item[\Huge{$\square$}] $x \wedge y$
+\item[\Huge{$\square$}] $x\equiv \top$?
+\item[\Huge{$\square$}] $x\equiv y$?
+\item[\Huge{$\square$}] $(x \rightarrow y) \wedge (y \rightarrow
+    x)$?
+\item[\Huge{$\square$}] $x\equiv \bot$?
+\item[\Huge{$\square$}] $y \wedge \neg x$
+\item[\Huge{$\square$}] $x \rightarrow \bot$?
+\item[\Huge{$\square$}] $\alpha \models x$?
+\item[\Huge{$\square$}] $\alpha \models \neg x$?
+\item[\Huge{$\square$}] $\neg \alpha \models x$?
+\item[\Huge{$\square$}] $x \rightarrow \alpha$?
+\item[\Huge{$\square$}] $y \rightarrow x$?
+\item[\Huge{$\square$}] $x \rightarrow y$?
+\item[\Huge{$\square$}] $(x \rightarrow y) \wedge (x\not \equiv
+    y)$?
+\end{itemize}

symbolic_encoding/1015.tex

@@ -0,0 +1,14 @@
+\item \self Find a symbolic binary encoding for
+$X = \{ 0,1, \ldots, 31 \}$.
+Use it to compute formulas in propositional logic that symbolically represent the following sets. 
+
+  \begin{itemize}
+    \item $B =\{4, 5, 12, 13, 20, 21, 28, 29 \}$
+    \item $C =\{1, 2, 13, 14 \}$
+  \end{itemize}
+  
+  Compute the characteristic functions of the following sets by symbolic operations, using your results from before.
+  \begin{enumerate}
+    \item $D = B \cup C$
+    \item $E = X \setminus D$
+  \end{enumerate}

symbolic_encoding/1016.tex

@@ -0,0 +1,15 @@
+\item \self Find a symbolic binary encoding for
+$X = \{ 0,1, \ldots, 31 \}$.
+Use it to compute formulas in propositional logic that symbolically represent the following sets. 
+\begin{itemize}
+  \item $B =\{x \in X \mid \text{x is even}\}$
+  \item $C =\{x \in X \mid \text{x is odd}\}$
+  \item $D =\{0,1,2,3,4,5,6,7\}$
+\end{itemize}
+
+Compute the characteristic functions of the following sets by symbolic operations, using your results from before.
+\begin{enumerate}
+  \item $E = B \cup D$
+  \item $F = C \cap E$
+  \item $G = E \setminus F$
+\end{enumerate}

symbolic_encoding/1017.tex

@@ -0,0 +1,15 @@
+\item \self Find a symbolic binary encoding for
+$X = \{ 0,1, \ldots, 31 \}$.
+Use it to compute formulas in propositional logic that symbolically represent the following sets. 
+
+\begin{itemize}
+  \item $B =\{8, 9, 10, 11, 12, 13, 14, 15\}$
+  \item $C =\{x \in X \mid 0 \leq x \leq 15\}$
+\end{itemize}
+
+Compute the characteristic functions of the following sets by symbolic operations, using your results from before.
+\begin{enumerate}
+  \item $D = B \cup C$
+  \item $E = B \cap C$
+  \item $F = C \setminus B$
+\end{enumerate}

symbolic_encoding/1018.tex

@@ -0,0 +1,4 @@
+\item \self Assume you are given the formulas $a$ and
+    $b$, which symbolically represent the sets $A$ and $B$,
+    respectively. Give the formula
+    $c$, which symbolically represents the set $C= A \setminus B$.

symbolic_encoding/1019.tex

@@ -0,0 +1,5 @@
+\item \self Assume you are given the formulas $a$ and
+    $b$, which symbolically represent the sets $A$ and $B$,
+    respectively. What would you have to check on $a, b$ to test
+    whether or not $A$ is a strict subset of $B$, i.e., $A \subset
+    B$?

symbolic_encoding/1020.tex

@@ -0,0 +1,3 @@
+\item \self Given a state space of size $|S| = 64$. Find a symbolic binary encoding for this state space and compute the formulas that symbolically represent the sets  $$B = \{s_{32}, s_{33}, s_{34}, ..., s_{63}\} \enspace \text{and} \enspace C = \{s_{16}, s_{17},, s_{18}, ..., s_{40}\}.$$
+
+Following, compute the formulas that represent the sets $D = B \cap C$, $E = B \cup C$, $F = B \setminus C$ and $G = C \setminus B$.

symbolic_encoding/1021.tex

@@ -0,0 +1,2 @@
+\item \self 
+  Given a state space of size $|S| = 64$, find a symbolic binary encoding for this state space and compute the formulas that symbolically represent the sets of states $$B = \{s_{16}, s_{17}, s_{18}, ..., s_{32}\} \enspace \text{and} \enspace C = \{s_{24}, s_{25},, s_{26}, ..., s_{64}.\}$$ Then compute the formulas that symbolically represent the sets $D = B \cap C$ and $E = B \cup C$. 

symbolic_encoding/multiple_choice/2_1_symbrep_lect.tex

@@ -0,0 +1,8 @@
+\item \lect Given a state space of the size $|S| = 2^5 = 32$ and assuming $v_4$ is the most significant bit and $v_0$ is the least significant bit, which of the following symbolic representations or the given states is correct?
+
+\begin{enumerate}
+    \item[$\square$] State $s_{15}$ is represented by: $\lnot v_4 \land v_3 \land v_3, \land v_1 \land \lnot v_0$
+    \item[$\square$] State $s_{31}$ is represented by: $v_4 \land v_3 \land v_3 \land v_1 \land v_0$
+    \item[$\square$] State $s_{4}$ is represented by: $\lnot v_4 \land \lnot v_3 \land \lnot v_2, \land v_1 \land \lnot v_0$
+    \item[$\square$] State $s_{10}$ is represented by: $v_4 \land v_3 \land \lnot v_3 \land \lnot v_1 \land v_0$
+\end{enumerate}

symbolic_encoding/multiple_choice/2_1_symbrep_self.tex

@@ -0,0 +1,8 @@
+\item \self Given a state space of the size $|S| = 2^9 = 512$ and assuming $v_8$ is the most significant bit and $v_0$ is the least significant bit, which of the following symbolic representations or the given states is correct?
+
+\begin{enumerate}
+    \item[$\square$] State $s_{214}$ is represented by: $\lnot v_8 \land v_7 \land v_6 \land \lnot v_5 \land v_4 \land \lnot v_3 \land v_2 \land v_1 \land \lnot v_0$
+    \item[$\square$] State $s_{501}$ is represented by: $v_8 \land v_7 \land v_6 \land v_5 \land v_4 \land \lnot v_3 \land v_2 \land \lnot v_1 \land v_0$
+    \item[$\square$] State $s_{0}$ is represented by: $\lnot v_8 \land \lnot v_7 \land \lnot v_6 \land \lnot v_5 \land \lnot v_4 \land \lnot v_3 \land \lnot v_2 \land \lnot v_1 \land \lnot v_0$
+    \item[$\square$] State $s_{448}$ is represented by: $v_8 \land v_7 \land v_6 \land \lnot v_5 \land \lnot v_4 \land \lnot v_3 \land \lnot v_2 \land  v_1 \land \lnot v_0$
+\end{enumerate}

symbolic_encoding/practical_questions/1_10_kripke_self.tex

@@ -0,0 +1,8 @@
+\item \self
+
+Consider the example of a controller for a lamp.
+
+Initially the light is off. Pressing the button once turns on the light and the light glows white. From this state, any short-lasting pressure of the button causes the light to switch its color randomly between white, red, green, blue, and yellow.
+At any state, pressing the button for a longer time turns the light off.
+
+Model the lamp controller as \emph{transition system}.

symbolic_encoding/practical_questions/1_1_kripke_lect.tex

@@ -0,0 +1,4 @@
+\item \lect Draw the graph for a \emph{transition system} $\mathcal{T}$ with: 
+$S = \{s_1, s_2, s_3, s_4\}, $\\
+$S_0 = \{s_2\}, $\\
+$R = \{\{s_1, s_2\}, \{s_1, s_1\}, \{s_2, s_4\}, \{s_2, s_3\}, \{s_3, s_1\}, \{s_4, s_2\}, \{s_4, s_3\}\}, $

symbolic_encoding/practical_questions/1_1_kripke_lect_sol.tex

@@ -0,0 +1,29 @@
+\begin{center}
+\begin{tikzpicture}[scale=0.2]
+\tikzstyle{every node}+=[inner sep=0pt]
+\draw [black] (11,-26.9) circle (3);
+\draw (11,-26.9) node {$s_2$};
+\draw [black] (23.6,-15) circle (3);
+\draw (23.6,-15) node {$s_1$};
+\draw [black] (23.6,-38) circle (3);
+\draw (23.6,-38) node {$s_4$};
+\draw [black] (36.7,-26.9) circle (3);
+\draw (36.7,-26.9) node {$s_3$};
+\draw [black] (21.42,-17.06) -- (13.18,-24.84);
+\fill [black] (13.18,-24.84) -- (14.11,-24.65) -- (13.42,-23.93);
+\draw [black] (3.7,-26.9) -- (8,-26.9);
+\fill [black] (8,-26.9) -- (7.2,-26.4) -- (7.2,-27.4);
+\draw [black] (25.89,-36.06) -- (34.41,-28.84);
+\fill [black] (34.41,-28.84) -- (33.48,-28.98) -- (34.12,-29.74);
+\draw [black] (34.48,-24.88) -- (25.82,-17.02);
+\fill [black] (25.82,-17.02) -- (26.08,-17.93) -- (26.75,-17.18);
+\draw [black] (14,-26.9) -- (33.7,-26.9);
+\fill [black] (33.7,-26.9) -- (32.9,-26.4) -- (32.9,-27.4);
+\draw [black] (21.35,-36.02) -- (13.25,-28.88);
+\fill [black] (13.25,-28.88) -- (13.52,-29.79) -- (14.18,-29.04);
+\draw [black] (13.25,-28.88) -- (21.35,-36.02);
+\fill [black] (21.35,-36.02) -- (21.08,-35.11) -- (20.42,-35.86);
+\draw [black] (25.456,-12.658) arc (169.34618:-118.65382:2.25);
+\fill [black] (26.59,-15.05) -- (27.28,-15.69) -- (27.47,-14.7);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/practical_questions/1_1_kripke_self.tex

@@ -0,0 +1,5 @@
+\item \self Draw the graph for a \emph{transition system} $\mathcal{T}$ with:
+
+$S = \{s_0, s_1, s_2\}, $\\
+$S_0 = \{s_0, s_1\}, $\\
+$R = \{\{s_0, s_0\}, \{s_0, s_1\}, \{s_0, s_2\}, \{s_1, s_0\}, \{s_1, s_1\}, \{s_1, s_2\}, \{s_2, s_0\}, \{s_2, s_1\}, \{s_2, s_2\}\}$.

symbolic_encoding/practical_questions/1_1_symbrep.tex

@@ -0,0 +1 @@
+\item \self Given a state space of size $|S| = 2^4 = 16$. Give the symbolic encoding for the following states: (a) $s_{4}$, (b) $s_{9}$, and (c) $s_{13}$.

symbolic_encoding/practical_questions/1_1_symbrep_self.tex

@@ -0,0 +1 @@
+\item \self Given a state space of size $|S| = 2^4 = 16$. Give the symbolic encoding for the following states: (a) $s_{4}$, (b) $s_{9}$, and (c) $s_{13}$.

symbolic_encoding/practical_questions/1_2_kripke_lect.tex

@@ -0,0 +1,38 @@
+
+\item \lect 
+Consider the example of an elevator.
+Initially, the elevator is in the ground floor. 
+From the ground floor, it can either go basement, stay there for a while,
+and then go back to the ground floor, or it can go from the ground floor 
+to the second floor, stay there for a while, and go back to the ground floor.
+While traveling between ground floor to second floor, the elevator passes the first floor, but it cannot stop there.
+
+Model this elevator as \emph{transition system}.
+
+
+
+
+\iffalse
+\item \lect 
+  Given the following graph representation of a \textit{transition system}, state the contents of the sets $S, S_0 and R$
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=20pt}]
+    \node[node] (s0) {$s0$};
+    \node (s0s2) [below= 1cm of s0] {};
+    \node[node] (s1) [right of=s0s2] {$s1$};
+    \node[node] (s2) [below of=s0]   {$s2$};
+    \node (start1) [left=1cm of s0] {};
+    
+    \path[->] (start1) edge (s0);
+    \path[->] (s1) edge (s0);
+    \path[->] (s0) edge [bend left] (s2);
+    \path[->] (s2) edge (s1);
+    \path[->] (s2) edge [bend left] (s0);
+    \path[->] (s0.30) edge[bend right=90, looseness=15, out=240, in=300] (s0.60);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+\end{tikzpicture}
+\end{center}
+\fi

symbolic_encoding/practical_questions/1_2_kripke_lect_sol.tex

@@ -0,0 +1,44 @@
+We use the following states:
+%ground floor, basement,second floor, passing ground floor
+\begin{itemize}
+\item $s_g$ indicates that the elevator is on the ground floor.
+\item $s_b$ indicates that the elevator is in the basement.
+\item $s_s$ indicates that the elevator is on the second floor.
+\item $s_f$ indicates that the elevator is passing the first floor.
+\end{itemize}
+The transition system is then given by: $\mathcal{T}$  = ($S$, $S_0$, $R$) with $S$ = $\{s_g, s_b, s_s, s_f\}$, $S_0$ = $\{s_g\}$, $R$ = $\{(s_g, s_g), (s_g, s_b), (s_b, s_b), (s_b, s_g), (s_g, s_f), (s_f, s_s), (s_s, s_s), (s_s, s_f), (s_f, s_g)\}$\\
+
+
+\begin{center}
+\begin{tikzpicture}[scale=0.2]
+\tikzstyle{every node}+=[inner sep=0pt]
+\draw [black] (13.9,-26) circle (3);
+\draw (13.9,-26) node {$s_g$};
+\draw [black] (27.8,-35.6) circle (3);
+\draw (27.8,-35.6) node {$s_b$};
+\draw [black] (27.8,-26) circle (3);
+\draw (27.8,-26) node {$s_f$};
+\draw [black] (27.8,-15.8) circle (3);
+\draw (27.8,-15.8) node {$s_s$};
+\draw [black] (4.8,-26) -- (10.9,-26);
+\fill [black] (10.9,-26) -- (10.1,-25.5) -- (10.1,-26.5);
+\draw [black] (11.501,-24.218) arc (261.13116:-26.86884:2.25);
+\fill [black] (13.85,-23.01) -- (14.47,-22.3) -- (13.48,-22.14);
+\draw [black] (16.37,-27.7) -- (25.33,-33.9);
+\fill [black] (25.33,-33.9) -- (24.96,-33.03) -- (24.39,-33.85);
+\draw [black] (26.957,-38.467) arc (11.34364:-276.65636:2.25);
+\fill [black] (25.01,-36.67) -- (24.13,-36.34) -- (24.33,-37.32);
+\draw [black] (25.33,-33.9) -- (16.37,-27.7);
+\fill [black] (16.37,-27.7) -- (16.74,-28.57) -- (17.31,-27.75);
+\draw [black] (16.9,-26) -- (24.8,-26);
+\fill [black] (24.8,-26) -- (24,-25.5) -- (24,-26.5);
+\draw [black] (27.8,-23) -- (27.8,-18.8);
+\fill [black] (27.8,-18.8) -- (27.3,-19.6) -- (28.3,-19.6);
+\draw [black] (28.483,-12.891) arc (194.52895:-93.47105:2.25);
+\fill [black] (30.52,-14.57) -- (31.42,-14.86) -- (31.17,-13.89);
+\draw [black] (27.8,-18.8) -- (27.8,-23);
+\fill [black] (27.8,-23) -- (28.3,-22.2) -- (27.3,-22.2);
+\draw [black] (24.8,-26) -- (16.9,-26);
+\fill [black] (16.9,-26) -- (17.7,-26.5) -- (17.7,-25.5);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/practical_questions/1_2_kripke_self.tex

@@ -0,0 +1,25 @@
+\item \self
+  Find a \textit{symbolic encoding} for the set of initial states and the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v1 \wedge \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    %\path[->] (s0) edge (s1);
+    \path[->] (s0) edge (s2);
+    \path[->] (s0) edge (s3);
+    \path[->] (s1) edge (s2);
+    \path[->] (s1) edge (s3);
+    %\path[->] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    %\path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    %\path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+    %\path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/practical_questions/1_2_symbrep_self.tex

@@ -0,0 +1,2 @@
+\item \self Given is the  set of states $S=\{s_0,\dots,s_7\}$.
+Find formulas in propositional logic that symbolically represent the sets $A = \{s_0, s_2,s_4, s_6 \}$, $B=\{s_0, s_1,s_2, s_3\}$, and $C=\{s_7, s_1\}$.

symbolic_encoding/practical_questions/1_3_kripke_self.tex

@@ -0,0 +1,25 @@
+\item \self
+  Find a \textit{symbolic encoding} for the set of initial states and the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v1 \wedge \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    \path[<->] (s0) edge (s1);
+    \path[<->] (s0) edge (s2);
+    \path[<->] (s0) edge (s3);
+    \path[<->] (s1) edge (s2);
+    \path[<->] (s1) edge (s3);
+    %\path[<->] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    \path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+    \path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/practical_questions/1_4_kripke_self.tex

@@ -0,0 +1,25 @@
+\item \self
+Find a \textit{symbolic encoding} for the set of initial states and the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v1 \wedge \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    \path[<->] (s0) edge (s2);
+    %\path[<->] (s0) edge (s2);
+    %\path[<->] (s0) edge (s3);
+    %\path[<->] (s1) edge (s2);
+    \path[<->] (s1) edge (s3);
+    %\path[<->] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    \path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+    \path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/practical_questions/2_05_symbrep_self.tex

@@ -0,0 +1,32 @@
+\item \self Let $x$ and $y$ be the symbolic representations of the sets $X$ and
+    $Y$ respectively, and let $\alpha$ be the symbolic encoding of an
+    element $a$. Consider the following operations and relations
+    between $x$, $y$, and $\alpha$:
+
+\begin{enumerate}[A.]
+\item $x \rightarrow y$
+\item $\alpha \models x$ ?
+\item $x \wedge \neg y$
+\item $\alpha \nvDash y$ ?
+\item $x \equiv \bot$ ?
+\item $x \vee y$
+\end{enumerate}
+
+For each of the following items, state which of the above operations
+symbolically performs the respective set operation or answers the
+respective set-specific question. Write the letters of the
+corresponding operation/question into the boxes of the items below.
+Note that some of the items below do not correspond to any of the
+above operations or questions. Put a ``--'' in the box of these
+items.
+
+\begin{itemize}
+\item[\Huge{$\square$}] Union: $X \cup Y$
+\item[\Huge{$\square$}] Intersection: $X \cap Y$
+\item[\Huge{$\square$}] Set Difference: $X \setminus Y$
+\item[\Huge{$\square$}] Containment: $a \in X$?
+\item[\Huge{$\square$}] Subset: $X \subseteq Y$?
+\item[\Huge{$\square$}] Strict Subset: $X \subset Y$?
+\item[\Huge{$\square$}] Emptiness: $X=\emptyset$?
+\item[\Huge{$\square$}] Equality: $X=Y$?
+\end{itemize}

symbolic_encoding/practical_questions/2_10_symbrep_lect.tex

@@ -0,0 +1,39 @@
+
+\item \lect
+Consider the domain $A=\{Spain, France, Italy, Germany\}$ and
+    the two different symbolic encodings for $A$ given below.
+    Which one gives a shorter symbolic representation for the set
+    $B=\{France, Germany\}$? Illustrate your answer by giving the
+    representing formulas for $B$ in both encodings.
+
+\vspace{.5cm}
+
+%  \begin{minipage}
+    \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 1}} \\
+    \hline
+    Element & $v_1$ & $v_0$ \\
+    \hline
+    Spain & $0$ & $0$ \\
+    France & $1$ & $0$ \\
+    Italy & $0$ & $1$ \\
+    Germany & $1$ & $1$
+   \end{tabular}
+%\end{minipage}
+%\begin{minipage}
+\hspace{3cm}
+   \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 2}} \\
+    \hline
+    Element & $v_1$ & $v_0$ \\
+    \hline
+    Spain & $0$ & $0$ \\
+    France & $1$ & $0$ \\
+    Italy & $1$ & $1$ \\
+    Germany & $0$ & $1$
+   \end{tabular}
+%\end{minipage}
+  
+  

symbolic_encoding/practical_questions/2_10_symbrep_lect_sol.tex

@@ -0,0 +1,7 @@
+Using encoding 1, we end up in the following formula:\\
+$b = v_1$\\\\
+Using encoding 2, we end up in the following formula:\\
+$b = (v_1 \land \lnot v_0) \lor (\lnot v_1 \land v_0)$\\
+
+Encoding 1 gives a shorter symbolic representation
+for the set $B=\{France, Germany\}$.

symbolic_encoding/practical_questions/2_10_symbrep_self.tex

@@ -0,0 +1 @@
+\item \self Given a state space of size $|S| = 16$, find a symbolic binary encoding for this state space and compute the characteristic function for the sets of states $$C = \{s_{0}, s_{2}, s_{8}, s_{10}\} \enspace \text{and} \enspace C = \{s_{0}, s_{2}, s_{8}, s_{10}\}$$ Then compute the characteristic function for the sets $D = C \subseteq B$, $E = B \setminus C$ and $F = S \setminus E$. If possible, simplify the formulas.

symbolic_encoding/practical_questions/2_11_symbrep_lect.tex

@@ -0,0 +1,9 @@
+\item  \lect 
+Find a symbolic binary encoding for $X = \{ 0,1, \ldots, 31 \}$.
+  Use it to find formulas that symbolically represent the sets $A$ and $B$ and simplify the formulas:
+  \begin{itemize}
+    \item $A =\{ 12, 13, 14, 15, 28, 29, 30, 31 \}$
+    \item $B =\{x \in X \mid 0 \leq x \leq 15\}$
+  \end{itemize}
+  Furthermore, give the formulas representing the sets
+  $C=A\cap B$ and $D = A \cup B$.

symbolic_encoding/practical_questions/2_11_symbrep_lect_sol.tex

@@ -0,0 +1,4 @@
+We use 5 Boolean variables, \{$v_4$, . . . , $v_0$\}, for the encoding.\\
+
+$A = (v_2 \land v_3)$\\
+$B = \lnot v_4$

symbolic_encoding/practical_questions/2_11_symbrep_self.tex

@@ -0,0 +1,21 @@
+\item \self Find a symbolic binary encoding for
+$X = \{ 0,1, \ldots, 31 \}$.
+Use it to find characteristic functions for the following sets. If possible, simplify the formulas.
+
+  \begin{enumerate}
+    \item $B =\{x \in X \mid 24 \leq x < 32\}$
+    \item 
+      $C = \left\{ x \in X \middle| \enspace
+      \begin{array}{lll}
+      4 & \leq \enspace x & < 8 \\
+      12  & \leq \enspace x & < 16  \\
+      20 & \leq \enspace x & < 24
+      \end{array}
+      \right\}$
+  \end{enumerate}
+  
+  Compute the characteristic functions of the following sets by symbolic operations, using your results from before and simplify your formulas.
+  \begin{enumerate}
+    \item $D = B \cup C$
+    \item $E = B \cap C$
+  \end{enumerate}

symbolic_encoding/practical_questions/2_12_symbrep_lect.tex

@@ -0,0 +1,7 @@
+\item \self Define the \textit{transition system} from the following symbolically encoded transition relations and draw the
+  corresponding graph:
+  \begin{align*}
+    (\lnot v_1 \land \lnot v_0 \land  v'_1 \land v'_0)  & \enspace \lor  \\
+    (\neg v_1 \land v_0 \land \lnot v'_1 \land  \lnot v'_0) & \enspace \lor  \\
+     (\lnot v_1 \land \lnot v_0 \land  \lnot v'_1 \land \lnot v'_0)    &
+  \end{align*}

symbolic_encoding/practical_questions/2_12_symbrep_self.tex

@@ -0,0 +1,15 @@
+\item \self Find a symbolic binary encoding for
+$X = \{ 0,1, \ldots, 31 \}$.
+Use it to compute formulas in propositional logic that symbolically represent the following sets. 
+
+\begin{itemize}
+  \item $B =\{8, 9, 10, 11, 12, 13, 14, 15\}$
+  \item $C =\{x \in X \mid 0 \leq x \leq 15\}$
+\end{itemize}
+
+Compute the characteristic functions of the following sets by symbolic operations, using your results from before.
+\begin{enumerate}
+  \item $D = B \cup C$
+  \item $E = B \cap C$
+  \item $F = C \setminus B$
+\end{enumerate}

symbolic_encoding/practical_questions/2_13_symbrep_lect.tex

@@ -0,0 +1,24 @@
+\item \lect Find a \textit{symbolic encoding} for the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v1 \wedge \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    \path[->] (s0) edge (s1);
+    \path[<->] (s0) edge (s2);
+    \path[<->] (s0) edge (s3);
+    \path[<->] (s1) edge (s2);
+    \path[<->] (s1) edge (s3);
+    \path[<->] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    %\path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+    \path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/practical_questions/2_13_symbrep_lect_sol.tex

@@ -0,0 +1,15 @@
+Using the variables $v_1$ and $v_0$, we can define the transition relation using the following formula:\\
+\begin{center}
+    $\lnot v_1 \land \lnot v_0 \land (\lnot v'_1 \land \lnot v'_0 \lor \lnot v'_1 \land v'_0 \lor v'_1 \land \lnot v'_0 \lor v'_1 \land v'_0) \ \lor$\\
+    $\lnot v_1 \land v_0 \land (\lnot v'_1 \land v'_0 \lor v'_1 \land \lnot v'_0 \lor v'_1 \land v'_0) \ \lor$\\
+    $v_1 \land \lnot v_0 \land (\lnot v'_1 \land \lnot v'_0 \lor \lnot v'_1 \land v'_0 \lor v'_1 \land v'_0) \ \lor$\\
+    $v_1 \land v_0 \land (\lnot v'_1 \land \lnot v'_0 \lor \lnot v'_1 \land v'_0 \lor v'_1 \land \lnot v'_0 \lor v'_1 \land v'_0)$
+\end{center}
+
+We can further simplify the formula to:
+\begin{center}
+    $\lnot v_1 \land \lnot v_0 \lor$\\ 
+    $\lnot v_1 \land v_0 \land (v'_0 \lor v'_1 \land \lnot v'_0) \ \lor$\\
+    $v_1 \land \lnot v_0 \land (\lnot v'_1 \lor v'_1 \land v'_0) \ \lor$\\
+    $v_1 \land v_0$
+\end{center}

symbolic_encoding/practical_questions/2_13_symbrep_self.tex

@@ -0,0 +1,15 @@
+\item \self Find a symbolic binary encoding for
+$X = \{ 0,1, \ldots, 31 \}$.
+Use it to compute formulas in propositional logic that symbolically represent the following sets. 
+\begin{itemize}
+  \item $B =\{x \in X \mid \text{x is even}\}$
+  \item $C =\{x \in X \mid \text{x is odd}\}$
+  \item $D =\{0,1,2,3,4,5,6,7\}$
+\end{itemize}
+
+Compute the characteristic functions of the following sets by symbolic operations, using your results from before.
+\begin{enumerate}
+  \item $E = B \cup D$
+  \item $F = C \cap E$
+  \item $G = E \setminus F$
+\end{enumerate}

symbolic_encoding/practical_questions/2_17_symbrep_self.tex

@@ -0,0 +1,33 @@
+\item \self Find a \textit{symbolic encoding} for the \textit{transition relation} of the following \textit{Kripke structure} and
+simplify your formulas. The labels of the states symbolize their symbolic encoding.
+E.g. "110" = "$x_2 \land x_1 \land \lnot x_0$", i.e. the least significant bit corresponds to $x_0$,
+the second-least significant bit to $x_1$, and so forth.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=2cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$000$};
+    \node[node] (s1) [above right of=s0] {$001$};
+    \node[node] (s2) [right of=s1] {$010$};
+    \node[node] (s3) [below right of=s2] {$011$};
+    \node[node] (s4) [below of=s3] {$100$};
+    \node[node] (s5) [below left of=s4] {$101$};
+    \node[node] (s6) [left of=s5] {$110$};
+    \node[node] (s7) [above left of=s6] {$111$};
+
+
+
+    \path[->] (s0) edge (s1);
+    \path[->] (s1) edge (s2);
+    \path[<->] (s2) edge (s3);
+    \path[->] (s3) edge (s4);
+    \path[->] (s4) edge (s7);
+    \path[->] (s7) edge (s6);
+    \path[<->] (s6) edge (s5);
+    \path[->] (s3) edge (s6);
+    \path[->] (s5) edge (s0);
+    \path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    \path[->] (s4.30) edge[bend right=90, looseness=15, out=240, in=300] (s4.60);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/practical_questions/2_18_symbrep_self.tex

@@ -0,0 +1,7 @@
+\item \self Define the \textit{transition system} from the following symbolically encoded transition relations and draw the
+  corresponding graph:
+  \begin{align*}
+    (v_1 \land v_0 \land \lnot v'_1 \land \neg v'_0)  & \enspace \lor  \\
+    (\neg v_1 \land v_0 \land \lnot v'_1 \land  v'_0) & \enspace \lor  \\
+     (v_1 \land v_0 \land  v'_1 \land v'_0)    &
+  \end{align*}

symbolic_encoding/practical_questions/2_1_symbrep_lect.tex

@@ -0,0 +1 @@
+\item \lect Given a state space of size $|S| = 2^4 = 16$, give the symbolic encoding for the following states: (a) $s_{7}$, (b) $s_{15}$, and (c) $s_{10}$.

symbolic_encoding/practical_questions/2_1_symbrep_lect_sol.tex

@@ -0,0 +1,6 @@
+For the symbolic encoding we need 4 Boolean variables, \{$v_3$, . . . , $v_0$\}.
+Let $v_3$ be the most significant bit, and $v_0$ the least significant bit.\\
+
+(a) $s_{7}$ = $\lnot v_3 \land v_2 \land v_1 \land v_0$\\
+(b) $s_{15}$ = $v_3 \land v_2 \land v_1 \land v_0$\\
+(c) $s_{10}$ = $v_3 \land \lnot v_2 \land v_1 \land \lnot v_0$

symbolic_encoding/practical_questions/2_1_symbrep_self.tex

@@ -0,0 +1,7 @@
+\item \self Given a state space of the size $|S| = 2^{11} = 2048$, give the symbolic encoding for the following states:
+
+\begin{enumerate}
+	\item $s_{435}$
+    \item $s_{1467}$
+    \item $s_{2022}$
+\end{enumerate}

symbolic_encoding/practical_questions/2_20_symbrep_self.tex

@@ -0,0 +1,9 @@
+\item \self Build a \textit{Kripke structure} from the following symbolically encoded transition relations and draw the
+  corresponding graph:
+
+  \begin{align*}
+    (\lnot x_2 \land \lnot x_1 \land \lnot x_0) \land (\lnot x_2' \land x_0')          &  \enspace \lor \\
+    (\lnot x_2 \land x_1)                       \land ((x_2' \oplus x_1') \land x_0')  &  \enspace \lor \\
+    (x_2 \land (x_1 \leftrightarrow x_0))       \land (x_2' \land (x_1' \oplus x_0'))  &  \enspace \lor \\
+    (x_2 \land x_1 \land \lnot x_0)             \land (x_2' \land (x_1' \lor x_0'))    & 
+  \end{align*}

symbolic_encoding/practical_questions/2_2_symbrep_lect.tex

@@ -0,0 +1,2 @@
+\item \lect Given is the  set of states $S=\{s_0,\dots,s_7\}$.
+Find formulas in propositional logic that symbolically represent the sets $A = \{s_7, s_6,s_3, s_2 \}$, $B=\{s_1, s_3,s_5, s_7\}$, and $C=\{s_7, s_6,s_0, s_1\}$.

symbolic_encoding/practical_questions/2_2_symbrep_lect_sol.tex

@@ -0,0 +1,6 @@
+$A = \{s_7, s_6,s_3, s_2 \}$ = $(v_2 \land v_1 \land v_0) \lor (v_2 \land v_1 \land \lnot v_0) \lor (\lnot v_2 \land v_1 \land v_0) \lor (\lnot v_2 \land v_1 \land \lnot v_0)$\\ 
+\hspace*{0,34cm} = $v_1$\\\\
+$B=\{s_1, s_3,s_5, s_7\}$ = $(\lnot v_2 \land \lnot v_1 \land v_0) \lor (\lnot v_2 \land v_1 \land v_0) \lor (v_2 \land \lnot v_1 \land v_0) \lor (v_2 \land v_1 \land v_0)$\\ 
+\hspace*{0,34cm} = $v_0$\\\\
+$C=\{s_7, s_6,s_0, s_1\}$ = $(v_2 \land v_1 \land v_0) \lor (v_2 \land v_1 \land \lnot v_0) \lor (\lnot v_2 \land \lnot v_1 \land \lnot v_0) \lor (\lnot v_2 \land \lnot v_1 \land v_0)$ \\ 
+\hspace*{0,34cm} = $(v_2 \land v_1) \lor (\lnot v_2 \land \lnot v_1)$

symbolic_encoding/practical_questions/2_2_symbrep_self.tex

@@ -0,0 +1,8 @@
+\item \self Consider the following set of states defined by the valuations: 
+\begin{align*}
+    (v_0 \imp \top, v_1 \imp \top, v_2 \imp \top, v_3 \imp \bot),
+    (v_0 \imp \top, v_1 \imp \bot, v_2 \imp \top, v_3 \imp \bot),\\
+    (v_0 \imp \top, v_1 \imp \bot, v_2 \imp \bot, v_3 \imp \bot),
+    (v_0 \imp \top, v_1 \imp \top, v_2 \imp \bot, v_3 \imp \bot),
+\end{align*}
+Represent this set of states symbolically using a propositional formula and simplify this formula as best as possible.

symbolic_encoding/practical_questions/2_3_symbrep_lect.tex

@@ -0,0 +1,40 @@
+\item \lect Consider the domain $A=\{Maths, \;English, \;Biology, \;Physical \; Education\}$ and
+    the two different symbolic encodings for $A$ given below.
+    Which one gives a shorter characteristic function for the following sets?
+    \begin{enumerate}
+	     \item $B = \{Maths, \; Biology\}$
+        \item $C = \{English, \; Biology\}$
+        \item $D = \{Maths, \;Physical \; Education\}$
+   \end{enumerate}
+    Illustrate your answer by giving the
+    characteristic function for $B$, $C$ and $D$ in both encodings.
+
+\vspace{.5cm}
+
+%\begin{minipage}
+    \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 1}} \\
+    \hline
+    Element & x & y \\
+    \hline
+    Maths & $0$ & $0$ \\
+    English & $1$ & $0$ \\
+    Biology & $0$ & $1$ \\
+    Physical Education & $1$ & $1$
+   \end{tabular}
+%\end{minipage}
+%\begin{minipage}
+\hspace{3cm}
+   \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 2}} \\
+    \hline
+    Element & x & y \\
+    \hline
+    Maths & $0$ & $0$ \\
+    English & $1$ & $0$ \\
+    Biology & $1$ & $1$ \\
+    Physical Education & $0$ & $1$
+   \end{tabular}
+%\end{minipage}

symbolic_encoding/practical_questions/2_3_symbrep_self.tex

@@ -0,0 +1,48 @@
+\item \self Consider the domain $A=\{Asparagus, \;Bell \; Pepper, \;Cabbage, \;Tomato, \\ \; Onion, \;Zucchini, \;Eggplant, \; Mushroom\}$ and
+    the two different symbolic encodings for $A$ given below.
+    Which one gives a shorter characteristic function for the following sets?
+    \begin{enumerate}
+	     \item $B = \{Asparagus, \; Mushroom\}$
+        \item $C = \{Bell \; Pepper, \; Onion, \; Zucchini, \; Eggplant\}$
+        \item $D = \{Cabbage, \; Tomato\}$
+   \end{enumerate}
+    Illustrate your answer by giving the
+    characteristic function for $B$, $C$ and $D$ in both encodings.
+
+\vspace{.5cm}
+
+%\begin{minipage}
+    \begin{tabular}{l|l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 1}} \\
+    \hline
+    Element & x & y & z\\
+    \hline
+    Asparagus & $0$ & $0$ & $0$ \\
+    Bell Pepper & $0$ & $0$ & $1$ \\
+    Cabbage & $0$ & $1$ & $0$ \\
+    Tomato & $0$ & $1$ & $1$  \\
+    Onion & $1$ & $0$ & $0$ \\
+    Zucchini & $1$ & $0$ & $1$  \\
+    Eggplant & $1$ & $1$ & $0$ \\
+    Mushroom & $1$ & $1$ & $1$
+   \end{tabular}
+%\end{minipage}
+%\begin{minipage}
+\hspace{3cm}
+   \begin{tabular}{l|l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 2}} \\
+    \hline
+    Element & x & y & z \\
+    \hline
+    Asparagus & $1$ & $1$ & $0$ \\
+    Bell Pepper & $1$ & $0$ & $1$ \\
+    Cabbage & $0$ & $1$ & $0$ \\
+    Tomato & $1$ & $1$ & $1$  \\
+    Onion & $0$ & $0$ & $0$ \\
+    Zucchini & $0$ & $0$ & $1$  \\
+    Eggplant & $1$ & $0$ & $0$ \\
+    Mushroom & $0$ & $1$ & $1$
+   \end{tabular}
+%\end{minipage}

symbolic_encoding/practical_questions/2_4_symbrep_lect.tex

@@ -0,0 +1,15 @@
+\item \self The following table shows eight students and their means of transportation. Find a symbolic encodings representing the list of students.
+For this encoding, give the symbolic representation of the set $B$ of all students that go by \emph{bike}, and the set $C$ of all students that go by \emph{car}.
+
+  \begin{tabular}{l|l}
+    Name  & Transportation    \\
+    \hline
+    Alice & Car     \\
+    Bob   & Bike     \\
+    Carl  & Tram     \\
+    David & Bike     \\
+    Eve   & Tram     \\
+    Frank & Bike     \\
+    Greg  & Tram     \\
+    Hank  & Bike     \\
+  \end{tabular}

symbolic_encoding/practical_questions/2_4_symbrep_self.tex

@@ -0,0 +1,17 @@
+\item \self 
+Listed are the participants of a seminar as well as their choice of snacks. Find a symbolic encodings for the participants.
+For for this encoding, give the symbolic representation of the set $B$ of all participants that ordered \emph{bananas}, and the set $C$ of all participants that ordered cake.
+
+  \begin{tabular}{l|l}
+    Name  & Snack    \\
+    \hline
+    Alice & banana     \\
+    Bob   & cake     \\
+    Carl  & banana     \\
+    David & banana     \\
+    Eve   & cake     \\
+    Frank & cake     \\
+    Greg  & orange     \\
+    Hank  & cake     \\
+  \end{tabular}
+

symbolic_encoding/practical_questions/2_5_symbrep_self.tex

@@ -0,0 +1,36 @@
+\item \self 
+Consider the domain $A=\{Spain, France, Italy, Germany\}$ and
+    the two different symbolic encodings for $A$ given below.
+    Which one gives a shorter symbolic representation for the set
+    $B=\{France, Italy\}$? Illustrate your answer by giving the
+    representing formulas for $B$ in both encodings.
+
+\vspace{.5cm}
+
+%  \begin{minipage}
+    \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 1}} \\
+    \hline
+    Element & $v_1$ & $v_0$ \\
+    \hline
+    Spain & $0$ & $0$ \\
+    France & $1$ & $0$ \\
+    Italy & $0$ & $1$ \\
+    Germany & $1$ & $1$
+   \end{tabular}
+%\end{minipage}
+%\begin{minipage}
+\hspace{3cm}
+   \begin{tabular}{l|l|l}
+    \hline
+    \multicolumn{3}{c}{\textbf{Encoding 2}} \\
+    \hline
+    Element & $v_1$ & $v_0$ \\
+    \hline
+    Spain & $0$ & $0$ \\
+    France & $1$ & $0$ \\
+    Italy & $1$ & $1$ \\
+    Germany & $0$ & $1$
+   \end{tabular}
+%\end{minipage}

symbolic_encoding/practical_questions/2_6_symbrep_lect.tex

@@ -0,0 +1,2 @@
+\item \self 
+  Given a state space of size $|S| = 64$, find a symbolic binary encoding for this state space and compute the formulas that symbolically represent the sets of states $$B = \{s_{16}, s_{17}, s_{18}, ..., s_{32}\} \enspace \text{and} \enspace C = \{s_{24}, s_{25},, s_{26}, ..., s_{64}.\}$$ Then compute the formulas that symbolically represent the sets $D = B \cap C$ and $E = B \cup C$. 

symbolic_encoding/practical_questions/2_6_symbrep_self.tex

@@ -0,0 +1,41 @@
+\item \self Consider the following set operations and relations between
+    two sets $X$ and $Y$, and an element $a$:
+\begin{enumerate}
+\item Union: $X \cup Y$
+\item Intersection: $X \cap Y$
+\item Set Difference: $X \setminus Y$
+\item Containment: $a \in X$?
+\item Subset: $X \subseteq Y$?
+\item Strict Subset: $X \subset Y$?
+\item Emptiness: $X=\emptyset$?
+\item Equality: $X=Y$?
+\end{enumerate}
+Let $x$ and $y$ be the symbolic representations of $X$ and $Y$
+respectively, and let $\alpha$ be the symbolic encoding of element
+$a$. For each of the following items, state which of the above
+operations is performed, or which of the above questions is answered.
+Write the letters of the corresponding operation/question into the
+boxes of the items below. Note that some of the items below do not
+perform any of the above operations or answer any of the above
+questions. Put a ``--'' in the box of these items. Also note that
+some of the items below might do the same computation or answer the
+same question.
+\begin{itemize}
+\item[\Huge{$\square$}] $\neg x \vee y$
+\item[\Huge{$\square$}] $x \wedge y$
+\item[\Huge{$\square$}] $x\equiv \top$?
+\item[\Huge{$\square$}] $x\equiv y$?
+\item[\Huge{$\square$}] $(x \rightarrow y) \wedge (y \rightarrow
+    x)$?
+\item[\Huge{$\square$}] $x\equiv \bot$?
+\item[\Huge{$\square$}] $y \wedge \neg x$
+\item[\Huge{$\square$}] $x \rightarrow \bot$?
+\item[\Huge{$\square$}] $\alpha \models x$?
+\item[\Huge{$\square$}] $\alpha \models \neg x$?
+\item[\Huge{$\square$}] $\neg \alpha \models x$?
+\item[\Huge{$\square$}] $x \rightarrow \alpha$?
+\item[\Huge{$\square$}] $y \rightarrow x$?
+\item[\Huge{$\square$}] $x \rightarrow y$?
+\item[\Huge{$\square$}] $(x \rightarrow y) \wedge (x\not \equiv
+    y)$?
+\end{itemize}

symbolic_encoding/practical_questions/2_7_symbrep_lect.tex

@@ -0,0 +1,14 @@
+\item \self Find a symbolic binary encoding for
+$X = \{ 0,1, \ldots, 31 \}$.
+Use it to compute formulas in propositional logic that symbolically represent the following sets. 
+
+  \begin{itemize}
+    \item $B =\{4, 5, 12, 13, 20, 21, 28, 29 \}$
+    \item $C =\{1, 2, 13, 14 \}$
+  \end{itemize}
+  
+  Compute the characteristic functions of the following sets by symbolic operations, using your results from before.
+  \begin{enumerate}
+    \item $D = B \cup C$
+    \item $E = X \setminus D$
+  \end{enumerate}

symbolic_encoding/practical_questions/2_7_symbrep_self.tex

@@ -0,0 +1,37 @@
+\item \self Consider the set $\mathbb{N}_{16} = \{0,1,2,3,\dots,14,15\}$.
+    Let $x_0$, $x_1$, $x_2$, and $x_3$ be propositional variables,
+    used for symbolic encoding of the elements of $\mathbb{N}_{16}$,
+    using standard binary encoding, with $x_0$ being the least
+    significant ($2^0$) bit, and $x_3$ being the most significant
+    ($2^3$) bit.
+
+    Now, consider the following subsets of $\mathbb{N}_{16}$.
+
+\begin{itemize}
+\item $A = \{0, 1, 2, 3\}$
+\item $B = \{0, 1, 2, 3, 4, 5, 6, 7\}$
+\item $C = \{0, 2, 4, 6, 8, 10, 12, 14\}$
+\item $D = \{8, 10, 12, 14\}$
+\item $E = \{3, 10\}$
+\item $F = \{ \}$
+\end{itemize}
+
+In the following list of formulas, write the letter of the set that
+the formula encodes into the adjacent box. Note that some sets might
+be encoded by more than one formula. Also note that some formulas
+might not encode any of the above sets; write a ``--'' in the box of
+such formulas.
+
+\begin{itemize}
+\item[\Huge{$\square$}] $\bot$
+\item[\Huge{$\square$}] $\top$
+\item[\Huge{$\square$}] $(\neg x_3 \wedge \neg x_2 \wedge x_1
+    \wedge x_0) \vee (x_3 \wedge \neg x_2 \wedge x_1 \wedge \neg
+    x_0)$
+\item[\Huge{$\square$}] $x_0$
+\item[\Huge{$\square$}] $\neg x_0$
+\item[\Huge{$\square$}] $x_3$
+\item[\Huge{$\square$}] $\neg x_3$
+\item[\Huge{$\square$}] $\neg x_3 \vee \neg x_2$
+\item[\Huge{$\square$}] $\neg x_3 \wedge \neg x_2$
+\end{itemize}

symbolic_encoding/practical_questions/2_8_symbrep_self.tex

@@ -0,0 +1 @@
+\item \self Given a state space of size $|S| = 2048$, find a symbolic binary encoding for this state space and compute the characteristic function for the sets of states $$B = \{s_0, s_1, s_2, ..., s_{1023}\} \enspace \text{and} \enspace C = \{s_{512}, s_{513},, s_{514}, ..., s_{1535}.\}$$ Then compute the characteristic function for the sets $D = B \cup C$ and $E = B \setminus C$. If possible, simplify the formulas.

symbolic_encoding/practical_questions/2_9_symbrep_lect.tex

@@ -0,0 +1,24 @@
+\item \lect Find a \textit{symbolic encoding} for the \textit{transition relation} of the following \emph{transition system} and
+simplify your formulas. Use a binary encoding to encode the states, e.g.,
+encode the state $s_2$ with the formula $v1 \wedge \neg v_0$.
+
+\begin{center}
+\vspace{-2em}
+\begin{tikzpicture}[auto, node distance=3cm,shorten >=1pt,
+thick,node/.style={circle,draw,minimum size=25pt}]
+    \node[node] (s0) {$s_0$};
+    \node[node] (s1) [right of=s0] {$s_1$};
+    \node[node] (s2) [below of=s1] {$s_2$};
+    \node[node] (s3) [below of=s0] {$s_3$};
+    %\path[<->] (s0) edge (s1);
+    \path[->] (s0) edge (s2);
+    \path[<-] (s0) edge (s3);
+    %\path[<->] (s1) edge (s2);
+    \path[<->] (s1) edge (s3);
+    \path[<-] (s2) edge (s3);
+    \path[->] (s0.120) edge[bend right=90, looseness=15, out=240, in=300] (s0.150);
+    %\path[->] (s1.30) edge[bend right=90, looseness=15, out=240, in=300] (s1.60);
+    %\path[->] (s2.30) edge[bend right=90, looseness=15, out=240, in=300] (s2.60);
+   % \path[->] (s3.120) edge[bend right=90, looseness=15, out=240, in=300] (s3.150);
+\end{tikzpicture}
+\end{center}

symbolic_encoding/practical_questions/2_9_symbrep_lect_sol.tex

@@ -0,0 +1,13 @@
+Using the variables $v_1$ and $v_0$, we can define the transition relation using the following formula:\\
+\begin{center}
+    $\lnot v_1 \land \lnot v_0 \land (\lnot v'_1 \land  \lnot v'_0 \lor v'_1 \land \lnot v'_0) \ \lor$\\
+    $\lnot v_1 \land v_0 \land v'_1 \land v'_0 \ \lor$\\
+    $v_1 \land v_0 \land (\lnot v'_1 \land v'_0  \lor \lnot v'_1 \land \lnot v'_0 \lor v'_1 \land \lnot v'_0)$\\
+\end{center}
+
+We can further simplify the formula to:
+\begin{center}
+    $\lnot v_1 \land \lnot v_0 \land \lnot v'_0 \ \lor$\\
+    $\lnot v_1 \land v_0 \land v'_1 \land v'_0 \ \lor$\\
+    $v_1 \land v_0 \land (\lnot v'_1 \land v'_0  \lor  \lnot v'_0)$\\
+\end{center}

symbolic_encoding/practical_questions/2_9_symbrep_self.tex

@@ -0,0 +1,3 @@
+\item \self Given a state space of size $|S| = 64$. Find a symbolic binary encoding for this state space and compute the formulas that symbolically represent the sets  $$B = \{s_{32}, s_{33}, s_{34}, ..., s_{63}\} \enspace \text{and} \enspace C = \{s_{16}, s_{17},, s_{18}, ..., s_{40}\}.$$
+
+Following, compute the formulas that represent the sets $D = B \cap C$, $E = B \cup C$, $F = B \setminus C$ and $G = C \setminus B$.

symbolic_encoding/symbolic_encoding.tex

@@ -0,0 +1,128 @@
+\begin{questionSection}{Transition Systems}
+\question{symbolic_encoding/0001.tex}
+         {symbolic_encoding/0001_sol.tex}
+         {3cm}
+
+\question{symbolic_encoding/0002.tex}
+         {symbolic_encoding/0002_sol.tex}
+         {3cm}
+\question{symbolic_encoding/1001.tex}
+         {no_solution}
+         {3cm}
+\question{symbolic_encoding/1007.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1008.tex}
+         {no_solution}
+         {3cm}
+\end{questionSection}
+
+
+\begin{questionSection}{Symbolic Encoding}
+
+\question{symbolic_encoding/0003.tex}
+         {symbolic_encoding/0003_sol.tex}
+         {3cm}
+
+\question{symbolic_encoding/0004.tex}
+         {symbolic_encoding/0004_sol.tex}
+         {3cm}
+
+
+\question{symbolic_encoding/0008.tex}
+         {symbolic_encoding/0008_sol.tex}
+         {3cm}
+\question{symbolic_encoding/1002.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1003.tex}
+         {no_solution}
+         {3cm}
+
+
+\question{symbolic_encoding/0005.tex}
+         {symbolic_encoding/0005_sol.tex}
+         {3cm}
+
+\question{symbolic_encoding/0006.tex}
+         {symbolic_encoding/0006_sol.tex}
+         {3cm}
+\question{symbolic_encoding/1004.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1005.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1006.tex}
+         {no_solution}
+         {3cm}
+
+
+\question{symbolic_encoding/1009.tex}
+         {no_solution}
+         {3cm}
+
+
+\question{symbolic_encoding/1011.tex}
+         {no_solution}
+         {3cm}
+
+
+
+\question{symbolic_encoding/1015.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1016.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1017.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1018.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1019.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1020.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1021.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1000.tex}
+         {no_solution}
+         {3cm}
+\question{symbolic_encoding/1010.tex}
+         {no_solution}
+         {3cm}
+\question{symbolic_encoding/1012.tex}
+         {no_solution}
+         {3cm}
+
+\question{symbolic_encoding/1013.tex}
+         {no_solution}
+         {3cm}
+\question{symbolic_encoding/0007.tex}
+         {symbolic_encoding/0007_sol.tex}
+         {3cm}
+
+\end{questionSection}
+
+
+
+
+%\question{symbolic_encoding/1014.tex}
+%         {no_solution}
+%         {3cm}

symbolic_encoding/theory_questions/0_1_symb_lect.tex

@@ -0,0 +1 @@
+\item \lect Given the term \textit{state} within a reactive system, what does a \textit{state} do? \underline{rausnehmen?}

symbolic_encoding/theory_questions/0_1_symb_self.tex

@@ -0,0 +1 @@
+\item \self Given the term \textit{transition} within a reactive system, what does the term \textit{transition} mean? How can a \textit{transition} be determined?

symbolic_encoding/theory_questions/1_0_kripke_self.tex

@@ -0,0 +1,2 @@
+\item \self 
+  Give the definition of a \emph{transition system} $\mathcal{T}$ and an example.

symbolic_encoding/theory_questions/2_01_symbrep_self.tex

@@ -0,0 +1,3 @@
+\item \self What is the main advantage of symbolic set operations over
+    non-symbolic operations that enumerate all set elements
+    explicitly?

symbolic_encoding/theory_questions/2_1_symbrep_self.tex

@@ -0,0 +1 @@
+\item \self Considering the \textit{representation of a single state}, for a given valuation, we can write a formula that is true for exactly that evaluation. How  can in contrast  the \textit{representation of sets of states} be defined? Give a small example of such a representation. 

symbolic_encoding/theory_questions/2_2_symbrep_self.tex

@@ -0,0 +1,4 @@
+\item \self Assume you are given the formulas $a$ and
+    $b$, which symbolically represent the sets $A$ and $B$,
+    respectively. Give the formula
+    $c$, which symbolically represents the set $C= A \setminus B$.

symbolic_encoding/theory_questions/2_3_symbrep_lect.tex

@@ -0,0 +1 @@
+\item \lect Given a \textit{set of ordered pairs of states}, how can a transition relation of a \textit{transition system} be symbolically represented? Explain which sets of variables you need and give an example of such a representation.

symbolic_encoding/theory_questions/2_3_symbrep_self.tex

@@ -0,0 +1,5 @@
+\item \self Assume you are given the formulas $a$ and
+    $b$, which symbolically represent the sets $A$ and $B$,
+    respectively. What would you have to check on $a, b$ to test
+    whether or not $A$ is a strict subset of $B$, i.e., $A \subset
+    B$?