added temporal logic chapter

decidability/0001.tex

@@ -0,0 +1 @@
+\item TODO

decidability/0002.tex

@@ -0,0 +1 @@
+\item TODO

decidability/0003.tex

@@ -0,0 +1 @@
+\item TODO

decidability/0004.tex

@@ -0,0 +1 @@
+\item TODO

decidability/decidability.tex

@@ -0,0 +1,14 @@
+\begin{questionSection}{Decidability}
+\question{decidability/0001.tex}
+         {no_solution}
+         {3cm}
+\question{decidability/0002.tex}
+         {no_solution}
+         {3cm}
+\question{decidability/0003.tex}
+         {no_solution}
+         {3cm}
+\question{decidability/0004.tex}
+         {no_solution}
+         {3cm}
+\end{questionSection}

main.tex

@@ -5,6 +5,7 @@
 
 \input{util/packages}
 \input{util/math_macros}
+\input{util/ltl_macros}
 \input{util/styling_macros}
 \input{util/constants}
 \input{util/chapter}
@@ -127,6 +128,13 @@
   \pagebreak
 \fi
 
+\ifchapterdecidability
+  \setsectionnum{10}
+  \section{Decidability}
+  \input{decidability/decidability.tex}
+  \pagebreak
+\fi
+
 \ifchaptertemporal
   \setsectionnum{10}
   \section{Temporal Logic}

temporal_logic/0001.tex

@@ -0,0 +1,6 @@
+\item \lect
+Translate the following sentences in computation tree logic $CTL^\star$.
+\begin{itemize}
+    \item In every execution the system gives a grant infinitely often.
+    \item There exists an execution in which the system sends a request finitely often.
+\end{itemize}

temporal_logic/0001_sol.tex

@@ -0,0 +1,6 @@
+\begin{itemize}
+    \item The Boolean variable $g$ represents ``The system gives a grant.''\newline
+    $\varphi_1 \coloneqq AGF g$
+     \item The Boolean variable $r$ represents ``The system sends a request.'' \newline
+    $\varphi_2 \coloneqq EFG \neg r$
+\end{itemize}

temporal_logic/0002.tex

@@ -0,0 +1,24 @@
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{a\}, \{b\}, \{\}, \{a\}, \{a, b\}^\omega$
+    \item  $\varphi = \nex a \lor a \unt b$
+\end{itemize}
+
+  \begin{ltltabular}{7}
+    \ltlSteps{6}{1}
+    \ltlTrace{a}{0|1|1|0|0|1 |1}
+    \ltlTrace{b}{0|0|0|1|0|0 |1}
+    \hline
+    \hline
+    \ltlFormula{$\nex a$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$a \unt b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\nex a \lor a \unt b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular}
+  
+  

temporal_logic/0002_sol.tex

@@ -0,0 +1,21 @@
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{a\}, \{b\}, \{\}, \{a\}, \{a, b\}^\omega$
+    \item  $\varphi = \nex a \lor a \unt b$
+\end{itemize}
+\begin{ltltabular}{7}
+    \ltlSteps{6}{1}
+    \ltlTrace{a}{0|1|1|0|0|1 |1}
+    \ltlTrace{b}{0|0|0|1|0|0 |1}
+    \hline
+    \hline
+    \ltlFormula{$\nex a$}
+               {1|1|0|0|1 |1 | 1}
+    \ltlFormula{$a \unt b$}
+               {0|1|1|1|0 |1 | 1}
+    \ltlFormula{$\nex a \lor a \unt b$}
+               {1|1|1|1|1 |1 | 1}
+  \end{ltltabular}

temporal_logic/0003.tex

@@ -0,0 +1,26 @@
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{ \}, \{a,b,c\}, \{a\}, \{a,b\}, ( \{a\} ,  \{a,c\}, \{a,c\})^\omega$
+    \item  $\varphi = \glob a \imp (\event b \lor c )$
+\end{itemize}
+
+  \begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1|1}
+    \hline
+    \hline
+    \ltlFormula{$\glob a$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event b \vee c$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\glob a \imp (\event b \lor c)$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular}
+  

temporal_logic/0003_sol.tex

@@ -0,0 +1,27 @@
+
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{ \}, \{a,b,c\}, \{a\}, \{a,b\}, ( \{a\} ,  \{a,c\}, \{a,c\})^\omega$
+    \item  $\varphi = \glob a \imp (\event b \lor c )$
+\end{itemize}
+
+
+\begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1|1}
+    \hline
+    \hline
+    \ltlFormula{$\glob a$}
+              {0|1|0|1|1|1 |1|1|1}
+    \ltlFormula{$\event b$}
+               {0|1|0|1|1|1 |0|0|0}
+    \ltlFormula{$\event b \vee c$}
+                {0|1|0|1|1|1 |0|1|1}
+    \ltlFormula{$\glob a \imp (\event b \lor c)$}
+               {0|1|0|1|1|1 |0|1|1}
+  \end{ltltabular}

temporal_logic/0004.tex

@@ -0,0 +1,27 @@
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{ \}, \{a,b,c\}, \{a\}, \{a,b\}, ( \{a\} ,  \{a,c\}, \{a,c\})^\omega$
+    \item  $\varphi = \glob \event a \imp (\event  \glob \neg b \wedge c )$
+\end{itemize}
+
+  \begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1|1}
+    \hline
+    \hline
+    \ltlFormula{$\glob \event a$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event \glob \neg b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event \glob \neg b \wedge c$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\glob \event a \imp (\event  \glob \neg b \wedge c )$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular} 
+  \newline

temporal_logic/0004_sol.tex

@@ -0,0 +1,25 @@
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{ \}, \{a,b,c\}, \{a\}, \{a,b\}, ( \{a\} ,  \{a,c\}, \{a,c\})^\omega$
+    \item  $\varphi = \glob \event a \imp (\event  \glob \neg b \wedge c )$
+\end{itemize}
+\begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1|1}
+    \hline
+    \hline
+    \ltlFormula{$\glob \event a$}
+              {1|1|1|1|1|1 |1|1|1}
+    \ltlFormula{$\event \glob \neg b$}
+               {1|1|1|1|1|1 |1|1|1}
+    \ltlFormula{$\event \glob \neg b \wedge c$}
+              {0|0|0|1|0|0 |0|1|1}
+    \ltlFormula{$\glob \event a \imp (\event  \glob \neg b \wedge c )$}
+               {0|0|0|1|0|0 |0|1|1}
+  \end{ltltabular}

temporal_logic/0005.tex

@@ -0,0 +1,8 @@
+\item \lect
+Translate the following sentences in computation tree logic $CTL^\star$.
+\begin{itemize}
+    \item For any execution, it always holds that whenever the robot
+visits region A, it visits region C within the next two steps.
+    \item There exists an execution such that the robot visits
+region C within the next two steps after visiting region A.
+\end{itemize}

temporal_logic/0005_sol.tex

@@ -0,0 +1,11 @@
+We use the following Boolean variables:
+\begin{itemize}
+    \item $a$ represents ``The robot visits region A''
+    \item $b$ represents ``The robot visits region B''    
+    \item $c$ represents ``The robot visits region C''  
+\end{itemize}
+
+\begin{itemize}
+    \item $\varphi_1 \coloneqq AG (a \rightarrow Xc \vee XXc)$
+    \item $\varphi_2 \coloneqq EG (a \rightarrow Xc \vee XXc)$
+\end{itemize}

temporal_logic/0006.tex

@@ -0,0 +1,7 @@
+\item \lect
+Translate the following sentences in computation tree logic $CTL^\star$.
+\begin{itemize}
+    \item The robot can visit region A infinitely often and region C infinitely often
+    \item Always, the robot visits region A infinitely often and region C infinitely often.
+    \item If the robot visits region A infinitely often, it should also visit region C finitely often.
+\end{itemize}

temporal_logic/0006_sol.tex

@@ -0,0 +1,12 @@
+We use the following Boolean variables:
+\begin{itemize}
+    \item $a$ represents ``The robot visits region A''
+    \item $b$ represents ``The robot visits region B''    
+    \item $c$ represents ``The robot visits region C''  
+\end{itemize}
+
+\begin{itemize}
+    \item $\varphi_1 \coloneqq E(GF a \wedge GF c)$
+    \item $\varphi_2 \coloneqq A (GF a \wedge FG \neg c)$
+    \item $\varphi_3 \coloneqq A (GF a \rightarrow GF c)$
+\end{itemize}

temporal_logic/0007.tex

@@ -0,0 +1,12 @@
+\item \lect 
+Given the following Kripke structure $\mathcal{K}$. Does the initial state $s_0$ of $\mathcal{K}$ satisfy the following formulas?
+\begin{itemize}
+    \item $\varphi_1 \coloneqq EXX(a\wedge b)$
+    \item $\varphi_2 \coloneqq EXAX (a \wedge b)$
+\end{itemize}
+\begin{figure}[h!]
+	\centering
+  \includegraphics[width=0.63\textwidth]{figures/computation_tree.png}
+	\caption{Left: Kripke structure of Example 7, Right: Corresponding computation tree}
+	\label{fig3}
+\end{figure}

temporal_logic/0007_sol.tex

@@ -0,0 +1,5 @@
+\begin{itemize}
+    \item $s_0 \vDash EXX(a\wedge b)$
+    \item $s_0 \nvDash EXAX (a \wedge b)$
+\end{itemize}
+

temporal_logic/0008.tex

@@ -0,0 +1,12 @@
+\item \lect 
+Given the following Kripke structure $\mathcal{K}$. Does the initial state $s_0$ of $\mathcal{K}$ satisfy the following formulas?
+\begin{itemize}
+    \item $\varphi_1 \coloneqq EXp$
+    \item $\varphi_2 \coloneqq EG\neg p$
+\end{itemize}
+\begin{figure}[h!]
+	\centering
+  \includegraphics[width=0.23\textwidth]{figures/kripke5.png}
+	\caption{Kripke structure of Example 8}
+	\label{fig3}
+\end{figure}

temporal_logic/0008_sol.tex

@@ -0,0 +1,5 @@
+\begin{itemize}
+    \item $s_0 \vDash EXp$
+    \item $s_0 \vDash EG\neg p$
+\end{itemize}
+

temporal_logic/1000.tex

@@ -0,0 +1,6 @@
+\item \self
+Give the definition of a \emph{Kripke structure}.
+Explain the components of the tuple a Kripke structure consists of.
+Give an example of a Kripke structure in the representation of a graph. 
+
+

temporal_logic/1001.tex

@@ -0,0 +1,6 @@
+\item \self
+Give the definition of 
+\emph{paths} and \emph{words} of Kripke structures.
+Give an example in which you draw a graph representing a Kripke structure, 
+and give one possible infinite path and corresponding word. 
+

temporal_logic/1002.tex

@@ -0,0 +1,3 @@
+\item \self What does a \emph{computation tree} of a Kripke structure represent?
+Give an example in which you draw a graph representing a Kripke structure, 
+and draw the first 3 levels of the computation tree of this Kripke structure.

temporal_logic/1003.tex

@@ -0,0 +1,15 @@
+\item \self
+ 
+The temporal operators describe properties that hold along a given infinite path $\rho$
+through the computation tree of a Kripke structure. Given two formulas
+$\varphi$ and $\psi$ describing state properties.
+\begin{itemize}
+    \item Which are the properties that $\rho$ needs to satisfy such that
+    $\rho \vDash G\varphi$?
+    \item Which are the properties that $\rho$ needs to satisfy such that
+    $\rho \vDash F\varphi$?    
+    \item Which are the properties that $\rho$ needs to satisfy such that
+    $\rho \vDash X\varphi$?    
+    \item Which are the properties that $\rho$ needs to satisfy such that
+    $\rho \vDash \varphi U \psi$?      
+\end{itemize}

temporal_logic/1004.tex

@@ -0,0 +1,18 @@
+\item \self
+    Consider an ordinary traffic junction with incoming lanes from the north, south, east and west. We want to formulate relevant constraints that a traffic light system has to fulfill.
+
+    Give a set of propositional variables that model whether the north and south \emph{or} the east and the west get the
+    \begin{itemize}
+      \itemsep0em
+      \item green,
+      \item yellow or
+      \item red
+    \end{itemize}
+    light, respectively. \\
+
+    Formulate the following sentences using \emph{$CTL^\star$}:
+    \begin{enumerate}[label=(\alph*)]
+      \item The north/south lanes will never get the green light at same time as the east/west lanes.
+      \item Whenever the north/south lane receive the green light it will stay green until it changes to yellow.
+      \item When the east/west lane has the red light, it will eventually get the yellow and red light until the light switches to green.
+    \end{enumerate}

temporal_logic/1005.tex

@@ -0,0 +1,24 @@
+\item \self Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{\}, \{a,b\}, \{a \}, \{a,b\}, (\{a\}, \{a, b\}, \{a \})^\omega$
+    \item  $\varphi = \event \glob a \imp \event \glob b$
+\end{itemize}
+
+
+  \begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|1|0}
+    \hline
+    \hline
+    \ltlFormula{$\event \glob a$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event \glob b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event \glob a \imp \event \glob b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular}
+  \newline

temporal_logic/1006.tex

@@ -0,0 +1,25 @@
+\item \self
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+
+\begin{itemize}
+    \item $w = \{a\},\{a\},\{a\},\{b,c\},\{a\},\{a,b\}(\{a\},\{c\})^\omega$
+    \item  $\varphi =  a \unt c \lor \event b $
+\end{itemize}
+
+  \begin{ltltabular}{8}
+    \ltlSteps{6}{2}
+    \ltlTrace{a}{1|1|1|0|1|1 |1|0}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1}
+    \hline
+    \hline
+    \ltlFormula{$a \unt c$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$a \unt c \lor \event b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular}
+  \newline

temporal_logic/1007.tex

@@ -0,0 +1,2 @@
+\item \self Give the definition of the syntax of the computation tree logic $CTL^\star$.
+In particular, give the definition of state formulas and path formulas.

temporal_logic/1008.tex

@@ -0,0 +1,2 @@
+\item \self Give an intuitive explanation of the semantics of computation tree logic $CTL^\star$.
+Therefore, explain the semantics of the introduced path quantifiers and temporal operators with respect to the computation tree of a Kripke structure.

temporal_logic/practical_questions/6_1_ltl_example_lect.tex

@@ -0,0 +1,24 @@
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{a\}, \{b\}, \{\}, \{a\}, \{a, b\}^\omega$
+    \item  $\varphi = \nex a \lor a \unt b$
+\end{itemize}
+
+  \begin{ltltabular}{7}
+    \ltlSteps{6}{1}
+    \ltlTrace{a}{0|1|1|0|0|1 |1}
+    \ltlTrace{b}{0|0|0|1|0|0 |1}
+    \hline
+    \hline
+    \ltlFormula{$\nex a$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$a \unt b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\nex a \lor a \unt b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular}
+  
+  

temporal_logic/practical_questions/6_1_ltl_example_lect_sol.tex

@@ -0,0 +1,13 @@
+  \begin{ltltabular}{7}
+    \ltlSteps{6}{1}
+    \ltlTrace{a}{0|1|1|0|0|1 |1}
+    \ltlTrace{b}{0|0|0|1|0|0 |1}
+    \hline
+    \hline
+    \ltlFormula{$\nex a$}
+               {1|1|0|0|1 |1 | 1}
+    \ltlFormula{$a \unt b$}
+               {0|1|1|1|0 |1 | 1}
+    \ltlFormula{$\nex a \lor a \unt b$}
+               {1|1|1|1|1 |1 | 1}
+  \end{ltltabular}

temporal_logic/practical_questions/6_2_ltl_example_lect.tex

@@ -0,0 +1,27 @@
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{ \}, \{a,b,c\}, \{a\}, \{a,b\}, ( \{a\} ,  \{a,c\}, \{a,c\})^\omega$
+    \item  $\varphi = \glob \event a \imp (\event  \glob \neg b \wedge c )$
+\end{itemize}
+
+  \begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1|1}
+    \hline
+    \hline
+    \ltlFormula{$\glob \event a$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event \glob \neg b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event \glob \neg b \wedge c$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\glob \event a \imp (\event  \glob \neg b \wedge c )$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular} 
+  \newline

temporal_logic/practical_questions/6_2_ltl_example_lect_sol.tex

@@ -0,0 +1,16 @@
+  \begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1|1}
+    \hline
+    \hline
+    \ltlFormula{$\glob \event a$}
+              {1|1|1|1|1|1 |1|1|1}
+    \ltlFormula{$\event \glob \neg b$}
+               {1|1|1|1|1|1 |1|1|1}
+    \ltlFormula{$\event \glob \neg b \wedge c$}
+              {0|0|0|1|0|0 |0|1|1}
+    \ltlFormula{$\glob \event a \imp (\event  \glob \neg b \wedge c )$}
+               {0|0|0|1|0|0 |0|1|1}
+  \end{ltltabular} 

temporal_logic/practical_questions/6_3_weekdays_lect.tex

@@ -0,0 +1,6 @@
+\item \lect
+  \begin{itemize}
+    \item draw weekdays kripke structure
+    \item There will always eventually be weekend.
+    \item If it is Tuesday, the next day will be Wednesday.
+  \end{itemize}

temporal_logic/practical_questions/6_4_ltl_example_self.tex

@@ -0,0 +1,24 @@
+\item \self Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{\}, \{a,b\}, \{a \}, \{a,b\}, (\{a\}, \{a, b\}, \{a \})^\omega$
+    \item  $\varphi = \event \glob a \imp \event \glob b$
+\end{itemize}
+
+
+  \begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|1|0}
+    \hline
+    \hline
+    \ltlFormula{$\event \glob a$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event \glob b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event \glob a \imp \event \glob b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular}
+  \newline

temporal_logic/practical_questions/6_5_ltl_example_self.tex

@@ -0,0 +1,25 @@
+\item \self
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+
+\begin{itemize}
+    \item $w = \{a\},\{a\},\{a\},\{b,c\},\{a\},\{a,b\}(\{a\},\{c\})^\omega$
+    \item  $\varphi =  a \unt c \lor \event b $
+\end{itemize}
+
+  \begin{ltltabular}{8}
+    \ltlSteps{6}{2}
+    \ltlTrace{a}{1|1|1|0|1|1 |1|0}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1}
+    \hline
+    \hline
+    \ltlFormula{$a \unt c$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$a \unt c \lor \event b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular}
+  \newline

temporal_logic/practical_questions/6_6_ltl_example_self.tex

@@ -0,0 +1,26 @@
+\item \lect
+Given the following execution word $w$ of a Kripke structure. Evaluate the
+formula $\varphi$ on $w$. Evaluate each sub-formula for any execution step using the
+provided table.
+\begin{itemize}
+    \item $w = \{\}, \{a\}, \{ \}, \{a,b,c\}, \{a\}, \{a,b\}, ( \{a\} ,  \{a,c\}, \{a,c\})^\omega$
+    \item  $\varphi = \glob a \imp (\event b \lor c )$
+\end{itemize}
+
+  \begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1|1}
+    \hline
+    \hline
+    \ltlFormula{$\glob a$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event b$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\event b \vee c$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+    \ltlFormula{$\glob a \imp (\event b \lor c)$}
+               {\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}|\ws{1cm}}
+  \end{ltltabular}
+  

temporal_logic/practical_questions/6_6_ltl_example_self_sol.tex

@@ -0,0 +1,16 @@
+  \begin{ltltabular}{9}
+    \ltlSteps{6}{3}
+    \ltlTrace{a}{0|1|0|1|1|1 |1|1|1}
+    \ltlTrace{b}{0|0|0|1|0|1 |0|0|0}
+    \ltlTrace{c}{0|0|0|1|0|0 |0|1|1}
+    \hline
+    \hline
+    \ltlFormula{$\glob a$}
+              {0|1|0|1|1|1 |1|1|1}
+    \ltlFormula{$\event b$}
+               {0|1|0|1|1|1 |0|0|0}
+    \ltlFormula{$\event b \vee c$}
+                {0|1|0|1|1|1 |0|1|1}
+    \ltlFormula{$\glob a \imp (\event b \lor c)$}
+               {0|1|0|1|1|1 |0|1|1}
+  \end{ltltabular}

temporal_logic/practical_questions/6_7_robot_city_self.tex

@@ -0,0 +1,31 @@
+\item \self
+  Consider a robot moving in this small transition system. The robot has to fulfill tasks in the state $00$ and $10$. \\
+  Let the state labelling be as follows: $(x_1 x_0)$.
+
+  \begin{figure}[h!]
+    \centering
+    \begin{conflictgraph}
+      %\tikzset{every loop/.style={min distance=10mm,in=0,out=60,looseness=10}}
+      \node[base node] (00)               {$00$};
+      \node[base node] (01) [right of=00] {$01$};
+      \node[base node] (10) [below of=01] {$10$};
+      \node[base node] (11) [below of=00] {$11$};
+      \path[->] (00) edge  [loop above] node {} ();
+      \path[->] (10) edge  [loop right] node {} ();
+      \path[->] (11) edge  [loop below] node {} ();
+      \path[]
+        (00) edge  [bend left] node {} (01)
+        (01) edge  [bend left] node {} (00)
+        (01) edge  [bend left] node {} (10)
+        (10) edge  [bend left] node {} (01)
+        (00) edge  []          node {} (11)
+        (10) edge  []          node {} (11);
+    \end{conflictgraph}
+  \end{figure}
+
+  Formulate the following using \emph{Linear Temporal Logic}:
+  \begin{enumerate}[(a)]
+    \item The robot will infinitely often see state $00$ and $10$.
+    \item The robot will never visit state $11$.
+    \item After visiting either state $00$ or $10$ the robot will visit state $01$ within the next three time steps.
+  \end{enumerate}

temporal_logic/practical_questions/6_8_traffic_light_lect.tex

@@ -0,0 +1,18 @@
+\item \self
+    Consider an ordinary traffic junction with incoming lanes from the north, south, east and west. We want to formulate relevant constraints that a traffic light system has to fulfill.
+
+    Give a set of propositional variables that model whether the north and south \emph{or} the east and the west get the
+    \begin{itemize}
+      \itemsep0em
+      \item green,
+      \item yellow or
+      \item red
+    \end{itemize}
+    light, respectively. \\
+
+    Formulate the following sentences using \emph{$CTL^\star$}:
+    \begin{enumerate}[(a)]
+      \item The north/south lanes will never get the green light at same time as the east/west lanes.
+      \item Whenever the north/south lane receive the green light it will stay green until it changes to yellow.
+      \item When the east/west lane has the red light, it will eventually get the yellow and red light until the light switches to green.
+    \end{enumerate}

temporal_logic/practical_questions/9_1_lect.tex

@@ -0,0 +1,6 @@
+\item \lect
+Translate the following sentences in computation tree logic $CTL^\star$.
+\begin{itemize}
+    \item In every execution the system gives a grant infinitely often.
+    \item There exists an execution in which the system sends a request finitely often.
+\end{itemize}

temporal_logic/practical_questions/9_1_lect_sol.tex

@@ -0,0 +1,6 @@
+\begin{itemize}
+    \item The Boolean variable $g$ represents ``The system gives a grant.''\newline 
+    $\varphi_1 \coloneqq AGF g$
+     \item The Boolean variable $r$ represents ``The system sends a request.'' \newline 
+    $\varphi_2 \coloneqq EGF \neg r$   
+\end{itemize}

temporal_logic/practical_questions/9_2_lect.tex

@@ -0,0 +1,8 @@
+\item \lect
+Translate the following sentences in computation tree logic $CTL^\star$.
+\begin{itemize}
+    \item For any execution, it always holds that whenever the robot
+visits region A, it visits region C within the next two steps.
+    \item There exists an execution such that the robot visits
+region C within the next two steps after visiting region A.
+\end{itemize}

temporal_logic/practical_questions/9_2_lect_sol.tex

@@ -0,0 +1,11 @@
+We use the following Boolean variables:
+\begin{itemize}
+    \item $a$ represents ``The robot visits region A''
+    \item $b$ represents ``The robot visits region B''    
+    \item $c$ represents ``The robot visits region C''  
+\end{itemize}
+
+\begin{itemize}
+    \item $\varphi_1 \coloneqq AG (a \rightarrow Xc \vee XXc)$
+    \item $\varphi_2 \coloneqq EG (a \rightarrow Xc \vee XXc)$
+\end{itemize}

temporal_logic/practical_questions/9_3_lect.tex

@@ -0,0 +1,7 @@
+\item \lect
+Translate the following sentences in computation tree logic $CTL^\star$.
+\begin{itemize}
+    \item The robot can visit region A infinitely often and region C infinitely often
+    \item Always, the robot visits region A infinitely often and region C infinitely often.
+    \item If the robot visits region A infinitely often, it should also visit region C finitely often.
+\end{itemize}

temporal_logic/practical_questions/9_3_lect_sol.tex

@@ -0,0 +1,12 @@
+We use the following Boolean variables:
+\begin{itemize}
+    \item $a$ represents ``The robot visits region A''
+    \item $b$ represents ``The robot visits region B''    
+    \item $c$ represents ``The robot visits region C''  
+\end{itemize}
+
+\begin{itemize}
+    \item $\varphi_1 \coloneqq E(GF a \wedge GF c)$
+    \item $\varphi_2 \coloneqq A (GF a \wedge FG \neg c)$
+    \item $\varphi_3 \coloneqq A (GF a \rightarrow GF c)$
+\end{itemize}

temporal_logic/practical_questions/9_4_lect.tex

@@ -0,0 +1,12 @@
+\item \lect 
+Given the following Kripke structure $\mathcal{K}$. Does the initial state $s_0$ of $\mathcal{K}$ satisfy the following formulas?
+\begin{itemize}
+    \item $\varphi_1 \coloneqq EXX(a\wedge b)$
+    \item $\varphi_2 \coloneqq EXAX (a \wedge b)$
+\end{itemize}
+\begin{figure}[h!]
+	\centering
+  \includegraphics[width=0.63\textwidth]{figures/computation_tree.png}
+	\caption{Left: Kripke structure of Example 7, Right: Corresponding computation tree}
+	\label{fig3}
+\end{figure}

temporal_logic/practical_questions/9_4_lect_sol.tex

@@ -0,0 +1,5 @@
+\begin{itemize}
+    \item $s_0 \vDash EXX(a\wedge b)$
+    \item $s_0 \nvDash EXAX (a \wedge b)$
+\end{itemize}
+

temporal_logic/practical_questions/9_5_lect.tex

@@ -0,0 +1,12 @@
+\item \lect 
+Given the following Kripke structure $\mathcal{K}$. Does the initial state $s_0$ of $\mathcal{K}$ satisfy the following formulas?
+\begin{itemize}
+    \item $\varphi_1 \coloneqq EXp$
+    \item $\varphi_2 \coloneqq EG\neg p$
+\end{itemize}
+\begin{figure}[h!]
+	\centering
+  \includegraphics[width=0.23\textwidth]{figures/kripke5.png}
+	\caption{Kripke structure of Example 8}
+	\label{fig3}
+\end{figure}

temporal_logic/practical_questions/9_5_lect_sol.tex

@@ -0,0 +1,5 @@
+\begin{itemize}
+    \item $s_0 \vDash EXp$
+    \item $s_0 \vDash EG\neg p$
+\end{itemize}
+

temporal_logic/temporal_logic.tex

@@ -0,0 +1,64 @@
+\begin{questionSection}{Temporal Logic}
+
+\question{temporal_logic/1000.tex}
+         {no_solution}
+         {3cm}
+\question{temporal_logic/1001.tex}
+         {no_solution}
+         {3cm}
+\question{temporal_logic/1002.tex}
+         {no_solution}
+         {3cm}
+\question{temporal_logic/1003.tex}
+         {no_solution}
+         {3cm}
+
+\question{temporal_logic/1007.tex}
+         {no_solution}
+         {3cm}
+
+\question{temporal_logic/1008.tex}
+         {no_solution}
+         {3cm}
+
+\question{temporal_logic/0001.tex}
+         {temporal_logic/0001_sol.tex}
+         {3cm}
+
+\question{temporal_logic/0005.tex}
+         {temporal_logic/0005_sol.tex}
+         {3cm}
+
+\question{temporal_logic/0006.tex}
+         {temporal_logic/0006_sol.tex}
+         {3cm}
+
+
+\mcquestion{temporal_logic/0002.tex}
+           {temporal_logic/0002_sol.tex}
+
+\mcquestion{temporal_logic/0003.tex}
+           {temporal_logic/0003_sol.tex}
+
+\mcquestion{temporal_logic/0004.tex}
+           {temporal_logic/0004_sol.tex}
+
+\mcquestion{temporal_logic/1005.tex}
+           {temporal_logic/1005.tex}
+
+\mcquestion{temporal_logic/1006.tex}
+           {temporal_logic/1006.tex}
+
+\question{temporal_logic/0007.tex}
+         {temporal_logic/0007_sol.tex}
+         {3cm}
+
+\question{temporal_logic/0008.tex}
+         {temporal_logic/0008_sol.tex}
+         {3cm}
+
+\question{temporal_logic/1004.tex}
+         {no_solution}
+         {3cm}
+
+\end{questionSection}

temporal_logic/theory_questions/6_1_self.tex

@@ -0,0 +1,6 @@
+\item \self
+Give the definition of a \emph{Kripke structure}.
+Explain the components of the tuple a Kripke structure consists of.
+Give an example of a Kripke structure in the representation of a graph. 
+
+

temporal_logic/theory_questions/6_2_self.tex

@@ -0,0 +1,15 @@
+\item \self
+ 
+The temporal operators describe properties that hold along a given infinite path $\rho$
+through the computation tree of a Kripke structure. Given two formulas
+$\varphi$ and $\psi$ describing state properties.
+\begin{itemize}
+    \item Which are the properties that $\rho$ needs to satisfy such that
+    $\rho \vDash G\varphi$?
+    \item Which are the properties that $\rho$ needs to satisfy such that
+    $\rho \vDash F\varphi$?    
+    \item Which are the properties that $\rho$ needs to satisfy such that
+    $\rho \vDash X\varphi$?    
+    \item Which are the properties that $\rho$ needs to satisfy such that
+    $\rho \vDash \varphi U \psi$?      
+\end{itemize}

temporal_logic/theory_questions/6_3_eventually_semantics_lect.tex

@@ -0,0 +1,2 @@
+\item \lect
+  Let $\pi$ be a trace over a transition system. Give the definition of the \emph{Eventually} operator from \emph{LTL}, i.e. when does $\pi\models\event\varphi$?

temporal_logic/theory_questions/6_3_self.tex

@@ -0,0 +1,3 @@
+\item \self What does a \emph{computation tree} of a Kripke structure represent?
+Give an example in which you draw a graph representing a Kripke structure, 
+and draw the first 3 levels of the computation tree of this Kripke structure.

temporal_logic/theory_questions/6_4_until_semantics_lect.tex

@@ -0,0 +1,2 @@
+\item \lect
+  Let $\pi$ be a trace over a transition system. Give the definition of the \emph{Until} operator from \emph{LTL}, i.e. when does $\pi\models\varphi_1\unt\varphi_2$?

temporal_logic/theory_questions/6_5_self.tex

@@ -0,0 +1,6 @@
+\item \self
+Give the definition of 
+\emph{paths} and \emph{words} of Kripke structures.
+Give an example in which you draw a graph representing a Kripke structure, 
+and give one possible infinite path and corresponding word. 
+

temporal_logic/theory_questions/6_6_self.tex

@@ -0,0 +1,2 @@
+\item \self Give the definition of the syntax of the computation tree logic $CTL^\star$.
+In particular, give the definition of state formulas and path formulas.

temporal_logic/theory_questions/6_7_self.tex

@@ -0,0 +1,2 @@
+\item \self Give an intuitive explanation of the semantics of computation tree logic $CTL^\star$.
+Therefore, explain the semantics of the introduced path quantifiers and temporal operators with respect to the computation tree of a Kripke structure.

util/version.tex

@@ -16,6 +16,7 @@
 \newif\ifchaptereqchecking
 \newif\ifchaptersymbenc
 \newif\ifchaptertemporal
+\newif\ifchapterdecidability
 
 \newif\ifassignmentsheet