[TI] Compact: Almost complete

semester3/ti-compact/parts/04_computability.tex

@@ -39,9 +39,9 @@ Since $M$ is a Turing Machine in the canonical ordering of all Turing Machines,
 
 This however leads to a contradiction, as $w_i \in L_\text{diag} \Longleftrightarrow d_{ii} = 0 \Longleftrightarrow w_i \notin L(M_i)$.
 
-In other words, $w_i$ is in $L_\text{diag}$ if and only if $w_i$ is not in $L(M_i)$, which contradicts our statement above, in which we assumed that $L_\text{diag} \in \cL_{RE}$
+In other words, $w_i$ is in $L_\text{diag}$ if and only if $w_i$ is not in $L(M_i)$, which contradicts our statement above, in which we assumed that $L_\text{diag} \in \cL_{RE}$.
 
-In other, more different, words, $w_i$ being in $L_\text{diag}$ implies (from the definition) that $d_{ii} = 0$, which from its definition implies that $w_i \notin L(M_i)$
+In other, more different, words, $w_i$ being in $L_\text{diag}$ implies (from the definition) that $d_{ii} = 0$, which from its definition implies that $w_i \notin L(M_i)$.
 
 \setLabelNumber{theorem}{3}
 \inlinetheorem $L_\text{diag} \notin \cL_{RE}$
@@ -50,7 +50,12 @@ In other, more different, words, $w_i$ being in $L_\text{diag}$ implies (from th
 
 
 \subsection{Reductions}
-This is the start of the topics that are part of the endterm.
+\label{sec:reductions}
+This is the start of the topics that are explicitly part of the endterm.
+
+For a language to be in $\cL_R$, in contrast to $L \in \cL_{RE}$, the TM has to halt also for \texttt{no} instances, i.e. it has to be an algorithm.
+In other words: A TM $A$ can enumerate all valid strings of a \textit{recursively enumerable language} ($L \in \cL_{RE}$),
+where for \textit{recursive languages}, it has to be able to difinitively answer for both \texttt{yes} and \texttt{no} and thus halt in finite time for both.
 
 First off, a list of important languages for this and the next section:
 \begin{itemize}

semester3/ti-compact/parts/05_complexity.tex

@@ -179,7 +179,10 @@ A few languages commonly used to show $NP$-completeness:
     \item $VC = \{ (G, k) \divides G \text{ is an undirected graph with a vertex cover of size $\leq k$ } \}$
     \item $3SAT = \{ \Phi \divides \Phi \text{ is a satisfiable formula in CNF with all clauses containing \textit{at most} three literals} \}$
 \end{itemize}
-where a clique is \TODO Add.
+where a $k$-clique is a complete subgraph consisting of $k$ vertices in $G$, with $k \leq |V|$.
 and a vertex cover is any set $U \subseteq V$ where all edges $\{ u, v \} \in E$ have at least one endpoint $u, v \in U$
 
-We have $SAT \leq_p \text{CLIQUE}$, $SAT \leq 3SAT$, $\text{CLIQUE} \leq VC$
+We have $SAT \leq_p \text{CLIQUE}$, $SAT \leq_p 3SAT$, $\text{CLIQUE} \leq_p VC$
+
+Additionally, $\text{MAX-SAT}$ and $\text{MAX-CL}$, the problem to determine the maximum number of fulfillable clauses in a formula $\Phi$
+and the problem to determine the maximum clique, respectively, are $NP$-hard

semester3/ti-compact/ti-compact.tex

@@ -56,6 +56,12 @@ It also lacks some formalism and is only intended to give some intuition, six pa
 
 As general recommendations, try to substitute possibly ``weird'' definitions in multiple choice to see a definition from the book.
 
+All content up to Chapter \ref{sec:reductions} is relevant for the midterm directly.
+
+The content for the endterm exam as of HS2025 starts in Chapter \ref{sec:reductions}.
+All prior content is still relevent to the extent that you need an understanding of the concepts treated there
+
+
 
 \input{parts/01_words-alphabets.tex}
 \input{parts/02_finite-automata.tex}