From 6012261f77154426dfa2bc7c4ac536f300cb5478 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 9 Dec 2025 10:58:20 +0100 Subject: [PATCH] [TI] Compact: Almost complete --- .../ti-compact/parts/04_computability.tex | 11 ++++++++--- semester3/ti-compact/parts/05_complexity.tex | 7 +++++-- semester3/ti-compact/ti-compact.pdf | Bin 378963 -> 380186 bytes semester3/ti-compact/ti-compact.tex | 6 ++++++ 4 files changed, 19 insertions(+), 5 deletions(-) diff --git a/semester3/ti-compact/parts/04_computability.tex b/semester3/ti-compact/parts/04_computability.tex index d9191b7..8c23fa2 100644 --- a/semester3/ti-compact/parts/04_computability.tex +++ b/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} diff --git a/semester3/ti-compact/parts/05_complexity.tex b/semester3/ti-compact/parts/05_complexity.tex index 9430469..53d96cd 100644 --- a/semester3/ti-compact/parts/05_complexity.tex +++ b/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 diff --git a/semester3/ti-compact/ti-compact.pdf b/semester3/ti-compact/ti-compact.pdf index 26290b365d8e27a26c27aeab7875bbcae7cd79e6..c832fa49cf5caabc06c730c883cb57983f7fbecb 100644 GIT binary patch delta 55658 zcmZs?V{j!-(C{1EcCxW;+qUhE?UQ6FSrvBb|y5LQ1!?4>I- z$A5$&$=RedWhyeXJ=8w)d0B^mt34+9_^&^|cG<*Sp-IW~z00w_^nCrxrJ%-00h2=Kp_Ie0 zvF=5N*x{0nL&qH1@H@0Pv))0Kh90u)`)ZDLK{<)XpnCyp(Xvf|39HfIwV*thbOl}%(G~lr|LD?OIqEzSuxq^uaAUopV$wfvsXDgOZks2t-aOl={KLbY|KqPGZ4PY1$x-w5A3uiFg;$FO$gKBXVDnLJmPm5VO+4!IBkN^v)1`0dFnA zmoyl6El)<0JB=k3Plgg90h>8hw4or*CE>%_$p9yk#OnTB4tHKYz*h&uQIwomMo7SZ zOWL59>+>PIv9{LQR;Lt)XY)LE`?^cg8iXgOgi6aaL_UMI`C&E1zAo0 z+Y~0Dxxw8w-(@hW| 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