diff --git a/semester3/ti-compact/parts/03_turing-machines.tex b/semester3/ti-compact/parts/03_turing-machines.tex index c1746a4..93a3bc3 100644 --- a/semester3/ti-compact/parts/03_turing-machines.tex +++ b/semester3/ti-compact/parts/03_turing-machines.tex @@ -9,6 +9,11 @@ and repeat until we only have the new symbol, at which point we accept, or there The Turing Machines have an accepting $\qacc$ and a rejecting state $\qrej$ and a configuration is an element of $\{ \{ \cent \}\cdot \Gamma^* \cdot Q \cdot \Gamma^+ \cup Q \cdot \{ \cent \} \cdot \Gamma^+ \}$ with $\cdot$ being the concatenation and $\cent$ the marker of the start of the band. +\rmvspace +\begin{align*} + \cL_{RE} & = \{ L(M) \divides M \text{ is a TM} \}\\ + \cL_R & = \{ L(M) \divides M \text{ is a TM and it always halts} \} +\end{align*} \subsection{Multi-tape TM and Church's Thesis} diff --git a/semester3/ti-compact/parts/04_computability.tex b/semester3/ti-compact/parts/04_computability.tex index 1bfd353..df03a2a 100644 --- a/semester3/ti-compact/parts/04_computability.tex +++ b/semester3/ti-compact/parts/04_computability.tex @@ -22,6 +22,9 @@ The following objects are uncountable: $[0, 1]$, $\R$, $\cP(\wordbool)$ \inlinecorollary $|\text{KodTM}| < |\cP(\wordbool)|$ and thus there exist infinitely many not recursively enumerable languages over $\alphabetbool$ +\setLabelNumber{theorem}{3} +\inlinetheorem $L_\text{diag} \notin \cL_{RE}$ + \fhlc{Cyan}{Proof of $L$ (not) recursively enumerable} Proving that a language \textit{is} recursively enumerable is as difficult as providing a Turing Machine that accepts it. @@ -43,12 +46,26 @@ In other words, $w_i$ is in $L_\text{diag}$ if and only if $w_i$ is not in $L(M_ 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 $L_\text{diag} \notin \cL_{RE}$ +\drmvspace +\proven + + +Another result (not formally proven in the script, but there is a proof by intimidation) that can come in useful, especially when trying to show $L \notin \cL_{RE}$ is: +\rmvspace +\begin{align*} + L, L^C \in \cL_{RE} \Longleftrightarrow L \in \cL_R +\end{align*} + +\drmvspace +Additionally, as a reminder, $\cL_{RE} = \{ L(M) \divides M \text{ is a TM} \}$, so to prove that a language $L \notin \cL_{RE}$, +we only need to show that there exists no TM $M$, for which $L(M) \in \cL_{RE}$. + + % ──────────────────────────────────────────────────────────────────── +\newpage \subsection{Reductions} \label{sec:reductions} This is the start of the topics that are explicitly part of the endterm. @@ -70,7 +87,7 @@ First off, a list of important languages for this and the next section: \item $L_{H, \lambda} = \{ \text{Kod}(M) \divides M \text{ halts on } \lambda \}$ ($\in \cL_{RE}$, but $\notin \cL_R$) \end{itemize} -\newpage + \setLabelNumber{definition}{3} \fancydef{Recursively reducible languages} $L_1 \leq_R L_2$ ($L_1$ reducible into $L_2$), if $L_2 \in \cL_R \Rightarrow L_1 \in \cL_R$ @@ -138,7 +155,6 @@ given that the transformed word is also in $L_2$. % ──────────────────────────────────────────────────────────────────── -\newpage \subsection{Rice's Theorem} \setLabelNumber{definition}{7} \inlinedef $L$ is called a \bi{semantically non-trivial decision problem}, if these conditions apply: @@ -159,13 +175,6 @@ or the condition can be restated such that only $L(M)$ is described by it. For a more formal proof of that condition, simply show that the implication holds -Another result (not formally proven in the script, but there is a proof by intimidation) that can come in useful, especially when trying to show $L \notin \cL_{RE}$ is: -\rmvspace -\begin{align*} - L, L^C \in \cL_{RE} \Longleftrightarrow L \in \cL_R -\end{align*} - - \stepcounter{subsection} \subsection{The method of the Kolmogorov-Complexity} \inlinetheorem The problem of computing the Kolmogorov-Complexity $K(x)$ for each $x$ is algorithmically unsolvable. diff --git a/semester3/ti-compact/parts/05_complexity.tex b/semester3/ti-compact/parts/05_complexity.tex index 53d96cd..b2059f0 100644 --- a/semester3/ti-compact/parts/05_complexity.tex +++ b/semester3/ti-compact/parts/05_complexity.tex @@ -175,14 +175,19 @@ $A$ is called a polynomial reduction of $L_1$ into $L_2$ A few languages commonly used to show $NP$-completeness: \begin{itemize} \item $SAT = \{ \Phi \divides \Phi \text{ is a satisfiable formula in CNF} \}$ + \item $3SAT = \{ \Phi \divides \Phi \text{ is a satisfiable formula in CNF with all clauses containing \textit{at most} three literals} \}$ \item $\text{CLIQUE} = \{ (G, k) \divides G \text{ is an undirected graph that contains a $k$-clique } \}$ \item $VC = \{ (G, k) \divides G \text{ is an undirected graph with a vertex cover of size $\leq k$ } \}$ - \item $3SAT = \{ \Phi \divides \Phi \text{ is a satisfiable formula in CNF with all clauses containing \textit{at most} three literals} \}$ + \item $SCP = \{ (X, \cS, k) \divides X \text{ has a set cover $\cC \subseteq \cS$ such that $|\cC| \leq k$ } \}$ + \item $DS = \{ (G, k) \divides G \text{ has a dominating set $D$ such that } |D| \leq k \}$ \end{itemize} -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 $u, v \in U$ +where a $k$-clique is a complete subgraph consisting of $k$ vertices in $G$, with $k \leq |V|$; +where a subset $\cC \subseteq \cS$ is a \textit{set cover} of $X$ if $X = \bigcup_{S \in \cC}$; +where a \textit{dominating} set is is a set $D \subseteq V$ such that for every vertex $v \in V$, $v \in D$ or exists $w \in D$ such that $\{ v, w \} \in E$ +and where 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 $SAT \leq_p \text{CLIQUE}$, $SAT \leq_p 3SAT$, $\text{CLIQUE} \leq_p VC$ +We have $SAT \leq_p \text{CLIQUE}$, $SAT \leq_p 3SAT$, $\text{CLIQUE} \leq_p VC$, $VC \leq_p SCP$ and $SCP \leq_p DS$. +Logically, we also have $SAT \leq_p DS$, etc, since $\leq_p$ is transitive (in fact, all reductions are transitive) 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 623c2c5..9c1ed27 100644 Binary files a/semester3/ti-compact/ti-compact.pdf and b/semester3/ti-compact/ti-compact.pdf differ