From c9d38f502419ee86206b492c3480575b76ea3a00 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 30 Oct 2025 13:56:15 +0100 Subject: [PATCH] [TI] Diagnolaization section done --- .../ti/parts/03_turing_machines/00_intro.tex | 2 + .../ti/parts/04_computability/00_intro.tex | 50 ++++++++++++++++++ .../parts/04_computability/01_reduction.tex | 1 + semester3/ti/ti-summary.pdf | Bin 959023 -> 967548 bytes semester3/ti/ti-summary.tex | 2 + 5 files changed, 55 insertions(+) create mode 100644 semester3/ti/parts/04_computability/01_reduction.tex diff --git a/semester3/ti/parts/03_turing_machines/00_intro.tex b/semester3/ti/parts/03_turing_machines/00_intro.tex index 48cf517..7876d76 100644 --- a/semester3/ti/parts/03_turing_machines/00_intro.tex +++ b/semester3/ti/parts/03_turing_machines/00_intro.tex @@ -28,6 +28,8 @@ Formaler: Ein \bi{Schritt von $M$} ist eine Relation $\bigvdash{M}{}$ auf der Menge der Konfigurationen, also $\bigvdash{M}{} \subseteq \text{Konf}(M) \times \text{Konf}(M)$. + $\mathcal{L}_{RE} = \{ L(M) \divides M \text{ ist eine Turingmaschine} \}$ + Der Rest der Definition findet sich auf Seiten 96 - 98 (= Seiten 110 - 112 im PDF) \end{definition} Turingmaschinen, die immer halten, repräsentieren Algorithmen, die immer terminieren und die richtige Ausgabe liefern. diff --git a/semester3/ti/parts/04_computability/00_intro.tex b/semester3/ti/parts/04_computability/00_intro.tex index e69de29..7f61fdc 100644 --- a/semester3/ti/parts/04_computability/00_intro.tex +++ b/semester3/ti/parts/04_computability/00_intro.tex @@ -0,0 +1,50 @@ +\subsection{Diagonalisierung} +Wir definieren $\text{KodTM}$ als die \textit{Menge der binären Kodierungen aller Turingmaschinen}. +Wir haben $\text{KodTM} \subseteq \wordbool$ und die obere Schranke der Kardinalität ist $|\wordbool|$, da es unendlich viele Turingmaschinen gibt. + +Im Folgenden wird wieder Cantor's Diagonalisierungsmethode verwendet +\begin{recall}[]{Cantor's Diagonalization Argument} + % TODO: Finish + \TODO Finish +\end{recall} + +\inlinedef $A$ und $B$ sind Mengen. Dann ist $|A| \leq |B|$ falls eine \textit{injektive} Funktion $f$ von $A$ nach $B$ existiert; +$|A| = |B|$ falls $|A| \leq |B|$ und $|B| \leq |A|$ (es existiert eine Bijektion); +$|A| < |B|$ falls $|A| \leq |B|$ und keine injektive Abbildung von $B$ nach $A$ existiert. + +Um zu zeigen, dass es nicht rekursiv aufzählbare (also von Turingmaschinen nicht erkennbare) Sprachen gibt. +Also müssen wir laut Definition \ref{definition:5-1} nur zeigen, dass keine Injektion von $\wordbool$ nach $\mathcal{L}_{RE}$ existiert. + +\fancydef{Abzählbarkeit} $A$ heisst abzählbar, falls $A$ endlich ist oder $|A| = |\N|$ + +\inlinelemma Sei $\Sigma$ ein beliebiges Alphabet. Dann ist $\word$ abzählbar + +\inlinetheorem Die Menge $\text{KodTM}$ der Turingmaschinenkodierungen ist abzählbar + +\inlinelemma $(\N - \{ 0 \}) \times (\N - \{ 0 \})$ ist abzählbar. Die Idee ist dieselbe wie für $|Q^+| = |\N|$, nämlich, dass wir jedem Element einen Index zuordnen können. + +\inlinetheorem $\Q^+$ ist abzählbar. Die Idee für den Beweis ist eine Bijektion nach obiger Menge zu finden. + +\inlinetheorem $[0, 1] \subseteq \R$ ist nicht abzählbar. Dies kann mit Cantor's Diagonalization Argument bewiesen werden. + +\inlinetheorem $\mathcal{P}(\wordbool)$ ist nicht abzählbar + +\inlinecorollary $|\text{KodTM}| < |\mathcal{P}(\wordbool)|$ und es existieren also unendlich viele nicht rekursiv aufzählbare Funktionen. + +Um für eine spezifische Sprache zu beweisen, dass sie rekursiv aufzählbar ist, können wir einfach eine Turingmaschine konstruieren. +Für eine Beweis dafür, dass eine Sprache nicht rekursiv aufzählbar ist können wir folgende Methode verwenden. Sei dazu mit $d_{ij} = 1 \Longleftrightarrow M_i \text{ akzeptiert } w_j$ +\begin{align*} + L_{\text{diag}} & = \{ w \in \wordbool \divides w = w_i \text{ für ein } i \in \N - \{ 0 \} \text{ und } M_i \text{ akzeptiert } w_i \text{ nicht} \} \\ + & = \{ w \in \wordbool \divides w = w_i \text{ für ein } i \in \N - \{ 0 \} \text{ und } d_{ii} = 0 \} +\end{align*} + +\inlinetheorem $L_{\text{diag}} \notin \mathcal{L}_{RE}$ +\inlineproof Zum Widerspruch nehmen wir an, dass $L_\text{diag} \in \mathcal{L}_{RE}$. +Dann gilt, dass $L_\text{diag} = L(M)$ für eine Turingmaschine $M$. +$M$ ist eine Turingmaschine in der kanonischen Ordnung der Turingmaschinen, also existiert ein $i \in \N - \{ 0 \}$, so dass $M = M_i$. + +Dies führt zu einem Widerspruch, denn $L_\text{diag}$ kann nicht gleich $L(M_i)$ sein, da +\begin{align*} + w_i \in L_\text{diag} \Longleftrightarrow d_{ii} \Longleftrightarrow w_i \notin L(M_i) +\end{align*} +also ist $w_i$ genau dann in $L_\text{diag}$ wenn $w_i$ \textit{nicht} in $L(M_i)$ ist und umgekehrt. 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