From 5528b6863adc0a8ed2e1a1c186d8ddf6d9604069 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 15 Oct 2025 17:09:03 +0200 Subject: [PATCH] [TI] Turing machine intro --- .../ti/parts/03_turing_machines/00_intro.tex | 48 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 925411 -> 934427 bytes semester3/ti/ti-summary.tex | 2 + 3 files changed, 50 insertions(+) diff --git a/semester3/ti/parts/03_turing_machines/00_intro.tex b/semester3/ti/parts/03_turing_machines/00_intro.tex index f3ac851..703353d 100644 --- a/semester3/ti/parts/03_turing_machines/00_intro.tex +++ b/semester3/ti/parts/03_turing_machines/00_intro.tex @@ -1,2 +1,50 @@ % Starts on Page 94 (= 108 in PDF) \subsection{Das Modell der Turingmaschine} +Eine Turingmaschine (oft auch Turing-Maschine geschrieben) besteht informell aus +\begin{enumerate}[label=(\roman*)] + \item einer endlichen Kontrolle, die das Programm enthält + \item einem Arbeitsband unendlicher Länge (das es Erlaubt, im Vergleich zum EA, Daten zu speichern) + \item einem Lese-/Schreibkopf, der sich in beide Richtungen auf dem Band bewegen kann +\end{enumerate} +Formaler: +\begin{definition}[]{Turingmaschine (TM)} + Eine \bi{Turingmaschine} ist eine $7$-Tupel $M = (Q, \Sigma, \Gamma, \delta, q_0, \qacc, \qrej)$, wobei: + \begin{enumerate}[label=(\roman*)] + \item $Q$ ist die \bi{Zustandsmenge} + \item $\Sigma$ ist das \bi{Eingabealphabet} mit $\cent, \textvisiblespace \notin \Sigma$ + \item $\Gamma$ ist das \bi{Arbeitsalphabet} mit $\Sigma \subseteq \Gamma$, $\cent, \textvisiblespace \in \Gamma$ und $\Gamma \cap Q = \emptyset$ + ($\cent$ = Startmarker, $\textvisiblespace$ = Blanksymbol) + \item $\delta : (Q - \{ \qacc, \qrej \}) \times \Gamma \longrightarrow Q \times \Gamma \times \{ L, R, N \}$ ist die \bi{Übergangsfunktion von $M$}, + wobei $\{ L, R, N \}$ die möglichen Bewegunsrichtungen des Lese-/Schreibkopfs sind + und hat die Eigenschaft $\delta(q, \cent) \in Q \times \{ \cent \} \times \{ R, N \}$ für alle $q \in Q$ + \item $q_0$ ist der \bi{Anfangszustand} + \item $\qacc$ ist der \bi{akzeptierende Zustand} (genau einer in jedem $M$) + \item $\qrej$ ist der \bi{verwerfende Zustand} (genau einer in jedem $M$) + \end{enumerate} + Eine \bi{Konfiguration} $C$ von $M$ ist ein Element aus $\text{Konf}(M) = \{ \cent \} \cdot \Gamma^* \cdot Q \cdot \Gamma^+ \cup Q \cdot \{ \cent \} \Gamma^+$ + (wobei $\cdot$ die Konkatenation ist) + + Eine \bi{Startkonfiguration} für ein Eingabewort $x$ ist $q_0\cent x$ + + Ein \bi{Schritt von $M$} ist eine Relation $\bigvdash{M}{}$ auf der Menge der Konfigurationen, also $\bigvdash{M}{} \subseteq \text{Konf}(M) \times \text{Konf}(M)$. + + 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. +Rekursive Sprachen und entscheidbare Entscheidungsprobleme sind algorithmisch erkennbar, respektive lösbar. + +Es gibt auch definitionen der TM, die ohne Startmarker $\cent$ auskommen, bei denen ist das Arbeitsband in beide Richtungen unendlich. + +Graphisch stellt man Turingmaschinen folgendermassen dar: +Wir haben wieder einen Graphen mit gerichteten Kanten. +Für $\delta(q, a) = (p, b, X)$ mit $q, p \in Q$, $a, b \in \Sigma$ und $X \in \{ L, R, N \}$ werden die Kanten mit folgendem Format beschriftet: +$q \rightarrow a, X$. + +Mit einem TM die Sprache $\{ 0^n 1^n \divides n \in \N \}$ erkennen kann man nun, indem man jeweils das linkeste und rechteste Symbol durch ein anderes Symbol ersetzt, +beispielsweise, wenn das Eingabealphabet $\Sigma = \{ 0, 1 \}$, dann könnte man $\Gamma$ das Symbol $2$ hinzufügen, mit dem man jedes bearbeitete Symbol ersetzt. + +Im Buch wird als Beispiel auf Seite 99ff (= Seite 114ff im PDF) ein komplexeres Wort gewählt, +bei welchem ein zeichen $a \in \alphabetbool$ durch $(a, B) \in \alphabetbool \times \{ A, B \}$ ersetzt wird, +da wir zwei Phasen haben und zwischen denen 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