From 3cf40eb21694e55204d78136d290ed112d77623a Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Wed, 22 Oct 2025 16:58:08 +0200 Subject: [PATCH] [TI] Multi-Band TM --- .../01_multi-band-church-thesis.tex | 30 ++++++++++++++++-- semester3/ti/ti-summary.pdf | Bin 460113 -> 940783 bytes 2 files changed, 27 insertions(+), 3 deletions(-) diff --git a/semester3/ti/parts/03_turing_machines/01_multi-band-church-thesis.tex b/semester3/ti/parts/03_turing_machines/01_multi-band-church-thesis.tex index 4b6858c..9e19a0a 100644 --- a/semester3/ti/parts/03_turing_machines/01_multi-band-church-thesis.tex +++ b/semester3/ti/parts/03_turing_machines/01_multi-band-church-thesis.tex @@ -1,9 +1,33 @@ \newsection \subsection{Mehrband-Turingmaschinen und Church'sche These} -Die Turingmaschinen sind das Standardmodell der Berechenbarkeitstheorie, aber benötigen einige Modifikationen, um wirklich geeignet zu sein +Die Turingmaschinen sind das Standardmodell der Berechenbarkeitstheorie, aber benötigen einige Modifikationen, um wirklich geeignet zu sein (da das Von-Neumann Modell physisch unterschiedliche CPU, Eingabemedium und Speicher für Programme und Daten fordert, aber die TM ein gemeinsames Eingabemedium und Speicher hat). -Eine $k$-Ban-Turingmaschine (für $k \in \N_0$) hat folgende Komponenten: +Eine $k$-Band-Turingmaschine (für $k \in \N_0$) hat folgende Komponenten: \begin{itemize} - \item + \item eine endliche Kontrolle (= Programm) + \item ein endliches Band mit einem Lesekopf + \item $k$ Arbeitsbänder, jedes mit eigenem Lese-/Schreibkopf \end{itemize} +Zu Beginn ist die MTM in folgender Situation: +\begin{itemize} + \item Das Eingabeband enthält $\cent w \$$, wobei $\cent$ und $\$$ die linke / rechte Seite der Eingabe markieren + \item Der Lesekopf des Eingabebands zeigt auf $\cent$ + \item Alle Arbeitsbänder beinhalten $\cent \text{\textvisiblespace\textvisiblespace} \ldots$ und deren Lese-/Schreibköpfe zeigen auf $\cent$ + \item Die endliche Kontrolle ist im Anfangszustand $q_0$ +\end{itemize} +Alle $k + 1$ Köpfe dürfen sich während der Berechnung in beide Richtungen bewegen (solange das nicht out-of-bounds geht). +Zudem darf der Lesekopf nicht schreiben, also beleibt der Inhalt des Eingabebands gleich. + +Gleich wie bei einer TM ist das Arbeitsalphabet der Arbeitsbänder $\Gamma$ und alle Felder der Arbeitsbänder sind von links nach rechts nummeriert, wobei $0$ bei $\cent$ liegt. + +Eine Konfiguration einer $k$-Band-TM $M$ ist $(q, w, i, u_1, i_1, u_2, i_2, \ldots, u_k, i_k) \in Q \times \word \times \N \times (\Gamma^* \times \N)^k$, +wobei $q$ der Zustand ist, der Inhalt des Eingabebands ist $\cent w \$$, der Lesekopf zeigt auf das $i$-te Feld, +für $j \in \{ 1, 2, \ldots, k \}$ ist der Inhalt des $j$-ten Bandes $\cent u_k \text{\textvisiblespace} \ldots$ und $i_j \leq |u_j|$ ist die Position des Feldes. + +Ein Berechnungsschritt von $M$ kann mit $\delta: Q \times (\Sigma \cup \{ \cent, \$ \}) \times \Gamma^k \rightarrow Q \times \{ L, R, N \} \times (\Gamma \times \{ L, R, N \})^k$ + dargestellt werden, wobei die Argumente $(q, a, b_1, \ldots, b_k)$ der aktuelle Zustand $q$, das gelesene Eingabesymbol $a$ und die $k$ Symbole $b_i \in \Gamma$, +auf welchen die Köpfe der Arbeitsbänder stehen. + +Die Eingabe $w$ wird von $M$ akzeptiert, falls $M$ den Zustand $\qacc$ erreicht und falls $M$ den Zustand $\qrej$ erreicht oder nicht terminiert, wird die Eingabe verworfen. +% Page 120 diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index acb1ff4079b2234893ddcb920425ac721b379b8a..88de9266178b4a80cd054e595bf804c908b33ca9 100644 GIT binary patch delta 478164 zcmce;1yq&M);5ZuAPA_GfTVOtcT0DNG$P#~A^CzxBi-E%(j}#v?(S}oP*VC|_?>gl z|K0oFKgK=Zbqod$*zC35wdQ)}GoSgaJrYcCSi|l8`9U@Ot{);R8#gOAXY>PP$^aaH zc<_Y)z8-?FM|C*KwN)@s;)T^UK6=*{5%*np zPfwxy`N5>q_S9G6i~u|lVd43Oh10}0W4G}1(UfvozO^|k)t=MC5idXT9GQI7(!wtt zW}iW=nmOFN!5LhjMavhRi%~8(+?^Gx>72^=vh1&e74)YHgM25YCGithab)Sn*HS 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