From 7be19c839db88d3e37fb38dbfa8d4eece70cdbb9 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 4 Oct 2025 10:08:43 +0200 Subject: [PATCH] [TI] Finish finite automata representation section --- .../02_kolmogorov-complexity.tex | 2 +- .../02_finite-automata/00_representation.tex | 33 ++++++++++++++++-- .../02_finite-automata/01_simulations.tex | 4 +++ semester3/ti/ti-summary.pdf | Bin 834892 -> 836110 bytes semester3/ti/ti-summary.tex | 1 + 5 files changed, 37 insertions(+), 3 deletions(-) create mode 100644 semester3/ti/parts/02_finite-automata/01_simulations.tex diff --git a/semester3/ti/parts/01_languages-problems/02_kolmogorov-complexity.tex b/semester3/ti/parts/01_languages-problems/02_kolmogorov-complexity.tex index f199e23..9209db6 100644 --- a/semester3/ti/parts/01_languages-problems/02_kolmogorov-complexity.tex +++ b/semester3/ti/parts/01_languages-problems/02_kolmogorov-complexity.tex @@ -98,4 +98,4 @@ Lemma 2.6 zeigt nicht nur, dass es unendlich viele Primzahlen geben muss, sonder \end{align*} \end{theorem} -Der Beweis hierfür ist sehr ausführlich ab Seite 42 im Buch erklärt +Der Beweis hierfür ist sehr ausführlich ab Seite 42 (= 57 im PDF) im Buch erklärt diff --git a/semester3/ti/parts/02_finite-automata/00_representation.tex b/semester3/ti/parts/02_finite-automata/00_representation.tex index a31ea76..d7dd767 100644 --- a/semester3/ti/parts/02_finite-automata/00_representation.tex +++ b/semester3/ti/parts/02_finite-automata/00_representation.tex @@ -90,9 +90,38 @@ Also gilt $L(M) = \{ w \in \Sigma^* \divides (q_0, w) \bigvdash{M}{*} (p, \lambd \inlinelemma $L(M) = \{ w \in \{ 0, 1 \}^* \divides |w|_0 + |w|_1 \equiv 0 \text{ mod } 2 \}$ -Jeder EA teilt die Menge $\Sigma^*$ in $|Q|$ Klassen +Jeder EA teilt die Menge $\Sigma^*$ in $|Q|$ Klassen $\text{Kl}[p] = \{ w \in \Sigma^* \divides \hat{\delta}(q_0, w) = p \} = \{ w \in \Sigma^* \divides (q_0, w) \bigvdash{M}{*} (p, \lambda) \}$ und entsprechend $\bigcup_{p \in Q} \text{Kl}[p] = \Sigma^*$ und $\text{Kl}[p] \text{Kl}[q] = \emptyset \smallhspace \forall p \neq q \in Q$. -In dieser Terminologie gilt dann $L(M) = \bigcup_{p \in F} \text{Kl}[p]$ +In dieser Terminologie gilt dann $L(M) = \bigcup_{p \in F} \text{Kl}[p]$. +Die Notation $|w|_i$ bedeutet die Länge der Buchstaben $i$ in $w$. Wir können $L(M)$ mit Klassen bestimmen und haben eine Äquivalenzrelation $x R_\delta y \Leftrightarrow \hat{\delta}(q_0, x) = \hat{\delta}(q_0, y)$ auf $\Sigma^*$. +Man beweist die Korrektheit der gewählten Klassen oft mithilfe von Induktion über die Länge der Wörter. +Wir beginnen mit der Länge an Wörtern der Länge kleiner gleich zwei und erhöhen dies dann während unseres Induktionsschrittes. + +Die Klassen bestimmen wir vor dem Beginn der Induktion auf und jede Klasse repräsentiert einen der Zustände. +\begin{wrapfigure}[5]{l}{0.3\textwidth} + \begin{tables}{ccc}{Zustand & 0 & 1} + $q_0$ & $q_2$ & $q_1$ \\ + $q_1$ & $q_3$ & $q_0$ \\ + $q_2$ & $q_0$ & $q_3$ \\ + $q_3$ & $q_1$ & $q_2$ \\ + \end{tables} +\end{wrapfigure} +Haben wir einen EA mit nebenstehender Tabelle, so sind die Klassen für unseren EA $M$ sind $\class[q_0], \ldots, \class[q_3]$, definiert durch: +\rmvspace +\begin{align*} + \class[q_0] & = \{ w \in \wordbool \divides |w|_0 \text{ und } |w|_1 \text{ sind gerade} \} \\ + \class[q_1] & = \{ w \in \wordbool \divides |w|_0 \text{ ist gerade, } |w|_1 \text{ ist ungerade} \} \\ + \class[q_2] & = \{ w \in \wordbool \divides |w|_0 \text{ ist ungerade, } |w|_1 \text{ ist gerade} \} \\ + \class[q_3] & = \{ w \in \wordbool \divides |w|_0 \text{ und } |w|_1 \text{ sind ungerade} \} +\end{align*} +% +Falls ein EA $A$ genügend anschaulich und strukturiert dargestellt ist, kann man die Sprache $L(A)$ auch ohne Beweis bestimmen. + +Idealerweise konstruieren wir einen EA so, dass wir die Menge aller Wörter aus $\Sigma^*$ so in Klassen aufteilen, +sodass Wörter mit denselben Eigenschaften in derselben Klasse liegen und wir dann Übergangsfunktionen zu anderen Klassen finden, +die nur einen Buchstaben aus $\Sigma$ zum Wort hinzufügen + +\inlineex Das Buch enthält einige zwei gute Beispiele (Beispiel 3.1 und 3.2) mit ausführlichen Erklärungen ab Seite 58 (= Seite 73 im PDF). diff --git a/semester3/ti/parts/02_finite-automata/01_simulations.tex 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z-+HbmcGcWaA-x!fysg}>qSk2Fr$61C+dFP-q-_1tDpM7nAjRS nIXjvd*!&0C8Ct?PIU6`SyE~eg!f-G%v$3(kkdcYViNgFpC>rP9 diff --git a/semester3/ti/ti-summary.tex b/semester3/ti/ti-summary.tex index e2c5474..b1fcb2f 100644 --- a/semester3/ti/ti-summary.tex +++ b/semester3/ti/ti-summary.tex @@ -71,6 +71,7 @@ \section{Endliche Automaten} \stepcounter{subsection} \input{parts/02_finite-automata/00_representation.tex} +\input{parts/02_finite-automata/01_simulations.tex} \end{document}