From 3c8394440bb03c651fb015e81b6e04a98effe30f Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Thu, 9 Oct 2025 19:02:31 +0200 Subject: [PATCH] [TI] Exercise Session Notes --- .../02_finite-automata/00_representation.tex | 22 ++++++++++++++---- .../02_proofs-of-nonexistance.tex | 21 ++++++++++++++--- semester3/ti/ti-summary.pdf | Bin 857451 -> 860837 bytes 3 files changed, 36 insertions(+), 7 deletions(-) diff --git a/semester3/ti/parts/02_finite-automata/00_representation.tex b/semester3/ti/parts/02_finite-automata/00_representation.tex index fbc5b2e..e27c5c8 100644 --- a/semester3/ti/parts/02_finite-automata/00_representation.tex +++ b/semester3/ti/parts/02_finite-automata/00_representation.tex @@ -34,6 +34,7 @@ Heute verwendet man meist einen gerichteten Graphen $G(A)$: \item Jeder Knoten hat den Ausgangsgrad $|\Sigma|$ (wir müssen alle Fälle abdecken) \end{itemize} +% TODO: Clean up (make it look less crowded) \begin{definition}[]{Endlicher Automat} Ist eine Quitupel $M = (Q, \Sigma, \delta, q_0, F)$: \begin{enumerate}[label=\textit{(\roman*)}] @@ -65,6 +66,7 @@ Heute verwendet man meist einen gerichteten Graphen $G(A)$: \end{itemize} \end{definition} Die Übergangsfunktion kann auch gut graphisch oder tabellarisch (wie eine Truth-Table) dargestellt werden. +$M$ ist in der Konfiguration $(q, w) \in Q \times \word$, wenn $M$ in Zustand $q$ ist und noch das Suffix $w$ zu lesen hat (also auf dem Eingabeband hinter dem Zeiger noch $w$ steht) \begin{definition}[]{Reflexive und transitive Hülle} Sei $M = (Q, \Sigma, \delta, q_0, F)$ ein endlicher Automat. Die reflexive und transitive Hülle $\bigvdash{M}{*}$ der Schrittrelation $\bigvdash{M}{}$ von $M$ als @@ -85,14 +87,19 @@ Die Übergangsfunktion kann auch gut graphisch oder tabellarisch (wie eine Truth \end{multicols} \end{definition} Und $(q, w) \bigvdash{M}{*} (p, u)$ bedeutet, dass es eine Berechnung von $M$ gibt, die von der Konfiguration $(q, w)$ zu $(p, u)$ führt, -während $\hat{\delta}(q, w) = p$ bedeutet einfach $(q, w) \bigvdash{M}{*} (p, \lambda)$, also falls $M$ im Zustand $q$ das Wort $w$ zu lesen beginnt, $M$ im Zustand $p$ endet. -Also gilt $L(M) = \{ w \in \Sigma^* \divides (q_0, w) \bigvdash{M}{*} (p, \lambda) \text{ mit } p \in F \} = \{ w \in \Sigma^* \divides \hat{\delta}(q_0, w) \in F \}$ +während $\hdelta(q, w) = p$ bedeutet einfach $(q, w) \bigvdash{M}{*} (p, \lambda)$, also falls $M$ im Zustand $q$ das Wort $w$ zu lesen beginnt, $M$ im Zustand $p$ endet. +Also gilt $L(M) = \{ w \in \Sigma^* \divides (q_0, w) \bigvdash{M}{*} (p, \lambda) \smallhspace \forall p \in F \} = \{ w \in \Sigma^* \divides \hdelta(q_0, w) \in F \}$ + +Intuition $\bigvdash{M}{*}$: Transitivität, also es existieren Zwischenschritte, so dass die Relation erfüllt ist. +Oder noch viel einfacher: Es gibt irgendwieviele Zwischenschritte zwischen dem linken und rechten Zustand + +Intuition $\hdelta$: Der letzte Zustand in der Berechnung ausgehend vom gegebenen Zustand \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 -$\text{Kl}[p] = \{ w \in \Sigma^* \divides \hat{\delta}(q_0, w) = p \} = \{ w \in \Sigma^* \divides (q_0, w) \bigvdash{M}{*} (p, \lambda) \}$ -und entsprechend $\bigcup_{p \in Q} \text{Kl}[p] = \Sigma^*$ und $\text{Kl}[p] \text{Kl}[q] = \emptyset \smallhspace \forall p \neq q \in Q$. +$\text{Kl}[p] = \{ w \in \Sigma^* \divides \hdelta(q_0, w) = p \} = \{ w \in \Sigma^* \divides (q_0, w) \bigvdash{M}{*} (p, \lambda) \}$ +und entsprechend $\bigcup_{p \in Q} \class[p] = \Sigma^*$ und $\class[p] \cup \class[q] = \emptyset \smallhspace \forall p \neq q \in Q$. \shade{Cyan}{Intuition}: Die Klassen sind Mengen, die hier Wörter mit gewissen Eigenschaften, die der EA bestimmt hat, wenn er in Zustand $q_i$ endet, enthalten. Diese Eigenschaften sind beispielsweise, dass alle Wörter, für die der EA in Zustand $q_i$ endet mit einer gewissen Sequenz enden, sie einen gewissen Zahlenwert haben, etc. @@ -129,3 +136,10 @@ sodass Wörter mit denselben Eigenschaften in derselben Klasse liegen und wir da 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). + +% TODO: Move to correct place +Produktautomaten erstellt man, in dem man die (meist zwei) Automaten als einen Gridgraph aufschreibt und eine Art Graph-Layering betreibt, +so dass der eine Graph horizontal und der andere Graph vertikal orientiert ist. +Dann werden die Übergänge folgendermassen definiert: +Für jeden Eingang liefert der Graph, der horizontal ausgerichtet ist, ob wir nach links oder rechts gehen (oder bleiben), +während der vertikal ausgerichtete Graph entscheidet, ob wir nach oben oder unten gehen (oder bleiben). diff --git a/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex b/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex index 01d2a60..6c20713 100644 --- a/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex +++ b/semester3/ti/parts/02_finite-automata/02_proofs-of-nonexistance.tex @@ -21,6 +21,7 @@ so heisst das für uns von jetzt an, dass $A$ nicht zwischen $x$ und $y$ untersc \end{align*} \end{lemma} Das obenstehende Lemma 3.3 ist ein Spezialfall einer Eigenschaft, die für jedes (deterministische) Rechnermodell gilt. +Es besagt eigentlich nichts anderes, als dass wenn das Wort $xz$ akzeptiert wird, so wird auch das Wort $yz$ Mithilfe von Lemma 3.3 kann man für viele Sprachen deren Nichtregularität beweisen. @@ -30,10 +31,24 @@ Wir müssen hier nur formal ausdrücken, dass das Zählen benötigt wird, dass $ Dazu benutzen wir einen indirekten Beweis. Sei $A$ ein EA über $\alphabets{bool}$ und $L(A) = L$. Wir betrachten die Wörter $0^1, 0^2, \ldots, 0^{|Q| + 1}$. -Weil wir $|Q| + 1$ Wörter haben, existiert $i, j \in \{ 1, 2, \ldots, |Q| + 1 \}$, so dass $\hdelta_A(q_0, 0^i) = \hdelta_A(q_0, 0^j)$, -also gilt nach Lemma $0^i z \in L \Leftrightarrow 0^j z \in L \smallhspace \forall z \in \wordbool$. +Weil wir $|Q| + 1$ Wörter haben, existiert $i, j \in \{ 1, 2, \ldots, |Q| + 1 \}$, so dass $\hdelta_A(q_0, 0^i) = \hdelta_A(q_0, 0^j)$, +also gilt nach Lemma $0^i z \in L \Leftrightarrow 0^j z \in L \smallhspace \forall z \in \wordbool$. Dies gilt jedoch nicht, weil für jedes $z = 1^i$ zwar jedes $0^i 1^i \in L$ gilt, aber $0^j 1^j \notin L$ -Um die Nichtregularität konkreter Sprachen zu beweisen, sucht man nach einfach verifizierbaren Eigenschaften, +Um die Nichtregularität konkreter Sprachen zu beweisen, sucht man nach einfach verifizierbaren Eigenschaften, denn wenn eine Sprache eine dieser Eigenschaften \textit{nicht} erfüllt, so ist sie nicht regulär. + +% TODO: For Kolmogorov complexity elaborate some more, i.e. how to do proofs properly / how to derive a word more easily +% -> TA Slides explain that really well + +Eine Methode zum Beweis von Aussagen $L \notin \mathcal{L}_{\text{EA}}$ nennt sich \bi{Pumping} und basiert auf folgender Idee: +Wenn für ein Wort $x$ und einen Zustand $p$ gilt, dass $(p, x) \bigvdash{A}{*} (p, \lambda)$, so gilt auch für alle $i \in \N$, dass $(p, x^i) \bigvdash{A}{*} (p, \lambda)$. +Also kann $A$ nicht zwischen $x$ und $x^i$ unterscheiden, oder in anderen Worten, wie viele $x$ er gelesen hat, +also akzeptiert $A$ entweder alle Wörter der Form $yx^iz$ (für $i \in \N$) oder keines davon + +\begin{lemma}[]{Pumping-Lemma für reguläre Sprachen} + Sei $L$ regulär. 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