From ff4eb22f130cd63be8e237290c06f9bfecfa2caf Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 10 Oct 2025 21:31:26 +0200 Subject: [PATCH] [TI] Non-determinism: Done up to T3.2, i.e. current state --- .../02_finite-automata/03_non-determinism.tex | 75 +++++++++++++++++- semester3/ti/ti-summary.pdf | Bin 909494 -> 2356 bytes 2 files changed, 71 insertions(+), 4 deletions(-) diff --git a/semester3/ti/parts/02_finite-automata/03_non-determinism.tex b/semester3/ti/parts/02_finite-automata/03_non-determinism.tex index 1499034..97e8c7d 100644 --- a/semester3/ti/parts/02_finite-automata/03_non-determinism.tex +++ b/semester3/ti/parts/02_finite-automata/03_non-determinism.tex @@ -15,17 +15,84 @@ Wir notieren das Ganze in graphischer Darstellung so, dass wir aus einem Zustand \item \bi{Anfangszustand:} $q_0 \in Q$ \item \bi{Akzeptierende Zustände:} $F \subseteq Q$ \end{enumerate} - Der Rest der Eigenschaften ist sehr ähnlich wie die des deterministischen EA, mit der bedeutenden Ausnahme, - dass ein Schritt in der $\delta$-Notation nicht $\delta(q, a) = p$, sondern $p \in \delta(q, a)$ ist, da die Übergangsfunktion ja jetzt ins - Powerset von $Q$ anstelle von nach $Q$ direkt mapped. Die komplette Definition des Schritts ist also: + Ein \bi{Schritt} in der $\delta$-Notation ist im Vergleich zum deterministischen EA nicht $\delta(q, a) = p$, + sondern $p \in \delta(q, a)$ ist, da die Übergangsfunktion ja jetzt ins Powerset von $Q$, + anstelle von nach $Q$ direkt mapped. Die komplette Definition des Schritts ist also: \begin{align*} (q, w) \bigvdash{M}{} (p, x) \Longleftrightarrow w = ax \text{ für ein } a \in \Sigma \text{ und } p \in \delta(q, a) \end{align*} + + Eine \bi{Berechnung von $M$} ist eine endliche Folge $D_1, D_2, \ldots, D_k$ von Konfigurationen, + wobei $D_i \bigvdash{M}{} D_{i + 1}$ für $i = 1, \ldots, k - 1$ + + Eine \bi{Berechnung von $M$ auf $x$} hingegen ist eine Berechnung $C_0, C_1, \ldots, C_m$ von $M$, + wobei $C_0 = (q_0, x)$ und entweder $C_m \in Q \times \{ \lambda \}$ oder $C_M = (q, ay)$ für ein $a \in \Sigma, y \in \word$ und $q \in Q$, + so dass $\delta(q, a) = \emptyset$. + + $C_0, \ldots, C_m$ ist \bi{akzeptierend} falls $C_m = (p, \lambda)$ für ein $p \in F$ + + Die Sprache $L(M) = \{ w \in \word \divides (q_0, w) \bigvdash{M}{*} (p, \lambda) \text{ für ein } p \in F \}$ Für die $\hdelta$-Funktion, gilt nun $\hdelta(q, \lambda) = \{ q \}$ für jedes $q \in Q$ und wir definieren: \begin{align*} \hdelta(q, wa) & = \{ p \in Q \divides \text{es existiert ein } r \in \hdelta(q, w), \text{ so dass } p \in \delta(r, a) \} \\ & = \bigcup_{r \in \hdelta(q, w)} \delta(r, a) \smallhspace \forall q \in Q, a \in \Sigma, w \in \word \end{align*} - % Page 92 in PDF currently \end{definition} +Ein Wort ist in $L(M)$, falls $M$ mindestens eine akzeptierende Berechnung auf $x$ hat. + +Bei einer akzeptierenden Berechnung \textit{auf} $x$ wird wie beim EA gefordert, dass das ganze Wort $x$ gelesen worden ist und $M$ nach dem Lesen in einem akzeptierenden Zustand ist. + +Bei NEA kann eine nicht akzeptierende Berechnung auch vor Beendung des Lesevorgangs enden, da wir hier nicht vorschreiben, +dass es für jedes Symbol des Eingabealphabets eine definierte Übergangsfunktion gibt, es ist also erlaubt, dass bspw. $\delta(q, a) = \emptyset$. + +Zudem haben wir aus der Definition von $\hdelta$ eine alternative Definition der von $M$ akzeptierten Sprache: +$L(M) = \{ w \in \word \divides \hdelta(q_0, w) \cap F \neq \emptyset \}$ + +Für NEA kann man einen \bi{Berechnungsbaum $\mathcal{B}_M(x)$ von $M$ auf $x$} erstellen, der dann anschaulich alle möglichen Enden der Berechnung darstellt. +Wir beginnen den Baum mit Konfiguration $(q_0, x)$ und führen dann mit den Kanten alle möglichen Berechnungen aus, die mit dem ersten Symbol des Wortes möglich sind. + +Wir erreichen so also zum Beispiel die Konfiguration $(q_1, x_1)$, wobei $x_1$ $x$ ohne das erste Zeichen ist. + + +\fancylemma{NEA aus Abbildung 3.15 im Buch} Sei $M$ der NEA aus Abbildung 3.15 im Buch (auf Seite 77 (= 92 im PDF) zu finden). +Dann ist $L(M) = \{ x11y \divides x, y \in \wordbool \}$ + +Der Beweis für eine solche Aussage läuft oft über Teilmengen (also mit $X \subseteq Y \land Y \subseteq X \Leftrightarrow X = Y$). + + +Eine zentrale Frage dieses Kapitels ist es, ob $\mathcal{L}_{\text{NEA}} = \mathcal{L}_{\text{EA}}$, wobei $\mathcal{L}_{\text{NEA}} = \{ L(M) \divides M \text{ ist ein NEA} \}$. +In anderen Worten: Können EA die Arbeit von NEA simulieren? + +Ja, es ist möglich und gilt allgemein, dass die Simulation von Nichtdeterminismus durch Determinismus nur dann realisierbar ist, +wenn es möglich ist, alle nichtdeterministischen Berechnungen durch deterministische Berechnungen nachzuahmen. + +Bei EA (nennen einen $A$ im Folgenden) basiert diese Idee auf BFS der Berechnungsbäume von $M$. +Die Idee ist dann, dass alle Knoten mit Entfernung $i$ von der Wurzel die ersten $i$ Symbole von $x$ gelesen haben. +Da NEA endlich viele Konfigurationen bei Entfernung $i$ haben ist es möglich, die Transformation durchzuführen. +Wenn es zwei Knoten $u \neq v$ identisch sind, so müssen wir nur in einem der Teilbäume nach einer akzeptierenden Berechnung suchen. + +\bi{Potenzmengenkonstruktion:} Ein Zustand $\langle P \rangle$ von $A$ für $P \subseteq Q$ erhält die Bedeutung, +dass nach der gegebenen Anzahl an Berechnungsschritten genau die Zustände aus $P$ in den Berechnungen von $M$ auf der gegebenen Ebene erreichbar sind, also $P = \hdelta(q_0, z)$. +Ein Berechnungsschritt in $A$ aus einem Zustand $\langle P \rangle$ für ein gelesenes Symbol $a$ bedeutet die Bestimmung der Menge $\bigcup_{p \in P} \delta(p, a)$, +also aller Zustände, die aus irgendeinem Zustand $p \in P$ beim Lesen von $a$ erreichbar sind. + +Dabei benutzen wir $\langle P \rangle$ statt $P$, um zu verdeutlichen, dass wir eine Zustand von $A$ und nicht die Menge der Zustände von $M$ bezeichnen. + +\inlinetheorem Zu jedem NEA $M$ existiert ein EA $A$, so dass $L(M) = L(A)$ + +Um $L(M) = L(A)$ zu zeigen, müssen wir folgende Äquivalenz beweisen: +\rmvspace +\begin{align*} + \forall x \in \word : \hdelta_M(q_0, x) = P \Longleftrightarrow \hdelta(q_{0 A}, x) = \langle P \rangle +\end{align*} + +\rmvspace +Wir können dies über einen Induktionsbeweis tun und ein vollständiger Beweis findet sich unten auf Seite 82 (= Seite 97 im PDF) im Buch. + +Wir sagen, dass zwei Automaten \bi{äquivalent} sind, falls $L(A) = L(B)$. + +Eine Folge von Satz 3.2 ist eben, 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