From 61ef0e2af5c5b5a561a682d024ebc313cb7c2a10 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 10 Oct 2025 18:22:40 +0200 Subject: [PATCH] [TI] Start section on non-determinism, done up to definition 3.3 --- .../02_finite-automata/03_non-determinism.tex | 29 ++++++++++++++++++ semester3/ti/ti-summary.pdf | Bin 897861 -> 909494 bytes 2 files changed, 29 insertions(+) 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 c643462..1499034 100644 --- a/semester3/ti/parts/02_finite-automata/03_non-determinism.tex +++ b/semester3/ti/parts/02_finite-automata/03_non-determinism.tex @@ -1,2 +1,31 @@ \newsection \subsection{Nichtdeterminismus} +Einfach gesagt werden hier Automaten behandelt, die zufällige (genannt \bi{nichtdeterministische}) Entscheidungen treffen. +Beispielsweise für ein Entscheidungsproblem $(\Sigma, L)$ bedeutet dies, dass ein nichtdeterministischer EA $A$ eine Sprache $L$ akzeptiert, +falls für jedes $x \in L$ mindestens eine akzeptierende Berechnung von $A$ auf $x$ existiert und für $y \in \word - L$ keine solve existiert. + +Wir notieren das Ganze in graphischer Darstellung so, dass wir aus einem Zustand mehrere Übergänge mit dem gleichen Eingabesymbol erlauben. + +\begin{definition}[]{nichtdeterministischer Endlicher Automat (NEA)} + Ein NEA ist eine Quitupel $M = (Q, \Sigma, \delta, q_0, F)$: + \begin{enumerate}[label=\textit{(\roman*)}] + \item \bi{Zustandsmenge:} $Q$ ist eine endliche Menge + \item \bi{Eingabealphabet:} $\Sigma$ ist ein Alphabet + \item \bi{Übergangsfunktion:} $\delta : Q \times \Sigma \rightarrow \mathcal{P}(Q)$. $\mathcal{P}(Q)$ ist das Powerset hierbei + \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: + \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*} + + 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} diff --git a/semester3/ti/ti-summary.pdf b/semester3/ti/ti-summary.pdf index 751c930a3fc029f997f33d4ae91df26c85115e3e..e28588976e30fe27ec345e811b0387b1dbf3988d 100644 GIT binary patch delta 51545 zcmV(wK#SRM8&I7HZPK@t(=OZt2{L;VPnf)M_C-QW%lcIpFBZs@Q zGqW?Z^RYBpCNz0<_WaG+H!rzK3Q}mUlQ%bsEQqE$$qPw}TqSQ7$=meJZME4|UEN-v 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