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z5zoUXQTg0I;`A0|7MSJflFmT7$e~B^J~c86iS@ipR1# zyWX`eC(onw#RC1);l&MQBe|}S3R7m$O zhn+9RFPGQ|m^VXWJHcu?A(C=A!FxC%Bj0LK=zcKoX+oL3GHXlbDCM_dkB<|Bz)}^Y X*%gBmAOhHUI5|1tC@IAiCE)%SA^;); diff --git a/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex b/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex index fd5fd47..f54b59a 100644 --- a/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex +++ b/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex @@ -96,15 +96,30 @@ \begin{parts} \part Eine Regel $(l\rightarrow r)$ einer Grammatik $G=(V,\sum,P,S)$ heißt rechtslinear, falls ... \begin{solution} + immer das an der am weitesten rechts stehende Nicht-Terminal in ein Terminal umgewandelt wird. + Dazu muss $l\in V$ und $r\in \sum V\cup {\epsilon}$. \end{solution} + \part Ein NFA ist ein Tupel $M=(...)$ \begin{solution} + ein nichtdeterministischer endlicher Automat $M$ ist ein 5-Tupel $M=(Z,\sum,S,\delta,E)$ mit + \begin{itemize} + \item $Z$ ist eine endliche Menge von Zuständen + \item $\sum$ ist das Eingabealphabet + \item $S\subseteq Z$ die Menge der Startzustände (können mehrere sein) + \item $\delta: Z \times \sum \rightarrow P(Z)$ ist die (Menge der) Überführungs/Übergangsfunktion + \item $E\subseteq Z$ die Menge der Endzustände + \end{itemize} \end{solution} + \part Die von einem PDA $M=(Z,\sum, \Gamma, \delta, z_0, \#)$ akzeptierten Sprache ist $L(M)=...$ \begin{solution} + $L(M)=\{x\in\sum^* | \text{ es gibt } z\in Z \text{ mit } (z_0, x, \#) [...] ^*(z,\epsilon, \epsilon)\}$ \end{solution} + \part Sei $M=(Z,\sum,z_0,\delta, E)$ ein DFA. Die Zustände $z,z'\in Z$ heißen erkennungsäquivalent, wenn \begin{solution} + Zwei Zustände $z,z'\in Z$ heißen erkennungsäquivalent ($z\equiv z'$) wenn für jedes Wort $w\in \sum^*$ gilt: $\hat{\sigma}(z,w)\in E \leftrightarrow \hat{\sigma}(z',w)\in E$. \end{solution} \end{parts} @@ -112,12 +127,29 @@ \begin{parts} \part Sei $L\supseteq \sum^*$ eine Sprache. Dann sind äquivalent: 1) L ist regulär (d.h. wird von einem DFA akzeptiert), 2)..., 3)... \begin{solution} + \begin{enumerate} + \item L ist regulär (d.h. von einem DFA akzeptiert) + \item L wird von einem NFA akzeptiert + \item L ist rechtslinear (d.h. von einer Typ-3 Grammatik erzeugt) + \end{enumerate} \end{solution} + \part Der Satz von Myhill-Nerode besagt,... \begin{solution} + Sei L eine Sprache. L ist regulär $\leftrightarrow index(R_L)< \infty$ (d.h. nur wenn die Myhill-Nerode-Äquivalenz endliche Klassen hat). \end{solution} + \part Das Pumping-Lemma für kontextfreie Sprachen ... \begin{solution} + Man versucht auszunutzen, daß eine kontextfreie Sprache von einer Grammatik mit endlich vielen Nichtterminalen erzeugt werden muss. Das bedeutet auch: wenn ein Ableitungsbaum ausreichend tief ist, so gibt es einen Ast, der ein Nichtterminal mehrfach enthält. Die durch diese zwei Vorkommen bestimmten Teilbäume werden ,,gepumpt''. + + Wenn $L$ eine kontextfreie Sprache ist, dann gibt es $n>= 1$ derart, dass für alle $z$ in $L$ mit $|z| >= n$ gilt: es gibt Wörter $u, v , w , x, y$ in SUM mit + \begin{enumerate} + \item $z = uvwxy$, + \item $|vwx| <= n$, + \item $|vx| >= 1$ und + \item $uv^i wx^i y \in L$ für alle $i >= 0$ + \end{enumerate} \end{solution} \end{parts} @@ -181,6 +213,11 @@ \end{solution} \part Der Satz von Rice lautet... \begin{solution} + dass es unmöglich ist, eine beliebige nicht-triviale Eigenschaft der erzeugten Funktion einer Turing-Maschine (oder eines Algorithmus in einem anderen Berechenbarkeitsmodell) algorithmisch zu entscheiden. + + Es sei $\mathcal{P}$ die Menge aller partiellen Turing-berechenbaren Funktionen und $\mathcal{S}\subsetneq\mathcal{P}$ eine nicht-leere, echte Teilmenge davon. Außerdem sei eine effektive Nummerierung vorausgesetzt, die einer natürlichen Zahl $n\in\mathbb{N}$ die dadurch codierte Turing-Maschine $M_{n}$ zuordnet. Dann ist die Menge $\mathcal{C}(\mathcal{S})=\{n\mid \text{die von } M_n \text{ berechnete Funktion liegt in }\mathcal{S}\}$ nicht entscheidbar. + + ,,Sei U eine nicht-triviale Eigenschaft der partiellen berechenbaren Funktionen, dann ist die Sprache $L_U=\{⟨M⟩\mid \text{ M berechnet } f\in U\}$ nicht entscheidbar.'' \end{solution} \end{parts} @@ -208,23 +245,32 @@ \begin{parts} \part EXPSPACE ? EXPTIME \begin{solution} + EXPSPACE $\geq$ EXPTIME \end{solution} \part NP ? P \begin{solution} - \end{solution} - \part NPSPACE ? EXPTIME - \begin{solution} + NP $\geq$ P \end{solution} \part NP ? NPSPACE \begin{solution} + NP $\leq$ NPSPACE \end{solution} \part NPSPACE ? PSPACE \begin{solution} + NPSPACE $=$ PSPACE \end{solution} \end{parts} - \question NP-vollständiges Problem: Gebe zwei NP-vollständige Probleme an (als Menge oder Eingabe-Frage-Paar). + \question NP-vollständiges Problem: Gebe (mind. zwei) NP-vollständige Probleme an (als Menge oder Eingabe-Frage-Paar). \begin{solution} + + Hamilton Kreis + \begin{itemize} + \item Eingabe: Graph(V,E) + \item Frage: Kann der Graph so durchlaufen werden, dass jeder Knoten genau ein mal besucht/abgelaufen wird? + \end{itemize} + + \end{solution} \end{questions} \end{document} \ No newline at end of file