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Komplexität - Prüfungsvorbereitung.tex +++ b/Automaten, Sprachen und Komplexität - Prüfungsvorbereitung.tex @@ -317,17 +317,20 @@ \end{tikzpicture} \end{center} \begin{solution} - * erster Schritt, (*) zweiter Schritt + * erster Schritt: Paare mit mind. einem Endzustand markieren, + (*) zweiter Schritt: leere Paare die in Endzustandspaar überführen markieren, + dritter Schritt: unmarkierte Paare verschmelzen \begin{tabular}{c|c|c|c|c|c} - 2 & * \\\hline - 3 & * & * \\\hline - 4 & (*) & * & * \\\hline - 5 & * & * & * & * \\\hline + 2 & * \\ + 3 & * & * \\ + 4 & (*) & * & * \\ + 5 & * & * & * & * \\ 6 & (*) & * & * & (*) & * \\\hline & 1 & 2 & 3 & 4 & 5 \end{tabular} + Keine unmarkierten Paare $\rightarrow$ nicht minimierbar bzw schon minimal \end{solution} @@ -357,6 +360,14 @@ \end{tikzpicture} \end{center} \begin{solution} + \begin{tabular}{c|c|c|c|c|c} + 2 & * \\ + 3 & * & * \\ + 4 & * & * & \\ + 5 & * & * & (*) & * \\ + 6 & * & * & * & * & * \\\hline + & 1 & 2 & 3 & 4 & 5 + \end{tabular} \end{solution} \end{parts} @@ -381,12 +392,111 @@ \begin{parts} \part Sei $G$ die kontextfreie Grammatik mit Startsymbol S und der Regelmenge $S\rightarrow AB$, $A\rightarrow aBS|a$ und $B\rightarrow bBa|b|\epsilon$. Überführe G in eine äquivalente Grammatik in Chomsky Normalform. \begin{solution} + Chomsky Normalform hat auf rechter Ableitungsseite nur ein Terminal oder zwei Nicht-Terminale + \begin{enumerate} + \item Startzustand \\$S\rightarrow AB$, $A\rightarrow aBS|a$, $B\rightarrow bBa|b|\epsilon$ + \item $\epsilon$-Regel: Menge $M=\{B\}$ der $epsilon$ Terminal-Überführungen kompensieren; \\ + $S\rightarrow AB|A$, $A\rightarrow aBS|a|aS$, $B\rightarrow bBa|b|ba$ + \item Kettenregel: Menge $M=\{(S,A), (S,S), (A,A), (B,B)\}$ von Ketten (Ableitungen auf ein Nicht-Terminal)\\ + $S\rightarrow AB|aBS|a|aS$, $A\rightarrow aBS|a|aS$, $B\rightarrow bBa|b|ba$ + \item Terminale und Nicht-Terminal trennen: \\ $S\rightarrow AB|CBS|C|CS$, $A\rightarrow CBS|C|CS$, $B\rightarrow DBC|b|DC$, $C\rightarrow a$, $D\rightarrow b$ + \item Längen verkürzen: \\$S\rightarrow AB|CX|C|CS$, $A\rightarrow CX|C|CS$, $B\rightarrow DY|b|DC$, $C\rightarrow a$, $D\rightarrow b$, $X\rightarrow BS$, $Y\rightarrow BC$ + \end{enumerate} \end{solution} - \part Sei $G'$ die kontextfreie Grammatik mit Startsymbol $S$ und der Regelmenge $S\rightarrow AB$, $A\rightarrow CD|CF$, $F\rightarrow AD$, $B\rightarrow c|EB$, $C\rightarrow a$, $D\rightarrow b$, $E\rightarrow c$. Entscheide mit dem CYK-Algorithmus, ob die Wörter $w_1=aaabbbcc$ oder $w_2=aaabbccc$ von $G'$ erzeugt werden. + + \part Sei $G'$ die kontextfreie Grammatik mit Startsymbol $S$ und der Regelmenge \\\begin{center} + $S\rightarrow AB$, $A\rightarrow CD|CF$, $F\rightarrow AD$, $B\rightarrow c|EB$, $C\rightarrow a$, $D\rightarrow b$, $E\rightarrow c$ \\\end{center} Entscheide mit dem CYK-Algorithmus, ob die Wörter $w_1=aaabbbcc$ oder $w_2=aaabbccc$ von $G'$ erzeugt werden. \begin{solution} + + $w_1=aaabbbcc$ + + \begin{tabular}{c|c|c|c|c|c|c|c} + a & a & a & b & b & b & c & c \\\hline + C & C & C & D & D & D & B,E & B,E \\ + & & A & & & & B \\ + & & F & & & \\ + & A & & & \\ + & F & & \\ + A & & \\ + S & \\ + S + \end{tabular} + $\Rightarrow w_1$ wird von $G'$ erzeugt + + $w_2=aaabbccc$ + + \begin{tabular}{c|c|c|c|c|c|c|c} + a & a & a & b & b & c & c & c \\\hline + C & C & C & D & D & B,E & B,E & B,E \\ + & & A & & & B & B \\ + & & F & & & B \\ + & A & & & \\ + & F & & \\ + A & & \\ + S & \\ + \end{tabular} + $\Rightarrow w_2$ wird nicht von $G'$ erzeugt \end{solution} + \part Gebe für die Wörter aus b), die von $G'$ erzeugt werden, den Ableitungsbaum an. \begin{solution} + + $w_1=aaabbbcc$ + \begin{center} + \begin{tikzpicture}[node distance = 3cm, on grid, auto] + \node (q1) [state] {S}; + + \node (q2) [state, below left = of q1] {A}; + \node (q3) [state, below right=2cm and 5cm = of q1] {B}; + \node (q4) [state, below left = of q3] {E}; + \node (q6) [below = of q4] {c}; + \node (q5) [state, below right = of q3] {B}; + \node (q7) [below = of q5] {c}; + \node (q8) [state, below left = of q2] {C}; + \node (q9) [below = of q8] {a}; + \node (q10) [state, below right = of q2] {F}; + \node (q11) [state, below left = of q10] {A}; + \node (q12) [state, below right = of q10] {D}; + \node (q13) [below = of q12] {b}; + \node (q14) [state, below left = of q11] {C}; + \node (q15) [below = of q14] {a}; + \node (q16) [state, below right = of q11] {F}; + \node (q17) [state, below left = of q16] {A}; + \node (q18) [state, below right = of q16] {D}; + \node (q19) [below = of q18] {b}; + \node (q20) [state, below left = of q17] {C}; + \node (q21) [state, below right = of q17] {D}; + \node (q22) [below = of q20] {a}; + \node (q23) [below = of q21] {b}; + + \path [-stealth, thick] + (q1) edge (q2) + (q1) edge (q3) + (q2) edge (q8) + (q2) edge (q10) + (q3) edge (q4) + (q3) edge (q5) + (q4) edge (q6) + (q5) edge (q7) + (q8) edge (q9) + (q10) edge (q11) + (q10) edge (q12) + (q11) edge (q14) + (q11) edge (q16) + (q12) edge (q13) + (q14) edge (q15) + (q16) edge (q17) + (q16) edge (q18) + (q17) edge (q20) + (q17) edge (q21) + (q18) edge (q19) + (q20) edge (q22) + (q21) edge (q23) + ; + \end{tikzpicture} + \end{center} + + %$w_2=aaabbccc$ \end{solution} \end{parts} @@ -394,26 +504,63 @@ \begin{parts} \part Ein While Programm ist von der Form... \begin{solution} + \begin{itemize} + \item $x_i=c, x_i=x_j+c, x_i=x_j-c$ mit $c\in\{0,1\}$ und $i,j\geq 1$ (Wertzuweisung) oder + \item $P_1,P_2$, wobei $P_1$ und $P_2$ bereits While Programme sind (sequentielle Komposition) oder + \item while $x_i\not = 0$ do P end, wobei P ein While Programm ist und $i\geq 1$. + \end{itemize} + \end{solution} + + \part Ein Loop-Programm ist von der Form + \begin{solution} + \begin{itemize} + \item $x_i := c, x_i := x_j + c, x_i := x_j \div c$ mit $c\in\{0, 1\}$ und $i, j$ (Wertzuweisung) oder + \item $P_1 ; P_2$, wobei $P_1$ und $P_2$ Loop-Programme sind (sequentielle Komposition) oder + \item loop $x_i$ do P end, wobei P ein Loop-Programm ist und $i_1$. + \end{itemize} \end{solution} \part Eine Turingmaschine ist ein 7-Tupel $M=(Z,\sum,\Gamma,\delta,z_0,\Box, E)$, wobei... \begin{solution} + \begin{itemize} + \item 7-Tupel $M=(Z,\sum, \Phi, \delta, z_o, \Box, E)$ + \item $\sum$ das Eingabealphabet + \item $\Phi$ mit $\Phi\supseteq\sum$ und $\Phi\cap Z\not= 0$ das Arbeits- oder Bandalphabet, + \item $z_0\in Z$ der Startzustand, + \item $\delta:Z\times\Phi\rightarrow(Z\times\Phi\times\{L,N,R\})$ die Überführungsfunktion + \item $\Box\in\Phi/\sum$ das Leerzeichen oder Blank und + \item $E\subseteq Z$ die Menge der Endzustände ist + \end{itemize} \end{solution} \part Die von einer Turingmaschine $M$ akzeptierte Sprache ist $L(M)=...$ \begin{solution} + $L(M)=\{ w\in\sum^* | \text{es gibt akzeptierte Haltekonfiguration mit } z_0w\Box\vdash_M^* k\}$. \end{solution} - \part Seien $A\supseteq \sum^*$ und B$\supseteq \Gamma^*$. Eine Reduktion von A auf B ist ... + \part Seien $A\subseteq \sum^*$ und B$\subseteq \Gamma^*$. Eine Reduktion von A auf B ist ... \begin{solution} + Eine Reduktion von A auf B ist eine totale und berechenbare Funktion $f:\sum^*\rightarrow\Gamma^*$, so dass für alle $w\in\sum^*$ gilt: $w\in A\leftrightarrow f(x)\in B$. A heißt auf B reduzierbar (in Zeichen $A\leq B$), falls es eine Reduktion von A auf B gibt. \end{solution} \part Eine Sprache $L$ heißt rekursiv aufzählbar, falls ... \begin{solution} + \begin{itemize} + \item L ist semi-entscheidbar + \item L wird von einer Turing-Maschine akzeptiert + \item L ist vom Typ 0 (d.h. von Grammatik erzeugt) + \item L ist Bild berechenbarer partiellen Funktion $\sum^*\rightarrow\sum^*$ + \item L ist Bild berechenbarer totalen Funktion $\sum^*\rightarrow\sum^*$ + \item L ist Definitionsbereich einer berechenbaren partiellen Funktion $\sum^*\rightarrow\sum^*$ + \end{itemize} \end{solution} \part Sei $f:N\rightarrow N$ eine monotone Funktion. Die Klasse $TIME(f)$ besteht aus allen Sprachen L, für die es eine Turingmaschine $M$ gibt mit ... \begin{solution} + \begin{itemize} + \item M berechnet die charakteristische Funktion von L. + \item Für jede Eingabe $w\in\sum^*$ erreicht M von der Startkonfiguration $z_0 w\Box$ aus nach höchstens $f(|w|)$ Rechenschritten eine akzeptierende Haltekonfiguration (und gibt 0 oder 1 aus, je nachdem ob $w\not\in L$ oder $w\in L$ gilt). + \end{itemize} \end{solution} \end{parts} @@ -421,6 +568,11 @@ \begin{parts} \part Zu jeder Mehrband-Turingmaschine $M$ gibt es ... \begin{solution} + eine Turingmaschine M' die diesselbe Funktion löst + \begin{itemize} + \item Simulation mittels Einband-Turingmaschine durch Erweiterung des Alphabets: Wir fassen die übereinanderliegenden Bandeinträge zu einem Feld zusammen und markieren die Kopfpositionen auf jedem Band durch $\ast$. + \item Alphabetsymbol der Form $(a,\ast,b,\diamond,c,\ast,...)\in(\Phi\times\{\ast,\diamond\})^k$ bedeutet: 1. und 3. Kopf anwesend ($\ast$ Kopf anwesend, $\diamond$ Kopf nicht anwesend) + \end{itemize} \end{solution} \part Sei $f:N^k\rightarrow\mathbb{N}$ eine Funktion für ein $k\in\mathbb{N}$. Die folgenden Aussagen sind äquivalent: 1) $f$ ist Turing-berechenbar, 2)..., 3)..., 4)... @@ -445,18 +597,58 @@ \begin{parts} \part Gebe ein Loop-Programm an, das die Funktion $n\rightarrow n^2-n$ berechnet \begin{solution} + \begin{lstlisting} + h= 1 + for(i= 0; i < 2; i++) do { + h= h * n + } + h= h - 1; + return h + \end{lstlisting} \end{solution} \part Gebe ein Loop Programm an, das die Funktion $f:\mathbb{N}\rightarrow \mathbb{N}$ mit $f(n_1,n_2)=2n_1n_2$ berechnet. Verwende nur elementare Anweisungen und keine Abkürzungen. \begin{solution} + \begin{lstlisting} + h= 1 + for (i=0; i<2; i++) do { + h = h * n_i + } + h = 2 * h + return h + \end{lstlisting} \end{solution} \part Gebe ein GoTo Programm an, das die Funktion $g:\mathbb{N}\rightarrow\mathbb{N}$ mit $g(n_1,n_2)=|n_1-n_2|$ berechnet. Verwende nur elementare Anweisungen und keine Abkürzungen. \begin{solution} + \begin{lstlisting} + start: + h = n_1 - n_2 + if h > 0: + goto end + h = -h + end: + return h + \end{lstlisting} \end{solution} \part Gebe eine deterministische Turingmaschine $M$ für das Eingabealphabet $\{0,1\}$ an, das folgende Funktion berechnet: Für Eingabe $a_1a_2...a_{n-1}a_n$ berechnet M die Ausgabe $a_na_1...a_{n-1}$ (letzte Symbol der Eingabe an erste Stelle). \begin{solution} + $\sum=\{0,1\}$ + + $z_0$ Zahlenende finden: $\delta(z_0,0)=(z_0,0,R), \delta(z_0,1)=(z_0,1,R), \delta(z_0,\Box)=(z_1,\Box,L)$ + + $z_1$ letzte Zahl löschen: $\delta(z_1,0)=(z_2,\Box,L), \delta(z_1,1)=(z_3,\Box,L), \delta(z_1,\Box)=(z_2,\Box,N)$ + + $z_2$ zurück zum Anfang bei $a_n=0$: $\delta(z_2,0)=(z_2,0,L), \delta(z_2,1)=(z_2,1,L), \delta(z_2,\Box)=(z_4,\Box,R)$ + + $z_3$ zurück zum Anfang bei $a_n=1$: $\delta(z_2,0)=(z_2,0,L), \delta(z_2,1)=(z_2,1,L), \delta(z_2,\Box)=(z_5,\Box,N)$ + + $z_4$ $a_n=0$ an Anfang schreiben: $\delta(z_4,\Box)=(z_e,0,N)$ + + $z_5$ $a_n=1$ an Anfang schreiben: $\delta(z_4,\Box)=(z_e,1,N)$ + + $z_e$ Endzustand: $\delta(z_e, 0)=(z_e,0,N), \delta(z_e,1)=(z_e,1,N), \delta(z_e,\Box)?(z_e,\Box,N)$ \end{solution} \end{parts} @@ -492,6 +684,18 @@ \question Unentscheidbare Probleme: Gebe (mind vier) unterscheidbare Probleme an (als Menge oder als Eingabe-Frage-Paar). \begin{solution} + + das spezielle Halteproblem + + das allgemeine Halteproblem + + das Halteproblem auf leerem Band + + das allgemeine Wortproblem $A=\{(G,w) | \text{ G ist Grammatik mit } w\in L(G)\}$ + + Posts Korrespondenzproblem + + das Schnitt- und verwandte Probleme über kontextfreie Sprachen \end{solution} \question NP-vollständiges Problem: Gebe (mind. zwei) NP-vollständige Probleme an (als Menge oder Eingabe-Frage-Paar). @@ -503,7 +707,6 @@ \item Frage: Kann der Graph so durchlaufen werden, dass jeder Knoten genau ein mal besucht/abgelaufen wird? \end{itemize} - \end{solution} \question Polynomialzeitreduktion: Betrachte das Problem 4C, also die Menge der ungerichteten Graphen die sich mit vier Farben färben lassen.