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0 HcmV?d00001 diff --git a/Logik und Logikprogrammierung - Prüfungsvorbereitung copy.tex b/Logik und Logikprogrammierung - Prüfungsvorbereitung copy.tex new file mode 100644 index 0000000..1c839f0 --- /dev/null +++ b/Logik und Logikprogrammierung - Prüfungsvorbereitung copy.tex @@ -0,0 +1,311 @@ +\documentclass[10pt, a4paper]{exam} +\printanswers % Comment this line to hide the answers +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage[ngerman]{babel} +\usepackage{listings} +\usepackage{float} +\usepackage{graphicx} +\usepackage{color} +\usepackage{listings} +\usepackage[dvipsnames]{xcolor} +\usepackage{tabularx} +\usepackage{geometry} +\usepackage{color,graphicx,overpic} +\usepackage{amsmath,amsthm,amsfonts,amssymb} +\usepackage{tabularx} +\usepackage{listings} +\usepackage[many]{tcolorbox} +\usepackage{multicol} +\usepackage{hyperref} +\usepackage{pgfplots} +\usepackage{bussproofs} +\usepackage{tikz} +\usetikzlibrary{automata, arrows.meta, positioning} +\renewcommand{\solutiontitle}{\noindent\textbf{Antwort}: } +\SolutionEmphasis{\small} +\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm} + +\pdfinfo{ + /Title (Logik und Logikprogrammierung - Prüfungsvorbereitung) + /Creator (TeX) + /Producer (pdfTeX 1.40.0) + /Author (Robert Jeutter) + /Subject () +} +\title{Logik und Logikprogrammierung - Prüfungsvorbereitung} +\author{} +\date{} + +% Don't print section numbers +\setcounter{secnumdepth}{0} + +\newtcolorbox{myboxii}[1][]{ + breakable, + freelance, + title=#1, + colback=white, + colbacktitle=white, + coltitle=black, + fonttitle=\bfseries, + bottomrule=0pt, + boxrule=0pt, + colframe=white, + overlay unbroken and first={ + \draw[red!75!black,line width=3pt] + ([xshift=5pt]frame.north west) -- + (frame.north west) -- + (frame.south west); + \draw[red!75!black,line width=3pt] + ([xshift=-5pt]frame.north east) -- + (frame.north east) -- + (frame.south east); + }, + overlay unbroken app={ + \draw[red!75!black,line width=3pt,line cap=rect] + (frame.south west) -- + ([xshift=5pt]frame.south west); + \draw[red!75!black,line width=3pt,line cap=rect] + (frame.south east) -- + ([xshift=-5pt]frame.south east); + }, + overlay middle and last={ + \draw[red!75!black,line width=3pt] + (frame.north west) -- + (frame.south west); + \draw[red!75!black,line width=3pt] + (frame.north east) -- + (frame.south east); + }, + overlay last app={ + \draw[red!75!black,line width=3pt,line cap=rect] + (frame.south west) -- + ([xshift=5pt]frame.south west); + \draw[red!75!black,line width=3pt,line cap=rect] + (frame.south east) -- + ([xshift=-5pt]frame.south east); + }, +} + +\begin{document} +\begin{myboxii}[Disclaimer] + Aufgaben aus dieser Vorlage stammen aus der Vorlesung \textit{Logik und Logikprogrammierung} und wurden zu Übungszwecken verändert oder anders formuliert! Für die Korrektheit der Lösungen wird keine Gewähr gegeben. +\end{myboxii} + +%########################################## +\begin{questions} + \question Definitionen und Sätze + \begin{parts} + \part Der Korrektheitssatz der Aussagenlogik für den Wahrheitswertebereich $B$ lautet... + \begin{solution} + \end{solution} + + \part Eine Menge von Formeln $\Gamma$ heißt erfüllbar, wenn... + \begin{solution} + \end{solution} + + \part Zwei Formeln $\alpha$ und $\beta$ heißen äquivalent, wenn... + \begin{solution} + \end{solution} + + \part Der Kompaktheitssatz der Aussagenlogik lautet... + \begin{solution} + \end{solution} + + \part Eine Horn Klausel ist eine Formel der Form + \begin{solution} + \end{solution} + \end{parts} + + \question Wahrheitswertebereiche + \begin{parts} + \part Werte die Formel $\varpi_a=\lnot p \wedge \lnot\lnot p$ im Heytingschen Wahrheitswertebereich $H_{\mathbb{R}}$ aus für die $H_{\mathbb{R}}$-Belegung $B$ mit $B(p)=\mathbb{R}\backslash \{0\}$ + \begin{solution} + \end{solution} + + \part Überprüfe ob die Formel $\varphi_B=(\lnot p\rightarrow \lnot p)\rightarrow p$ eine $K_3$-Tautologie ist. Ist $\varphi_b$ eine $B_{\mathbb{R}}$ Tautologie? + \begin{solution} + \end{solution} + + \part Überprüfe ob die semantische Folgeung $\{p\rightarrow q, q\rightarrow r\}\Vdash_B r\rightarrow\lnot p$ gilt. + \begin{solution} + \end{solution} + \end{parts} + + \question Erfüllbarkeit + \begin{parts} + \part Überprüfe mittels Markierungsalgorithmus, ob die Formel $\varphi_a=(\lnot p\vee q)\wedge(t\vee \lnot s)\wedge(\lnot r\vee s\vee \lnot q)\wedge r\wedge (\lnot p\vee t)\wedge \lnot s \wedge (\lnot r\vee p)$ erfüllbar ist. + \begin{solution} + \end{solution} + + \part Überprüfe mittels SLD Resolution, ob die Formel $\varphi_b=(r\wedge p)\vee\lnot t\vee (p\wedge \lnot q)\vee \lnot p\vee (\lnot r\wedge q \wedge t)$ eine Tautologie ist + \begin{solution} + \end{solution} + \end{parts} + + \question Monotone Formeln: Eine aussagenlogische Formel $\varphi$ heißt monoton, falls für alle zu $\varphi$ passenden $B$-Belegungen $B_1,B_2$ mit $B_1(p_i)\leq B_2(p_i)$ für alle $i\in\mathbb{N}$ gilt $B_1(\varphi)\leq B_2(\varphi)$. Beispielsweise sind $p_1\wedge p_2$ und $\lnot\lnot p_1$ monoton. + \begin{parts} + \part Entscheide, welche der Formeln $\varphi=p_1\wedge(p_2\rightarrow p_3)$, $\psi=\lnot p_1\rightarrow p_2$ monoton sind. + \begin{solution} + \end{solution} + + \part Zeige per vollständiger Induktion über den Formelaufbau, dass aussagenlogische Formeln in denen weder $\lnot$ noch $\rightarrow$ vorkommen, monoton sind. + \begin{solution} + \end{solution} + \end{parts} + + \question Definitionen und Sätze: Sei $\sum$ eine Signatur. Verfollständige die folgenden Definitionen und Sätze. + \begin{parts} + \part Es gilt $\Delta\vdash \varphi$ für eine $\sum$-Formel $\varphi$ und eine Menge $\Delta$ von $\sum$-Formeln, falls + \begin{solution} + \end{solution} + + \part Der Vollständigkeitssatz der Prädikatenlogik lautet... + \begin{solution} + \end{solution} + + \part Der Satz von Löwenheim-Skolem lautet... + \begin{solution} + \end{solution} + + \part Die (elementare) Theorie einer $\sum$-Struktur $A$ ist + \begin{solution} + \end{solution} + \end{parts} + + \question Natürliches Schließen + \begin{parts} + \part Gebe die Regeln $(\forall-I)$, $(\exists-E)$ und $(GfG)$ inklusive Bedingung an + \begin{solution} + \end{solution} + + \part Zeige, dass $\forall x\exists y(f(x)=y)$ ein Theorem ist, indem du eine entsprechende Deduktion angibst + \begin{solution} + \end{solution} + + \part Zeige, dass $\exists x\forall y(f(x)=y)$ nicht allgemeingültig ist + \begin{solution} + \end{solution} + + \part Zeige, dass die Formel aus c) erfüllbar ist + \begin{solution} + \end{solution} + \end{parts} + + \question Prädikatenlogische Definierbarkeit: Betrachte im folgenden Graphen als $\sum$-Struktur, wobei $\sum$ eine Signatur mit einem zweistelligen Relationssymbol $E$ ist. + \begin{parts} + \part Betrachte den (kommt noch) Graphen und die $\sum$-Formel $\varphi_a=\forall x\exists y\exists z(((E(x,y)\wedge E(y,z))\vee(E(y,x)\wedge E(z,x)))\wedge y\not =z)$. Gebe eine Kante an, sodass $G$ mit dieser zusätzlichen Kante als $\sum_a$-Struktur ein Modell der Formel $\varphi_a$ ist. Begründe deine Antwort. + \begin{solution} + \end{solution} + + \part Betrachte die folgenden (kommen noch) Graphen $G_1$ und $G_2$. Gebe einen $\sum$-Satz $\varphi_b$ an, so dass $G_1\Vdash \varphi_b$ und $G_2\not\Vdash\varphi_b$ gilt. + \begin{solution} + \end{solution} + + \part Gebe einen $\sum$-Satz $\varphi_c$ an, so dass für alle $\sum$-Strukturen $A$ genau dann $A\Vdash \varphi_c$ gilt, wenn $E^A$ eine Äquivalenzrelation ist (d.h. reflexiv, symmetrisch und transitiv). + \begin{solution} + \end{solution} + \end{parts} + + \question Normalformeln und Unifikatoren + \begin{parts} + \part Betrachte die Formel $\varphi=\forall x(\exists y(R(x,y)\wedge \lnot \exists x(R(y,x))))$. Gebe eine Formel $\psi_1$ in Pränexform an, die äquivalent zu $\varphi$ ist und eine Formel $\psi_2$ in Skolemform, die erfüllbarkeitsäquivalent zu $\varphi$ ist. + \begin{solution} + \end{solution} + + \part Sei $\sum$ eine Signatur mit zweistelligem Relationssymbol $R$, zweistelligem Funktionssymbol $f$, einstelligem Funktionssymbol $g$ und Konstanten $a,b$. Ermittle mit dem Unifikationsalgorithmus, welche der folgenden Paare atomarer Formeln unifizierbar sind und gebe einen allgemeinsten Unifikator an, falls dieser existiert.\\ + i) $(R(x, f(y,g(a))), R(a,f(g(x),y)))$\\ + ii) $(R(f(g,x),y),g(y), R(f(y,z),z))$ + \begin{solution} + \end{solution} + \end{parts} + + \question Gegeben sei folgende Wissensbasis:\begin{itemize} + \item über(rot, orange). + \item über(orange, gelb). + \item über(gelb, grün). + \item über(grün, blau). + \item über(blau, violett). + \item top(X):~über(\_, X), !, fail. + \item top(\_). + \item oben(X):-über(X,\_),top(X). + \end{itemize} Wie antwortet ein Prolog System mit dieser Wissensbasis auf die folgenden Fragen: + \begin{parts} + \part ?-top(grün). + \begin{solution} + \end{solution} + + \part ?-top(X). + \begin{solution} + \end{solution} + + \part ?-top(rot). + \begin{solution} + \end{solution} + + \part ?-oben(grün). + \begin{solution} + \end{solution} + + \part ?-oben(X). + \begin{solution} + \end{solution} + + \part ?-oben(rot). + \begin{solution} + \end{solution} + \end{parts} + + \question Man implementiere folgende Prädikate in Form von Prolog Klauseln + \begin{parts} + \part Das Prädikat $parition(L,E,Kl,Gr)$ soll eine gegebene Liste ganzer Zahlen $L$ in zwei Teillisten partitionieren.\begin{itemize} + \item die Liste $Kl$ aller Elemente aus $L$, welche kleiner oder gleich $E$ sind und + \item die Liste $Gr$ aller Elemente aus $L$, welche größer als $E$ sind + \end{itemize} + Beispiel: ?-parition([1,2,3,4,5,6], 3, Kl, Gr). \begin{itemize} + \item $Kl=[1,2,3]$ + \item $Gr=[4,5,6]$ + \end{itemize} + \begin{solution} + \end{solution} + + \part Das Prädikat $merge(L1,L2,L)$ soll zwei sortierte Listen mit ganzen Zahlen $L1$ und $L2$ zu einer sortierten Liste $L$ verschmelzen. + \begin{solution} + \end{solution} + + \part Das Prädikat $listmerge(ListenListe, L)$ bekommt eine Liste sortierter Listen $ListenListe$ und soll sie zu einer sortierten Liste L verschmelzen. Das in Aufgabe b) definierte Prädikat $merge$ kann dabei verwendet werden. + \begin{solution} + \end{solution} + + \part Das Prädikat $am\_groesten(L, Max)$ soll das größte Element $Max$ einer Zahlenliste $L$ ermitteln. Falls $L$ leer ist, soll ,,nein'' geantwortet werden. + \begin{solution} + \end{solution} + + \part Das Prädikat $am\_kuerzesten(ListenListe, L)$ soll aus einer Liste von $ListenListe$ die kürzeste Liste $L$ ermitteln. Dies soll möglichst effizient geschehen: \begin{itemize} + \item Gestalte die Prozedur rechtsrekursiv + \item Sehe davon ab, Listenlängen explizit zu ermitteln. Ermittle diese mit einem Hilfsprädikat $kuerzer\_als(L1,L2)$ + \item Höre mit der Suche auf, sobald eine leere Liste gefunden wurde. Kürzer geht nicht + \end{itemize} + Falls $ListenListe$ leer ist, soll ,,nein'' geantwortet werden. + \begin{solution} + \end{solution} + \end{parts} + + \question Ein binärer Suchbaum mit natürlichen Zahlen in den Knoten sei in Prolog wie folgt als strukturierter Term repräsentiert:\begin{itemize} + \item leerer Baum: nil + \item nichtlerer Baum: baum(Wurzel, LinkerUnterbaum, RechterUnterbaum) + \end{itemize} + Beispiel Baum mit Wurzel 6, Wurzel 4 im linken Unterbaum, Wurzel 7 im rechten Unterbaum und 2,4 und 9 als Blätter: $baum(6,baum(4, baum(2,nil,nil), baum(5, nil, nil)), baum(7, nil, baum(0, nil, nil)))$.\\ + Man implementiere folgende Prädikate in Prolog + \begin{parts} + \part Das Prädikat $enthalten(Baum, Zahl)$ bekommt einen binären Suchbaum $Baum$ sowie eine Zahl $Zahl$ und soll entscheiden, ob diese Zahl in Baum enthalten ist und die Antwort ,,ja'' oder ,,nein'' liefern. + \begin{solution} + \end{solution} + + \part Das Prädikat $flattern(Baum,Liste)$ soll aus einem gegebenen Suchbaum $Baum$ die Liste $Liste$ aller der im Baum enthaltenen Zahlen in aufsteigender sortierter Reihenfolge liefern. + \begin{solution} + \end{solution} + \end{parts} + +\end{questions} +\end{document} \ No newline at end of file