From 4d3ae3a79aa5f11d2bfac248166026dd44a8ff2f Mon Sep 17 00:00:00 2001
From: Robert Jeutter <robert.jeutter@gmx.de>
Date: Tue, 3 Aug 2021 13:35:37 +0200
Subject: [PATCH] markdown to tex

---
 ...k und Logikprogrammierung - Cheatsheet.pdf |  Bin 0 -> 1732463 bytes
 ...k und Logikprogrammierung - Cheatsheet.tex | 5437 +++++++++++++++++
 2 files changed, 5437 insertions(+)
 create mode 100644 Logik und Logikprogrammierung - Cheatsheet.pdf
 create mode 100644 Logik und Logikprogrammierung - Cheatsheet.tex

diff --git a/Logik und Logikprogrammierung - Cheatsheet.pdf b/Logik und Logikprogrammierung - Cheatsheet.pdf
new file mode 100644
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diff --git a/Logik und Logikprogrammierung - Cheatsheet.tex b/Logik und Logikprogrammierung - Cheatsheet.tex
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+\fancyhead[L]{Logik und Logikprogrammierung}
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+  /Title (Logik und Logikprogrammierung - Cheatsheet)
+  /Creator (TeX)
+  /Producer (pdfTeX 1.40.0)
+  /Author (Robert Jeutter)
+  /Subject ()
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+
+\begin{document}
+
+\raggedright
+\begin{multicols}{3}\scriptsize
+  % multicol parameters
+  % These lengths are set only within the two main columns
+  %\setlength{\columnseprule}{0.25pt}
+  \setlength{\premulticols}{1pt}
+  \setlength{\postmulticols}{1pt}
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+  \setlength{\columnsep}{2pt}
+
+  \subsubsection{Probleme mit natürlicher Sprache}
+
+  \begin{enumerate}
+    \item Zuordnung von Wahrheitswerten zu natürlichsprachigen Aussagen ist problematisch. (Ich habe nur ein bißchen getrunken.)
+    \item Natürliche Sprache ist oft schwer verständlich.
+    \item Natürliche Sprache ist mehrdeutig.
+    \item Natürliche Sprache hängt von Kontext ab.
+  \end{enumerate}
+
+  \section{Aussagenlogik}
+  In der Aussagenlogik gehen wir von ``Aussagen'' aus, denen wir (zumindest prinzipiell) Wahrheitswerte zuordnen können.
+
+  Für Aussagen verwenden wir die atomaren Formeln $p,q,r$ bzw.
+  $p_0,p_1,...$
+
+  Die Aussagen werden durch ``Operatoren'' verbunden. Beispiele - \ldots{}
+  und\ldots{} - \ldots{} oder\ldots{} - nicht\ldots{} - wenn\ldots{}
+  dann\ldots{} - entweder\ldots{} oder\ldots{} , aber nicht beide. - mehr
+  als die Hälfte der Aussagen \ldots{} gilt.
+
+  Für solche zusammengesetzten Aussagen verwenden wir $\varphi,\psi$ usw.
+
+  Durch die Wahl der erlaubten Operatoren erhält man unterschiedliche
+  ``Logiken''.
+
+  Da der Wahrheitswert einer zusammengesetzten Aussage nur vom
+  Wahrheitswert der Teilaussagen abhängen soll, sind Operatoren wie
+  ``weil'' oder ``obwohl'' nicht zulässig.
+
+  \subsection{Syntax der Aussagenlogik}\label{syntax-der-aussagenlogik}
+
+  Eine atomare Formel hat die Form $p_i$ (wobei
+  $i\in\mathbb{N}=\{0,1,...\}$). Formeln werden durch folgenden induktiven
+  Prozess definiert: 1. Alle atomaren Formeln und $\bot$ sind Formeln. 2.
+  Falls $\varphi$ und $\psi$ Formeln sind, sind auch
+  $(\varphi\wedge\psi),(\varphi\wedge\psi)$($\varphi \rightarrow\psi$)und
+  $\lnot\varphi$Formeln. 3. Nichts ist Formel, was sich nicht mittels der
+  obigen Regeln erzeugen läßt.
+
+  Beispielformel: $\lnot((\lnot p_4 \vee p_1)\wedge\bot)$
+
+  Bezeichnungen: - Falsum: $\bot$ - Konjunktion: $\wedge$ - Disjunktion:
+  $\vee$ - Implikation: $\rightarrow$ - Negation: $\lnot$
+
+  Abkürzungen $p,q,r...$ statt $p_0,p_1,p_2...$
+
+  $(\bigvee_{i=1}^n \varphi_i$ statt
+  $(...((\varphi_1\vee\varphi_2)\vee\varphi_3)\vee...\vee\varphi_n)$
+
+  $(\bigwedge_{i=1}^n \varphi_i)$ statt
+  $(...((\varphi_1\wedge\varphi_2)\wedge\varphi_3)\wedge...\wedge\varphi_n)$
+
+  $(\varphi \leftrightarrow \psi)$ statt
+  $((\varphi\rightarrow\psi)\wedge(\psi\rightarrow\varphi))$
+
+  Präzedenz der Operatoren: - $\leftrightarrow$ bindet am schwächsten -
+  $\rightarrow$\ldots{} - $\vee$\ldots{} - $\wedge$\ldots{} - $\lnot$
+  bindet am stärksten
+
+  Es gilt also z.B.:
+  $(\alpha\leftrightarrow\beta\vee\lnot\gamma\rightarrow\sigma\wedge\lnot\eta) = (\alpha\leftrightarrow ((\beta\vee\lnot\gamma)\rightarrow(\sigma\wedge\lnot\eta)))$
+
+  Dennoch: Zu viele Klammern schaden i.A. nicht.
+
+  \subsection{Natürliches Schließen}\label{natuxfcrliches-schlieuxdfen}
+
+  Ein (mathematischer) Beweis zeigt, wie die Behauptung aus den
+  Voraussetzungen folgt.\\Analog zeigt ein ``Beweisbaum'' (=
+  ``Herleitung'' = ``Deduktion''), wie eine Formel der Aussagenlogik aus
+  Voraussetzungen (ebenfalls Formeln der Aussagenlogik) folgt.\\Diese
+  ``Deduktionen'' sind Bäume, deren Knoten mit Formeln beschriftet sind: -
+  an der Wurzel steht die Behauptung (= Konklusion $\varphi$) - an den
+  Blättern stehen Voraussetzungen (= Hypothesen oder Annahmen aus
+  $\Gamma$) - an den inneren Knoten stehen ``Teilergebnisse'' und
+  ``Begründungen''
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-deduktionsbaum.png}
+
+  \subsection{Konstruktion von
+    Deduktionen}\label{konstruktion-von-deduktionen}
+
+  Aus der Annahme der Aussage $\varphi$ folgt $\varphi$ unmittelbar: eine
+  triviale Deduktion
+
+  $\varphi$ mit Hypothesen $\{\varphi\}$ und Konklusion $\varphi$.
+
+  Folgend werden wir - überlegen, wie aus ``einfachen mathematischen
+  Beweisen'' umfangreichere entstehen können und - parallel
+  dazudefinieren, wie aus einfachen Deduktionen umfangreichere konstruiert
+  werden können.
+
+  \subsubsection{Konjunktion}\label{konjunktion}
+
+  \paragraph{Konjunktionseinführung in math.
+    Beweisen}\label{konjunktionseinfuxfchrung-in-math.-beweisen}
+
+  Ein mathematischer Beweis einer Aussage ``$\varphi$ und $\psi$'' sieht
+  üblicherweise so aus: - ``Zunächst zeige ich $\varphi$: \ldots{} (hier
+  steckt die eigentliche Arbeit)\\- Jetzt zeige ich $\psi$: \ldots{}
+  (nochmehr eigentliche Arbeit)\\- Also haben wir''$\varphi$ und $\psi$"
+  gezeigt. qed"
+
+  \paragraph{Konjunktionseinführung
+    (ausführlich)}\label{konjunktionseinfuxfchrung-ausfuxfchrlich}
+
+  Ist D eine Deduktion von $\varphi$ mit Hypothesen aus $\Gamma$ und ist E
+  eine Deduktion von $\psi$ mit Hypothesen aus $\Gamma$, so ergibt sich
+  die folgende Deduktion von $\varphi\wedge\psi$ mit Hypothesen aus
+  $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-konjunktionseinführung.png}
+
+  Kurzform: $\frac{\varphi\quad\psi}{\varphi\wedge\psi} (\wedge I)$
+
+  \paragraph{Konjunktionselimination
+    (ausführlich)}\label{konjunktionselimination-ausfuxfchrlich}
+
+  Ist D eine Deduktion von $\varphi\wedge\psi$ mit Hypothesen aus
+  $\Gamma$, so ergeben sich die folgenden Deduktionen von $\varphi$ bzw.
+  von $\psi$ mit Hypothesen aus $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Konjunktionselimination.png}
+
+  Kurzform:
+  $\frac{\varphi\wedge\psi}{\varphi} (\wedge E_1) \quad\quad \frac{\varphi\wedge\psi}{\psi} (\wedge E_2)$
+
+  \paragraph{Beispiel}\label{beispiel}
+
+  Wir zeigen $\varphi\wedge\psi$ unter der Hypothese
+  $\psi\wedge\varphi$:\ldots{}
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Deduktionsbeispiel.png}
+
+  Dies ist eine Deduktion mit Konklusion $\varphi\wedge\psi$ und Hypothese
+  $\psi\wedge\varphi$ (zweimal verwendet).
+
+  \subsubsection{Implikation}\label{implikation}
+
+  \paragraph{Implikationseinführung in math.
+    Beweisen}\label{implikationseinfuxfchrung-in-math.-beweisen}
+
+  Ein mathematischer Beweis einer Aussage ``Aus $\varphi$ folgt $\psi$''
+  sieht üblicherweise so aus:\\- ``Angenommen, $\varphi$ gilt. - Dann
+  \ldots{} (hier steckt die eigentliche Arbeit).\\- Damit gilt $\psi$.\\-
+  Also haben wir gezeigt, dass $\psi$ aus $\varphi$ folgt. qed''
+
+  Die Aussage $\varphi$ ist also eine ``temporäre Hypothese''.
+
+  \paragraph{Implikationseinführung
+    (ausführlich)}\label{implikationseinfuxfchrung-ausfuxfchrlich}
+
+  Ist D eine Deduktion von $\psi$ mit Hypothesen aus
+  $\Gamma\cup\{\varphi\}$, so ergibt sich die folgende Deduktion von
+  $\varphi\rightarrow\psi$ mit Hypothesen aus $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Implikationseinführung.png}
+
+  Kurzform $[\varphi]$ $\vdots$
+  $\frac{\psi}{\varphi\rightarrow\psi} (\rightarrow I)$
+
+  Beispiel: \ldots{} Dies ist eine Deduktion von
+  $\varphi\rightarrow\varphi$ ohne Hypothesen.
+
+  \paragraph{Implikationselimination in math.
+    Beweisen}\label{implikationselimination-in-math.-beweisen}
+
+  Ein mathematischer Beweis einer Aussage ``$\psi$ gilt'' über eine
+  Hilfsaussage sieht so aus: - ``Zunächst zeigen wir, dass $\varphi$ gilt:
+  \ldots{} - Dann beweisen wir, dass $\psi$ aus $\varphi$ folgt: \ldots{}
+  - Also haben wir $\psi$ gezeigt. qed''
+
+  \paragraph{Implikationselimination oder modus ponens
+    (ausführlich)}\label{implikationselimination-oder-modus-ponens-ausfuxfchrlich}
+
+  Ist D eine Deduktion von $\varphi$ mit Hypothesen aus $\Gamma$ und ist E
+  eine Deduktion von $\varphi\rightarrow\psi$ mit Hypothesen aus $\Gamma$,
+  so ergibt sich die folgende Deduktion von $\psi$ mit Hypothesen aus
+  $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Implikationselimination.png}
+
+  Kurzform:
+  $\frac{\varphi\quad \varphi\rightarrow\psi}{\psi} (\rightarrow E)$
+
+  \paragraph{Beispiel}\label{beispiel-1}
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Implikationseleimination-beispiel.png}
+
+  Bemerkung: die Indizes 1, 2 und 3 machen deutlich, welche Hypothese bei
+  welcher Regelanwendung gestrichen wurde. Deduktionen können recht groß
+  werden.
+
+  Diese Deduktion hat keine Hypothesen!
+
+  \subsubsection{Disjunktion}\label{disjunktion}
+
+  \paragraph{Disjunktionselimination oder Fallunterscheidung in math.
+    Beweisen}\label{disjunktionselimination-oder-fallunterscheidung-in-math.-beweisen}
+
+  Ein mathematischer Beweis einer Aussage ``$\sigma$ gilt'' mittels
+  Fallunterscheidung sieht üblicherweise so aus: - ``Zunächst zeigen wir,
+  dass $\varphi\vee\psi$ gilt: \ldots{} - Gilt $\varphi$, so gilt
+  $\sigma$, denn \ldots{} - Gilt $\psi$, so gilt ebenfalls $\sigma$, denn
+  \ldots{} - Also haben wir gezeigt, dass $\sigma$ gilt. qed''
+
+  Die Aussagen $\varphi$ und $\psi$ sind also wieder ``temporäre
+  Hypothesen''.
+
+  \paragraph{Disjunktionselimination oder Fallunterscheidung
+    (ausführlich)}\label{disjunktionselimination-oder-fallunterscheidung-ausfuxfchrlich}
+
+  Ist D eine Deduktion von $\varphi\vee\psi$ mit Hypothesen aus $\Gamma$,
+  ist E eine Deduktion von $\sigma$ mit Hypothesen aus
+  $\Gamma\cup\{\varphi\}$und ist F eine Deduktion von $\sigma$ mit
+  Hypothesen aus $\Gamma\cup\{\psi\}$, so ergibt sich die folgende
+  Deduktion von $\sigma$ mit Hypothesen aus $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Disjunktionselimination.png}
+
+  Disjunktionselimination Kurzform: $\quad [\psi] \quad[\varphi]$
+  $\quad \vdots \quad\vdots$
+  $\frac{\varphi\vee\psi \quad\sigma \quad\sigma}{\sigma} (\vee E)$
+
+  Disjunktionseinführung (Kurzform)
+  $\frac{\varphi}{\varphi\vee\psi} (\vee I_1) \quad \frac{\psi}{\varphi\vee\psi} (\vee I_2)$
+
+  \subsubsection{Negation}\label{negation}
+
+  \paragraph{Negationseinführung in math.
+    Beweisen}\label{negationseinfuxfchrung-in-math.-beweisen}
+
+  Ein mathematischer Beweis einer Aussage ``$\varphi$ gilt nicht'' sieht
+  so aus:\\- ``Angenommen,$\varphi$gilt.\\- Dann folgt $0=1$, denn
+  \ldots{}. Mit anderen Worten, dies führt zu einem Widerspruch. - Also
+  haben wir gezeigt, dass $\varphi$ nicht gilt. qed''\\- Die Aussage
+  $\varphi$ ist also wieder eine ``temporäre Hypothese''.
+
+  \paragraph{Negationseinführung
+    (ausführlich)}\label{negationseinfuxfchrung-ausfuxfchrlich}
+
+  Ist D eine Deduktion von $\bot$ mit Hypothesen aus
+  $\Gamma\cup\{\varphi\}$, so ergibt sich die folgende Deduktion von
+  $\lnot\varphi$ mit Hypothesen aus $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Negationseinführung.png}
+
+  Kurzform: $[\varphi]$ $\vdots$ $\frac{\bot}{\lnot\varphi} (\lnot I)$
+
+  \paragraph{Negationselimination
+    (ausführlich)}\label{negationselimination-ausfuxfchrlich}
+
+  Ist D eine Deduktion von $\lnot\varphi$ mit Hypothesen aus $\Gamma$ und
+  ist E eine Deduktion von $\varphi$ mit Hypothesen aus $\gamma$, so
+  ergibt sich die folgende Deduktion von $\bot$ mit Hypothesen aus
+  $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Negationselimination.png}
+
+  Kurzform: $\frac{\lnot\varphi \quad \varphi}{\bot} (\lnot E)$
+
+  \subsubsection{Falsum}\label{falsum}
+
+  Hat man ``$0=1$'' bewiesen, so ist man bereit, alles zu glauben: ex
+  falso sequitur quodlibet
+
+  ausführlich: Ist D eine Deduktion von $\bot$ mit Hypothesen aus
+  $\Gamma$, so ergibt sich die folgende Deduktion von $\varphi$ mit
+  Hypothesen aus $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-Falsumeinführung.png}
+
+  Kurzform: $\frac{\bot}{\varphi} (\bot)$
+
+  \paragraph{math. Widerspruchsbeweis}\label{math.-widerspruchsbeweis}
+
+  Ein indirekter Beweis einer Aussage ``$\varphi$ gilt'' sieht
+  üblicherweise so aus: - ``Angenommen, $\varphi$ gilt nicht, d.h.
+  $\lnot\varphi$ gilt. - Dann folgt $0=1$, d.h. ein Widerspruch.\\- Also
+  haben wir gezeigt, dass $\varphi$ gilt. qed''
+
+  Die Aussage $\lnot\varphi$ ist also wieder eine ``temporäre Hypothese''.
+
+  \paragraph{reductio ad absurdum
+    (ausführlich)}\label{reductio-ad-absurdum-ausfuxfchrlich}
+
+  Ist D eine Deduktion von $\bot$ mit Hypothesen aus
+  $\Gamma\cup\{\lnot\varphi\}$, so ergibt sich die folgende Deduktion von
+  $\varphi$ mit Hypothesen aus $\Gamma$:
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-reductio-ad-absurdum.png}
+
+  Kurzform: $[\lnot\varphi]$ $\vdots$ $\frac{\bot}{\varphi} (raa)$
+
+  \subsection{Regeln des natürlichen
+    Schließens}\label{regeln-des-natuxfcrlichen-schlieuxdfens}
+
+  \begin{quote}
+    Definition
+
+    Für eine Formelmenge $\Gamma$ und eine Formel $\varphi$ schreiben wir
+    $\Gamma\Vdash\varphi$ wenn es eine Deduktion gibt mit Hypothesen aus
+    $\Gamma$ und Konklusion $\varphi$. Wir sagen ``$\varphi$ ist eine
+    syntaktische Folgerung von $\Gamma$''.
+
+    Eine Formel $\varphi$ ist ein Theorem, wenn $\varnothing\Vdash\varphi$
+    gilt.
+  \end{quote}
+
+  \subsubsection{Bemerkung}\label{bemerkung}
+
+  $\Gamma\Vdash\varphi$ sagt (zunächst) nichts über den Inhalt der Formeln
+  in $\Gamma\cup\{\varphi\}$ aus, sondern nur über die Tatsache, dass
+  $\varphi$ mithilfe des natürlichen Schließens aus den Formeln aus
+  $\Gamma$ hergeleitet werden kann.
+
+  Ebenso sagt ``$\varphi$ ist Theorem'' nur, dass $\varphi$ abgeleitet
+  werden kann, über ``Wahrheit'' sagt dieser Begriff (zunächst) nichts
+  aus.
+
+  \subsubsection{Satz}\label{satz}
+
+  Für alle Formeln $\varphi$ und $\psi$ gilt
+  $\{\lnot(\varphi\vee\psi)\}\Vdash\lnot\varphi\wedge\lnot\psi$.
+
+  Beweis: Wir geben eine Deduktion an\ldots{} -
+  $\{\lnot\varphi\wedge\lnot\psi\}\Vdash\lnot(\varphi\vee\psi)$
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-1.png} -
+  $\{\lnot\varphi\vee\lnot\psi\}\Vdash\lnot(\varphi\wedge\psi)$
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-2.png} -
+  $\{\varphi\vee\psi\} \Vdash \psi\vee\varphi$
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-3.png}
+
+  \subsubsection{Satz}\label{satz-1}
+
+  Für jede Formel $\varphi$ ist $\lnot\lnot\varphi\rightarrow\varphi$ ein
+  Theorem.
+
+  Beweis: Wir geben eine Deduktion mit Konklusion
+  $\lnot\lnot\varphi\rightarrow\varphi$ ohne Hypothesen an\ldots{}
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-5.png}
+
+  \subsubsection{Satz}\label{satz-2}
+
+  Für jede Formel $\varphi$ ist $\varphi\vee\lnot\varphi$ ein Theorem.
+
+  Beweis: Wir geben eine Deduktion mit Konklusion
+  $\varphi\vee\lnot\varphi$ ohne Hypothesen an\ldots{}
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-6.png}
+
+  Bemerkung: Man kann beweisen, dass jede Deduktion der letzten beiden
+  Theoreme die Regel (raa) verwendet, sie also nicht ``intuitionistisch''
+  gelten.
+
+  \subsubsection{Satz}\label{satz-3}
+
+  $\{\lnot(\varphi\wedge\psi)\}\Vdash\lnot\varphi\vee\lnot\psi$
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-4.png}
+
+  \subsection{Semantik}\label{semantik}
+
+  Formeln sollen Verknüpfungen von Aussagen widerspiegeln, wir haben dies
+  zur Motivation der einzelnen Regeln des natürlichen Schließens genutzt.
+  Aber die Begriffe ``syntaktische Folgerung'' und ``Theorem'' sind rein
+  syntaktisch definiert.
+
+  Erst die jetzt zu definierende ``Semantik'' gibt den Formeln
+  ``Bedeutung''.
+
+  Idee der Semantik: wenn man jeder atomaren Formel $p_i$ einen
+  Wahrheitswertzuordnet, so kann man den Wahrheitswert jeder Formel
+  berechnen.
+
+  Es gibt verschiedene Möglichkeiten, Wahrheitswerte zu definieren: -
+  zweiwertige oder Boolesche Logik $B=\{0,1\}$: Wahrheitswerte ``wahr''=1
+  und ``falsch''= 0 - dreiwertige Kleene-Logik $K_3=\{0,\frac{1}{2},1\}$:
+  zusätzlicher Wahrheitswert ``unbekannt''$=\frac{1}{2}$ - Fuzzy-Logik
+  $F=[0,1]$: Wahrheitswerte sind ``Grad der Überzeugtheit'' - unendliche
+  Boolesche Algebra $B_R$= Menge der Teilmengen von $\mathbb{R}$;
+  $A\subseteq\mathbb{R}$ ist ``Menge der Menschen, die Aussage für wahr
+  halten'' - Heyting-Algebra $H_R$= Menge der offenen Teilmengen von
+  $\mathbb{R}$ - Erinnerung: $A\subseteq\mathbb{R}$ offen, wenn
+  $\forall a\in A\exists\epsilon >0:(a-\epsilon,a+\epsilon)\subseteq A$,
+  d.h., wenn $A$ abzählbare Vereinigung von offenen Intervallen $(x,y)$
+  ist.
+
+  Beispiele: - offen:
+  $(0,1), \mathbb{R}_{>0}, \mathbb{R}\backslash\{0\}, \mathbb{R}\backslash\mathbb{N}$
+  - nicht offen:
+  $[1,2), \mathbb{R}_{\geq 0}, \mathbb{Q}, \mathbb{N}, \{\frac{1}{n} | n\in\mathbb{N}\}, \mathbb{R}\backslash\mathbb{Q}$
+
+  Sei W eine Menge von Wahrheitswerten.\textbackslash{} Eine W-Belegung
+  ist eine Abbildung $B:V\rightarrow W$, wobei
+  $V\subseteq\{p_0 ,p_1 ,...\}$ eine Menge atomarer Formeln ist.
+
+  Die W-Belegung $B:V\rightarrow W$ paßt zur Formel $\phi$, falls alle
+  atomaren Formeln aus $\phi$ zu V gehören.
+
+  Sei nun B eine W-Belegung. Was ist der Wahrheitswert der Formel
+  $p_0\vee p_1$ unter der Belegung B?
+
+  Zur Beantwortung dieser Frage benötigen wir eine Funktion
+  $\vee_W :W\times W\rightarrow W$ (analog für
+  $\wedge,\rightarrow,\lnot$).
+
+  \subsection{Wahrheitswertebereiche}\label{wahrheitswertebereiche}
+
+  \begin{quote}
+    Definition: Sei W eine Menge und $R\subseteq W\times W$ eine binäre
+    Relation. - R ist reflexiv, wenn $(a,a)\in R$ für alle $a\in W$ gilt. -
+    R ist antisymmetrisch, wenn $(a,b),(b,a)\in R$ impliziert, dass $a=b$
+    gilt (für alle $a,b\in W$). - R ist transitive, wenn $(a,b),(b,c)\in R$
+    impliziert, dass $(a,c)\in R$ gilt (für alle $a,b,c\in W$). - R ist eine
+    Ordnungsrelation, wenn R reflexiv, antisymmetrisch und transitiv ist. In
+    diesem Fall heißt das Paar $(W,R)$ eine partiell geordnete Menge.
+  \end{quote}
+
+  Beispiel 1. Sei $\leq$ übliche Ordnung auf $\mathbb{R}$und
+  $W\subseteq\mathbb{R}$. Dann ist $(W,\leq)$ partiell geordnete Menge. 2.
+  Sei $X$ eine Menge und $W\subseteq P(X)$. Dann ist $(W,\subseteq)$
+  partiell geordnete Menge. 3. Sei $W=P(\sum *)$ und $\leq_p$ die Relation
+  ``es gibt Polynomialzeitreduktion'' (vgl. ``Automaten, Sprachen und
+  Komplexität''). Diese Relation ist reflexiv, transitiv, aber nicht
+  antisymmetrisch (denn $3-SAT\leq_{p} HC$ und $HC\leq_{p} 3-SAT$).
+
+  \begin{quote}
+    Definition: Sei $(W,\leq)$ partiell geordnete Menge, $M\subseteq W$ und
+    $a\in W$. - a ist obere Schranke von $M$, wenn $m\leq a$ für alle
+    $m\in M$ gilt. - a ist kleinste obere Schranke oder Supremum von $M$,
+    wenn $a$ obere Schranke von $M$ ist und wenn $a\leq b$ für alle oberen
+    Schranken $b$ von $M$ gilt. Wir schreiben in diesem Fall $a=sup \ M$. -
+    a ist untere Schranke von $M$, wenn $a\leq m$ für alle $m\in M$ gilt. -
+    a ist größte untere Schranke oder Infimum von $M$, wenn a untere
+    Schranke von $M$ ist und wenn $b\leq a$ für alle unteren Schranken $b$
+    von $M$ gilt. Wir schreiben in diesem Fall $a=inf\ M$.
+  \end{quote}
+
+  Beispiel 1. betrachte $(W,\leq)$ mit $W=\mathbb{R}$ und $\leq$ übliche
+  Ordnung auf $\mathbb{R}$. - Dann gelten $sup[0,1] = sup(0,1) =1$. -
+  $sup\ W$ existiert nicht (denn $W$ hat keine obere Schranke). 2.
+  betrachte $(W,\subseteq)$ mit $X$ Menge und $W =P(X)$. -
+  $sup\ M=\bigcup_{A\in M} A$ für alle $M\subseteq W$ 3. betrachte
+  $(W,\subseteq)$ mit $W=\{\{0\},\{1\},\{0,1,2\},\{0,1,3\}\}$. -
+  $sup\{\{0\},\{0,1,2\}\}=\{0,1,2\}$ - $\{0,1,2\}$ und $\{0,1,3\}$ sind
+  die oberen Schranken von $M=\{\{0\},\{1\}\}$, aber $M$ hat kein Supremum
+
+  \begin{quote}
+    Definition: Ein (vollständiger) Verband ist eine partiell geordnete
+    Menge $(W,\leq)$, in der jede Menge $M\subseteq W$ ein Supremum $sup\ M$
+    und ein Infimum $inf\ M$ hat. In einem Verband $(W,\leq)$ definieren
+    wir: - $0_W = inf\ W$ und $1_W= sup\ W$ - $a\wedge_W b= inf\{a,b\}$ und
+    $a\vee_W b= sup\{a,b\}$ für $a,b\in W$
+  \end{quote}
+
+  Bemerkung: In jedem Verband $(W,\leq)$ gelten $0_W= sup\ \varnothing$
+  und $1_W= inf\ \varnothing$ (denn jedes Element von $W$ ist obere und
+  untere Schranke von $\varnothing$).
+
+  \begin{quote}
+    Definition: Ein Wahrheitswertebereich ist ein Tupel
+    $(W,\leq,\rightarrow W,\lnot W)$, wobei $(W,\leq)$ ein Verband und
+    $\rightarrow W:W^2 \rightarrow W$ und $\lnot W:W\rightarrow W$
+    Funktionen sind.
+  \end{quote}
+
+  \subsubsection{Beispiel}\label{beispiel-2}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Der Boolesche Wahrheitswertebereich B ist definiert durch die
+          Grundmenge $B=\{0,1\}$, die natürliche Ordnung $\leq$ und die
+          Funktionen $\lnot_B (a) = 1-a$, $\rightarrow_B(a,b) = max(b, 1 -a)$.
+          Hier gelten:
+
+          \begin{itemize}
+            \item
+                  $0_B=0$, $1_B= 1$,
+            \item
+                  $a\wedge_B b= min(a,b)$, $a\vee_B b= max(a,b)$
+          \end{itemize}
+    \item
+          Der Kleenesche Wahrheitswertebereich $K_3$ ist definiert durch die
+          Grundmenge $K_3=\{0,\frac{1}{2},1\}$ mit der natürlichen Ordnung
+          $\leq$ und durch die Funktionen $\lnot_{K_3} (a) = 1 -a $,
+          $\rightarrow_{K_3} (a,b) = max(b, 1-a)$. Hier gelten:
+    \item
+          $\lnot_{K_3} = 0$, $1_{K_3} = 1$
+    \item
+          $a\wedge_{K_3} b= min(a,b)$, $a\vee_{K_3} b= max(a,b)$
+    \item
+          Der Wahrheitswertebereich F der Fuzzy-Logik ist definiert durch die
+          Grundmenge $F=[0,1]\subseteq\mathbb{R}$ mit der natürlichen Ordnung
+          $\leq$ und durch die Funktionen $\lnot_F (a) = 1-a$,
+          $\rightarrow_F (a,b) = max(b, 1-a)$. Hier gelten:
+    \item
+          $0_F= 0$, $1_F= 1$
+    \item
+          $a\wedge_F b= min(a,b)$, $a\vee_F b= max(a,b)$
+    \item
+          Der Boolesche Wahrheitswertebereich $B_R$ ist definiert durch die
+          Grundmenge $B_R=\{A|A\subseteq \mathbb{R}\}$ mit der Ordnung
+          $\subseteq$ und durch die Funktionen
+          $\lnot_{B_R} (A) =\mathbb{R}\backslash A$,
+          $\rightarrow_{B_R} (A,B) = B\cup\mathbb{R}\backslash A$. Hier gelten:
+    \item
+          $0_{B_R}=\varnothing$, $1_{B_R}=\mathbb{R}$
+    \item
+          $A\wedge_{B_R} B=A\cap B$, $A\vee_{B_R} B=A\cup B$
+    \item
+          Der Heytingsche Wahrheitswertebereich $H_R$ ist definiert durch die
+          Grundmenge
+          $H_{mathbb{R}} =\{A\subseteq\mathbb{R} | \text{A ist offen}\}$, die
+          Ordnung $\subseteq$ und durch die Funktionen
+          $\lnot_{H_R} (A) = Inneres(\mathbb{R}\backslash A)$,
+          $\rightarrow_{H_R} (A,B) =Inneres(B\cup \mathbb{R}\backslash A)$. Hier
+          gelten:
+    \item
+          $0_{H_R}=\varnothing$, $1_{H_R}=\mathbb{R}$
+    \item
+          $A\wedge_{H_R} B= A\cap B$, $A\vee_{H_R} B=A\cup B$
+    \item
+          Erinnerung:
+          $Inneres(A) =\{a\in A|\exists \epsilon > 0 : (a-\epsilon,a+\epsilon)\subseteq A\}$
+    \item
+          Beispiele:
+          $Inneres((0,1))=(0,1)=Inneres([0,1]),Inneres(N)=\varnothing,Inneres(\mathbb{R}_{\geq 0}) = \mathbb{R}_{> 0}$
+  \end{itemize}
+
+  Sei W ein Wahrheitswertebereich und B eine W-Belegung. Induktiv über den
+  Formelaufbau definieren wir den Wahrheitswert $\hat{B}(\phi)\in W$ jeder
+  zu $B$ passenden Formel $\phi$: - $\hat{B}(\bot) = 0_W$ -
+  $\hat{B}(p) = B(p)$ falls $p$ eine atomare Formel ist -
+  $\hat{B}((\phi\wedge \psi )) = \hat{B}(\phi)\wedge_W \hat{B}(\psi )$ -
+  $\hat{B}((\phi\vee \psi )) = \hat{B}(\phi)\vee_W \hat{B}(\psi )$ -
+  $\hat{B}((\phi\rightarrow \psi )) = \rightarrow W(\hat{B}(\phi),\hat{B}(\psi ))$
+  - $\hat{B}(\lnot\phi) = \lnot W(\hat{B}(\phi))$
+
+  Wir schreiben im folgenden $B(\phi)$ anstatt $\hat{B}(\phi)$.
+
+  Beispiel: Betrachte die Formel
+  $\phi= ((p\wedge q)\rightarrow (q\wedge p))$. - Für eine beliebige
+  B-Belegung $B:\{p,q\}\rightarrow B$ gilt
+  $B((p\wedge q)\rightarrow (q\wedge p)) = max(B(q\wedge p), 1 -B(p\wedge q)) = max(min(B(q),B(p)), 1 -min(B(p),B(q))) = 1 = 1_B$
+  - Für die $K_3$-Belegung $B:\{p,q\}\rightarrow K_3$ mit
+  $B(p) =B(q) = \frac{1}{2}$\} gilt
+  $B((p\wedge q)\rightarrow (q\wedge p)) = max(B(q\wedge p), 1 -B(p\wedge q))= max(min(B(q),B(p)), 1 -min(B(p),B(q))) = \frac{1}{2} \not= 1_{K_3}$
+  - analog gibt es eine F-Belegung $B:\{p,q\}\rightarrow F$, so dass
+  $B((p\wedge q)\rightarrow (q\wedge p)) \not = 1_F$ gilt. - Für eine
+  beliebige $H_{mathbb{R}}$-Belegung $B:\{p,q\}\rightarrow H_R$ gilt
+  $B((p\wedge q)\rightarrow (q\wedge p)) = Inneres(B(q\wedge p)\cup \mathbb{R}\backslash B(p\wedge q)) = Inneres((B(q)\cap B(p))\cup \mathbb{R}\backslash (B(p)\cap B(q))) = Inneres(\mathbb{R}) = \mathbb{R} = 1_{H_R}$
+
+  \subsection{Folgerung und Tautologie}\label{folgerung-und-tautologie}
+
+  Sei W ein Wahrheitswertebereich. Eine Formel $\phi$ heißt eine
+  W-Folgerung der Formelmenge $\Gamma$, falls für jede W-Belegung B, die
+  zu allen Formeln aus $\Gamma \cup\{\phi\}$ paßt, gilt:
+  $inf\{B(\gamma )|\gamma \in \Gamma \}\leq B(\phi)$
+
+  Wir schreiben $\Gamma \Vdash W\phi$, falls $\phi$ eine W-Folgerung von
+  $\Gamma$ ist.
+
+  Bemerkung: Im Gegensatz zur Beziehung $\Gamma \vdash \phi$, d.h. zur
+  syntaktischen Folgerung, ist $\Gamma \Vdash W \phi$ eine semantische
+  Beziehung.
+
+  Eine W-Tautologie ist eine Formel $\phi$ mit $\varnothing \Vdash W\phi$,
+  d.h. $B(\phi) = 1_W$ für alle passenden W-Belegungen B (denn
+  $inf\{\hat{B}(\gamma )|\gamma \in \varnothing \}= inf \varnothing = 1_W)$.
+
+  Wahrheitstafel für den Booleschen Wahrheitswertebereich B:
+
+  \begin{tabular}{llllllll}
+    RL & AK & BK & $AK\vee BK$ & $AK\rightarrow BK$ & $(BK\wedge RL)\rightarrow\lnot AK$ & RL & $\lnot AK$ \\\hline
+    0  & 0  & 0  & 0           & 1                  & 1                                  & 0  & 1          \\
+    0  & 0  & 1  & 1           & 1                  & 1                                  & 0  & 1          \\
+    0  & 1  & 0  & 1           & 0                  & 1                                  & 0  & 0          \\
+    0  & 1  & 1  & 1           & 1                  & 1                                  & 0  & 0          \\
+    1  & 0  & 0  & 0           & 1                  & 1                                  & 1  & 1          \\
+    1  & 0  & 1  & 1           & 1                  & 1                                  & 1  & 1          \\
+    1  & 1  & 0  & 1           & 0                  & 1                                  & 1  & 0          \\
+    1  & 1  & 1  & 1           & 1                  & 0                                  & 1  & 0          \\
+  \end{tabular}
+
+  Wir erhalten also
+  $\{(AK\vee BK),(AK\rightarrow BK), ((BK\wedge RL)\rightarrow \lnot AK),RL\} \Vdash_B \lnot AK$
+  und können damit sagen:
+
+  ``Wenn die Aussagen''Bauteil A oder Bauteil B ist kaputt" und ``daraus,
+  dass Bauteil A kaputt ist, folgt, dass Bauteil B kaputt ist''
+  und\ldots{} wahr sind, \ldots{} dann kann man die Folgerung ziehen: die
+  Aussage ``das Bauteil A ist heil'' ist wahr."
+
+  Erinnerung aus der ersten Vorlesung:
+  $\{(AK\vee BK),(AK\rightarrow BK), ((BK\wedge RL)\rightarrow \lnot AK),RL\} \vdash  \lnot AK$
+
+  Beispiel Sei $\phi$ beliebige Formel mit atomaren Formeln in V. - Sei
+  $B:V\rightarrow B$ eine B-Belegung. Dann gilt
+
+  \begin{verbatim}
+$B(\lnot\lnot\phi\rightarrow\phi) = \rightarrow B(\lnot B\lnot B(B(\phi)),B(\phi)) = max(B(\phi), 1 -( 1 -( 1 -B(\phi)))) = max(B(\phi), 1 -B(\phi)) = 1 = 1_B$.
+
+Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den Wahrheitswertebereich $B_R$).
+\end{verbatim}
+
+  \begin{itemize}
+    \item
+          Sei $B:V\rightarrow H_R$ eine $H_R$-Belegung mit
+          $B(\phi) =R\backslash\{0\}$. Dann gelten
+
+          \begin{itemize}
+            \item
+                  $B(\lnot\phi) = Inneres(\mathbb{R}\backslash B(\phi)) = Inneres(\{0\}) =\varnothing$
+            \item
+                  $B(\lnot\lnot\phi) = Inneres(\mathbb{R}\backslash B(\lnot\phi)) = Inneres(\mathbb{R}) = \mathbb{R}$
+            \item
+                  $B(\lnot\lnot\phi\rightarrow\phi) = \rightarrow_{H_R} (B(\lnot\lnot\phi),B(\phi)) = \rightarrow_{H_R} (\mathbb{R},\mathbb{R}\backslash \{0\}) = Inneres(\mathbb{R}\backslash\{0\}\cup\mathbb{R}\backslash\mathbb{R}) = \mathbb{R}\backslash\{0\}\not =\mathbb{R}= 1_{H_R}$
+          \end{itemize}
+
+          Also ist $\lnot\lnot\phi\rightarrow\phi$ keine $H_R$-Tautologie (gilt
+          ebenso für die Wahrheitswertebereiche $K_3$ und $F$).
+    \item
+          Sei $B:V\rightarrow B$ eine B-Belegung. Dann gilt
+
+          $B(\phi\vee\lnot\phi) = max(B(\phi), 1 -B(\phi)) = 1 = 1_B$.
+
+          Also ist $\phi\vee\lnot\phi$ eine B-Tautologie (gilt ebenso für den
+          Wahrheitswertebereich $B_R$).
+    \item
+          Sei $B:V\rightarrow H_R$ eine $H_R$-Belegung mit
+          $B(\phi)=\mathbb{R}\backslash\{0\}$. Dann gilt
+          $B(\phi\vee\lnot\phi) = B(\phi)\cup B(\lnot\phi) = \mathbb{R}\backslash\{0\}\cup \varnothing \not= 1_{H_R}$.
+
+          Also ist $\phi\vee\lnot\phi$ keine $H_R$-Tautologie (gilt ebenso für
+          die Wahrheitswertebereiche $K_3$ und $F$).
+    \item
+          Sei $B:V\rightarrow B$ eine B-Belegung. Dann gilt
+
+          $B(\lnot\phi\rightarrow\bot) = \rightarrow_B(B(\lnot\phi),B(\bot)) = max(0,1-B(\lnot \phi)) = 1 -( 1 -B(\phi)) =B(\phi)$.
+
+          Also haben wir $\{\lnot\phi\rightarrow\bot\}\Vdash B\phi$ und
+          $\{\phi\}\Vdash B\lnot \phi\rightarrow\bot$.
+    \item
+          Ebenso erhält man:
+
+          \begin{itemize}
+            \item
+                  $\{\lnot\phi\rightarrow\bot\}\Vdash_{K_3} \phi$
+            \item
+                  $\{\phi\}\Vdash_{K_3} \lnot\phi\rightarrow\bot$
+            \item
+                  $\{\lnot\phi\rightarrow\bot\}\Vdash_F\phi$
+            \item
+                  $\{\phi\}\Vdash F\lnot\phi\rightarrow\bot$
+          \end{itemize}
+    \item
+          Sei $B:D\rightarrow H_R$ eine $H_R$-Belegung mit
+          $B(\phi) =\mathbb{R}\backslash\{0\}$. Dann gilt
+          $B(\lnot\phi\rightarrow\bot) = Inneres(B(\bot )\cup \mathbb{R}\backslash B(\lnot\phi))= Inneres(\varnothing \cup \mathbb{R}\backslash\varnothing)= \mathbb{R} \not\supseteq B(\phi)$.
+
+          also $\{\lnot\phi\rightarrow\bot\}\not\Vdash_{H_R} \phi$.
+
+          Es gilt aber $\{\phi\}\Vdash_{H_R}\lnot \phi\rightarrow\bot$.
+  \end{itemize}
+
+  Zusammenfassung der Beispiele
+
+  \begin{tabular}{lllllll}
+                                                       & B & $B_R$ & $K_3$ & F & $H_R$ & \\\hline
+    $\varnothing\Vdash_W\lnot\lnot\phi\rightarrow\phi$ & Y & Y     & -     & - & -     &
+    $\varnothing\vdash \lnot\lnot\phi\rightarrow\phi$                                    \\
+    $\varnothing\Vdash_W\phi\vee\lnot\phi$             & Y & Y     & -     & - & -     &
+    $\varnothing\vdash\phi\vee\lnot\phi$                                                 \\
+    $\{\lnot\phi\rightarrow\bot\}\Vdash_W\phi$         & Y & Y     & Y     & Y & -     &
+    $\{\lnot\phi\rightarrow\bot\}\vdash\phi$                                             \\
+    $\{\phi\}\Vdash_W\lnot\phi\rightarrow\bot$         & Y & Y     & Y     & Y & Y     &
+    $\{\phi\}\vdash\lnot\phi\rightarrow\bot$                                             \\
+  \end{tabular}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          $Y$ in Spalte W:W-Folgerung gilt
+    \item
+          $-$ in Spalte W:W-Folgerung gilt nicht
+  \end{itemize}
+
+  \begin{quote}
+    Überblick: Wir haben definiert - $\Gamma\vdash\phi$ syntaktische
+    Folgerung - Theorem (``hypothesenlos ableitbar'') -
+    $\Gamma\Vdash_W \phi$ (semantische) W-Folgerung - W-Tautologie (``wird
+    immer zu $1_W$ ausgewertet'')
+  \end{quote}
+
+  Frage: Was ist die Beziehung zwischen diesen Begriffen, insbes. zwischen
+  ``Theorem'' und ``W-Tautologie''? Da z.B. B-Folgerung
+  $\not =K_3$-Folgerung, hängt die Anwort von W ab.
+
+  \subsection{Korrektheit}\label{korrektheit}
+
+  Können wir durch mathematische Beweise zu falschen Aussagenkommen?
+
+  Können wir durch das natürliche Schließen zu falschen Aussagen kommen?
+
+  Existiert eine Menge $\Gamma$ von Formeln und eine Formel $\varphi$ mit
+  $\Gamma\vdash\varphi$ und $\Gamma\not\Vdash_W \varphi$? Für welche
+  Wahrheitswertebereiche W?
+
+  Frage für diese Vorlesung: Für welche Wahrheitswertebereiche W gilt
+  $\Gamma\vdash\varphi\Rightarrow\Gamma\vdash_W \varphi$ bzw. $\varphi$
+  ist Theorem $\Rightarrow\varphi$ ist W-Tautologie?
+
+  Beispiel: Betrachte den Kleeneschen Wahrheitswertebereich $K_3$. - Sei
+  $p$ atomare Formel. $\frac{[p]^4}{p\rightarrow p}$ Also gilt
+  $\varnothing\vdash p\rightarrow p$, d.h. $p\rightarrow p$ ist Theorem. -
+  Sei $B$ $K_3$-Belegung mit $B(p)=\frac{1}{2}$. Dann gilt
+  $B(p\rightarrow p) = max(B(p), 1-B(p)) =\frac{1}{2}$, also
+  $inf\{B(\gamma)|\gamma\in\varnothing\}= 1 >\frac{1}{2} = B(p\rightarrow p)$.
+  Damit haben wir gezeigt $\varnothing\not\Vdash_{K_3} p\rightarrow p$.
+
+  Die Implikation $\Gamma\vdash\varphi\Rightarrow\Gamma\vdash_W \varphi$
+  gilt also NICHT für den Kleeneschen Wahrheitswertebereich $W=K_3$ und
+  damit auch NICHT für den Wahrheitswertebereich der Fuzzy-Logik $F$.
+
+  \begin{quote}
+    Korrektheitslemma für nat. Schließen \& Wahrheitswertebereich B
+
+    Sei $D$ eine Deduktion mit Hypothesen in der Menge $\Gamma$ und
+    Konklusion $\varphi$. Dann gilt $\Gamma\vdash_B \varphi$, d.h.
+    $inf\{B(\gamma)|\gamma\in\Gamma\}\leq B(\varphi)$ für alle passenden
+    B-Belegungen $B$.
+  \end{quote}
+
+  Beweis: Induktion über die Größe der Deduktion $D$ (d.h. Anzahl der
+  Regelanwendungen). - I.A.: die kleinste Deduktion $D$ hat die Form
+  $\varphi$ mit Hypothese $\varphi$ und Konklusion $\varphi$. Sei $B$
+  passendeB-Belegung. Hypothesen von $D$ in
+  $\Gamma\Rightarrow\varphi\in\Gamma\Rightarrow inf\{B(\gamma)|\gamma\in\Gamma\}\leq B(\varphi)\Rightarrow\Gamma\vdash_B \varphi$
+  - I.V.: Behauptung gelte für alle Deduktionen, die kleiner sind als $D$.
+  - I.S.: Wir unterscheiden verschiedene Fälle, je nachdem, welche Regel
+  als letzte angewandt wurde. - $(\wedge I)$ Die Deduktion hat die Form
+  $\frac{\alpha\quad\beta}{\alpha\wedge\beta}$ mit
+  $\varphi=\alpha\wedge\beta$. Sei $B$ passende B-Belegung. Nach IV gelten
+  $inf\{B(\gamma)|\gamma\in\Gamma\}\leq B(\alpha)$ und
+  $inf\{B(\gamma)|\gamma\in\Gamma\}\leq B(\beta)$ und damit
+  $inf\{B(\gamma)|\gamma\in\Gamma\}\leq B(\alpha)\wedge_B B(\beta)=B(\alpha\wedge\beta) =B(\varphi)$.
+  Da $B$ beliebig war, haben wir $\Gamma\vdash_B \varphi$ gezeigt. -
+  $(\vee E)$ Die Deduktion $D$ hat die Form
+  $\frac{\alpha\vee\beta\quad\phi\quad\phi}{\phi}$ Also gibt es Deduktion
+  $E$ mit Hypothesen in $\Gamma$ und Konklusion $\alpha\vee\beta$ und
+  Deduktionen $F$ und $G$ mit Hypothesen in $\Gamma\cup\{\alpha\}$ bzw.
+  $\Gamma\cup\{\beta\}$ und Konklusion $\varphi$. Sei $B$ passende
+  B-Belegung. Nach IV gelten
+  $inf\{B(\gamma)|\gamma\in\Gamma\}\leq B(\alpha\vee\beta)$ (1)
+  $inf\{B(\gamma)|\gamma\in\Gamma\cup\{\alpha\}\}\leq B(\varphi)$ (2)
+  $inf\{B(\gamma)|\gamma\in\Gamma\cup\{\beta\}\}\leq B(\varphi)$ (3) Wir
+  unterscheiden zwei Fälle: - $B(\alpha)\leq B(\beta)$:
+  $inf\{B(\gamma)|\gamma\in\Gamma\}\leq B(\alpha\vee\beta) =B(\alpha)\vee_B B(\beta) =B(\beta)$
+  impliziert
+  $inf\{B(\gamma)|\gamma\in\Gamma\}= inf\{B(\gamma)|\gamma\in\Gamma\cup\{\beta\}\}\leq B(\varphi)$
+  - $B(\alpha)>B(\beta)$: analog Da $B$ beliebig war, haben wir
+  $\Gamma\vdash_B \varphi$ gezeigt. - $(\rightarrow I)$ Die DeduktionDhat
+  die Form $\frac{\beta}{\alpha\rightarrow\beta}$ mit
+  $\varphi=\alpha\rightarrow\beta$. Sei $B$ eine passende B-Belegung. Nach
+  IV gilt $inf\{B(\gamma)|\gamma\in\Gamma\cup\{\alpha\}\}\leq B(\beta)$
+  Wir unterscheiden wieder zwei Fälle: -
+  $B(\alpha)=0:inf\{B(\gamma)|\gamma\in\Gamma\}\leq 1 =\rightarrow_B(B(\alpha),B(\beta)) = B(\alpha\rightarrow\beta) =B(\varphi)$
+  -
+  $B(\alpha)=1:inf\{B(\gamma)|\gamma\in\Gamma\}=inf\{B(\gamma)|\gamma\in\Gamma\cup\{\alpha\}\}\leq B(\beta) =\rightarrow_B (B(\alpha),B(\beta)) = B(\alpha\rightarrow\beta) =B(\varphi)$
+  Da $B$ beliebig war, habe wir $\Gamma\vdash_B \varphi$ gezeigt. -
+  $(raa)$ Die DeduktionDhat die Form $\frac{\bot}{\phi}$ Sei $B$ eine
+  passende B-Belegung. Nach IV gilt
+  $inf\{B(\gamma)|\gamma\in\Gamma\cup\{\lnot\varphi\}\}\leq B(\bot) = 0$.
+  Wir unterscheiden wieder zwei Fälle: -
+  $inf\{B(\gamma)|\gamma\in\Gamma\}=0$: dann gilt
+  $inf\{B(\gamma)|\gamma\in\Gamma\}\leq B(\varphi)$. -
+  $inf\{B(\gamma)|\gamma\in\Gamma\}=1$: Wegen
+  $inf\{B(\gamma)|\gamma\in\Gamma\cup\{\lnot\varphi\}\}=0$ folgt
+  $0 =B(\lnot\varphi)=\lnot_B (B(\varphi))$ und daher
+  $B(\varphi)=1\geq inf\{B(\gamma)|\gamma\in\Gamma\}$. Da $B$ beliebig
+  war, haben wir $\Gamma\vdash_B \varphi$ gezeigt.
+
+  Ist die letzte Schlußregel in der Deduktion $D$ von der Form
+  $(\wedge I), (\vee E), (\rightarrow I)$ oder $(raa)$, so haben wir die
+  Behauptung des Lemmas gezeigt. Analog kann dies für die verbleibenden
+  Regeln getan werden.
+
+  \begin{quote}
+    Korrektheitssatz für natürliches Schließen \& Wahrheitswertebereich $B$
+
+    Für jede Menge von Formeln $\Gamma$ und jede Formel $\varphi$ gilt
+    $\Gamma\vdash\varphi\Rightarrow\Gamma\vdash_B\varphi$.
+  \end{quote}
+
+  Beweis: Wegen $\Gamma\vdash\varphi$ existiert eine Deduktion $D$ mit
+  Hypothesen in $\Gamma$ und Konklusion $\varphi$. Nach dem
+  Korrektheitslemma folgt $\Gamma\vdash_B \varphi$.
+
+  \begin{quote}
+    Korollar: Jedes Theorem ist eine B-Tautologie.
+  \end{quote}
+
+  \begin{quote}
+    Korrektheitssatz für natürliches Schließen \& Wahrheitswertebereich $B$
+
+    Für jede Menge von Formeln $\Gamma$ und jede Formel $\varphi$ gilt
+    $\Gamma\vdash\varphi\Rightarrow\Gamma\vdash_{B_\mathbb{R}}\varphi$.
+  \end{quote}
+
+  Beweis: 1. Variante: verallgemeinere den Beweis von Korrektheitslemma
+  und Korrektheitssatz für $B$ auf $B_\mathbb{R}$ (Problem: wir haben
+  mehrfach ausgenutzt, dass $B=\{0,1\}$ mit $0<1$) 2. Variante: Folgerung
+  aus Korrektheitssatz für $B$.
+
+  \begin{quote}
+    Korollar: Jedes Theorem ist eine $B_\mathbb{R}$-Tautologie.
+  \end{quote}
+
+  \begin{quote}
+    Korrektheitslemma für nat. Schließen \& Wahrheitswertebereich
+    $H_{mathbb{R}}$
+
+    Sei $D$ eine Deduktion mit Hypothesen in der Menge $\Gamma$ und
+    Konklusion $\varphi$, die die Regel $(raa)$ nicht verwendet. Dann gilt
+    $\Gamma\vdash_{H_\mathbb{R}}\varphi$.
+  \end{quote}
+
+  Beweis: ähnlich zum Beweis des Korrektheitslemmas für den
+  Wahrheitswertebereich B. Nur die Behandlung der Regel $(raa)$ kann nicht
+  übertragen werden.
+
+  Beispiel: Sei $p$ eine atomare Formel.
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-7.png} Also gilt
+  $\{\lnot\lnot p\}\vdash p$, d.h. $p$ ist syntaktische Folgerung von
+  $\lnot\lnot p$. - Sei $B$ $H_{mathbb{R}}$-Belegung mit
+  $B(p)=\mathbb{R}\backslash\{0\}$. -
+  $\Rightarrow B(\lnot\lnot p) =\mathbb{R}\not\subseteq \mathbb{R}\backslash\{0\}=B(p)$
+  - $\Rightarrow\lnot\lnot p\not\Vdash_{H_{mathbb{R}}} p$, d.h. $p$ ist
+  keine $H_{mathbb{R}}$ -Folgerung von $\lnot\lnot p$.
+
+  \begin{quote}
+    Korrektheitssatz für nat. Schließen \& Wahrheitswertebereich
+    $H_{mathbb{R}}$
+
+    Für jede Menge von Formeln $\Gamma$ und jede Formel $\varphi$ gilt
+    $\Gamma\vdash\varphi$ ohne $(raa)$
+    $\Rightarrow\Gamma\vdash_{H_{mathbb{R}}}\varphi$.
+  \end{quote}
+
+  \begin{quote}
+    Korollar: Jedes $(raa)$-frei herleitbare Theorem ist eine
+    $H_{mathbb{R}}$-Tautologie.
+  \end{quote}
+
+  Folgerung: Jede Deduktion der Theoreme
+  $\lnot\lnot\varphi\rightarrow\varphi$ und $\varphi\vee\lnot\varphi$ ohne
+  Hypothesen verwendet $(raa)$.
+
+  \subsection{Vollständigkeit}\label{vollstuxe4ndigkeit}
+
+  Können wir durch mathematische Beweise zu allen korrekten Aussagen
+  kommen?
+
+  Können wir durch das natürliche Schließen zu allen korrekten Aussagen
+  kommen?
+
+  Existiert eine Menge $\Gamma$ von Formeln und eine Formel $\varphi$ mit
+  $\Gamma\vdash_W\varphi$ und $\Gamma\not\vdash\varphi$? Für welche
+  Wahrheitswertebereiche $W$?
+
+  Frage für diese Vorlesung: Für welche Wahrheitswertebereiche $W$ gilt
+  $\Gamma\vdash_W \varphi\Rightarrow\Gamma\vdash\varphi$ bzw. $\varphi$
+  ist $W$-Tautologie $\Rightarrow\varphi$ ist Theorem?
+
+  \subsubsection{Plan}\label{plan}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Sei $W$ einer der Wahrheitswertebereiche
+          $B,K_3 ,F ,B_\mathbb{R}, H_{mathbb{R}}$.
+    \item
+          z.z. ist $\Gamma\vdash_W\varphi\Rightarrow\Gamma\vdash\varphi$.
+    \item
+          dies ist äquivalent zu
+          $\Gamma\not\vdash\varphi\Rightarrow\Gamma\not\Vdash_W \varphi$.
+    \item
+          hierzu gehen wir folgendermaßen vor:
+    \item
+          $\Gamma \not\Vdash_W\varphi$
+    \item
+          $\Leftrightarrow$ $\Gamma\cup\{\lnot\varphi\}$ konsistent
+    \item
+          $\Rightarrow$ $\exists\Delta\subseteq\Gamma\cup\{\lnot\varphi\}$
+          maximal konsistent
+    \item
+          $\Rightarrow$ $\Delta$ erfüllbar
+    \item
+          $\Rightarrow$ $\Gamma\cup\{\lnot\varphi\}$ erfüllbar
+    \item
+          $\Leftrightarrow$ $\Gamma\not\Vdash_B \varphi$
+    \item
+          $\Rightarrow$ $\Gamma\not\Vdash\varphi$
+  \end{itemize}
+
+  \subsubsection{Konsistente Mengen}\label{konsistente-mengen}
+
+  \begin{quote}
+    Definition
+
+    Sei $\Gamma$ eine Menge von Formeln. $\Gamma$ heißt inkonsistent, wenn
+    $\Gamma\vdash\bot$ gilt. Sonst heißt $\Gamma$ konsistent.
+  \end{quote}
+
+  \begin{quote}
+    Lemma
+
+    Sei $\Gamma$ eine Menge von Formeln und $\varphi$ eine Formel. Dann gilt
+    $\Gamma\not\vdash\varphi \Leftrightarrow \Gamma\cup\{\lnot\varphi\}$
+    konsistent.
+  \end{quote}
+
+  Beweis: Wir zeigen
+  ``$\Gamma\vdash\varphi\Leftrightarrow \Gamma\cup\{\lnot\varphi\}$
+  inkonsistent'': - Richtung ``$\Rightarrow$'', gelte also
+  $\Gamma \vdash \varphi$. - $\Rightarrow$ es gibt Deduktion $D$ mit
+  Hypothesen in $\Gamma$ und Konklusion $\varphi$ - $\Rightarrow$ Wir
+  erhalten die folgende Deduktion mit Hypothesen in
+  $\Gamma\cup\{\lnot\varphi\}$ und Konklusion $\bot$:
+  $\frac{\lnot\varphi\quad\varphi}{\bot}$ -
+  $\Rightarrow\Gamma\cup\{\lnot\varphi\}\vdash\bot$,
+  d.h.$\Gamma\cup\{\lnot\varphi\}$ ist inkonsistent. - Richtung
+  ``$\Leftarrow$'', sei also $\Gamma\cup\{\lnot\varphi\}$ inkonsistent. -
+  $\Rightarrow$ Es gibt Deduktion $D$ mit Hypothesen in
+  $\Gamma\cup\{\lnot\varphi\}$ und Konklusion $\bot$. - $\Rightarrow$ Wir
+  erhalten die folgende Deduktion mit Hypothesen in $\Gamma$ und
+  Konklusion $\varphi$: $\frac{\bot}{\varphi}$ - $\Gamma\vdash\varphi$
+
+  \subsubsection{Maximal konsistente
+    Mengen}\label{maximal-konsistente-mengen}
+
+  \begin{quote}
+    Definition
+
+    Eine Formelmenge $\Delta$ ist maximal konsistent, wenn sie konsistent
+    ist und wenn gilt ``$\sum\supseteq\Delta$ konsistent
+    $\Rightarrow\sum = \Delta$''.
+  \end{quote}
+
+  \begin{quote}
+    Satz
+
+    Jede konsistente Formelmenge $\Gamma$ ist in einer maximal konsistenten
+    Formelmenge $\Delta$ enthalten.
+  \end{quote}
+
+  Beweis: Sei $\varphi_1,\varphi_2,...$ eine Liste aller Formeln (da wir
+  abzählbar viele atomare Formeln haben, gibt es nur abzählbar viele
+  Formeln)
+
+  Wir definieren induktiv konsistente Mengen $\Gamma_n$: - Setze
+  $\Gamma_1 = \Gamma$ - Setze
+  $\Gamma_{n+1}= \begin{cases} \Gamma_n\cup\{\varphi_n\}\quad\text{ falls diese Menge konsistent} \\ \Gamma_n \quad\text{sonst}\end{cases}$
+
+  Setze nun $\Delta =\bigcup_{n\geq 1} \Gamma_n$. 1. Wir zeigen indirekt,
+  dass $\Delta$ konsistent ist: Angenommen, $\Delta\vdash\bot$. -
+  $\Rightarrow$ Es gibt Deduktion $D$ mit Konklusion $\bot$ und endlicher
+  Menge von Hypothesen $\Delta'\subseteq\Delta$. - $\Rightarrow$ Es gibt
+  $n\geq 1$ mit $\Delta'\subseteq\Gamma_n$ -
+  $\Rightarrow \Gamma_n\vdash\bot$, zu $\Gamma_n$ konsistent. Also ist
+  $\Delta$ konsistent. 2. Wir zeigen indirekt, dass $\Delta$ maximal
+  konsistent ist. Sei also $\sum\supseteq\Delta$ konsistent. Angenommen,
+  $\sum\not=\Delta$. - $\Rightarrow$ es gibt $n\in N$ mit
+  $\varphi_n\in\sum\backslash\Delta$ -
+  $\Rightarrow \Gamma_n\cup\{\varphi_n\}\subseteq\Delta\cup\sum= \sum$
+  konsistent. - $\Rightarrow \varphi_n \in\Gamma_{n+1}\subseteq \Delta$,
+  ein Widerspruch, d.h. $\Delta$ ist max. konsistent.
+
+  \begin{quote}
+    Lemma 1
+
+    Sei $\Delta$ maximal konsistent und gelte $\Delta\vdash\varphi$. Dann
+    gilt $\varphi\in\Delta$.
+  \end{quote}
+
+  Beweis: 1. Zunächst zeigen wir indirekt, dass $\Delta\cup\{\varphi\}$
+  konsistent ist: - Angenommen, $\Delta\cup\{\varphi\}\vdash\bot$. -
+  $\Rightarrow$ $\exists$ Deduktion $D$ mit Hypothesen in
+  $\Delta\cup\{\varphi\}$ und Konklusion $\bot$. -
+  $\Delta\vdash \varphi \Rightarrow \exists$ Deduktion $E$ mit Hypothesen
+  in $\Delta$ und Konklusion $\varphi$. - $\Rightarrow$ Wir erhalten die
+  folgende Deduktion: $\frac{\Delta \frac{\Delta}{\varphi}}{\bot}$
+
+  \begin{verbatim}
+Also $\Delta\vdash\bot$, ein Widerspruch zur Konsistenz von $\Delta$. Also ist $\Delta\cup\{\varphi\}$ konsistent.
+\end{verbatim}
+
+  \begin{enumerate}
+    \setcounter{enumi}{1}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Da $\Delta\cup\{\varphi\}\supseteq\Delta$ konsistent und $\Delta$
+          maximal konsistent ist, folgt $\Delta=\Delta\cup\{\varphi\}$, d.h.
+          $\varphi\in\Delta$.
+  \end{enumerate}
+
+  \begin{quote}
+    Lemma 2
+
+    Sei $\Delta$ maximal konsistent und $\varphi$ Formel. Dann gilt
+    $\varphi\not\in\Delta\Leftrightarrow\lnot\varphi\in\Delta$.
+  \end{quote}
+
+  Beweis: - Zunächst gelte $\lnot\varphi\in\Delta$. Angenommen,
+  $\varphi\in\Delta$. Dann haben wir die Deduktion
+  $\frac{\lnot\varphi\quad\varphi}{\bot}$ und damit $\Delta\vdash\bot$,
+  was der Konsistenz von $\Delta$ widerspricht. - Gelte nun
+  $\varphi\not\in\Delta$. - $\Rightarrow$
+  $\Delta(\Delta\cup\{\varphi\}\Rightarrow\Delta\cup\{\varphi\}$
+  inkonsistent (da $\Delta$ max. konsistent) - $\Rightarrow$ Es gibt
+  Deduktion $D$ mit Hypothesen in $\Delta\cup\{\varphi\}$ \&Konklusion
+  $\bot$. - $\Rightarrow$ Wir erhalten die folgende Deduktion:
+  $\frac{\bot}{\lnot\varphi}$ - $\Rightarrow$
+  $\Delta\vdash\lnot\varphi\Rightarrow\lnot\varphi\in\Delta$ (nach Lemma
+  1)
+
+  \subsection{Erfüllbare Mengen}\label{erfuxfcllbare-mengen}
+
+  \begin{quote}
+    Definition
+
+    Sei $\Gamma$ eine Menge von Formeln. $\Gamma$ heißt erfüllbar, wenn es
+    eine passende B-Belegung $B$ gibt mit $B(\gamma) = 1_B$ für alle
+    $\gamma\in\Gamma$.
+  \end{quote}
+
+  Bemerkung - Die Erfüllbarkeit einer endlichen Menge $\Gamma$ ist
+  entscheidbar: - Berechne Menge $V$ von in $\Gamma$ vorkommenden atomaren
+  Formeln - Probiere alle B-Belegungen $B:V\rightarrow B$ durch - Die
+  Erfüllbarkeit einer endlichen Menge $\Gamma$ ist NP-vollständig (Satz
+  von Cook)
+
+  \begin{quote}
+    Satz Sei $\Delta$ eine maximal konsistente Menge von Formeln. Dann ist
+    $\Delta$ erfüllbar.
+  \end{quote}
+
+  Beweis: Definiere eine B-Belegung $B$ mittels
+  $B(p_i) = \begin{cases} 1_B \quad\text{ falls } p_i\in\Delta \\ 0_B \quad\text{ sonst. } \end{cases}$
+  Wir zeigen für alle Formeln
+  $\varphi: B(\varphi) = 1_B \Leftarrow\Rightarrow\varphi\in\Delta$ (*)
+
+  Der Beweis erfolgt per Induktion über die Länge von $\varphi$.
+
+  \begin{enumerate}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          I.A.: hat $\varphi$ die Länge 1, so ist $\varphi$ atomare Formel. Hier
+          gilt (*) nach Konstruktion von $B$.
+    \item
+          I.V.: Gelte (*) für alle Formeln der Länge $<n$.
+    \item
+          I.S.: Sei $\varphi$ Formel der Längen$>1$. $\Rightarrow$ Es gibt
+          Formeln $\alpha$ und $\beta$ der Länge$<n$ mit
+          $\varphi\in\{\lnot\alpha,\alpha\wedge\beta,\alpha\vee\beta,\alpha\rightarrow\beta\}$.
+
+          \begin{itemize}
+            \item
+                  Wir zeigen (*) für diese vier Fälle einzeln auf den folgenden
+                  Folien.
+            \item
+                  Zur Erinnerung: $\Delta$ max. konsistent, $\varphi$ Formel
+            \item
+                  Lemma 1: $\Delta\vdash\varphi\Rightarrow\varphi\in\Delta$
+            \item
+                  Lemma 2:
+                  $\varphi\not\in\Delta\Leftarrow\Rightarrow\lnot\varphi\in\Delta$
+          \end{itemize}
+    \item
+          $\varphi =\lnot\alpha$.
+          $B(\varphi) = 1_B \Leftarrow\Rightarrow B(\alpha) = 0_B \Leftarrow\Rightarrow \alpha\not\in\Delta\Leftarrow\Rightarrow \Delta \owns\lnot\alpha =\varphi$
+    \item
+          $\varphi =\alpha\wedge\beta$.
+  \end{enumerate}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          $B(\varphi) = 1_B \Rightarrow B(\alpha) =B(\beta) = 1_B \Rightarrow\alpha,\beta\in\Delta\Rightarrow\Delta\vdash\varphi$
+          denn $\frac{\alpha\quad\beta}{\alpha\wedge\beta}$ ist Deduktion
+          $\Rightarrow\varphi\in\Delta$.
+    \item
+          $\varphi$ $\in$ $\Delta$ $\Rightarrow\Delta\vdash\alpha$ und
+          $\Delta\vdash\beta$ denn $\frac{\varphi}{\alpha}$ und
+          $\frac{\varphi}{\beta}$ sind Deduktionen.
+          $\Rightarrow\alpha,\beta\in\Delta\Rightarrow B(\alpha),B(\beta) = 1_B=\Rightarrow B(\varphi) = 1_B$
+  \end{itemize}
+
+  \begin{enumerate}
+    \setcounter{enumi}{2}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          $\varphi =\alpha\vee\beta$.
+  \end{enumerate}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          $B(\varphi) = 1_B \Rightarrow B(\alpha) = 1_B$ oder $B(\beta) = 1_B$
+
+          \begin{itemize}
+            \item
+                  angenommen,
+                  $B(\alpha) = 1_B \Rightarrow\alpha\in\Delta\Rightarrow\Delta\vdash\varphi$
+                  denn $\frac{\alpha}{\varphi}$ ist Deduktion
+                  $\Rightarrow\varphi\in\Delta$
+            \item
+                  angenommen, $B(\alpha) = 0_B \Rightarrow B(\beta) = 1_B$. weiter
+                  analog.
+          \end{itemize}
+    \item
+          $\varphi\in\Delta$. Dann gilt
+          $\Delta\cup\{\lnot\alpha ,\lnot\beta\}\vdash \bot$ aufgrund der
+          Deduktion \includegraphics[width=\linewidth]{Assets/Logik-beispiel-8.png} Da $\Delta$
+          konsistent ist, folgt
+          $\Delta\not=\Delta\cup\{\lnot\alpha,\lnot\beta\}$ und damit
+          $\lnot\alpha\in\Delta$ oder $\lnot\beta\in\Delta$.
+          $\Rightarrow\alpha\in\Delta$ oder $\beta\in\Delta$ nach Lemma 2
+          $\Rightarrow B(\alpha) = 1_B$ oder $B(\beta) =1_B$
+          $\Rightarrow B(\varphi) = 1_B$.
+  \end{itemize}
+
+  \begin{enumerate}
+    \setcounter{enumi}{3}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          $\varphi = \alpha\rightarrow\beta$.
+  \end{enumerate}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          $B(\varphi) = 1_B \Rightarrow B(\alpha) = 0_B$ oder
+          $B(\beta) = 1_B \Rightarrow\lnot\alpha\in\Delta$ oder $\beta\in\Delta$
+          Aufgrund nebenstehender Deduktionen gilt in beiden Fällen
+          $\Delta\vdash\alpha\rightarrow\beta\Rightarrow\varphi\in\Delta$
+          \includegraphics[width=\linewidth]{Assets/Logik-beispiel-9.png}
+    \item
+          $\varphi\in\Delta$. Angenommen,
+          $B(\varphi) = 0_B = \Rightarrow B(\alpha) = 1_B, B(\beta) = 0_B$
+          $\Rightarrow\alpha\in\Delta, \beta\not\in\Delta \Rightarrow \lnot\beta\in\Delta$
+          Aufgrund der nebenstehenden Deduktion gilt $\Delta\vdash\bot$, d.h.
+          $\Delta$ ist inkonsistent, im Widerspruch zur Annahme.
+          $\Rightarrow B(\varphi) = 1_B$
+          \includegraphics[width=\linewidth]{Assets/Logik-beispiel-10.png}
+  \end{itemize}
+
+  \begin{quote}
+    Lemma
+
+    Sei $\Gamma$ eine Menge von Formeln und $\varphi$ eine Formel. Dann gilt
+    $\Gamma\not\Vdash_B\varphi\Leftarrow\Rightarrow\Gamma\cup\{\lnot \varphi\}$
+    erfüllbar.
+  \end{quote}
+
+  Beweis: $\Gamma\not\Vdash_B\varphi$ $\Leftarrow\Rightarrow$ es gibt
+  passende B-Belegung $B$ mit
+  $inf\{B(\gamma)|\gamma\in\Gamma\} \not\leq_B B(\varphi)$
+  $\Leftarrow\Rightarrow$ es gibt passende B-Belegung $B$ mit
+  $inf\{B(\gamma)|\gamma\in\Gamma\}= 1_B$ und $B(\varphi)=0_B$
+  $\Leftarrow\Rightarrow$ es gibt passende B-Belegung $B$ mit
+  $B(\gamma) = 1_B$ für alle $\gamma\in\Gamma$ und $B(\lnot\varphi) = 1_B$
+  $\Leftarrow\Rightarrow \Gamma\cup\{\lnot\varphi\}$ erfüllbar
+
+  \begin{quote}
+    Beobachtung: Sei $W$ einer der Wahrheitswertebereiche $B, K_3, F, H_R$
+    und $B_R,\Gamma$ eine Menge von Formeln und $\varphi$ eine Formel. Dann
+    gilt $\Gamma\Vdash W\varphi\Rightarrow\Gamma\Vdash B\varphi$.
+  \end{quote}
+
+  Beweis: Sei $B$ beliebige B-Belegung, die zu jeder Formel in
+  $\Gamma\cup\{\varphi\}$ paßt. definiere W-Belegung $B_W$ durch
+  $B_W(pi) = \begin{cases} 1_W \quad\text{ falls } B(p_i) = 1_B \\ 0_W \quad\text{ sonst} \end{cases}$.
+  per Induktion über die Formelgröße kann man für alle Formeln $\psi$, zu
+  denen $B$ paßt, zeigen:
+  $B_W(\psi) = \begin{cases} 1_W \quad\text{ falls } B(\psi) = 1_B \\ 0_W \quad\text{ sonst.} \end{cases}$
+  (*)
+
+  Wir unterscheiden zwei Fälle: -
+  $inf\{B(\gamma)|\gamma\in\Gamma\}= 1_B \Rightarrow inf\{B_W(\gamma)|\gamma\in\Gamma\} = 1_W$
+  (wegen (\emph{)) $\Rightarrow 1_W = B_W(\varphi)$ (wegen
+    $\Gamma\Vdash_W\varphi$) $\Rightarrow 1_B = B(\varphi)$ (wegen (}))
+  $\Rightarrow inf\{B(\gamma)|\gamma\in\Gamma\} = 1_B \leq B(\varphi)$ und
+  -
+  $inf\{B(\gamma)|\gamma\in\Gamma\} \not= 1_B \Rightarrow inf\{B(\gamma)|\gamma\in\Gamma\}= 0_B$
+  $\Rightarrow inf\{B(\gamma)|\gamma\in\Gamma\}= 0_B \leq B(\varphi)$.
+
+  Da $B$ beliebig war, gilt $\Gamma\Vdash_B \varphi$.
+
+  \begin{quote}
+    Satz (Vollständigkeitssatz)
+
+    Sei $\Gamma$ eine Menge von Formeln, $\varphi$ eine Formel und $W$ einer
+    der Wahrheitswertebereiche $B,K_3 , F, B_R$ und $H_R$. Dann gilt
+    $\Gamma\Vdash_W\varphi \Rightarrow \Gamma\vdash\varphi$. Insbesondere
+    ist jede W-Tautologie ein Theorem.
+  \end{quote}
+
+  Beweis: indirekt - $\Gamma\not\Vdash$ - $\Gamma\cup\{\lnot\varphi\}$
+  konsistent - $\exists\Delta\supseteq\Gamma\cup\{\lnot\varphi\}$ maximal
+  konsistent - $\Rightarrow\Delta$ erfüllbar -
+  $\Gamma\cup\{\lnot\varphi\}$ erfüllbar - $\Gamma\not\Vdash_B \varphi$\\-
+  $\Gamma\not\Vdash_W \varphi$
+
+  \subsection{Vollständigkeit und
+    Korrektheit}\label{vollstuxe4ndigkeit-und-korrektheit}
+
+  \begin{quote}
+    Satz
+
+    Seien $\Gamma$ eine Menge von Formeln und $\varphi$ eine Formel. Dann
+    gilt $\Gamma\vdash\varphi\Leftarrow\Rightarrow\Gamma\Vdash_B \varphi$.
+    Insbesondere ist eine Formel genau dann eine B-Tautologie, wenn sie ein
+    Theorem ist.
+  \end{quote}
+
+  Beweis: Folgt unmittelbar aus Korrektheitssatz und Vollständigkeitssatz.
+
+  \begin{quote}
+    Bemerkung: - gilt für jede ``Boolesche Algebra'', z.B. $B_R$ -
+    $\Gamma\vdash\varphi$ ohne ($raa$)
+    $\Leftarrow\Rightarrow\Gamma\Vdash_{H_R} \varphi$ (Tarksi 1938)
+  \end{quote}
+
+  \subsubsection{Folgerung 1:
+    Entscheidbarkeit}\label{folgerung-1-entscheidbarkeit}
+
+  \begin{quote}
+    Satz: die Menge der Theoreme ist entscheidbar.
+  \end{quote}
+
+  Beweis: Sei $\varphi$ Formel und $V$ die Menge der vorkommenden atomaren
+  Formeln. Dann gilt $\varphi$ Theorem - $\Leftarrow\Rightarrow\varphi$
+  B-Tautologie - $\Leftarrow\Rightarrow$ für alle Abbildungen
+  $B:V\rightarrow\{0_B, 1_B\}$ gilt $B(\varphi) = 1_B$
+
+  Da es nur endlich viele solche Abbildungen gibt und $B(\varphi)$
+  berechnet werden kann, ist dies eine entscheidbare Aussage.
+
+  \subsubsection{Folgerung 2: Äquivalenzen und
+    Theoreme}\label{folgerung-2-uxe4quivalenzen-und-theoreme}
+
+  \begin{quote}
+    Definition
+
+    Zwei Formeln $\alpha$ und $\beta$ heißen äquivalent
+    $(\alpha\equiv\beta)$, wenn für alle passenden B-Belegungen $B$ gilt:
+    $B(\alpha) =B(\beta)$.
+  \end{quote}
+
+  \begin{quote}
+    Satz: Es gelten die folgenden Äquivalenzen: 1.
+    $p_1 \vee p_2 \equiv p_2 \vee p_1$ 2.
+    $(p_1 \vee p_2 )\vee p_3 \equiv p_1 \vee (p_2 \vee p_3 )$ 3.
+    $p_1 \vee (p_2 \wedge p_3 )\equiv (p_1 \vee p_2 )\wedge (p_1 \vee p_3 )$
+    4. $\lnot(p_1 \vee p_2 )\equiv\lnot p_1 \wedge\lnot p_2$ 5.
+    $p_1 \vee p_1 \equiv p_1$ 6.
+    $(p_1 \wedge \lnot p_1 )\vee p_2 \equiv p_2$ 7.
+    $\lnot\lnot p_1\equiv p_1$ 8. $p_1 \wedge\lnot p_1 \equiv\bot$ 9.
+    $p_1 \vee\lnot p_1 \equiv\lnot\bot$ 10.
+    $p_1 \rightarrow p_2 \equiv \lnot p_1 \vee p_2$
+  \end{quote}
+
+  Beweis: Wir zeigen nur die Äquivalenz (3): Sei $B$ beliebige B-Belegung,
+  die wenigstens auf $\{p_1, p_2, p_3\}$ definiert ist. Dazu betrachten
+  wir die Wertetabelle: \textbar{} $B(p_1)$ \textbar{} $B(p_2)$ \textbar{}
+  $B(p_3)$ \textbar{} $B(p_1\vee(p_2\wedge p_3))$ \textbar{}
+  $B((p_1\vee p_2)\wedge(p_1 \vee p_3 ))$ \textbar{} \textbar{} --------
+  \textbar{} -------- \textbar{} -------- \textbar{}
+  --------------------------- \textbar{}
+  --------------------------------------- \textbar{} \textbar{} $0_B$
+  \textbar{} $0_B$ \textbar{} $0_B$ \textbar{} $0_B$ \textbar{} $0_B$
+  \textbar{} \textbar{} $0_B$ \textbar{} $0_B$ \textbar{} $1_B$ \textbar{}
+  $0_B$ \textbar{} $0_B$ \textbar{} \textbar{} $0_B$ \textbar{} $1_B$
+  \textbar{} $0_B$ \textbar{} $0_B$ \textbar{} $0_B$ \textbar{} \textbar{}
+  $0_B$ \textbar{} $1_B$ \textbar{} $1_B$ \textbar{} $1_B$ \textbar{}
+  $1_B$ \textbar{} \textbar{} $1_B$ \textbar{} $0_B$ \textbar{} $0_B$
+  \textbar{} $1_B$ \textbar{} $1_B$ \textbar{} \textbar{} $1_B$ \textbar{}
+  $0_B$ \textbar{} $1_B$ \textbar{} $1_B$ \textbar{} $1_B$ \textbar{}
+  \textbar{} $1_B$ \textbar{} $1_B$ \textbar{} $0_B$ \textbar{} $1_B$
+  \textbar{} $1_B$ \textbar{} \textbar{} $1_B$ \textbar{} $1_B$ \textbar{}
+  $1_B$ \textbar{} $1_B$ \textbar{} $1_B$ \textbar{}
+
+  Die anderen Äquivalenzen werden analog bewiesen.
+
+  Aus dieser Liste von Äquivalenzen können weitere hergeleitet werden:
+
+  Beispiel: Für alle Formeln $\alpha$ und $\beta$ gilt
+  $\lnot(\alpha\wedge\beta)\equiv\lnot\alpha\vee\lnot\beta$.
+
+  Beweis:
+  $\lnot(\alpha\wedge\beta) \equiv \lnot(\lnot\lnot\alpha\wedge\lnot\lnot\beta) \equiv \lnot\lnot(\lnot\alpha\vee\lnot\beta) \equiv \lnot\alpha\vee\lnot\beta$
+
+  \begin{quote}
+    Bemerkung Mit den üblichen Rechenregeln für Gleichungen können aus
+    dieser Liste alle gültigen Äquivalenzen hergeleitet werden.
+  \end{quote}
+
+  \paragraph{Zusammenhang zw. Theoremen und
+    Äquivalenzen}\label{zusammenhang-zw.-theoremen-und-uxe4quivalenzen}
+
+  \begin{quote}
+    Satz
+
+    Seien $\alpha$ und $\beta$ zwei Formeln. Dann gilt
+    $\alpha\equiv\beta\Leftarrow\Rightarrow(\alpha\leftrightarrow\beta)$ ist
+    Theorem.
+  \end{quote}
+
+  Beweis: $\alpha\equiv\beta$ - $\Leftarrow\Rightarrow$ für alle passenden
+  B-Belegungen $B$ gilt $B(\alpha)=B(\beta)$ -
+  $\Leftarrow\Rightarrow \{\alpha\}\Vdash_B\beta$ und
+  $\{\beta\}\Vdash_B \alpha$ -
+  $\Leftarrow\Rightarrow \{\alpha\}\vdash\beta$ und
+  $\{\beta\}\vdash\alpha$ (nach Korrektheits- und Vollständigkeitssatz)
+
+  es bleibt z.z., dass dies äquivalent zu
+  $\varnothing\vdash(\alpha\leftrightarrow\beta)$ ist. - $\Rightarrow$:
+  Wir haben also Deduktionen mit Hypothesen in $\{\alpha\}$ bzw. in
+  $\{\beta\}$ und Konklusionen $\beta$ bzw.$\alpha$. Es ergibt sich eine
+  hypothesenlose Deduktion von $\alpha\leftrightarrow\beta$:
+  \includegraphics[width=\linewidth]{Assets/Logik-deduktion-1.png} - $\Leftarrow$: Wir haben
+  also eine hypothesenlose Deduktion von $\alpha\leftrightarrow\beta$. Es
+  ergeben sich die folgenden Deduktionen mit Hypothesen $\beta$ bzw.
+  $\alpha$ und Konklusionen $\alpha$ bzw. $\beta$:
+  \includegraphics[width=\linewidth]{Assets/Logik-deduktion-2.png}
+
+  \begin{quote}
+    Satz
+
+    Sei $\alpha$ eine Formel. Dann gilt $\alpha$ ist Theorem
+    $\Leftarrow\Rightarrow\alpha\equiv\lnot\bot$.
+  \end{quote}
+
+  Beweis: $\alpha$ ist Theorem - $\Leftarrow\Rightarrow\alpha$ ist
+  B-Tautologie (Korrektheits- und Vollständigkeitssatz) -
+  $\Leftarrow\Rightarrow$ für alle passenden B-Belegungen $B$ gilt
+  $B(\alpha) = 1_B$ - $\Leftarrow\Rightarrow$ für alle passenden
+  B-Belegungen $B$ gilt $B(\alpha) =B(\lnot\bot)$ -
+  $\Leftarrow\Rightarrow\alpha\equiv\lnot\bot$
+
+  \subsubsection{Folgerung 3: Kompaktheit}\label{folgerung-3-kompaktheit}
+
+  \begin{quote}
+    Satz
+
+    Sei $\Gamma$ eine u.U. unendliche Menge von Formeln und $\varphi$ eine
+    Formel mit $\Gamma\Vdash_B\varphi$. Dann existiert
+    $\Gamma'\subseteq\Gamma$ endlich mit $\Gamma'\Vdash_B \varphi$.
+  \end{quote}
+
+  Beweis: $\Gamma\Vdash_B\varphi$ - $\Rightarrow\Gamma\vdash\varphi$ (nach
+  dem Vollständigkeitssatz) - $\Rightarrow$ es gibt Deduktion von
+  $\varphi$ mit Hypothesen $\gamma_1,...,\gamma_n\in\Gamma$ -
+  $\Rightarrow\Gamma'=\{\gamma_1,...,\gamma_n\}\subseteq\Gamma$ endlich
+  mit $\Gamma'\vdash\varphi$ - $\Rightarrow\Gamma'\Vdash_B\varphi$ (nach
+  dem Korrektheitssatz).
+
+  \begin{quote}
+    Folgerung (Kompaktheits- oder Endlichkeitssatz)
+
+    Sei $\Gamma$ eine u.U. unendliche Menge von Formeln. Dann gilt $\Gamma$
+    unerfüllbar $\Leftarrow\Rightarrow\exists\Gamma'\subseteq\Gamma$
+    endlich: $\Gamma'$ unerfüllbar
+  \end{quote}
+
+  Beweis: $\Gamma$ unerfüllbar -
+  $\Leftarrow\Rightarrow\Gamma\cup\{\lnot\bot\}$ unerfüllbar -
+  $\Leftarrow\Rightarrow\Gamma\Vdash_B\bot$ - $\Leftarrow\Rightarrow$ es
+  gibt $\Gamma'\subseteq\Gamma$ endlich: $\Gamma'\Vdash_B\bot$ -
+  $\Leftarrow\Rightarrow$ es gibt $\Gamma'\subseteq\Gamma$ endlich:
+  $\Gamma'\cup\{\lnot\bot\}$ unerfüllbar - $\Leftarrow\Rightarrow$ es gibt
+  $\Gamma'\subseteq\Gamma$ endlich: $\Gamma'$ unerfüllbar
+
+  \subsubsection{1. Anwendung des Kompaktheitsatzes:
+    Färbbarkeit}\label{anwendung-des-kompaktheitsatzes-fuxe4rbbarkeit}
+
+  \begin{quote}
+    Definition
+
+    Ein Graph ist ein Paar $G=(V,E)$ mit einer Menge $V$ und
+    $E\subseteq\binom{V}{2} =\{X\subseteq V:|V\Vdash 2\}$. Für $W\subseteq V$ sei $G\upharpoonright_W= (W,E\cap\binom{W}{2})$
+    der von $W$ induzierte Teilgraph. Der Graph G ist 3-färbbar, wenn es
+    eine Abbildung $f:V\rightarrow\{1,2,3\}$ mit $f(v)\not=f(w)$ für alle
+    $\{v,w\}\in E$.
+  \end{quote}
+
+  Bemerkung: Die 3-Färbbarkeit eines endlichen Graphen ist NP-vollständig
+
+  \begin{quote}
+    Satz Sei $G= (N,E)$ ein Graph. Dann sind äquivalent 1. $G$ ist
+    3-färbbar. 2. Für jede endliche Menge $W\subseteq N$ ist
+    $G\upharpoonright_W$ 3-färbbar.
+  \end{quote}
+
+  Beweis: - $1.\Rightarrow 2.$ trivial - $2.\Rightarrow 1.$ Sei nun, für
+  alle endlichen Menge $W\subseteq N$, der induzierte Teilgraph
+  $G\upharpoonright_W$ 3-färbbar.
+
+  Wir beschreiben zunächst mit einer unendlichen Menge $\Gamma$ von
+  Formeln, dass eine 3-Färbung existiert: - atomare Formeln $p_{n,c}$ für
+  $n\in N$ und $c\in\{1,2,3\}$ (Idee: der Knoten n hat die Farbe c) -
+  $\Gamma$ enthält die folgenden Formeln: - für alle
+  $n\in N:p_{n, 1} \vee p_{n, 2} \vee p_{n, 3}$ (der Knoten n ist gefärbt)
+  - für alle
+  $n\in N:\bigwedge_{1\leq c< d \leq 3} \lnot(p_{n,c} \wedge p_{n,d})$
+  (der Knoten n ist nur mit einer Farbe gefärbt) - für alle
+  $\{m,n\}\in E: \bigwedge_{1\leq c\leq 3} \lnot(p_{m,c} \wedge p_{n,c})$
+  (verbundene Knoten m und n sind verschieden gefärbt)
+
+  Behauptung: Jede endliche Menge $\Delta\subseteq\Gamma$ ist erfüllbar.
+
+  Begründung: - Da $\Delta$ endlich ist, existiert endliche Menge
+  $W\subseteq N$, so dass jede atomare Formel in $\Delta$ die Form
+  $p_{n,c}$ für ein $n\in W$ und ein $c\in\{1,2,3\}$ hat. - Nach Annahme
+  existiert $f_W:W\rightarrow\{1,2,3\}$ mit $f_W(m) \not=f(n)$ f.a.
+  $\{m,n\}\in E\cap\binom{W}{2}$. - Definiere
+  $B:\{p_{n,c}|n\in W, 1 \leq c\leq 3\}\rightarrow\{0,1\}$ durch
+  $B(p_{n,c}) = \begin{cases} 1 \quad\text{ falls } f_W(n) = c \\ 0 \quad\text{ sonst.} \end{cases}$
+  - Diese Belegung erfüllt $\Delta$, d.h. $\Delta$ ist erfüllbar, womit
+  die Behauptung gezeigt ist.
+
+  Nach dem Kompaktheitssatz ist also $\Gamma$ erfüllbar. Sei $B$
+  erfüllende Belegung. Für $n\in N$ existiert genau ein $c\in\{1,2,3\}$
+  mit $B(p_{n,c}) =1$. Setze $f(n) =c$. Dann ist $f$ eine gültige Färbung
+  des Graphen $G$.
+
+  \subsubsection{2. Anwendung des Kompaktheitsatzes:
+    Parkettierungen}\label{anwendung-des-kompaktheitsatzes-parkettierungen}
+
+  Idee: Gegeben ist eine Menge von quadratischen Kacheln mit gefärbten
+  Kanten. Ist es möglich, mit diesen Kacheln die gesamte Ebene zu
+  füllen,so dass aneinanderstoßende Kanten gleichfarbig sind?
+
+  Berühmtes Beispiel: Mit diesen 11 Kacheln kann die Ebene gefüllt werden,
+  aber dies ist nicht periodisch möglich.
+  \includegraphics[width=\linewidth]{Assets/Logik-parkettierung-1.png}
+
+  \begin{quote}
+    Definition
+
+    Ein Kachelsystem besteht aus einer endlichen Menge C von ``Farben'' und
+    einer Menge K von Abbildungen $\{N,O,S,W\}\rightarrow C$ von
+    ``Kacheln''. Eine Kachelung von $G\subseteq Z\times Z$ ist eine
+    Abbildung $f:G\rightarrow K$ mit - $f(i,j)(N) =f(i,j+ 1 )(S)$ für alle
+    $(i,j),(i,j+ 1 )\in G$ - $f(i,j)(O) =f(i+ 1 ,j)(W)$ für alle
+    $(i,j),(i+ 1 ,j)\in G$
+  \end{quote}
+
+  \begin{quote}
+    Satz
+
+    Sei $K$ ein Kachelsystem. Es existiert genau dann eine Kachelung von
+    $Z\times Z$, wenn für jedes $n\in N$ eine Kachelung von
+    $\{(i,j) :|i|,|j| \leq n\}$ existiert.
+  \end{quote}
+
+  Beweis: - $\Rightarrow$: trivial - $\Leftarrow$: Wir beschreiben
+  zunächst mit einer unendlichen Menge $\Gamma$ von Formeln, dass eine
+  Kachelung existiert: atomare Formeln $p_{k,i,j}$ für $k\in K$ und
+  $i,j\in Z$ (Idee: an der Stelle $(i,j)$ liegt die Kachel $k$, d.h.
+  $f(i,j) =k$) Für alle $(i,j)\in Z$ enthält $\Gamma$ die folgenden
+  Formeln: - eine der Kacheln aus $K$ liegt an der Stelle
+  $(i,j):\bigvee_{k\in K} p_{k,i,j}$ - es liegen nicht zwei verschiedene
+  Kacheln an der Stelle
+  $(i,j): \bigwedge_{k,k'\in K,k\not=k'} \lnot(p_{k,i,j}\wedge p_{k',i,j})$
+  - Kacheln an Stellen $(i,j)$ und $(i,j+1)$ ``passen übereinander'':
+  $\bigvee_{k,k'\in K,k(N)=k'(S)} (p_{k,i,j}\wedge p_{k',i,j+1})$ -
+  Kacheln an Stellen $(i,j)$ und $(i+1,j)$ ``passen nebeneinander'':
+  $\bigvee_{k,k'\in K,k(W)=k'(O)} (p_{k,i,j}\wedge p_{k',i+1,j})$
+
+  Sei nun $\Delta\subseteq\Gamma$ endlich. - $\Rightarrow$ es gibt
+  $n\in N$, so dass $\Delta$ nur atomare Formeln der Form $p_{k,i,j}$ mit
+  $|i|,|j|\leq n$ enthält. - Voraussetzung $\Rightarrow$ es gibt Kachelung
+  $g:\{(i,j) :|i|,|j| \leq n\}\rightarrow K$ für $k\in K$ und
+  $|i|,|j|\leq n$ definiere
+  $B(p_{k,i,j}) = \begin{cases} 1_B \quad\text{ falls } g(i,j) =k \\ 0_B \quad\text{ sonst} \end{cases}$
+  - $\Rightarrow B(\sigma) = 1_B$ für alle $\sigma\in\Delta$ (da $g$
+  Kachelung) - Also haben wir gezeigt, dass jede endliche Teilmenge von
+  $\Gamma$ erfüllbar ist. - Kompaktheitssatz $\Rightarrow$ es gibt
+  B-Belegung $B$ mit $B(\gamma) = 1_B$ für alle $\gamma\in\Gamma$ -
+  $\Rightarrow$ es gibt Abbildung $f:Z\times Z\rightarrow K$ mit
+  $f(i,j) =k \Leftarrow\Rightarrow B(p_{k,i,j}) = 1_B$. - Wegen
+  $B\Vdash\Gamma$ ist dies eine Kachelung.
+
+  Weitere Anwendungen des Kompaktheitsatzes - abz. partielle Ordnungen
+  sind linearisierbar - abz. Gleichungssystem über $\mathbb{Z}_2$ lösbar
+  $\Leftarrow\Rightarrow$ jedes endliche Teilsystem lösbar -
+  Heiratsproblem - Kőnigs Lemma (Übung) - \ldots{}
+
+  Bemerkung: Der Kompaktheitssatz gilt auch, wenn die Menge der atomaren
+  Formeln nicht abzählbar ist. Damit gelten die obigen Aussagen
+  allgemeiner: - 3-Färbbarkeit: beliebige Graphen - Linearisierbarkeit:
+  beliebige partielle Ordnungen - Lösbarkeit: beliebig große
+  Gleichungssysteme über $\mathbb{Z}_2$ - \ldots{}
+
+  \subsection{Erfüllbarkeit}\label{erfuxfcllbarkeit}
+
+  \begin{quote}
+    Erfüllbarkeitsproblem
+
+    Eingabe: Formel $\Gamma$
+
+    Frage: existiert eine B-Belegung $B$ mit $B(\Gamma) = 1_B$.
+  \end{quote}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          offensichtlicher Algorithmus: probiere alle Belegungen durch (d.h.
+          stelle Wahrheitswertetabelle auf)$\rightarrow$ exponentielle Zeit
+    \item
+          ``Automaten, Sprachen und Komplexität'': das Problem ist
+          NP-vollständig
+    \item
+          nächstes Ziel:spezielle Algorithmen für syntaktisch eingeschränkte
+          Formeln $\Gamma$
+    \item
+          Spätere Verallgemeinerung dieser Algorithmen (letzte Vorlesung des
+          Logik-Teils von ``Logik und Logikprogrammierung'') bildet Grundlage
+          der logischen Programmierung.
+  \end{itemize}
+
+  \subsubsection{Hornformeln (Alfred Horn,
+    1918-2001)}\label{hornformeln-alfred-horn-1918-2001}
+
+  \begin{quote}
+    Definition
+
+    Eine Hornklausel hat die Form
+    $(\lnot\bot\wedge p_1\wedge p_2\wedge ... \wedge p_n)\rightarrow q$ für
+    $n\geq 0$, atomare Formeln $p_1 ,p_2 ,... ,p_n$ und $q$ atomare Formel
+    oder $q=\bot$. Eine Hornformel ist eine Konjunktion von Hornklauseln.
+  \end{quote}
+
+  Schreib- und Sprechweise - $\{p_1,p_2 ,... ,p_n\}\rightarrow q$ für
+  Hornklausel
+  $(\lnot\bot\wedge p_1 \wedge p_2 \wedge ...\wedge p_n)\rightarrow q$
+  insbes. $\varnothing\rightarrow q$ für $\lnot\bot\rightarrow q$ -
+  $\{(M_i\rightarrow q_i)| 1 \leq i\leq n\}$ für Hornformel
+  $\bigwedge_{1 \leq i \leq n} (M_i\rightarrow q_i)$
+
+  Bemerkung, in der Literatur auch: -
+  $\{\lnot p_1,\lnot p_2 ,... ,\lnot p_n,q\}$ für
+  $\{p_1 ,... ,p_n\}\rightarrow q$ mit $q$ atomare Formel -
+  $\{\lnot p_1,\lnot p_2 ,... ,\lnot p_n\}$ für
+  $\{p_1 ,... ,p_n\}\rightarrow\bot$ - $\Box$ für
+  $\varnothing\rightarrow\bot$, die ``leere Hornklausel''
+
+  \subsubsection{Markierungsalgorithmus}\label{markierungsalgorithmus}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Eingabe: eine endliche Menge $\Gamma$ von Hornklauseln.
+  \end{itemize}
+
+  \begin{enumerate}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          while es gibt in $\Gamma$ eine Hornklausel $M\rightarrow q$, so dass
+          alle $p\in M$ markiert sind und $q$ unmarkierte atomare Formel ist: do
+          markiere $q$ (in allen Hornklauseln in $\Gamma$)
+    \item
+          if $\Gamma$ enthält eine Hornklausel der Form $M\rightarrow\bot$, in
+          der alle $p\in M$ markiert sind then return ``unerfüllbar'' else
+          return ``erfüllbar''
+  \end{enumerate}
+
+  Beweis einer Folgerung: Beispiel - Ziel ist es, die folgende Folgerung
+  zu zeigen:
+  $\{(AK\vee BK),(AK\rightarrow BK),(BK\wedge RL\rightarrow\lnot AK),RL\}\Vdash\lnot AK$
+  - Lemma: man muß Unerfüllbarkeit der folgenden Menge zeigen:
+  $\{(AK\vee BK),(AK\rightarrow BK),(BK\wedge RL\rightarrow \lnot AK),RL,\lnot\lnot AK\}$
+  - Dies ist keine Menge von Hornklauseln! - Idee: ersetze $BK$ durch
+  $\lnot BH$ in allen Formeln. - Ergebnis: - Aus $AK\vee BK$ wird
+  $\lnot BH\vee AK\equiv BH\rightarrow AK\equiv\{BH\}\rightarrow AK$. -
+  Aus $AK\rightarrow BK$ wird
+  $AK\rightarrow\lnot BH\equiv\lnot AK\vee\lnot BH\equiv AK\wedge BH\rightarrow\bot\equiv\{AK,BH\} \rightarrow\bot$.
+  - Aus $BK\wedge RL\rightarrow\lnot AK$ wird
+  $\lnot BH\wedge RL\rightarrow\lnot AK\equiv BH\vee\lnot RL\vee\lnot AK\equiv RL\wedge AK\rightarrow BH\equiv\{RL,AK\}\rightarrow BH$
+  - $RL\equiv (\varnothing\rightarrow RL)$ -
+  $\lnot\lnot AK\equiv (\varnothing\rightarrow AK)$ - Wir müssen also
+  zeigen, dass die folgende Menge von Hornklauseln unerfüllbar ist:
+  $\{\{BH\}\rightarrow AK,\{AK,BH\}\rightarrow\bot,\{RL,AK\}\rightarrow BH,\varnothing\rightarrow RL,\varnothing\rightarrow AK\}$
+
+  Der Markierungsalgorithmus geht wie folgt vor: 1. Runde: markiere $RL$
+  aufgrund der Hornklausel $\varnothing\rightarrow RL$ 2. Runde: markiere
+  $AK$ aufgrund der Hornklausel $\varnothing\rightarrow AK$ 3. Runde:
+  markiere $BH$ aufgrund der Hornklausel $\{RL,AK\}\rightarrow BH$
+
+  dann sind keine weiteren Markierungen möglich.
+
+  In der Hornklausel $\{AK,BH\}\rightarrow\bot$ sind alle atomaren Formeln
+  aus $\{AK,BH\}$ markiert. Also gibt der Algorithmus aus, dass die Menge
+  von Hornklauseln nicht erfüllbar ist.
+
+  Nach unserer Herleitung folgern wir, dass das Teil $A$ heil ist.
+
+  \begin{enumerate}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Der Algorithmus terminiert: in jedem Durchlauf der while-Schleife wird
+          wenigstens eine atomare Formel markiert. Nach endlich vielen Schritten
+          terminiert die Schleife also.
+    \item
+          Wenn der Algorithmus eine atomare Formelqmarkiert und wenn $B$ eine
+          B-Belegung ist, die $\Gamma$ erfüllt, dann gilt $B(q) = 1_B$. Beweis:
+          wir zeigen induktiv über $n$: Wenn $q$ in einem der ersten $n$
+          Schleifendurchläufe markiert wird, dann gilt $B(q) = 1_B$.
+  \end{enumerate}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          I.A. Die Aussage gilt offensichtlich für $n=0$.
+    \item
+          I.S. werde die atomare Formel $q$ in einem der ersten $n$
+          Schleifendurchläufe markiert. Dann gibt es eine Hornklausel
+          $\{p_1,p_2 ,... ,p_k\}\rightarrow q$, so dass $p_1 ,... ,p_k$ in den
+          ersten $n-1$ Schleifendurchläufen markiert wurden. Also gilt
+          $B(p_1)=...=B(p_k) = 1_B$ nach IV. Da $B$ alle Hornformeln aus
+          $\Gamma$ erfüllt, gilt insbesondere
+          $B(\{p_1 ,p_2 ,... ,p_k\}\rightarrow q) = 1_B$ und damit $B(q) = 1_B$.
+  \end{itemize}
+
+  \begin{enumerate}
+    \setcounter{enumi}{2}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Wenn der Algorithmus ``unerfüllbar'' ausgibt, dann ist $\Gamma$
+          unerfüllbar. Beweis: indirekt, wir nehmen also an, dass der
+          Algorithmus ``unerfüllbar'' ausgibt, $B$ aber eine B-Belegung ist, die
+          $\Gamma$ erfüllt. Sei $\{p_1 ,... ,p_k\}\rightarrow\bot$ die
+          Hornklausel aus $\Gamma$, die die Ausgabe ``unerfüllbar'' verursacht
+          (d.h. die atomaren Formeln $p_1 ,... ,p_k$ sind markiert). Nach 2.
+          gilt $B(p_1) =...=B(p_k) = 1_B$, also
+          $B(\{p_1 ,p_2 ,... ,p_k\}\rightarrow\bot) = 0_B$ im Widerspruch zur
+          Annahme, dass $B$ alle Hornklauseln aus $\Gamma$ erfüllt. Also kann es
+          keine erfüllende B-Belegung von $\Gamma$ geben.
+    \item
+          Wenn der Algorithmus ``erfüllbar'' ausgibt, dann erfüllt die folgende
+          B-Belegung alle Formeln aus $\Gamma$:
+          $B(p_i)=\begin{cases} 1_B \quad\text{ der Algorithmus markiert } p_i \\ 0_B \quad\text{ sonst} \end{cases}$
+          Beweis:
+
+          \begin{itemize}
+            \item
+                  Sei $M\rightarrow q$ eine beliebige Hornklausel aus $\Gamma$.
+            \item
+                  Ist ein $p\in M$ nicht markiert, so gilt
+                  $B(\bigwedge_{p\in M} p) = 0_B$ und damit $B(M\rightarrow q) = 1_B$.
+            \item
+                  Sind alle $p\in M$ markiert, so wird auch $q$ markiert, also
+                  $B(q) = 1_B$ und damit $B(M\rightarrow q) = 1_B$.
+            \item
+                  Gilt $q=\bot$, so existiert unmarkiertes $p\in M$ (da der
+                  Algorithmus sonst ``unerfüllbar'' ausgegeben hätte), also
+                  $B(M\rightarrow\bot) = 1_B$ wie im ersten Fall. Also gilt
+                  $B(M\rightarrow q) = 1_B$ für alle Hornklauseln aus $\Gamma$, d.h.
+                  $\Gamma$ ist erfüllbar.
+          \end{itemize}
+  \end{enumerate}
+
+  \begin{quote}
+    Satz
+
+    Sei $\Gamma$ endliche Menge von Hornklauseln. Dann terminiert der
+    Markierungsalgorithmus mit dem korrekten Ergebnis.
+  \end{quote}
+
+  Beweis: Die Aussagen 1.-4. beweisen diesen Satz.
+
+  Bemerkungen: - Mit einer geeigneten Implementierung läuft der
+  Algorithmusin linearer Zeit. - Wir haben sogar gezeigt, dass bei Ausgabe
+  von ``erfüllbar'' eine erfüllende B-Belegung berechnet werden kann.
+
+  \subsubsection{SLD-Resolution}\label{sld-resolution}
+
+  \begin{quote}
+    Definition
+
+    Sei $\Gamma$ eine Menge von Hornklauseln. Eine SLD-Resolution aus
+    $\Gamma$ ist eine Folge
+    $(M_0\rightarrow\bot,M_1\rightarrow\bot,... ,M_m\rightarrow\bot)$ von
+    Hornklauseln mit - $(M_0\rightarrow\bot)\in\Gamma$ - für alle
+    $0\leq n<m$ existiert $(N\rightarrow q)\in\Gamma$ mit $q\in M_n$ und
+    $M_{n+1} = M_n\backslash\{q\}\cup N$
+  \end{quote}
+
+  Beispiel: -
+  $\Gamma =\{\{BH\}\rightarrow AK,\{AK,BH\}\rightarrow\bot,\{RL,AK\}\rightarrow BH,\varnothing\rightarrow RL,\varnothing\rightarrow AK\}$
+  - $M_0 =\{AK,BH\}$ - $M_1 =M_0 \backslash\{BH\}\cup\{RL,AK\}=\{RL,AK\}$
+  - $M_2 =M_1 \backslash\{RL\}\cup\varnothing =\{AK\}$ -
+  $M_3 =M_2 \backslash\{AK\}\cup\varnothing =\varnothing$
+
+  \begin{quote}
+    Lemma A
+
+    Sei $\Gamma$ eine (u.U. unendliche) Menge von Hornklauseln und
+    $(M_0\rightarrow\bot, M_1\rightarrow\bot,... , M_m\rightarrow\bot)$ eine
+    SLD-Resolution aus $\Gamma$ mit $M_m=\varnothing$. Dann ist $\Gamma$
+    nicht erfüllbar.
+  \end{quote}
+
+  Beweis: - indirekt: angenommen, es gibt B-Belegung $B$ mit
+  $B(N\rightarrow q) = 1_B$ für alle $(N\rightarrow q)\in\Gamma$. - Wir
+  zeigen für alle $0\leq n\leq m$ per Induktion über n: es gibt $p\in M_n$
+  mit $B(p) = 0_B$ (4) - I.A.: $n=0:(M_0 \rightarrow\bot,...)$
+  SLD-Resolution $\Rightarrow(M_0\rightarrow\bot)\in\Gamma$ -
+  $\Rightarrow B(M_0\rightarrow\bot) = 1_B$ - $\Rightarrow$ es gibt
+  $p\in M_0$ mit $B(p) = 0_B$ - I.V.: sei $n<m$ und $p\in M_n$ mit
+  $B(p) = 0_B$ - I.S.:
+  $(... ,M_n\rightarrow\bot,M_{n+ 1}\rightarrow\bot,...)$ SLD-Resolution
+  $\Rightarrow$ es gibt $(N\rightarrow q)\in\Gamma$ mit
+  $M_{n+1} =M_n\backslash\{q\}\cup N$ und $q\in M_n$. Zwei Fälle werden
+  unterschieden: 1. $p\not=q$: dann gilt $p\in M_{n+1}$ mit $B(p) = 0_B$
+  2. $p=q:(N\rightarrow q)\in\Gamma\Rightarrow B(N\rightarrow q) = 1_B$ es
+  gibt $p'\in N\subseteq M_{n+1}$ mit $B(p')=0_B$ in jedem der zwei Fälle
+  gilt also (4) für $n+1$. - Damit ist der induktive Beweis von (4)
+  abgeschlossen. - Mit $m=n$ haben wir insbesondere ``es gibt $p\in M_m$
+  mit $B(p) = 0_B$'' im Widerspruch zu $M_m=\varnothing$. Damit ist der
+  indirekte Beweis abgeschlossen.
+
+  \begin{quote}
+    Lemma B
+
+    Sei $\Gamma$ eine (u.U. unendliche) unerfüllbare Menge von Hornklauseln.
+    Dann existiert eine SLD-Resolution
+    $(M_0\rightarrow\bot,...,M_m\rightarrow\bot)$ aus $\Gamma$ mit
+    $M_m=\varnothing$.
+  \end{quote}
+
+  Beweis: Endlichkeitssatz: es gibt $\Delta\subseteq\Gamma$ endlich und
+  unerfüllbar. Bei Eingabe von$\Delta$ terminiert Markierungsalgorithmus
+  mit ``unerfüllbar'' - $r\geq 0...$ Anzahl der Runden - $q_i...$
+  Atomformel, die in $i$ Runde markiert wird $(1\leq i\leq r)$
+
+  Behauptung: Es gibt $m\leq r$ und SLD-Resolution
+  $(M_0\rightarrow\bot,...,M_m\rightarrow\bot)$ aus $\Delta$ mit
+  $M_m=\varnothing$ und $M_n\subseteq\{q_1,q_2,... ,q_{r-n}\}$ f.a.
+  $0\leq n\leq m$. (5)
+
+  Beweis der Behauptung: Wir konstruieren die Hornklauseln
+  $M_i\rightarrow\bot$ induktiv: - I.A.: Da der Markierungsalgorithmus mit
+  ``unerfüllbar'' terminiert, existiert eine Hornklausel
+  $(M_0\rightarrow\bot)\in\Gamma$ mit $M_0\subseteq\{q_1,... ,q_{r- 0}\}$.
+  $(M_0\rightarrow\bot)$ ist SLD-Resolution aus $\Delta$, die (5) erfüllt.
+  - I.V.: Sei $n\leq r$ und $(M_0\rightarrow\bot,... ,M_n\rightarrow\bot)$
+  SLD-Resolution, so dass (5) gilt. - I.S.: wir betrachten drei Fälle: 1.
+  Fall $M_n=\varnothing$: mit $m:=n$ ist Beweis der Beh. abgeschlossen. 2.
+  Fall $n=r$: Nach (5) gilt $M_n\subseteq\{q_1,...,q_{r-n}\}=\varnothing$.
+  Mit $m:=n$ ist der Beweis der Beh. abgeschlossen. 3. Fall $n<r$ und
+  $M_n \not=\varnothing$. Sei $k$ maximal mit
+  $q_k\in M_n\subseteq\{q_1,q_2,... ,q_{r-n}\}$. Also existiert
+  $(N\rightarrow q_k)\in\Delta$, so dass $N\subseteq\{q_1,... ,q_{k-1}\}$.
+  Setze
+  $M_{n+1}=M_n\backslash\{q_k\}\cup N\subseteq\{q_1,... ,q_{k-1}\}\subseteq\{q_1,...,q_{r-(n+1)}\}$.
+
+  Damit ist der induktive Beweis der Beh. abgeschlossen, woraus das Lemma
+  unmittelbar folgt.
+
+  \begin{quote}
+    Satz
+
+    Sei $\Gamma$ eine (u.U. unendliche) Menge von Hornklauseln. Dann sind
+    äquivalent: 1. $\Gamma$ ist nicht erfüllbar. 2. Es gibt eine
+    SLD-Resolution
+    $(M_0\rightarrow\bot,M_1\rightarrow\bot,... ,M_m\rightarrow\bot)$ aus
+    $\Gamma$ mit $M_m=\varnothing$.
+  \end{quote}
+
+  Beweis: Folgt unmittelbar aus Lemmata A und B.
+
+  Beispiel:
+  $\Gamma=\{\{a,b\}\rightarrow\bot,\{a\}\rightarrow c, \{b\}\rightarrow c,\{c\}\rightarrow a,\varnothing\rightarrow b$;
+  alle SLD-Resolutionen aus$\Gamma$ kann man durch einen Baum beschreiben:
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-11.png}
+
+  Die Suche nach einer SLD-Resolution mit $M_m=\varnothing$ kann
+  grundsätzlich auf zwei Arten erfolgen: - Breitensuche: - findet
+  SLD-Resolution mit $M_m=\varnothing$ (falls sie existiert), da Baum
+  endlich verzweigend ist (d.h. die Niveaus sind endlich) - hoher
+  Platzbedarf, da ganze Niveaus abgespeichert werden müssen (in einem
+  Binärbaum der Tiefe $n$ kann es Niveaus der Größe $2^n$ geben) -
+  Tiefensuche: - geringerer Platzbedarf (in einem Binärbaum der Tiefe $n$
+  hat jeder Ast die Länge $\leq n$) - findet existierende SLD-Resolution
+  mit $M_m=\varnothing$ nicht immer (siehe Beispiel)
+
+  \subsection{Zusammenfassung
+    Aussagenlogik}\label{zusammenfassung-aussagenlogik}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Das natürliche Schließen formalisiert die ``üblichen'' Argumente in
+          mathematischen Beweisen.
+    \item
+          Unterschiedliche Wahrheitswertebereiche formalisieren unterschiedliche
+          Vorstellungen von ``Wahrheit''.
+    \item
+          Das natürliche Schließen ist vollständig und korrekt für den
+          Booleschen Wahrheitswertebereich.
+    \item
+          Der Markierungsalgorithmus und die SLD-Resolution sind praktikable
+          Verfahren, um die Erfüllbarkeit von Hornformeln zu bestimmen.
+  \end{itemize}
+
+  \section{Kapitel 2: Prädikatenlogik}\label{kapitel-2-pruxe4dikatenlogik}
+
+  Beispiel: Graphen
+  \includegraphics[width=\linewidth]{Assets/Logik-prädikatenlogik-graph.png} Um über diesen
+  Graphen Aussagen in der Aussagenlogik zu machen, verwenden wir Formeln
+  $\varphi_{i,j}$ für $1\leq i,j\leq 9$ mit
+  $\varphi_{i,j}=\begin{cases} \lnot\bot\quad\text{ falls} (v_i,v_j) Kante\\ \bot\quad\text{ sonst}\end{cases}$
+  - Die aussagenlogische Formel $\bigvee_{1\leq i,j\leq 9} \varphi_{i,j}$
+  sagt aus, dass der Graph eine Kante enthält. - Die aussagenlogische
+  Formel $\bigwedge_{1\leq i\leq 9} \bigvee_{1\leq j\leq 9} \varphi_{i,j}$
+  sagt aus, dass jeder Knoten einen Nachbarn hat - Die aussagenlogische
+  Formel
+  $\bigvee_{1\leq i,j,k\leq 9 verschieden} \varphi_{i,j}\wedge\varphi_{j,k}\wedge\varphi_{k,i}$
+  sagt aus, dass der Graph ein Dreieck enthält. Man kann so vorgehen, wenn
+  der Graph bekannt und endlich ist. Sollen analoge Aussagen für einen
+  anderen Graphen gemacht werden, so ist die Kodierungsarbeit zu
+  wiederholen.
+
+  Beispiel: Datenbanken - Im folgenden reden wir über die Studenten und
+  die Lehrenden in Veranstaltungen zur Theoretischen Informatik in diesem
+  Semester. Betrachte die folgenden Aussagen: - Jeder ist Student oder
+  wissenschaftlicher Mitarbeiter oder Professor. - Dietrich Kuske ist
+  Professor. - Kein Student ist Professor. - Jeder Student ist jünger als
+  jeder Professor. - Es gibt eine Person, die an den Veranstaltungen
+  ``Logik und Logikprogrammierung'' und ``Algorithmen und
+  Datenstrukturen'' teilnimmt. - Es gibt eine Person, die kein
+  wissenschaftlicher Mitarbeiter ist und nicht an beiden Veranstaltungen
+  teilnimmt. - Jeder Student ist jünger als die Person, mit der er am
+  besten über Informatik reden kann. - Um sie in der Aussagenlogik machen
+  zu können, müssen wir atomare Aussagen für ``Hans ist Student'', ``Otto
+  ist jünger als Ottilie'' usw. einführen. Dies ist nur möglich, wenn 1.
+  alle involvierten Personen bekannt sind und fest stehen und 2. es nur
+  endlich viele involvierte Personen gibt. - Sollen analoge Aussagen für
+  das vorige oder das kommende Jahr gemacht werden, so ist die gesamte
+  Kodierungsarbeit neu zu machen.
+
+  \subsection{Kodierung in einer
+    ``Struktur''}\label{kodierung-in-einer-struktur}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Grundmenge: Die Studenten und die Lehrenden in Veranstaltungen zur
+          Theoretischen Informatik in diesem Sommersemester
+    \item
+          Teilmengen:
+    \item
+          $S(x)$ ``x ist Student''
+    \item
+          $LuLP(x)$ ``x nimmt an der Veranstaltung LuLP teil''
+    \item
+          $AuD(x)$ ``x nimmt an der Veranstaltung AuD teil''
+    \item
+          $Pr(x)$ ``x ist Professor''
+    \item
+          $WM(x)$ ``x ist wissenschaftlicher Mitarbeiter''
+    \item
+          Relationen:
+    \item
+          $J(x,y)$ ``x ist jünger als y''
+    \item
+          Funktion:
+    \item
+          $f(x)$ ist diejenige Person (aus dem genannten Kreis), mit der x am
+          besten über Informatik reden kann.
+    \item
+          Konstante:
+    \item
+          $dk$ Dietrich Kuske
+  \end{itemize}
+
+  Die in der Aussagenlogik nur schwer formulierbaren Aussagen werden nun -
+  Für alle $x$ gilt $S(x)\vee WM(x)\vee Pr(x)$ - $Pr(dk)$ - Für alle $x$
+  gilt $S(x)\rightarrow\lnot Pr(x)$ - Für alle $x$ und $y$ gilt
+  $(S(x)\wedge Pr(y))\rightarrow J(x,y)$ - Es gibt ein $x$ mit
+  $LuLP(x)\wedge AuD(x)$ - Es gibt ein $x$ mit
+  $((\lnot LuLP(x)\vee\lnot AuD(x))\wedge\lnot WM(x))$ - Für alle $x$ gilt
+  $S(x)\rightarrow J(x,f(x))$
+
+  Bemerkung: Diese Formulierungen sind auch brauchbar, wenn die Grundmenge
+  unendlich ist. Sie sind auch unabhängig vom Jahr (im nächsten Jahr
+  können diese Folien wieder verwendet werden).
+
+  Ziel - Wir wollen in der Lage sein, über Sachverhalte in ``Strukturen''
+  (Graphen, Datenbanken, relle Zahlen, Gruppen\ldots{} ) zu reden. - Dabei
+  soll es ``Relationen'' geben, durch die das Enthaltensein in einer
+  Teilmenge oder Beziehungen zwischen Objekten ausgedrückt werden können
+  (z.B. $S(x),J(x,y),...$ ) - Weiter soll es ``Funktionen'' geben, durch
+  die Objekte (oder Tupel von Objekten) auf andere Objekte abgebildet
+  werden (z.B. $f$) - Nullstellige Funktionen (ohne Argumente): Konstante
+  (z.B. $dk$)
+
+  Fragen - Nach welchen Regeln bildet man korrekte Formeln? - Was ist eine
+  Struktur? - Wann hat eine Aussage in einer Struktur eine Bedeutung (ist
+  ``sinnvoll'')? - Wann ``gilt'' eine Aussage in einer Struktur? - Gibt es
+  Formeln, die in allen Strukturen gelten? - Kann man solche Formeln
+  algorithmisch identifizieren? Gibt es einen Beweiskalkül wie das
+  natürliche Schließen oder die SLD-Resolution? - \ldots{}\ldots{}\ldots{}
+
+  \subsection{Syntax der
+    Prädikatenlogik}\label{syntax-der-pruxe4dikatenlogik}
+
+  Formeln machen Aussagen über Strukturen. Dabei hat es keinen Sinn zu
+  fragen, ob eine Formel, die über Studenten etc. redet, im Graphen $G$
+  gilt.
+
+  \begin{quote}
+    Definition
+
+    Eine Signatur ist ein Tripel $\sum=(\Omega, Rel,ar)$, wobei $\Omega$ und
+    $Rel$ disjunkte Mengen von Funktions- und Relationsnamen sind und
+    $ar:\Omega\cup Rel\rightarrow\mathbb{N}$ eine Abbildung ist.
+  \end{quote}
+
+  Beispiel: $\Omega=\{f,dk\}$ mit $ar(f) =1,ar(dk)=0$ und
+  $Rel=\{S,LuLP,AuD,Pr,WM,J\}$ mit
+  $ar(S) =ar(LuLP) =ar(AuD) =ar(Pr) =ar(WM) =1 undar(J) = 2$ bilden die
+  Signatur der Datenbank von vorhin. - typische Funktionsnamen:
+  $f, g, a, b...$ mit $ar(f),ar(g) > 0$ und $ar(a) =ar(b) = 0$ - typische
+  Relationsnamen: $R,S,...$
+
+  \begin{quote}
+    Definition
+
+    Die Menge der Variablen ist $Var=\{x_0,x_1 ,...\}$.
+  \end{quote}
+
+  \begin{quote}
+    Definition
+
+    Sei $\sum$ eine Signatur. Die Menge $T_{\sum}$ der $\sum$-Terme ist
+    induktiv definiert: 1. Jede Variable ist ein Term, d.h.
+    $Var\subseteq T_{\sum}$ 2. ist $f\in\Omega$ mit $ar(f)=k$ und sind
+    $t_1,...,t_k\in T_{\sum}$, so gilt $f(t_1,...,t_k)\in T_{\sum}$ 3.
+    Nichts ist $\sum$-Term, was sich nicht mittels der obigen Regeln
+    erzeugen läßt.
+  \end{quote}
+
+  Beispiel:In der Signatur der Datenbank von vorhin haben wir u.a. die
+  folgenden Terme: - $x_1$ und $x_8$ - $f(x_0)$ und $f(f(x_3))$ - $dk()$
+  und $f(dk())$ - hierfür schreiben wir kürzer $dk$ bzw. $f(dk)$
+
+  \begin{quote}
+    Definition
+
+    Sei $\sum$ Signatur. Die atomaren $\sum$-Formeln sind die Zeichenketten
+    der Form - $R(t_1,t_2,...,t_k)$ falls $t_1,t_2,...,t_k\in T_{\sum}$ und
+    $R\in Rel$ mit $ar(R)=k$ oder - $t_1=t_2$ falls $t_1,t_2\in T_{\sum}$
+    oder - $\bot$.
+  \end{quote}
+
+  Beispiel: In der Signatur der Datenbank von vorhin haben wir u.a. die
+  folgenden atomaren Formeln: - $S(x_1)$ und $LuLP(f(x4))$ - $S(dk)$ und
+  $AuD(f(dk))$
+
+  \begin{quote}
+    Definition
+
+    Sei $\sum$ Signatur. $\sum$-Formeln werden durch folgenden induktiven
+    Prozeß definiert: 1. Alle atomaren $\sum$-Formeln sind $\sum$-Formeln.
+    2. Falls $\varphi$ und $\Psi$ $\sum$-Formeln sind, sind auch
+    $(\varphi\wedge\Psi)$,$(\varphi\vee\Psi)$ und $(\varphi\rightarrow\Psi)$
+    $\sum$-Formeln. 3. Falls $\varphi$ eine $\sum$-Formel ist, ist auch
+    $\lnot\varphi$ eine $\sum$-Formel. 4. Falls $\varphi$ eine $\sum$-Formel
+    und $x\in Var$, so sind auch $\forall x\varphi$ und $\exists x\varphi$
+    $\sum$-Formeln. 5. Nichts ist $\sum$-Formel, was sich nicht mittels der
+    obigen Regeln erzeugen läßt.
+  \end{quote}
+
+  Ist die Signatur $\sum$ aus dem Kontext klar, so sprechen wir einfach
+  von Termen, atomaren Formeln bzw.Formeln.
+
+  Beispiel:In der Signatur der Datenbank von vorhin haben wir u.a. die
+  folgenden Formeln: - $\forall x_0 (S(x_0)\vee WM(x_0)\vee Pr(x_0))$ -
+  $Pr(dk)$ - $\forall x_3 (S(x_3)\rightarrow\lnot Pr(x_3))$ -
+  $\forall x_0 \forall x_2 ((S(x_0)\wedge Pr(x_2))\rightarrow J(x_0,x_2))$
+  - $\exists x_1 (LuLP(x_1)\wedge AuD(x_1))$ -
+  $\exists x_2 ((\lnot LuLP(x_2)\vee\lnot AuD(x_2))\wedge\lnot WM(x_2))$ -
+  $\forall x_1 (S(x_1)\rightarrow J(x_1,f(x_1)))$
+
+  Wir verwenden die aus der Aussagenlogik bekannten Abkürzungen, z.B.
+  steht $\varphi\leftrightarrow\Psi$ für
+  $(\varphi\rightarrow\Psi)\wedge(\Psi\rightarrow\varphi)$.
+
+  Zur verbesserten Lesbarkeit fügen wir mitunter hinter quantifizierten
+  Variablen einen Doppelpunkt ein, z.B. steht $\exists x\forall y:\varphi$
+  für $\exists x\forall y\varphi$
+
+  Ebenso verwenden wir Variablennamen $x$,$y$,$z$ u.ä.
+
+  Präzedenzen: $\lnot,\forall x,\exists x$ binden am stärksten
+
+  Beispiel:
+  $(\lnot\forall x:R(x,y)\wedge\exists z:R(x,z))\rightarrow P(x,y,z)$
+  steht für
+  $((\lnot(\forall x:R(x,y))\wedge\exists z:R(x,z))\rightarrow P(x,y,z))$
+
+  \subsection{Aufgabe}\label{aufgabe}
+
+  Im folgenden seien - $P$ ein-stelliges, $Q$ und $R$ zwei-stellige
+  Relationssymbole, - $a$ null-stelliges und $f$ ein-stelliges
+  Funktionssymbol und - $x,y$ und $z$ Variable.
+
+  Welche der folgenden Zeichenketten sind Formeln? \textbar{} \textbar{}
+  \textbar{} \textbar{} ---------------------------------------------
+  \textbar{} ---- \textbar{} \textbar{} $\forall x P(a)$ \textbar{} ja
+  \textbar{} \textbar{} $\forall x\exists y(Q(x,y)\vee R(x))$ \textbar{}
+  nein \textbar{} \textbar{} $\forall x Q(x,x)\rightarrow\exists x Q(x,y)$
+  \textbar{} ja \textbar{} \textbar{} $\forall x P(f(x))\vee\forall$ x
+  Q(x,x)\$ \textbar{} ja \textbar{} \textbar{}
+  $\forall x(P(y)\wedge\forall y P(x))$ \textbar{} ja \textbar{}
+  \textbar{} $P(x) \rightarrow\exists x Q(x,P(x))$ \textbar{} nein
+  \textbar{} \textbar{} $\forall f\exists x P(f(x))$ \textbar{} nein
+  \textbar{}
+
+  \begin{quote}
+    Definition
+
+    Sei $\sum$ eine Signatur. Die Menge $FV(\varphi)$ der freien Variablen
+    einer $\sum$-Formel $\varphi$ ist induktiv definiert: - Ist $\varphi$
+    atomare $\sum$-Formel, so ist $FV(\varphi)$ die Menge der in $\varphi$
+    vorkommenden Variablen. -
+    $FV(\varphi\Box\Psi) =FV(\varphi)\cup FV(\Psi)$ für
+    $\Box\in\{\wedge,\vee,\rightarrow\}$ - $FV(\lnot\varphi) =FV(\varphi)$ -
+    $FV(\exists x\varphi) =FV(\forall x\varphi) =FV(\varphi)\backslash\{x\}$.
+    Eine $\sum$-Formel $\varphi$ ist geschlossen oder ein $\sum$-Satz, wenn
+    $FV(\varphi)=\varnothing$ gilt.
+  \end{quote}
+
+  Was sind die freien Variablen der folgenden Formeln? Welche Formeln sind
+  Sätze? \textbar{} \textbar{} freie Variablen? \textbar{} Satz?
+  \textbar{} \textbar{}
+  -----------------------------------------------------------------------
+  \textbar{} ---------------- \textbar{} ----- \textbar{} \textbar{}
+  $\forall x P(a)$ \textbar{} nein \textbar{} ja \textbar{} \textbar{}
+  $\forall x Q(x,x)\rightarrow\exists x Q(x,y)$ \textbar{} y \textbar{}
+  nein \textbar{} \textbar{} $\forall x P(x)\vee\forall x Q(x,x)$
+  \textbar{} nein \textbar{} ja \textbar{} \textbar{}
+  $\forall x(P(y)\wedge\forall y P(x))$ \textbar{} y \textbar{} nein
+  \textbar{} \textbar{} $\forall x(\lnot\forall y Q(x,y)\wedge R(x,y))$
+  \textbar{} y \textbar{} nein \textbar{} \textbar{}
+  $\exists z(Q(z,x)\vee R(y,z))\rightarrow\exists y(R(x,y)\wedge Q(x,z))$
+  \textbar{} x,y,z \textbar{} nein \textbar{} \textbar{}
+  $\exists x(\lnot P(x)\vee P(f(a)))$ \textbar{} nein \textbar{} ja
+  \textbar{} \textbar{} $P(x)\rightarrow\exists x P(x)$ \textbar{} x
+  \textbar{} nein \textbar{} \textbar{}
+  $\exists x\forall y((P(y)\rightarrow Q(x,y))\vee\lnot P(x))$ \textbar{}
+  x \textbar{} nein \textbar{} \textbar{} $\exists x\forall x Q(x,x)$
+  \textbar{} nein \textbar{} ja \textbar{}
+
+  Semantik der Prädikatenlogik - Erinnerung: Die Frage ``Ist die
+  aussagenlogische Formel $\varphi$ wahr oder falsch?'' war sinnlos, denn
+  wir wissen i.a. nicht, ob die atomaren Aussagen wahr oder falsch sind. -
+  Analog: Die Frage ``Ist die prädikatenlogische Formel $\varphi$ wahr
+  oder falsch?'' ist sinnlos, denn wir wissen bisher nicht, über welche
+  Objekte, über welche ``Struktur'' $\varphi$ spricht.
+
+  \begin{quote}
+    Definition
+
+    Sei $\sum$ eine Signatur. Eine $\sum$-Struktur ist ein Tupel
+    $A=(U_A,(f^A)_{f\in\Omega},(R^A)_{R\in Rel})$, wobei - $U_A$ eine
+    nichtleere Menge, das Universum, - $R^A\supseteq U_A^{ar(R)}$ eine
+    Relation der Stelligkeit $ar(R)$ für $R\in Rel$ und -
+    $f^A:U_A^{ar(f)}\rightarrow U_A$ eine Funktion der Stelligkeit $ar(f)$
+    für $f\in\Omega$ ist.
+  \end{quote}
+
+  Bemerkung: $U_A^0=\{()\}$. - Also ist $a^A:U_A^0\rightarrow U_A$ für
+  $a\in\Omega$ mit $ar(a)=0$ vollständig gegeben durch $a^A(())\in U_A$.
+  Wir behandeln 0-stellige Funktionen daher als Konstanten. - Weiterhin
+  gilt $R^A=\varnothing$ oder $R^A=\{()\}$ für $R\in Rel$ mit $ar(R)=0$.
+
+  Beispiel: Graph - Sei $\sum=(\Omega ,Rel,ar)$ mit
+  $\Omega=\varnothing ,Rel=\{E\}$ und $ar(E)=2$ die Signatur der Graphen.
+  - Um den Graphen als $\sum$-Struktur $A=(UA,EA)$ zu betrachten, setzen
+  wir - $UA=\{v_1,v_2,...,v_9\}$ und -
+  $EA=\{(v_i,v_j)|(v_i,v_j) ist Kante\}$
+
+  Im folgenden sei $\sum$ eine Signatur, A eine $\sum$-Struktur und
+  $\rho:Var\rightarrow U_A$ eine Abbildung (eine Variableninterpretation).
+  Wir definieren eine Abbildung $\rho':T\sum\rightarrow U_A$ induktiv für
+  $t\in T_{\sum}$: - ist $t\in Var$, so setze $\rho'(t) =\rho(t)$ -
+  ansonsten existieren $f\in\Omega$ mit $ar(f)=k$ und
+  $t_1,...,t_k\in T_{\sum}$ mit $t=f(t_1,...,t_k)$. Dann setze
+  $\rho'(t) =f^A(\rho'(t_1),...,\rho'(t_k))$. Die Abbildung $\rho'$ ist
+  die übliche ``Auswertungsabbildung''. Zur Vereinfachung schreiben wir
+  auch $\rho(t)$ an Stelle von $\rho'(t)$.
+
+  Beispiel: - Seien $A=(R,f^A,a^A)$ mit $f^A$ die Subtraktion und $a$
+  nullstelliges Funktionssymbol mit $a^A=10$. Seien weiter $x,y\in Var$
+  mit $\rho(x)=7$ und $\rho(y)=-2$. Dann gilt
+  $\rho(f(a,f(x,y))) =\rho(a)-(\rho(x)-\rho(y)) =a^A-(\rho(x)-\rho(y)) = 1$
+  - Seien $A= (Z,f^A,a^A)$ mit $f^A$ die Maximumbildung, $a$ nullstelliges
+  Funktionssymbol mit $a^A=10$. Seien weiter $x,y\in Var$ mit $\rho(x)=7$
+  und $\rho(y)=-2$. In diesem Fall gilt
+  $\rho(f(a,f(x,y))) = max(\rho(a),max(\rho(x),\rho(y)) = max(a^A,max(\rho(x),\rho(y))) = 10$
+
+  Bemerkung: Wir müssten also eigentlich noch vermerken, in welcher
+  Struktur $\rho(t)$ gebildet wird - dies wird aber aus dem Kontext immer
+  klar sein.
+
+  Für eine $\sum$-Formel $\varphi$ definieren wir die Gültigkeit in einer
+  $\sum$-Struktur $A$ unter der Variableninterpretation $\rho$ (in
+  Zeichen: $A\Vdash_\rho\varphi$) induktiv: - $A\Vdash_\rho\bot$ gilt
+  nicht. - $A\Vdash_\rho R(t_1,...,t_k)$ falls
+  $(\rho(t_1),...,\rho(t_k))\in R^A$ für $R\in Rel$ mit $ar(R)=k$ und
+  $t_1,...,t_k\in T_{\sum}$. - $A\Vdash_\rho t_1 =t_2$ falls
+  $\rho(t_1) =\rho(t_2)$ für $t_1,t_2\in T_{\sum}$.
+
+  Für $\sum$-Formeln $\varphi$ und $\Psi$ und $x\in Var$: -
+  $A\Vdash_p \varphi\wedge\Psi$ falls $A\Vdash_p\varphi$ und
+  $A\Vdash_p \Psi$. - $A\Vdash_p \varphi\vee\Psi$ falls $A\Vdash_p\varphi$
+  oder $A\Vdash_p\Psi$ . - $A\Vdash_p \varphi\rightarrow\Psi$ falls nicht
+  $A\Vdash_p\varphi$ oder $A\Vdash_p\Psi$ . - $A\Vdash_p \lnot\varphi$
+  falls $A\Vdash_p \varphi$ nicht gilt. - $A\Vdash_p \exists x\varphi$
+  falls ??? - $A\Vdash_p \forall x\varphi$ falls ???
+
+  Für $x\in Var$ und $a\in U_A$ sei
+  $\rho[x\rightarrow a]:Var\rightarrow U_A$ die Variableninterpretation,
+  für die gilt
+  $(\rho[x\rightarrow a])(y) = \begin{cases} \rho(y) \quad\text{ falls } x\not=y \\ a \quad\text{ sonst } \end{cases}$
+  - $A\Vdash_p \exists x\varphi$ falls es $a\in U_A$ gibt mit
+  $A\Vdash_{p[x\rightarrow a]}\varphi$. - $A\Vdash_p \forall x\varphi$
+  falls $A\Vdash_{p[x\rightarrow a]}\varphi$ für alle $a\in U_A$.
+
+  \begin{quote}
+    Definition
+
+    Sei $\sum$ eine Signatur, $\varphi$ eine $\sum$-Formel, $\Delta$ eine
+    Menge von $\sum$-Formeln und $A$ eine $\sum$-Struktur. -
+    $A\Vdash\varphi$ ($A$ ist Modell von $\varphi$) falls $A\Vdash_p\varphi$
+    für alle Variableninterpretationen $\rho$ gilt. - $A\Vdash\Delta$ falls
+    $A\Vdash\Psi$ für alle $\Psi\in\Delta$.
+  \end{quote}
+
+  Aufgaben - Sei $A$ die Struktur, die dem vorherigen Graphen entspricht -
+  Welche der folgenden Formeln $\varphi$ gelten in $A$, d.h. für welche
+  Formeln gilt $A\Vdash_p\varphi$ für alle Variableninterpretationen
+  $\rho$? 1. $\exists x\exists y:E(x,y)$ 2. $\forall x\exists y:E(x,y)$ 3.
+  $\exists x\forall y:(x\not=y\rightarrow E(x,y))$ 4.
+  $\forall x\forall y:(x\not=y\rightarrow E(x,y))$ 5.
+  $\exists x\exists y\exists z:(E(x,y)\wedge E(y,z)\wedge E(z,x))$ - In
+  der Prädikatenlogik ist es also möglich, die Eigenschaften vom Anfang
+  des Kapitels auszudrücken, ohne den Graphen direkt in die Formel zu
+  kodieren.
+
+  \begin{quote}
+    Definition
+
+    Sei $\sum$ eine Signatur, $\varphi$ eine $\sum$-Formel, $\Delta$ eine
+    Menge von $\sum$-Formeln und $A$ eine $\sum$-Struktur. - $\Delta$ ist
+    erfüllbar, wenn es $\sum$-Struktur $B$ und Variableninterpretation
+    $\rho:Var\rightarrow U_B$ gibt mit $B\Vdash_\rho\Psi$ für alle
+    $\Psi\in\Delta$. - $\varphi$ ist semantische Folgerung von
+    $\Delta(\Delta\Vdash\varphi)$ falls für alle $\sum$-Strukturen $B$ und
+    alle Variableninterpretationen $\rho:Var\rightarrow U_B$ gilt: Gilt
+    $B\Vdash_\rho\Psi$ für alle $\Psi\in\Delta$, so gilt auch
+    $B\Vdash_\rho \varphi$. - $\varphi$ ist allgemeingültig, falls
+    $B\Vdash \rho\varphi$ für alle $\sum$-Strukturen $B$ und alle
+    Variableninterpretationen $\rho$ gilt.
+  \end{quote}
+
+  Bemerkung: $\varphi$ allgemeingültig gdw. $\varnothing\Vdash\varphi$
+  gdw. $\{\lnot\varphi\}$ nicht erfüllbar. Hierfür schreiben wir auch
+  $\Vdash\varphi$.
+
+  Beispiel: Der Satz
+  $\varphi=(\forall x:R(x)\rightarrow\forall x:R(f(x)))$ ist
+  allgemeingültig.
+
+  Beweis: Sei $\sum$ Signatur, so dass $\varphi$ $\sum$-Satz ist. Sei $A$
+  $\sum$-Struktur und $\rho$ Variableninterpretation. Wir betrachten zwei
+  Fälle: 1. Falls $A\not\Vdash_\rho\forall x R(x)$, so gilt
+  $A\Vdash_p\varphi$. 2. Wir nehmen nun $A\Vdash_p\forall x R(x)$ an. Sei
+  $a\in U_A$ beliebig und $b=f^A(a)$.
+  $A\Vdash_p\forall x R(x) \Rightarrow A\Vdash_{p[x\rightarrow b]} R(x) \Rightarrow RA\owns (p[x\rightarrow b])(x) = b = f^A(a) = (\rho[x\rightarrow a])(f(x)) \Rightarrow A\Vdash_{p[x\rightarrow a]}R(f(x))$.
+  Da $a\in U_A$ beliebig war, haben wir also $A\Vdash_p\forall x:R(f(x))$.
+  Also gilt auch in diesem Fall $A\Vdash_p\varphi$. Da $A$ und $\rho$
+  beliebig waren, ist $\varphi$ somit allgemeingültig.
+
+  Beispiel: - Der Satz $\varphi =\exists x(R(x)\rightarrow R(f(x)))$ ist
+  allgemeingültig. - Beweis: Sei $\sum$ Signatur, so dass $\varphi$
+  $\sum$-Satz ist. Sei $A$ $\sum$-Struktur und $\rho$
+  Variableninterpretation. Wir betrachten wieder zwei Fälle: 1.
+  Angenommen, $R^A=U_A$. Sei $a\in U_A$ beliebig. -
+  $\Rightarrow f^A(a)\in R^A$ -
+  $\Rightarrow A\Vdash_{p[x\rightarrow a]} R(f(x))$ -
+  $\Rightarrow A\Vdash_{p[x\rightarrow a]} R(x)\rightarrow R(f(x))$ -
+  $\Rightarrow A\Vdash_p\varphi$. 2. Ansonsten existiert
+  $a\in U_A\backslash R^A$. -
+  $\Rightarrow A\not\Vdash_{p[x\rightarrow a]} R(x)$ -
+  $\Rightarrow A\Vdash_{p[x\rightarrow a]} R(x)\rightarrow R(f(x))$ -
+  $\Rightarrow A\Vdash_p \varphi$. Da $A$ und $\rho$ beliebig waren, ist
+  $\varphi$ somit allgemeingültig.
+
+  Aufgabe \textbar{} \textbar{} allgemeingültig \textbar{} erfüllbar
+  \textbar{} unerfüllbar \textbar{} \textbar{}
+  --------------------------------------------------------------------
+  \textbar{} --------------- \textbar{} --------- \textbar{} -----------
+  \textbar{} \textbar{} $P(a)$ \textbar{} nein \textbar{} ja \textbar{}
+  nein \textbar{} \textbar{} $\exists x:(\lnot P(x)\vee P(a))$ \textbar{}
+  ja \textbar{} ja \textbar{} nein \textbar{} \textbar{}
+  $P(a)\rightarrow\exists x:P(x)$ \textbar{} ja \textbar{} ja \textbar{}
+  nein \textbar{} \textbar{} $P(x)\rightarrow\exists x:P(x)$ \textbar{} ja
+  \textbar{} ja \textbar{} nein \textbar{} \textbar{}
+  $\forall x:P(x)\rightarrow\exists x:P(x)$ \textbar{} ja \textbar{} ja
+  \textbar{} nein \textbar{} \textbar{}
+  $\forall x:P(x)\wedge\lnot\forall y:P(y)$ \textbar{} nein \textbar{}
+  nein \textbar{} ja \textbar{} \textbar{}
+  $\forall x:(P(x,x)\rightarrow\exists x\forall y:P(x,y))$ \textbar{} nein
+  \textbar{} ja \textbar{} nein \textbar{} \textbar{}
+  $\forall x\forall y:(x=y\rightarrow f(x) =f(y))$ \textbar{} ja
+  \textbar{} ja \textbar{} nein \textbar{} \textbar{}
+  $\forall x\forall y:(f(x) =f(y)\rightarrow x=y)$ \textbar{} nein
+  \textbar{} ja \textbar{} nein \textbar{} \textbar{}
+  $\exists x\exists y\exists z:(f(x) =y\wedge f(x) =z\wedge y \not=z)$
+  \textbar{} nein \textbar{} nein \textbar{} ja \textbar{} \textbar{}
+  $\exists x\forall x:Q(x,x)$ \textbar{} nein \textbar{} ja \textbar{}
+  nein \textbar{}
+
+  \subsection{Substitutionen}\label{substitutionen}
+
+  \begin{quote}
+    Definition
+
+    Eine Substitution besteht aus einer Variable $x\in Var$ und einem Term
+    $t\in T_{\sum}$, geschrieben $[x:=t]$.
+  \end{quote}
+
+  Die Formel $\varphi[x:=t]$ ist die Anwendung der Substitution $[x:=t]$
+  auf die Formel $\varphi$. Sie entsteht aus $\varphi$, indem alle freien
+  Vorkommen von $x$ durch $t$ ersetzt werden. Sie soll das über $t$
+  aussagen, was $\varphi$ über $x$ ausgesagt hat. Dazu definieren wir
+  zunächst induktiv, was es heißt, die freien Vorkommen von $x$ im Term
+  $s\in T_{\sum}$ zu ersetzen: - $x[x:=t] =t$ - $y[x:=t] =y$ für
+  $y\in Var\backslash\{x\}$ -
+  $(f(t_1 ,...,t_k))[x:=t] =f(t_1 [x:=t],...t_k[x:=t])$ für $f\in\Omega$
+  mit $ar(f) =k$ und $t_1,...,t_k\in T_{\sum}$
+
+  \begin{quote}
+    Lemma
+
+    Seien $\sum$ Signatur, $A$ $\sum$-Struktur, $\rho:Var\rightarrow U_A$
+    Variableninterpretation, $x\in Var$ und $s,t\in T_{\sum}$. Dann gilt
+    $\rho(s[x:=t])=\rho[x\rightarrow \rho(t)](s)$.
+  \end{quote}
+
+  Beweis: Induktion über den Aufbau des Terms $s$ (mit
+  $\rho'=\rho[x\rightarrow \rho(t)]$ ): -
+  $s=x:\rho(s[x:=t])=\rho(t) =\rho'(x) =\rho'(s)$ -
+  $s\in Var\backslash\{x\}:\rho(s[x:=t])=\rho(s) =\rho'(s)$ -
+  $s=f(t_1 ,...,t_k):\rho((f(t_1 ,...,t_k))[x:=t])= \rho(f(t_1[x:=t],...,t_k[x:=t]))= f^A(\rho(t_1[x:=t]),...,\rho(t_k[x:=t])) = f^A(\rho'(t_1),...,\rho'(t_k))= \rho'(f(t_1 ,...,t_k))=\rho'(s)$
+
+  Die Definition von $s[x:=t]$ kann induktiv auf $\sum$-Formeln
+  fortgesetzt werden: - $(t_1 =t_2 )[x:=t] = (t_1 [x:=t] =t_2 [x:=t])$ für
+  $t_1 ,t_2 \in T_{\sum}$ -
+  $(R(t_1 ,...,t_k))[x:=t] =R(t_1 [x:=t],...,t_k[x:=t])$ für $R\in Rel$
+  mit $ar(R) =k$ und $t_1 ,...,t_k\in T_{\sum}$ - $\bot[x:=t] =\bot$
+
+  Für $\sum$ -Formeln $\varphi$ und $\Psi$ und $y\in Var$: -
+  $(\varphi\Box\Psi)[x:=t]=\varphi [x:=t]\Box\Psi[x:=t]$ für
+  $\Box\in\{\wedge,\vee,\rightarrow\}$ -
+  $(\lnot\varphi)[x:=t] = \lnot(\varphi[x:=t])$ -
+  $(Qy\varphi)[x:=t] = \begin{cases} Qy\varphi[x:=t] \quad\text{ falls } x\not=y \\ Qy\varphi \quad\text{ falls } x=y \end{cases} \text{ für } Q\in\{\exists,\forall\}$.
+
+  Beispiel:
+  $(\exists x P(x,f(y))\vee\lnot\forall yQ(y,g(a,h(z))))[y:=f(u)] = (\exists x P(x,f(f(u)))\vee\lnot\forall yQ(y,g(a,h(z))))$
+
+  $\varphi [x:=t]$ ``soll das über $t$ aussagen, was $\varphi$ über $x$
+  ausgesagt hat.''
+
+  Gegenbeispiel: Aus $\exists y$ $Mutter(x) =y$ mit Substitution
+  $[x:=Mutter(y)]$ wird $\exists y$ Mutter$(Mutter(y)) =y$.
+
+  \begin{quote}
+    Definition
+
+    Sei $[x:=t]$ Substitution und $\varphi$ $\sum$-Formel. Die Substitution
+    $[x:=t]$ heißt für $\varphi$ zulässig, wenn für alle $y\in Var$ gilt:
+    $y$ Variable in $t\Rightarrow\varphi$ enthält weder $\exists y$ noch
+    $\forall y$
+  \end{quote}
+
+  \begin{quote}
+    Lemma
+
+    Sei $\sum$ Signatur, A $\sum$-Struktur, $\rho:Var\rightarrow U_A$
+    Variableninterpretation, $x\in Var$ und $t\in T_{\sum}$. Ist die
+    Substitution $[x:=t]$ für die $\sum$-Formel $\varphi$ zulässig, so gilt
+    $A\Vdash_p\varphi [x:=t]\Leftrightarrow  A\Vdash_{p[x\rightarrow \rho(t)]}\varphi$.
+  \end{quote}
+
+  Beweis: Induktion über den Aufbau der Formel $\varphi$ (mit
+  $\rho'=\rho[x\rightarrow \rho(t)])$: - $\varphi = (s_1 =s_2)$: -
+  $A\Vdash_p(s_1 =s_2)[x:=t] \Leftrightarrow A\Vdash_p s_1[x:=t] =s_2[x:=t]$
+  -
+  $\Leftrightarrow \rho(s_1[x:=t]) =\rho(s_2[x:=t])\Leftrightarrow \rho'(s_1) =\rho'(s_2)$
+  - $\Leftrightarrow A\Vdash_{p'} s_1 =s_2$ - andere atomare Formeln
+  analog - $\varphi =\varphi_1\wedge\varphi_2$: -
+  $A\Vdash_p(\varphi_1\wedge\varphi_2)[x:=t] \Leftrightarrow A\Vdash_p\varphi_1 [x:=t]\wedge\varphi_2[x:=t]$
+  - $\Leftrightarrow A\Vdash_p\varphi_1[x:=t]$ und
+  $A\Vdash_p\varphi_2[x:=t]$ - $\Leftrightarrow A\Vdash_{p'}\varphi_1$ und
+  $A\Vdash_{p'}\varphi_2$ -
+  $\Leftrightarrow A\Vdash_{p'}\varphi_1\wedge\varphi_2$ -
+  $\varphi=\varphi_1\vee\varphi_2,\varphi =\varphi_1\rightarrow\varphi_2$
+  und $\varphi=\lnot\psi$ analog - $\varphi=\forall y\psi$: - Wir
+  betrachten zunächst den Fall $x=y$: -
+  $A\Vdash_p(\forall x\psi)[x:=t]\Leftrightarrow A\Vdash_p\forall x\psi \Leftrightarrow A\Vdash_{p'}\forall x\psi$
+  (denn $x\not\in FV(\forall x\psi)$ ) - Jetzt der Fall $x\not=y$: - Für
+  $a\in U_A$ setze $\rho_a=\rho[y\rightarrow a]$. Da $[x:=t]$ für
+  $\varphi$ zulässig ist, kommt $y$ in $t$ nicht vor. Zunächst erhalten
+  wir - $\rho_a[x\rightarrow \rho_a(t)] = \rho_a[x\rightarrow \rho(t)]$ da
+  $y$ nicht in $t$ vorkommt -
+  $=\rho[y\rightarrow a][x\rightarrow \rho(t)] = \rho[x\rightarrow \rho(t)][y\rightarrow a]$
+  da $x\not=y$ - Es ergibt sich
+  $A\Vdash_p(\forall y\psi)[x:=t]\Leftrightarrow A\Vdash_p\forall y(\psi[x:=t])$
+  (wegen $x\not=y$) - $\Leftrightarrow A\Vdash_{pa}\psi[x:=t]$ für alle
+  $a\in U_A$ - $\Leftrightarrow A\Vdash_{pa[x\rightarrow \rho_a(t)]}\psi$
+  für alle $a\in U_A$ -
+  $\Leftrightarrow A\Vdash_{p[x\rightarrow \rho(t)][y\rightarrow a]}\psi$
+  für alle $a\in U_A$ -
+  $\Leftrightarrow A\Vdash_{p[x\rightarrow \rho(t)]}\forall y\psi$ -
+  $\varphi=\exists y\psi$ : analog
+
+  \subsection{Natürliches Schließen}\label{natuxfcrliches-schlieuxdfen-1}
+
+  Wir haben Regeln des natürlichen Schließens für aussagenlogische Formeln
+  untersucht und für gut befunden. Man kann sie natürlich auch für
+  prädikatenlogische Formeln anwenden.
+
+  Beispiel: Für alle $\sum$-Formel $\varphi$ und $\psi$ gibt es eine
+  Deduktion mit Hypothesen in $\{\lnot\varphi\wedge\lnot\psi\}$ und
+  Konklusion $\lnot(\varphi\vee\psi)$:
+  \includegraphics[width=\linewidth]{Assets/Logik-deduktion-konklusion.png}
+
+  \subsection{Korrektheit}\label{korrektheit-1}
+
+  Können wir durch mathematische Beweise zu falschen Aussagen kommen?
+  Können wir durch das natürliche Schließen zu falschen Aussagen kommen?
+  Existiert eine Menge $\Gamma$ von $\sum$-Formeln und eine $\sum$-Formel
+  $\varphi$ mit $\Gamma\vdash\varphi$ und $\Gamma\not\Vdash\varphi$?
+
+  Frage: Gilt $\Gamma\vdash\varphi\Rightarrow \Gamma\Vdash\varphi$ bzw.
+  $\varphi$ ist Theorem $\Rightarrow\varphi$ ist allgemeingültig?
+
+  Der Beweis des Korrektheitslemmas für das natürliche Schließen kann ohne
+  große Schwierigkeiten erweitert werden. Man erhält
+
+  \begin{quote}
+    Lemma V0
+
+    Sei $\sum$ eine Signatur, $\Gamma$ eine Menge von $\sum$-Formeln und
+    $\varphi$ eine $\sum$-Formel. Sei weiter $D$ eine Deduktion mit
+    Hypothesen in $\Gamma$ und Konklusion $\varphi$, die die Regeln des
+    natürlichen Schließens der Aussagenlogik verwendet. Dann gilt
+    $\Gamma\Vdash\varphi$.
+  \end{quote}
+
+  Umgekehrt ist nicht zu erwarten, dass aus $\Gamma\Vdash\varphi$ folgt,
+  dass es eine Deduktion mit Hypothesen in $\Gamma$ und Konklusion
+  $\varphi$ gibt, denn die bisher untersuchten Regeln erlauben keine
+  Behandlung von $=,\forall$ bzw. $\exists$. Solche Regeln werden wir
+  jetzt einführen.
+
+  Zunächst kümmern wir uns um Atomformeln der Form $t_1 =t_2$. Hierfür
+  gibt es die zwei Regeln $(R)$ und $(GfG)$:
+
+  \begin{quote}
+    Reflexivität (ausführlich)
+
+    Für jeden Termt ist $\frac{}{t=t}$ eine hypothesenlose Deduktion mit
+    Konklusion $t=t$. \includegraphics[width=\linewidth]{Assets/Logik-reflexivität-kurz.png}
+  \end{quote}
+
+  \begin{quote}
+    Gleiches-für-Gleiches in mathematischen Beweisen
+
+    ,,Zunächst zeige ich, dass $s$ die Eigenschaft $\varphi$ hat:\ldots{}
+    Jetzt zeige ich $s=t$:\ldots{} Also haben wir gezeigt, dass $t$ die
+    Eigenschaft $\varphi$ hat. qed''
+
+    Gleiches-für-Gleiches (ausführlich) Seien $s$ und $t$ Terme und
+    $\varphi$ Formel, so dass die Substitutionen $[x:=s]$ und $[x:=t]$ für
+    $\varphi$ zulässig sind. Sind $D$ und $E$ Deduktionen mit Hypothesen in
+    $\Gamma$ und Konklusionen $\varphi[x:=s]$ bzw. $s=t$, so ist das
+    folgende eine Deduktion mit Hypothesen in $\Gamma$ und Konklusion
+    $\varphi[x:=t]$:
+    \includegraphics[width=\linewidth]{Assets/Logik-gleiches-für-gleiches-ausführlich.png}
+
+    Gleiches-für-Gleiches (Kurzform)
+    \includegraphics[width=\linewidth]{Assets/Logik-gleiches-für-gleiches-kurz.png} Bedingung:
+    über keine Variable aus $s$ oder $t$ wird in $\varphi$ quantifiziert
+  \end{quote}
+
+  Die folgenden Beispiele zeigen, dass wir bereits jetzt die üblichen
+  Eigenschaften der Gleichheit (Symmetrie, Transitivität, Einsetzen)
+  folgern können.
+
+  Beispiel: Seien $x$ Variable, $s$ Term ohne $x$ und $\varphi=(x=s)$. -
+  Da $\varphi$ quantorenfrei ist, sind die Substitutionen $[x:=s]$ und
+  $[x:=t]$ für $\varphi$ zulässig. - Außerdem gelten
+  $\varphi[x:=s] = (s=s)$ und $\varphi[x:=t] = (t=s)$. - Also ist das
+  folgende eine Deduktion:
+  \includegraphics[width=\linewidth]{Assets/Logik-deduktion-beispiel.png} - Für alle
+  Termesundthaben wir also $\{s=t\}\vdash t=s$.
+
+  Beispiel: Seien $x$ Variable, $r,s$ und $t$ Terme ohne $x$ und
+  $\varphi=(r=x)$. - Da $\varphi$ quantorenfrei ist, sind die
+  Substitutionen $[x:=s]$ und $[x:=t]$ für $\varphi$ zulässig. - Außerdem
+  gelten $\varphi[x:=s]=(r=s)$ und $\varphi[x:=t]=(r=t)$. - Also ist das
+  folgende eine Deduktion:
+  \includegraphics[width=\linewidth]{Assets/Logik-deduktion-beispiel-2.png} - Für alle Terme
+  $r,s$ und $t$ haben wir also $\{r=s,s=t\}\vdash r=t$.
+
+  Beispiel: Seien $x$ Variable, $s$ und $t$ Terme ohne $x$,$f$
+  einstelliges Funktionssymbol und $\varphi=(f(s)=f(x))$. - Da $\varphi$
+  quatorenfrei ist, sind die Substitutionen $[x:=s]$ und $[x:=t]$ für
+  $\varphi$ zulässig. - Außerdem gelten $\varphi[x:=s]=(f(s)=f(s))$ und
+  $\varphi[x:=t]=(f(s)=f(t))$. - Also ist das folgende eine Deduktion:
+  \includegraphics[width=\linewidth]{Assets/Logik-deduktion-beispiel-3.png}
+
+  \begin{quote}
+    Lemma V1
+
+    Sei $\sum$ eine Signatur, $\Gamma$ eine Menge von $\sum$-Formeln und
+    $\varphi$ eine $\sum$-Formel. Sei weiter $D$ eine Deduktion mit
+    Hypothesen in $\Gamma$ und Konklusion $\varphi$, die die Regeln des
+    natürlichen Schließens der Aussagenlogik, $(R)$ und $(GfG)$ verwendet.
+    Dann gilt $\Gamma\Vdash\varphi$.
+  \end{quote}
+
+  Beweis: Wir erweitern den Beweis des Korrektheitslemmas bzw. des Lemmas
+  V0, der Induktion über die Größe der Deduktion $D$ verwendete. - Wir
+  betrachten nur den Fall, dass $D$ die folgende Form hat:
+  \includegraphics[width=\linewidth]{Assets/Logik-lemma-v1-beweis.png} - Da dies Deduktion
+  ist, sind die Substitutionen $[x:=s]$ und $[x:=t]$ für $\varphi$
+  zulässig, d.h. in $\varphi$ wird über keine Variable aus $s$ oder $t$
+  quantifiziert. - $E$ und $F$ kleinere Deduktionen
+  $\Rightarrow\Gamma\Vdash\varphi[x:=s]$ und $\Gamma\Vdash s=t$ - Seien A
+  $\sum$-Struktur und $\rho$ Variableninterpretation mit $A\Vdash_p\gamma$
+  für alle $\gamma\in\Gamma$. - $\Rightarrow A\Vdash_p\varphi[x:=s]$ und
+  $A\Vdash_p s=t$ - $\Rightarrow A\Vdash_{p[x\rightarrow \rho(s)]}\varphi$
+  und $\rho(s) =\rho(t)$ -
+  $\Rightarrow A\Vdash_{p[x\rightarrow \rho(t)]}\varphi$ -
+  $\Rightarrow A\Vdash_p \varphi[x:=t]$ - Da $A$ und $\rho$ beliebig waren
+  mit $A\Vdash_p\gamma$ für alle $\gamma\in\Gamma$ haben wir
+  $\Gamma\Vdash\varphi[x:=t]$ gezeigt.
+
+  \subsubsection{$\forall$ in math.
+    Beweisen}\label{forall-in-math.-beweisen}
+
+  Ein mathematischer Beweis einer Aussage ``für alle $x$ gilt $\varphi$''
+  sieht üblicherweise so aus: ``Sei $x$ beliebig, aber fest. Jetzt zeige
+  ich $\varphi$ (hier steckt die eigentliche Arbeit). Da $x$ beliebig war,
+  haben wird''für alle $x$ gilt $\varphi$" gezeigt. qed"
+
+  \begin{quote}
+    $\forall$ -Einführung
+
+    Sei $D$ eine Deduktion mit Hypothesen in $\Gamma$ und Konklusion
+    $\varphi$ und sei $x$ eine Variable, die in keiner Formel aus $\Gamma$
+    frei vorkommt. Dann ist das folgende eine Deduktion mit Hypothesen in
+    $\Gamma$ und Konklusion
+    $\forall x\varphi: \frac{\phi}{\forall x\varphi}$
+
+    Bedingung: $x$ kommt in keiner Hypothese frei vor
+  \end{quote}
+
+  \begin{quote}
+    Lemma V2
+
+    Sei $\sum$ eine Signatur, $\Gamma$ eine Menge von $\sum$-Formeln und
+    $\varphi$ eine $\sum$-Formel. Sei weiter $D$ eine Deduktion mit
+    Hypothesen in $\Gamma$ und Konklusion $\varphi$, die die Regeln des
+    natürlichen Schließens der Aussagenlogik, (R), (GfG) und ($\forall$ -I)
+    verwendet. Dann gilt $\Gamma\Vdash\varphi$.
+  \end{quote}
+
+  Beweis: Betrachte die folgende Deduktion $D$ - Insbesondere ist $x$
+  keine freie Variable einer Formel aus $\Gamma$ und es gilt nach IV
+  $\Gamma\Vdash\varphi$ - Sei nun $A$ $\sum$-Struktur und $\rho$
+  Variableninterpretation mit $A\Vdash_p y$ für alle $y\in\Gamma$. - Zu
+  zeigen ist $A\Vdash_p \forall x\varphi$: - Sei also $a\in U_A$ beliebig.
+  - $\Rightarrow$ für alle $y\in\Gamma$ gilt
+  $A\Vdash_{p[x\rightarrow a]} y$ da $x\not\in FV(y)$ und $A\Vdash_p y$ -
+  $\Rightarrow A\Vdash_{\rho[x\rightarrow a]}\varphi$ - Da $a\in U_A$
+  beliebig war, haben wir $A\Vdash_\rho\forall x\varphi$ gezeigt - Da $A$
+  und $\rho$ beliebig waren mit $A\Vdash_\rho\Gamma$ für alle
+  $\gamma\in\Gamma$ haben wir also $\Gamma\Vdash\forall x\varphi$ gezeigt.
+
+  \subsubsection{$\forall$ -Elimination in math.
+    Beweisen}\label{forall--elimination-in-math.-beweisen}
+
+  Ein mathematischer Beweis einer Aussage ``t erfüllt $\varphi$'' kann so
+  aussehen: ``Zunächst zeige ich $\forall x\varphi$ (hier steckt die
+  eigentliche Arbeit). Damit erfüllt insbesondere $t$ die Aussage$\varphi$
+  , d.h., wir haben''$t$ erfüllt $\varphi$" gezeigt. qed"
+
+  \begin{quote}
+    $\forall$ -Elimination
+
+    Sei $D$ eine Deduktion mit Hypothesen in $\Gamma$ und Konklusion
+    $\forall x\varphi$ und seit Term, so dass Substitution {[}x:=t{]} für
+    $\varphi$ zulässig ist. Dann ist das folgende eine Deduktion mit
+    Hypothesen in $\Gamma$ und Konklusion
+    $\varphi[x:=t]:\frac{\forall x\varphi}{\varphi[x:=t]}$
+
+    Bedingung: über keine Variable aus $t$ wird in $\varphi$ quantifiziert
+  \end{quote}
+
+  \begin{quote}
+    Lemma V3
+
+    Sei $\sum$ eine Signatur, $\Gamma$ eine Menge von $\sum$-Formeln und
+    $\varphi$ eine $\sum$-Formel. Sei weiter $D$ eine Deduktion mit
+    Hypothesen in $\Gamma$ und Konklusion $\varphi$, die die Regeln des
+    natürlichen Schließens der Aussagenlogik, (R), (GfG), ($\forall$-I) und
+    ($\forall$-E) verwendet. \textgreater{} Dann gilt $\Gamma\Vdash\varphi$.
+  \end{quote}
+
+  Beweis: Analog zum Beweis von Lemma V2.
+
+  \subsubsection{$\exists$ in math.
+    Beweisen}\label{exists-in-math.-beweisen}
+
+  Ein Beweis von ``$\sigma$ gilt'' kann so aussehen: ``Zunächst zeige ich
+  $\exists x\varphi$ (hier steckt Arbeit). Jetzt zeige ich, dass $\sigma$
+  immer gilt, wenn$\varphi$ gilt (mehr Arbeit). Damit gilt $\sigma$. qed''
+
+  \begin{quote}
+    $\exists$ -Elimination
+
+    Sei $\Gamma$ eine Menge von Formeln, die die Variable $x$ nicht frei
+    enthalten und enthalte die Formel $\sigma$ die Variabel $x$ nicht frei.
+    Wenn $D$ eine Deduktion mit Hypothesen in $\Gamma$ und Konklusion
+    $\exists x\varphi$ und $E$ eine Deduktion mit Hypothesen in
+    $\Gamma\cup\{\varphi\}$ und Konklusion $\sigma$ ist, dann ist das
+    folgende eine Deduktion mit Hypothesen in $\Gamma$ und Konklusion
+    $\sigma:\frac{\exists x\varphi \quad\quad \sigma}{\sigma}$
+
+    Bedingung: $x$ kommt in den Hypothesen und in $\sigma$ nicht frei vor
+  \end{quote}
+
+  \begin{quote}
+    Lemma V4 Sei $\sigma$ eine Signatur, $\Gamma$ eine Menge von
+    $\sum$-Formeln und $\varphi$ eine $\sigma$ -Formel. Sei weiter $D$ eine
+    Deduktion mit Hypothesen in $\Gamma$ und Konklusion $\varphi$, die die
+    Regeln des natürlichen Schließens der Aussagenlogik, (R), (GfG),
+    ($\forall$-I), ($\forall$-E) und ($\exists$-E) verwendet. Dann gilt
+    $\Gamma\Vdash\varphi$.
+  \end{quote}
+
+  Beweis: Sei $D$ die folgende Deduktion - Insbesondere kommt $x$ in den
+  Formeln aus $\Gamma\cup\{\sigma\}$ nicht frei vor. Außerdem gelten nach
+  IV $\Gamma\Vdash\exists x\varphi$ und
+  $\Gamma\cup\{\varphi\}\Vdash\sigma$. - Sei nun $A$ $\sigma$-Struktur und
+  $\rho$ Variableninterpretation mit $A\Vdash_\rho\Gamma$ für alle
+  $\gamma\in\Gamma$. - Zu zeigen ist $A\Vdash_\rho\sigma$: - Wegen
+  $A\Vdash_\rho\exists x\varphi$ existiert $a\in U_A$ mit
+  $A\Vdash_{\rho[x\rightarrow a]}\varphi$. - $x$ kommt in Formeln aus
+  $\Gamma$ nicht frei vor
+  $\Rightarrow A\Vdash_{\rho[x\rightarrow a]}\gamma$ für alle
+  $\gamma\in\Gamma$. - Aus $\Gamma\cup\{\varphi\}\Vdash\sigma$ folgt
+  $A\Vdash_{\rho[x\rightarrow a]}\sigma$. - Da $x\not\in FV(\sigma)$
+  erhalten wir $A\Vdash_\rho \sigma$. - Da $A$ und $\rho$ beliebig waren
+  mit $A\Vdash_\rho\gamma$ für alle $\gamma\in\Gamma$ haben wir also
+  $\Gamma\Vdash\sigma$ gezeigt.
+
+  \subsubsection{$\exists$ -Einführung in math.
+    Beweisen}\label{exists--einfuxfchrung-in-math.-beweisen}
+
+  Ein mathematischer Beweis einer Aussage ``es gibt ein $x$, das $\varphi$
+  erfüllt'' sieht üblicherweise so aus: ``betrachte dieses $t$ (hier ist
+  Kreativität gefragt). Jetzt zeige ich, daß $t\varphi$ erfüllt (u.U.
+  harte Arbeit). Also haben wir''es gibt ein $x$, das $\varphi$ erfüllt"
+  gezeigt. qed"
+
+  \begin{quote}
+    $\exists$ -Einführung
+
+    Sei die Substitution $[x:=t]$ für die Formel $\varphi$ zulässig. Sei
+    weiter $D$ eine Deduktion mit Hypothesen in $\Gamma$ und Konklusion
+    $\varphi[x:=t]$. Dann ist das folgende eine Deduktion mit Hypothesen in
+    $\Gamma$ und Konklusion
+    $\exists x\varphi:\frac{\varphi[x:=t]}{\exists x\varphi}$
+
+    Bedingung: über keine Variable in $t$ wird in $\varphi$ quantifiziert
+  \end{quote}
+
+  \begin{quote}
+    Korrektheitslemma für das natürliche Schließen in der Prädikatenlogik
+
+    Sei $\sigma$ eine Signatur, $\Gamma$ eine Menge von $\sum$-Formeln und
+    $\varphi$ eine $\sigma$ -Formel. Sei weiter $D$ eine Deduktion mit
+    Hypothesen in $\Gamma$ und Konklusion $\varphi$, die die Regeln des
+    natürlichen Schließens der Aussagenlogik, (R), (GfG), ($\forall$-I),
+    ($\forall$-E), ($\exists$ -E) und ($\exists$ -I) verwendet. Dann gilt
+    $\Gamma\Vdash\varphi$.
+  \end{quote}
+
+  Beweis: analog zu obigen Beweisen.
+
+  \subsubsection{Regeln des natürlichen Schließens
+    (Erweiterung)}\label{regeln-des-natuxfcrlichen-schlieuxdfens-erweiterung}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          ($R$): $\frac{}{t=t}$
+    \item
+          (GfG): $\frac{\varphi[x:=s] \quad\quad s=t}{\varphi[x:=t]}$ (über
+          keine Variable aus $s$ oder $t$ wird in $\varphi$ quantifiziert)
+    \item
+          ($\forall$-I): $\frac{\varphi}{\forall x\varphi}$ (x nicht frei in
+          Hypothesen)
+    \item
+          ($\forall$-E): $\frac{\forall x\varphi}{\varphi[x:=t]}$ (über keine
+          Variable aus $t$ wird in $\varphi$ quantifiziert)
+    \item
+          ($\exists$-I): $\frac{\varphi [x:=t]}{\exists x\varphi}$ (über keine
+          Variable aus $t$ wird in $\varphi$ quantifiziert)
+    \item
+          ($\exists$-I): $\frac{\exists x\varphi\quad\quad \sigma}{\sigma}$ ($x$
+          kommt in Hypothesen und $\sigma$ nicht frei vor)
+  \end{itemize}
+
+  \begin{quote}
+    Definition
+
+    Für eine Menge $\Gamma$ von $\sum$-Formeln und eine $\sum$-Formel
+    $\varphi$ schreiben wir $\Gamma\vdash\varphi$ wenn es eine Deduktion
+    gibt mit Hypothesen in $\Gamma$ und Konklusion $\varphi$. Wir sagen
+    ``$\varphi$ ist eine syntaktische Folgerung von $\Gamma$''. Eine Formel
+    $\varphi$ ist ein Theorem, wenn $\varnothing\vdash\varphi$ gilt.
+  \end{quote}
+
+  Bemerkung: $\Gamma\vdash\varphi$ sagt (zunächst) nichts über den Inhalt
+  der Formeln in $\Gamma\cup\{\varphi\}$ aus, sondern nur über den Fakt,
+  dass $\varphi$ mithilfe des natürlichen Schließens aus den Formeln aus
+  $\Gamma$ hergeleitet werden kann. Ebenso sagt ``$\varphi$ ist Theorem''
+  nur, dass $\varphi$ abgeleitet werden kann, über ``Wahrheit'' sagt
+  dieser Begriff (zunächst) nichts aus. Wir haben aber ``en passant'' das
+  folgende gezeigt:
+
+  \begin{quote}
+    Korrektheitssatz
+
+    Für eine Menge von $\sum$-Formeln $\Gamma$ und eine $\sum$-Formel
+    $\varphi$ gilt $\Gamma\vdash\varphi \Rightarrow \Gamma\Vdash\varphi$.
+  \end{quote}
+
+  Beispiel: Seien $\varphi$ Formel und $x$ Variable. Dann gelten
+  $\{\lnot\exists x\varphi\}\Vdash\forall x\lnot\varphi$ und
+  $\{\forall x\lnot\varphi\}\Vdash\lnot\exists x\varphi$. - Beweis:
+  \includegraphics[width=\linewidth]{Assets/Logik-korrekheitssatz.png}
+
+  Beispiel: Seien $\varphi$ Formel und $x$ Variable. Dann gelten
+  $\{\lnot\forall x\varphi\}\Vdash \exists x\lnot\varphi$ und
+  $\{\exists x\lnot\varphi\}\Vdash\lnot\forall x\varphi$. - Beweis:
+  \includegraphics[width=\linewidth]{Assets/Logik-beispiel-korrekheitssatz.png}
+
+  \subsection{Vollständigkeit}\label{vollstuxe4ndigkeit-1}
+
+  Können wir durch mathematische Beweise zu allen korrekten Aussagen
+  kommen? Können wir durch das natürliche Schließen zu allen korrekten
+  Aussagen kommen?
+
+  Existiert eine Menge $\Gamma$ von $\sum$-Formeln und eine $\sum$-Formel
+  $\varphi$ mit $\Gamma\Vdash\varphi$ und $\Gamma\not\vdash\varphi$?
+
+  Frage: Gilt $\Gamma\Vdash\varphi \Rightarrow \Gamma\vdash\varphi$ bzw.
+  $\varphi$ ist allgemeingültig $\Rightarrow\varphi$ ist Theorem?
+
+  Plan: - z.z. ist $\Gamma\Vdash\varphi \Rightarrow \Gamma\vdash\varphi$.
+  - dies ist äquivalent zu
+  $\Gamma\not\vdash\varphi \Rightarrow \Gamma\not\Vdash\varphi$. - hierzu
+  geht man folgendermaßen vor: - $\Gamma\not\vdash\varphi$ -
+  $\Leftrightarrow \Gamma\cup\{\lnot\varphi\}$ konsistent -
+  $\Rightarrow \exists\Delta\supseteq\Gamma\cup\{\lnot\varphi\}$ maximal
+  konsistent - $\Rightarrow \exists\Delta^+ \supseteq\Delta$ maximal
+  konsistent mit Konkretisierung - $\Rightarrow \Delta^+$ erfüllbar -
+  $\Rightarrow \Delta$ erfüllbar -
+  $\Rightarrow \Gamma\cup\{\lnot\varphi\}$ erfüllbar -
+  $\Leftrightarrow \Gamma\cup\{\lnot\varphi\}$
+
+  \begin{quote}
+    Definition
+
+    Eine Menge $\Delta$ von Formeln hat Konkretisierungen, wenn für alle
+    $\exists x\varphi\in\Delta$ ein variablenloser Term $t$ existiert mit
+    $\varphi[x:=t]\in\Delta$.
+  \end{quote}
+
+  \begin{quote}
+    Satz
+
+    Sei $\Delta$ eine maximal konsistente Menge von $\sum$-Formeln. Dann
+    existiert eine Signatur $\sum^+ \supseteq\sum$ und eine maximal
+    konsistente Menge von $\sum^+$-Formeln mit Konkretisierungen, so dass
+    $\Delta\subseteq\Delta^+$.
+  \end{quote}
+
+  Beweis: Wir konstruieren induktiv Signaturen $\sum_n$, maximal
+  konsistente Menge von $\sum_n$-Formeln $\Delta_n$ und konsistente Mengen
+  von $\sum_{n+1}$-Formeln $\Delta'_{n+1}$ mit -
+  $\sum =\sum_0 \subseteq\sum_1 \subseteq\sum_2...$ und -
+  $\Delta = \Delta_0 \subseteq \Delta'_1 \subseteq\Delta_1 \subseteq\Delta'_2...$
+  und setzen dann - $\sum^+ =\bigcup_{n\geq 0} \sum_n$ und
+  $\Delta^+ = \bigcup_{n\geq 0} \Delta_n$
+
+  \begin{enumerate}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          IA: $\sum_0 := \sum$ , $\Delta_0:=\Delta$
+    \item
+          IV: Sei $n\geq 0$ und $\Delta_n$ maximal konsistente Menge von
+          $\sum_n$-Formeln. $\psi=\exists x\varphi$, ein ``neues''
+          Konstantensymbol $c_{\psi}$
+    \item
+          IS: $\sum_{n+1}$: alle Symbole aus $\sum_n$ und, für jede Formel
+          $\psi\in\Delta_n$ der Form
+          $\Delta'_{n+1}:= \Delta_n\cup\{\varphi[x:=c_{\psi}]|\psi=\exists x\varphi\in\Delta_n\}$
+
+          \begin{itemize}
+            \item
+                  ohne Beweis: $\Delta'_{n+1}$ ist konsistent
+            \item
+                  Idee: Ist $\varphi$ $\sum_n$-Formel mit
+                  $\Delta'_{n+1}\vdash\varphi$, so gilt $\Delta_n\vdash\varphi$.
+            \item
+                  Konsistenz von $\Delta'_{n+1}$ folgt mit $\varphi=\bot$
+            \item
+                  Analog zum Satz aus Vorlesung 4 existiert
+                  $\Delta_{n+1}\supseteq \Delta'_{n+1}$ maximal konsistent
+          \end{itemize}
+  \end{enumerate}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Damit ist die Konstruktion der Signaturen $\sum_n$ und der maximal
+          konsistenten Mengen $\Delta_n$ von $\sum_n$-Formeln abgeschlossen.
+    \item
+          noch z.z.: $\Delta^+$ hat Konkretisierungen und ist maximal konsistent
+    \item
+          $\Delta^+$ hat Konkretisierungen: Sei
+          $\psi=\exists x\varphi\in\Delta^+$
+
+          \begin{itemize}
+            \item
+                  $\Rightarrow$ es gibt $n\geq 0$ mit $\psi\in\Delta_n$
+            \item
+                  $\Rightarrow \varphi[x:=c_{\psi}]\in\Delta'_{n+1}\subseteq \Delta_{n+1}\subseteq\Delta^+$.
+          \end{itemize}
+    \item
+          Konsistenz: (indirekt) angenommen, $\Delta^+\vdash\bot$
+
+          \begin{itemize}
+            \item
+                  Da jede Deduktion endlich ist, existiert $\Gamma\subseteq\Delta^+$
+                  endlich mit $\Gamma\vdash\bot$.
+            \item
+                  $\Rightarrow$ es gibt $n\geq 0$ mit $\Gamma\subseteq\Delta_n$
+            \item
+                  $\Rightarrow \Delta_n\vdash\bot$ - im Widerspruch zur Konsistenz von
+                  $\Delta_n$.
+          \end{itemize}
+    \item
+          maximale Konsistenz: (indirekt) angenommen, $\Delta^+$ ist nicht
+          maximal konsistent
+
+          \begin{itemize}
+            \item
+                  $\Rightarrow$ es gibt $\Gamma\not\subseteq\Delta^+$ konsistent
+            \item
+                  $\Rightarrow$ es gibt $\varphi\in\Gamma\backslash\Delta^+$
+            \item
+                  $\Rightarrow$ $\Delta^+\cup\{\varphi\}\subseteq\Gamma$ konsistent
+            \item
+                  $\varphi$ ist $\sum^+$-Formel $\Rightarrow$ es gibt $n\geq 0$, so
+                  dass $\varphi$ eine $\sum_n$-Formel ist.
+            \item
+                  $\Delta_n$ maximal konsistente Menge von $\sum_n$-Formeln
+            \item
+                  $\Rightarrow$ $\varphi\in\Delta_n\subseteq\Delta^+$ oder
+                  $\lnot\varphi\in\Delta_n\subseteq\Delta^+$
+            \item
+                  $\Rightarrow$ $\lnot\varphi\in\Delta^+\subseteq\Gamma$
+            \item
+                  Also $\varphi,\lnot\varphi\in\Gamma$, im Widerspruch zur Konsistenz
+                  von $\Gamma$.
+          \end{itemize}
+  \end{itemize}
+
+  \begin{quote}
+    Satz
+
+    Sei $\Delta^+$ maximal konsistente Menge von $\sum^+$-Formeln mit
+    Konkretisierungen. Dann ist $\Delta^+$ erfüllbar.
+  \end{quote}
+
+  Beweisidee: Sei $T$ die Menge der variablenlosen $\sum^+$-Terme. Auf $T$
+  definieren wir eine Äquivalenzrelation $\sim $ durch
+  $s\sim t\Leftrightarrow \Delta^+\vdash(s=t)\Leftrightarrow (s=t)\in\Delta^+$
+  Sei $A$ die folgende $\sum^+$-Struktur: - $U_A:=T/\sim $ ist die Menge der
+  $\sim $-Äquivalenzklassen -
+  $R^A=\{([t_1],...,[t_k])|t_1 ,...,t_k\in T,R(t_1,...,t_k)\in\Delta^+\}$
+  für alle Relationssymbole R aus $\sum^+$ -
+  $f^A([t_1],...,[t_k]) = [f(t_1,...,t_k)]$ für alle $t_1,...,t_k\in T$
+  und alle Funktionssymbole $f$ aus $\sum^+$ (Bemerkung: dies ist
+  wohldefiniert) Dann gilt tatsächlich $A\Vdash\Delta^+$.
+
+  \begin{quote}
+    Satz: Vollständigkeitssatz der Prädikatenlogik
+
+    Sei $\Gamma$ eine Menge von $\sum$-Formeln und $\varphi$ eine
+    $\sum$-Formel. Dann gilt
+    $\Gamma\Vdash\varphi \Rightarrow \Gamma\vdash\varphi$. Insbesondere ist
+    jede allgemeingültige Formel ein Theorem.
+  \end{quote}
+
+  Beweis:indirekt - $\Gamma\not\vdash\varphi$ -
+  $\Gamma\cup\{\lnot\varphi\}$ konsistent - $\Gamma\cup\{\lnot\varphi\}$
+  erfüllbar - $\exists\Delta\supseteq\Gamma\cup\{\lnot\varphi\}$ maximal
+  konsistent - $\exists\Delta^+\supseteq\Delta$ maximal konsistent mit
+  Konkretisierungen - $\Delta^+$ erfüllbar - $\Delta$ erfüllbar -
+  $\Gamma\not\Vdash\varphi$
+
+  Bemerkung - Dieser Satz ist (im wesentlichen) der berühmte Gödelsche
+  Vollständigkeitssatz von 1930. - Der obige Beweis wurde von Leon Henkin
+  1949 veröffentlicht.
+
+  Wir haben gleichzeitig gezeigt: \textgreater{} Satz \textgreater{}
+  \textgreater{} Sei $\Gamma$ höchstens abzählbar unendliche und
+  konsistente Menge von Formeln. Dann hat $\Gamma$ ein höchstens abzählbar
+  unendliches Modell.
+
+  Beweis: $\Gamma$ konsistent heißt $\Gamma\not\vdash\bot$. Obiger Beweis
+  gibt ein Modell $A$ von $\Gamma\cup\{\lnot\bot\}$ an. Wir zeigen, dass
+  diese Struktur $A$ höchstens abzählbar unendlich ist: - Sei $\sum$
+  Signatur der Relations- und Funktionssymbole aus $\Gamma$. -
+  $|\Gamma|\leq \mathbb{N}_0 \Rightarrow |\sum|\leq \mathbb{N}_0$ -
+  $\Rightarrow |\sum_n|\leq \mathbb{N}_0$ und
+  $|\Delta_n|\leq \mathbb{N}_0$ für alle $n\geq 0$ -
+  $\Rightarrow |\sum^+|,|\Delta^+| \leq\mathbb{N}_0$ -
+  $\Rightarrow |T| \leq\mathbb{N}_0$ - $\Rightarrow A$ hat
+  $\leq\mathbb{N}_0$ viele Elemente -
+  $\Rightarrow \Gamma\cup\{\lnot\bot\}$ hat ein höchstens abzählbar
+  unendliches Modell - $\Rightarrow \Gamma$ hat ein höchstens abzählbar
+  unendliches Modell
+
+  \subsection{Vollständigkeit und Korrektheit für die
+    Prädikatenlogik}\label{vollstuxe4ndigkeit-und-korrektheit-fuxfcr-die-pruxe4dikatenlogik}
+
+  \begin{quote}
+    Satz
+
+    Seien $\Gamma$ eine Menge von $\sum$-Formeln und $\varphi$ eine
+    $\sum$-Formel. Dann gilt
+    $\Gamma\vdash\varphi\Leftrightarrow \Gamma\Vdash\varphi$. Insbesondere
+    ist eine $\sum$-Formel genau dann allgemeingültig, wenn sie ein Theorem
+    ist.
+  \end{quote}
+
+  Beweis: Folgt unmittelbar aus Korrektheitssatz und Vollständigkeitssatz.
+
+  \subsubsection{Folgerung 1: Kompaktheit}\label{folgerung-1-kompaktheit}
+
+  \begin{quote}
+    Satz
+
+    Seien $\Gamma$ eine u.U. unendliche Menge von $\sum$-Formeln und
+    $\varphi$ eine $\sum$-Formel mit $\Gamma\Vdash\varphi$. Dann existiert
+    $\Gamma'\subseteq\Gamma$ endlich mit $\Gamma'\Vdash\varphi$.
+  \end{quote}
+
+  Beweis: $\Gamma\Vdash\varphi$ - $\Gamma\vdash\varphi$ (nach dem
+  Vollständigkeitssatz) - es gibt Deduktion von $\varphi$ mit Hypothesen
+  $\gamma_1,...,\gamma_n\in\Gamma$ -
+  $\Gamma'=\{\gamma_1,...,\gamma_n\}\subseteq\Gamma$ endlich mit
+  $\Gamma'\vdash\varphi$ - $\Gamma'\vdash\varphi$ (nach dem
+  Korrektheitssatz).
+
+  \begin{quote}
+    Folgerung (Kompaktheits- oder Endlichkeitssatz)
+
+    Sei $\Gamma$ eine u.U. unendliche Menge von $\sum$-Formeln. Dann gilt
+    $\Gamma$ erfüllbar $\Leftrightarrow \forall\Gamma'\subseteq\Gamma$
+    endlich: $\Gamma'$ erfüllbar
+  \end{quote}
+
+  Beweis: - $\Gamma$ unerfüllbar -
+  $\Leftrightarrow \Gamma\cup\{\lnot\bot\}$ unerfüllbar -
+  $\Leftrightarrow \Gamma\Vdash\bot$ - $\Leftrightarrow$ es gibt
+  $\Gamma'\subseteq\Gamma$ endlich: $\Gamma'\Vdash\bot$ -
+  $\Leftrightarrow$ es gibt $\Gamma'\subseteq\Gamma$ endlich:
+  $\Gamma'\cup\{\lnot\bot\}$ unerfüllbar - $\Leftrightarrow$ es gibt
+  $\Gamma'\subseteq\Gamma$ endlich: $\Gamma'$ unerfüllbar
+
+  \begin{quote}
+    Satz
+
+    Sei $\Delta$ eine u.U. unendliche Menge von $\sum$-Formeln, so dass für
+    jedes $n\in\mathbb{N}$ eine endliche Struktur $A_n$ mit $A_\Vdash\Delta$
+    existiert, die wenigstens $n$ Elemente hat. Dann existiert eine
+    unendliche Struktur $A$ mit $A\Vdash\Delta$.
+  \end{quote}
+
+  Beweis: für $n\in\mathbb{N}$ setze
+  $\varphi_n=\exists x_1 \exists x_2 ...\exists x_n \bigwedge_{1\leq i< j \leq n} x_i \not= x_j$
+  - und $\Gamma =\Delta\cup\{\varphi_n | n\geq 0\}$. - Für
+  $\Gamma'\subseteq\Gamma$ endlich existiert $n\in\mathbb{N}$ mit
+  $\varphi_m\not\in\Gamma'$ für alle $m\geq n$ -
+  $\Rightarrow A_n\Vdash\Gamma'$, d.h. jede endliche Teilmenge von
+  $\Gamma$ ist erfüllbar. - $\Rightarrow$ es gibt Struktur $A$ mit
+  $A\Vdash\Gamma$ - $\Rightarrow$ A hat $\geq n$ Elemente (für alle
+  $n\in\mathbb{N}$)
+
+  \subsubsection{Folgerung 2:
+    Löwenheim-Skolem}\label{folgerung-2-luxf6wenheim-skolem}
+
+  \begin{quote}
+    Frage: Gibt es eine Menge $\Gamma$ von $\sum$-Formeln, so dass für alle
+    Strukturen $A$ gilt:
+    $A\Vdash\Gamma \Leftrightarrow A\cong (\mathbb{R},+,*, 0 , 1 )$?
+  \end{quote}
+
+  \begin{quote}
+    Satz von Löwenheim-Skolem
+
+    Sei $\Gamma$ erfüllbare und höchstens abzählbar unendliche Menge von
+    $\sum$-Formeln. Dann existiert ein höchstens abzählbar unendliches
+    Modell von $\Gamma$.
+  \end{quote}
+
+  Beweis: - $Gamma$ erfüllbar $\Rightarrow \Gamma\not\Vdash\bot$ -
+  $\Rightarrow$ $\Gamma\not\vdash\bot$, d.h. $\Gamma$ konsistent -
+  $\Rightarrow \Gamma$ hat ein höchstens abzählbar unendliches Modell.
+
+  Die Frage auf der vorherigen Folie muß also verneint werden: -
+  angenommen, $\Gamma$ wäre eine solche Menge -
+  $\Rightarrow |\Gamma|\leq \mathbb{N}_0$ - $\Rightarrow \Gamma$ hat ein
+  höchstens abzählbar unendliches Modell $A$ -
+  $\Rightarrow A\not\cong (\mathbb{R},+,*, 0 , 1 )$
+
+  \subsubsection{Folgerung 3:
+    Semi-Entscheidbarkeit}\label{folgerung-3-semi-entscheidbarkeit}
+
+  \begin{quote}
+    Satz
+
+    Die Menge der allgemeingültigen $\sum$-Formeln ist semi-entscheidbar.
+  \end{quote}
+
+  Beweis:Sei$\varphi$ $\sum$-Formel. Dann gilt - $\varphi$ allgemeingültig
+  - $\Leftrightarrow \varphi$ Theorem - $\Leftrightarrow$ Es gibt
+  hypothesenlose Deduktion mit Konklusion $\varphi$
+
+  Ein Semi-Entscheidungsalgorithmus kann also folgendermaßen vorgehen:
+  Teste für jede Zeichenkette $w$ nacheinander, ob sie hypothesenlose
+  Deduktion mit Konklusion $\varphi$ ist. Wenn ja, so gib aus ``$\varphi$
+  ist allgemeingültig''. Ansonsten gehe zur nächsten Zeichenkette über.
+
+  \subsubsection{Der Satz von Church}\label{der-satz-von-church}
+
+  Jetzt zeigen wir, daß dieses Ergebnis nicht verbessert werden kann: Die
+  Menge der allgemeingültigen $\sum$-Formeln ist nicht entscheidbar. Wegen
+  $\varphi$ allgemeingültig $\Leftrightarrow\lnot\varphi$ unerfüllbar
+  reicht es zu zeigen, dass die Menge der erfüllbaren Sätze nicht
+  entscheidbar ist. Genauer zeigen wir dies sogar für ``Horn-Formeln'':
+
+  \begin{quote}
+    Definition
+
+    Eine Horn-Formel ist eine Konjunktion von $\sum$-Formeln der Form
+    $\forall x_1 \forall x_2 ...\forall x_n((\lnot\bot \wedge\alpha_1\wedge\alpha_2\wedge...\wedge\alpha_m)\rightarrow\beta)$,
+    wobei $\alpha_1,...,\alpha_m$ und $\beta$ atomare $\sum$-Formeln sind.
+  \end{quote}
+
+  Unser Beweis reduziert die unentscheidbare Menge PCP auf die Menge der
+  erfüllbaren Horn-Formeln.
+
+  Im folgenden sei also $I=((u_1,v_1),(u_2,v_2),...,(u_k,v_k))$ ein
+  Korrespondenzsystem und $A$ das zugrundeliegende Alphabet. Hieraus
+  berechnen wir eine Horn-Formel $\varphi_I$, die genau dann erfüllbar
+  ist, wenn $I$ keine Lösung hat. Wir betrachten die Signatur
+  $\sum= (\Omega,Rel,ar)$ mit - $\Omega=\{e\}\cup\{f_a|a\in A\}$ mit
+  $ar(e) =0$ und $ar(f_a) =1$ für alle $a\in A$. - $Rel=\{R\}$ mit
+  $ar(R)=2$.
+
+  Zur Abkürzung schreiben wir $f_{a_1 a_2 ...a_n} (x)$ für
+  $f_{a_1}(f_{a_2}(...(f_{a_n}(x))...))$ für alle $a_1,a_2,...,a_n\in A$
+  und $n\geq 0$ (insbes. steht $f_{\epsilon}(x)$ für $x$).
+
+  Zunächst betrachten wir die folgende Horn-Formel $\psi_I$: -
+  $\wedge \bigwedge_{1\leq j \leq k}^{R(e,e)} \forall x,y(R(x,y)\rightarrow R(f_{u_j}(x),f_{v_j}(y)))$
+  - $\wedge \bigwedge_{a\in A} \forall x(e=f_a(x)\rightarrow \bot)$
+
+  Beispiel: Betrachte die $\sum$-Struktur $A$ mit Universum $U_A=A^*$ -
+  $e^A=\epsilon$ - $f_a^A(u) =au$ -
+  $R^A=\{(u_{i1} u_{i2} ...u_{in},v_{i1} v_{i2} ...v_{in})|n\geq 0 , 1\geq i_1,i_2,...,i_n\geq k\}$
+  - Für $u,v\in A^*$ gilt $f_u^A(v) =uv$. - Dann gilt $A\Vdash \psi_I$.
+
+  \begin{quote}
+    Lemma
+
+    Angenommen, das Korrespondenzsystem $I$ hat keine Lösung. Dann ist die
+    Horn-Formel $\varphi_I=\psi_I \wedge \forall x(R(x,x)\rightarrow x=e)$
+    erfüllbar.
+  \end{quote}
+
+  Beweis: Sei $A$ die obige Struktur mit $A\Vdash\psi_I$. - Um
+  $A\Vdash\forall x(R(x,x)\rightarrow x=e)$ zu zeigen, sei $w\in U_A$
+  beliebig mit $(w,w)\in R^A$. - Die Definition von $R^A$ sichert die
+  Existenz von $n\geq 0$ und $1\leq i_1,i_2,...,i_n\leq k$ mit
+  $u_{i1} u_{i2}...u_{in}=w=v_{i1} v_{i2} ...v_{in}$. - Da $I$ keine
+  Lösung hat, folgt $n=0$ und damit $w=\epsilon$.
+
+  \begin{quote}
+    Lemma
+
+    Sei $B$ Struktur mit $B\Vdash\psi_I$. Dann gilt
+    $(f_{u_{i_1} u_{i_2} ...u_{i_n}}^B (e^B),f_{v_{i_1} v_{i_2}...v_{i_n}}^B(e^B))\in R^B$
+    für alle $n\geq 0, 1\leq i_1,i_2,...,i_n \leq k$.
+  \end{quote}
+
+  Beweis: per Induktion über $n\geq 0$. - IA: für $n=0$ gelten
+  $f_{u_{i_1} u_{i_2} ...u_{i_n}}^B(e^B) =e^B$ und
+  $f_{v_{i_1} v_{i_2}...v_{i_n}}^B(e^B) =e^B$ - und damit
+  $(f_{u_{i_1} u_{i_2} ...u_{i_n}}^B(e^B), f_{v_{i_1} v_{i_2}...v_{i_n}}^B(e^B) \in R^B$
+  - wegen $B\Vdash\psi_I$. - IS: Seien $n>0$ und
+  $1\leq i_1 ,i_2 ,...,i_n\leq k$. - Mit $u=u_{i2} u_{i3} ...u_{in}$ und
+  $v=v_{i2} v_{i3} ...v_{in}$ gilt nach IV
+  $(f_u^B(e^B),f_v^B(e^B))\in R^B$. Wegen $B\Vdash\psi_I$ folgt
+  $f_{u_{i_1} u_{i_2} ...u_{i_n}}^B(e^B), f_{v_{i_1} v_{i_2}...v_{i_n}}^B(e^B) = (f_{u_{i1}}^B (f_u^B(e^B)),f_{v_{i1}}^B (f_v^B(e^B)))\in RB$.
+
+  \begin{quote}
+    Lemma
+
+    Angenommen, $(i_1,...,i_n)$ ist eine Lösung von $I$. Dann ist die
+    $\sum$-Formel $\varphi_I$ unerfüllbar.
+  \end{quote}
+
+  \begin{quote}
+    Satz
+
+    Die Menge der unerfüllbaren Horn-Formeln ist nicht entscheidbar.
+  \end{quote}
+
+  Beweis: Die Abbildung $I\rightarrow\varphi_I$ ist berechenbar.
+
+  Nach den vorherigen Lemmata ist sie eine Reduktion von PCP auf die Menge
+  der unerfüllbaren Horn-Formeln. Da PCP unentscheidbar ist (vgl.
+  Automaten, Sprachen und Komplexität), ist die Menge der unerfüllbaren
+  Horn-Formeln unentscheidbar.
+
+  \begin{quote}
+    Folgerung (Church 1936)
+
+    Die Menge der allgemeingültigen $\sum$-Formeln ist nicht entscheidbar.
+  \end{quote}
+
+  Beweis: Eine $\sum$-Formel $\varphi$ ist genau dann unerfüllbar, wenn
+  $\lnot\varphi$ allgemeingültig ist. Also ist
+  $\varphi\rightarrow\lnot\varphi$ eine Reduktion der unentscheidbaren
+  Menge der unerfüllbaren $\sum$-Formeln auf die Menge der
+  allgemeingültigen $\sum$-Formeln, die damit auch unentscheidbar ist.
+
+  Allgemeingültige $\sum$-Formeln gelten in allen Strukturen. Was
+  passiert, wenn wir uns nur auf ``interessante'' StrukturenAeinschränken
+  (z.B. auf eine konkrete), d.h. wenn wir die Theorie $Th(A)$ von $A$
+  betrachten?
+
+  \subsection{Theorie der natürlichen
+    Zahlen}\label{theorie-der-natuxfcrlichen-zahlen}
+
+  \begin{quote}
+    Definition
+
+    Sei $A$ eine Struktur. Dann ist $Th(A)$ die Menge der
+    prädikatenlogischen $\sum$-Formeln $\varphi$ mit $A\Vdash\varphi$. Diese
+    Menge heißt die(elementare) Theorie von $A$.
+  \end{quote}
+
+  Beispiel: Sei $N= (N,\leq,+,*, 0 , 1 )$. Dann gelten -
+  $(\forall x\forall y:x+y=y+x)\in Th(N)$ -
+  $(\forall x\exists y:x+y= 0 )\not\in Th(N)$ - aber
+  $(\forall x\exists y:x+y= 0 )\in Th((Z,+, 0 ))$.
+
+  \begin{quote}
+    Lemma
+
+    Die Menge $Th(N)$ aller Sätze $\varphi$ mit $N\Vdash\varphi$ ist nicht
+    entscheidbar.
+  \end{quote}
+
+  \begin{quote}
+    Zahlentheoretisches Lemma
+
+    Für alle $n\in N,x_0,x_1,...,x_n\in N$ existieren $c,d\in N$, so dass
+    für alle $0\leq i\leq n$ gilt: $x_i=c\ mod ( 1 +d*(i+ 1 ))$.
+  \end{quote}
+
+  Beweis:Setze $m= max\{n,x_0,x_1 ,...,x_n\}$ und $d=2*3*4...(m+1)$. Dann
+  sind die Zahlen $1+d, 1+d*2,..., 1 +d*(n+1)$ paarweise teilerfremd. Nach
+  dem Chinesischen Restsatz folgt die Existenz einer natürlichen Zahl $c$.
+
+  Bemerkung: Es gibt $\sum$-Formeln - $mod(x_1,x_2 ,y)$ mit
+  $N\Vdash_{\alpha} mod \Leftrightarrow \alpha (x_1) mod\alpha (x_2) =\alpha (y)$.
+  - $\gamma(x_1 ,x_2 ,x_3 ,y)$ mit
+  $N\Vdash_{\alpha} \gamma\Leftrightarrow \alpha(x_1) mod(1+\alpha(x_2)*(\alpha (x3)+1)) =\alpha (y)$.
+
+  \begin{quote}
+    Satz
+
+    Sei $A$ eine Struktur, so dass $Th(A)$ semi-entscheidbar ist. Dann ist
+    $Th(A)$ entscheidbar.
+  \end{quote}
+
+  \begin{quote}
+    Korollar Die Menge $TH(N)$ der Aussagen $\varphi$ mit $N\Vdash\varphi$
+    ist nicht semi-entscheidbar.
+  \end{quote}
+
+  \begin{quote}
+    Korollar (1. Gödelscher Unvollständigkeitssatz)
+
+    Sei $Gamma$ eine semi-entscheidbare Menge von Sätzen mit $N\Vdash\gamma$
+    für alle $\gamma\in\Gamma$. Dann existiert ein Satz $\varphi$ mit
+    $\Gamma\not\vdash\varphi$ und $\Gamma\not\vdash\lnot\varphi$ (d.h.
+    ``$\Gamma$ ist nicht vollständig'').
+  \end{quote}
+
+  \subsection{2. Semi Entscheidungsverfahren für allgemeingültige
+    Formeln}\label{semi-entscheidungsverfahren-fuxfcr-allgemeinguxfcltige-formeln}
+
+  bekanntes Verfahren mittels natürlichem Schließen: Suche hypothesenlose
+  Deduktion mit Konklusion $\psi$.
+
+  Jetzt alternatives Verfahren, das auf den Endlichkeitssatz der
+  Aussagenlogik zurückgreift: - Berechne aus $\sum$-Formel $\psi$ eine
+  Menge E von aussagenlogischen Formeln mit $E$ unerfüllbar
+  $\Leftrightarrow\lnot\psi$ unerfüllbar $\Leftrightarrow\psi$
+  allgemeingültig - Suche endliche unerfüllbare Teilmenge $E'$ von $E$
+
+  Kern des Verfahrens ist es also, aus $\sum$-Formel $\varphi$ eine Menge
+  $E$ aussagenlogischer Formeln zu berechnen mit $\varphi$ unerfüllbar
+  $\Leftrightarrow$ E unerfüllbar.
+
+  Hierzu werden wir die Formel $\varphi$ zunächst in zwei Schritten
+  (Gleichungsfreiheit und Skolem-Form) vereinfachen, wobei die Formel
+  erfüllbar bzw unerfüllbar bleiben muss.
+
+  \begin{quote}
+    Definition
+
+    Zwei $\sum$-Formeln $\varphi$ und $\psi$ heißen
+    erfüllbarkeitsäquivalent, wenn gilt: $\varphi$ ist erfüllbar
+    $\Leftrightarrow\psi$ ist erfüllbar
+  \end{quote}
+
+  Unsere Vereinfachungen müssen also erfüllbarkeitsäquivalente Formeln
+  liefern.
+
+  \subsubsection{Elimination von
+    Gleichungen}\label{elimination-von-gleichungen}
+
+  \begin{quote}
+    Definition
+
+    Eine $\sum$-Formel ist gleichungsfrei, wenn sie keine Teilformel der
+    Form $s=t$ enthält.
+  \end{quote}
+
+  Ziel: aus einer $\sum$ Formel $\varphi$ soll eine
+  erfüllbarkeitsäquivalente gleichungsfreue Formel $\varphi'$ berechnet
+  werden
+
+  Bemerkung: Man kann i.a. keine äquivalente gleichungsfreie Formel
+  $\varphi'$ angeben, da es eine solche z.B. zu
+  $\varphi=(\forall x\forall y:x=y)$ nicht gibt.
+
+  Idee: Die Formel $\varphi'$ entsteht aus $\varphi$, indem alle
+  Teilformeln der Form $x=y$ durch $GI(x,y)$ ersetzt werden, wobei $GI$
+  ein neues Relationssymbol ist.
+
+  Notationen - Sei $\sum=(\Omega,Rel,ar)$ endliche Signatur und $\varphi$
+  $\sum$-Formel - $\sum_{GI} = (\Omega, Rel\bigcup^+\{GI\},ar_{GI})$ mit
+  $ar_{GI}(f)$ für alle $f\in\Omega\cup Rel$ und $ar_{GI}(GI)=2$ - Für
+  eine $\sum$-Formel $\varphi$ bezeichnet $\varphi_{GI}$ die
+  $\sum_{GI}$-Formel, die aus $\varphi$ entsthet, indem alle Vorkommen und
+  Teilformen $s=t$ durch $GI(s,t)$ ersetzt werden.
+
+  Behauptung: $\varphi$ erfüllbar $\Rightarrow \varphi_{GI}$ erfüllbar
+
+  Behauptung: es gilt nicht $\varphi$ erfüllbar $\Leftarrow\varphi_{GI}$
+  erfüllbar
+
+  \begin{quote}
+    Definition
+
+    Sei A eine $\sum$-Struktur und $\sim$ eine binäre Relation auf $U_A$.
+    Die Relation $\sim$ heißt Kongruenz auf A, wenn gilt: - $\sim$ ist eine
+    Äquivalentrelation (d.h. reflexiv, transitiv und symmetrisch) - für alle
+    $f\in\Omega$ mit $k=ar(f)$ und alle $a_1,b_1,...,a_k,b_k\in U_A$ gilt
+    $a_1\sim b_1,a_2\sim b_2,...,a_k\sim b_k\Rightarrow f^A(a_1,...,a_k)\sim f^A(b_1,...,b_k)$
+    - für alle $R\in Rel$ mit $k=ar(R)$ und alle
+    $a_1,b_1,...,a_k,b_k\in U_A$ gilt
+    $a_1\sim b_1,...,a_k\sim b_k,(a_1,...,a_k)\in R^A\Rightarrow (b_1,...,b_k)\in R^A$.
+  \end{quote}
+
+  \begin{quote}
+    Definition
+
+    Sei $A$ eine $\sum$-Struktur und $\sim$ eine Kongruenz auf A. 1. Für
+    $a\in U_A$ sei $[a]=\{b\in U_A|a\sim b\}$ die Äquivalenzklasse von a
+    bzgl $\sim$. 2. Dann definieren wir den Quotienten $B=A\backslash \sim$
+    von $A$ bzgl $\sim$ wie folgt: -
+    $U_B=U_A\backslash\sim = \{[a]|a\in U_A\}$ - Für jedes $f\in\Omega$ mit
+    $ar(f)=k$ und alle $a_1,...,a_k\in U_A$ setzten wir
+    $f^B([a_1],...,[a_k])=[f^A(a_1,...,a_k)]$ - für jede $R\in Rel$ mit
+    $ar(R)=k$ setzten wir
+    $R^B=\{([a_1],[a_2],...,[a_k])|(a_1,...,a_k)\in R^A\}$ 3. Sei
+    $p:Var\rightarrow U_A$ Variableninterpretation. Dann definieren die
+    Variableninterpretation
+    $p\backslash\sim: Var\rightarrow U_B:x\rightarrow[p(x)]$.
+  \end{quote}
+
+  Veranschaulichung:
+  \includegraphics[width=\linewidth]{Assets/Logik-variableninterpretation-beispiel.png}
+
+  \begin{quote}
+    Lemma 1
+
+    Sei A Struktur, $p:Var\rightarrow U_A$ Variableninterpretation und
+    $\sim$ Kongruenz. Seien weiter $B=A\backslash\sim$ und
+    $p_B=p\backslash\sim$. Dann gilt für jeden Term $:[p(t)]=p_B(t)$
+  \end{quote}
+
+  Beweis: per Induktion über den Aufbau des Terms t
+
+  \begin{quote}
+    Lemma 2
+
+    Sei $A$ $\sum$-Struktur, $\sim$ Kongruenz und $B=A\backslash\sim$. Dann
+    gilt für alle $R\in Rel$ mit $k=ar(R)$ und alle $c_1,...,c_k\in U_A$:
+    $([c_1],[c_2],...,[c_k])\in R^B\Leftrightarrow (c_1,c_2,...,c_k)\in R^A$
+  \end{quote}
+
+  \begin{quote}
+    Satz
+
+    Seien $A$ $\sum_{GI}$-Struktur und $p:Var\rightarrow U_A$
+    Variableninterpretation, so dass $\sim=GI^A$ Kongruenz auf A ist. Seien
+    $B=A\backslash\sim$ und $p_B=p\backslash\sim$. Dann gilt für alle
+    $\sum$-Formeln
+    $\varphi: A\Vdash_p \varphi_{GI} \Leftrightarrow B\Vdash_{p_B} \varphi$
+  \end{quote}
+
+  Beweis: per Induktion über den Aufbau der Formel $\varphi$
+
+  \begin{quote}
+    Lemma
+
+    Aus einer endlichen Signatur $\sum$ kann ein gleichungsfreuer Horn-Satz
+    $Kong_{\sum}$ über $\sum_{GI}$ berechnet werden, so dass für alle
+    $\sum_{GI}$-Strukturen $A$ gilt:
+    $A\Vdash Kong_{\sum} \Leftrightarrow GI^A$ ist eine Kongruenz
+  \end{quote}
+
+  \begin{quote}
+    Satz
+
+    Aus einer endlichen Signatur $\sum$ und einer $\sum$-Formel $\varphi$
+    kann eine gleichungsfreie und erfüllbarkeitsäquivalente
+    $\sum_{GI}$-Formel $\varphi'$ berechnet werden. Ist $\varphi$ Horn
+    Formel, so ist auch $\varphi'$ Horn Formel.
+  \end{quote}
+
+  Beweis: Setzte $\varphi' =\varphi_{GI}\wedge Kong_{\sum}$ und zeige:
+  $\varphi$ erfüllbar $\Leftrightarrow \varphi'$ erfüllbar.
+
+  \subsection{Skolemform}\label{skolemform}
+
+  Ziel: Jede $\sum$-Formel $\varphi$ ist erfüllbarkeitsäquivalent zu einer
+  $\sum$'-Formel $\varphi'=\forall x_1\forall x_2 ...\forall x_k \psi$,
+  wobei $\psi$ kein Quantoren enthält, $\varphi'$ heißt in Skolemform.
+
+  Bemerkung: Betrachte die Formel $\exists x\exists y E(x,y)$. Es gibt
+  keine Formel in Skolemform, die hierzu äquivalent ist.
+
+  2 Schritte: 1. Quantoren nach vorne (d.h. Pränexform) 2.
+  Existenzquantoren eliminieren
+
+  \begin{quote}
+    Definition
+
+    Zwei $\sum$-Formeln $\varphi$ und $\psi$ sind äquivalent
+    (kurz:$\varphi\equiv\psi$), wenn für alle $\sum$-Strukturen $A$ und alle
+    Variableninterpretationen $\rho$ gilt:
+    $A\Vdash_{\rho} \psi \Leftrightarrow A\Vdash_{\rho}\psi$.
+  \end{quote}
+
+  \begin{quote}
+    Lemma
+
+    Seien $Q\in\{\exists ,\forall\}$ und
+    $\oplus\in\{\wedge,\vee,\rightarrow,\leftarrow\}$. Sei
+    $\varphi= (Qx \alpha)\oplus\beta$ und sei $y$ eine Variable, die weder
+    in $\alpha$ noch in $\beta$ vorkommt. Dann gilt
+    $\varphi \equiv \begin{cases} Qy(\alpha[x:=y]\oplus\beta) \text{ falls } \oplus\in\{\wedge,\vee,\leftarrow\}\\ \forall y(\alpha[x:=y]\rightarrow\beta) \text{ falls } \oplus=\rightarrow,Q=\exists \\ \exists y(\alpha[x:=y]\rightarrow\beta) \text{ falls }\oplus=\rightarrow,Q=\forall\end{cases}$
+  \end{quote}
+
+  Notwendigkeit der Bedingung ``$y$ kommt weder in $\alpha$ noch in
+  $\beta$ vor'': -
+  $(\exists x:f(x) \not =f(y))\wedge\beta \not\equiv\exists y: (f(y) \not =f(y)\wedge\beta)$
+  -
+  $(\exists x:\lnot P(x))\wedge P(y)\not\equiv \exists y: (\lnot P(y) \wedge P(y))$
+
+  \begin{quote}
+    Lemma
+
+    Seien $Q\in\{\exists,\forall\}$ und
+    $\oplus\in\{\wedge,\vee,\rightarrow,\leftarrow\}$. Sei
+    $\varphi= (Qx\alpha)\oplus\beta$ und sei $y$ eine Variable, die weder in
+    $\alpha$ noch in $\beta$ vorkommt. Dann gilt
+    $\varphi\equiv\begin{cases} Qy(\alpha[x:=y]\oplus\beta) \text{ falls }\oplus\in\{\wedge,\vee,\leftarrow\} \\ \forall y(\alpha[x:=y]\rightarrow\beta) \text{ falls }\oplus=\rightarrow,Q=\exists \\ \exists y(\alpha[x:=y]\rightarrow\beta) \text{ falls }\oplus=\rightarrow,Q=\forall \end{cases}$
+  \end{quote}
+
+  Beweis: (für den Fall $Q=\exists$ und $\oplus=\wedge$) - Seien $A$
+  $\sum$-Struktur und $\rho$ Variableninterpretation. - Für $a\in U_A$
+  setze $\rho_a:=\rho[y\rightarrow a]$. - Dann gilt
+  $\rho_a[x\rightarrow \rho_a(y)](z) =\rho[x\rightarrow a](z)$ für alle
+  $z\not=y$
+
+  Wir erhalten also - $A\vdash_\rho (\exists x\alpha)\wedge\beta$ -
+  $\Leftrightarrow A\vdash_\rho (\exists x\alpha) $ und
+  $A\vdash_\rho \beta$ - $\Leftrightarrow$ (es gibt $a\in U_A$ mit
+  $A\vdash_{\rho[x\rightarrow a]}\alpha$) und (es gilt
+  $A\vdash_\rho \beta$) - $\Leftrightarrow$ es gibt $a\in U_A$ mit
+  ($A\vdash_{\rho[x\rightarrow a]}\alpha$ und $A\vdash_\rho \beta$) -
+  $\Leftrightarrow$ es gibt $a\in U_A$ mit -
+  $A\vdash_{\rho_a[x\rightarrow \rho_a(y)]}\alpha$ (da $y$ in $\alpha$
+  nicht vorkommt) - $A\vdash_{\rho_a} \beta$ (da $y$ in $\beta$ nicht
+  vorkommt) - $\Leftrightarrow$ es gibt $a\in U_A$ mit -
+  $A\vdash_{\rho_a} \alpha[x:=y]$ - $A\vdash_{\rho_a} \beta$ -
+  $\Leftrightarrow$ es gibt $a\in U_A$ mit
+  $A\vdash_{\rho[y\rightarrow a]}\alpha[x:=y]\wedge\beta$ -
+  $\Leftrightarrow A\vdash_\rho \exists y(\alpha[x:=y]\wedge\beta)$
+
+  \begin{quote}
+    Satz
+
+    Aus einer endlichen Signatur $\sum$ und einer $\sum$-Formel $\varphi$
+    kann eine äquivalente $\sum$-Formel
+    $\varphi'=Q_1 x_1 Q_2 x_2 ...Q_k x_k \psi$ (mit
+    $Q_i\in\{\exists,\forall\},\psi$ quantorenfrei und $x_i$ paarweise
+    verschieden) berechnet werden. Eine Formel $\varphi'$ dieser Form heißt
+    Pränexform. Ist $\varphi$ gleichungsfrei, so ist auch $\varphi'$
+    gleichungsfrei.
+  \end{quote}
+
+  Beweis: Der Beweis erfolgt induktiv über den Aufbau von $\varphi$: -
+  I.A. $\varphi$ ist atomare Formel: Setze $\varphi'=\varphi$. - I.S. -
+  $\varphi=\lnot\psi$ : Nach I.V. kann Formel in Pränexform
+  $\psi\equiv Q_1 x_1 Q_2 x_2 ...Q_m x_m \psi'$ berechnet werden. Mit
+  $\forall=\exists$ und $\exists=\forall$ setze
+  $\varphi'=Q_1 x_1 Q_2 x_2 ...Q_m x_m\lnot\psi'$. -
+  $\varphi=\exists x\psi$: Nach I.V. kann Formel in Pränexform
+  $\psi\equiv Q_1 x_1 Q_2 x_2 ...Q_m x_m \psi'$ berechnet werden. Setze
+  $\varphi'= \begin{cases} \exists x Q_1 x_1 Q_2 x_2 ...Q_m x_m\psi'\text{ falls }x\not\in\{x_1,x_2,...,x_m\}\\ Q_1 x_1 Q_2 x_2 ...Q_m x_m\psi'\text{ sonst}\end{cases}$
+  - $\varphi=\alpha\wedge\beta$: Nach I.V. können Formeln in Pränexform
+  $\alpha\equiv Q_1 x_1 Q_2 x_2 ...Q_mx_m \alpha_0; \beta\equiv Q_1'y_1 Q_2'y_2 ...Q_n'y_n \beta_0$
+  berechnet werden.
+
+  Ziel: Berechnung einer erfüllbarkeitsäquivalenten Formel in Skolemform
+
+  Idee: 1. wandle Formel in Pränexform um 2. eliminiere
+  $\exists$-Quantoren durch Einführen neuer Funktionssymbole
+
+  Konstruktion: Sei
+  $\varphi=\forall x_1\forall x_2...\forall x_m\exists y\psi$ Formel in
+  Pränexform (u.U. enthält $\psi$ weitere Quantoren). Sei $g\not\in\Omega$
+  ein neues m-stelliges Funktionssymbol. Setze
+  $\varphi'=\forall x_1\forall x_2...\forall x_m \psi[y:=g(x_1,...,x_m)]$.
+  Offensichtlich hat $\varphi$'einen Existenzquantor weniger als
+  $\varphi$. Außerdem ist $\varphi'$ keine $\sum$-Formel (denn sie
+  verwendet $g\not\in\Omega$), sondern Formel über einer erweiterten
+  Signatur.
+
+  \begin{quote}
+    Lemma
+
+    Die Formeln $\varphi$ und $\varphi'$ sind erfüllbarkeitsäquivalent.
+  \end{quote}
+
+  Beweis: ``$\Leftarrow$'' Sei $A'$ Struktur und $\rho'$
+  Variableninterpretation mit $A'\vdash_{\rho'}\varphi'$. Wir zeigen
+  $A'\vdash_{\rho'}\varphi$. Hierzu seien $a_1,...,a_m\in U_{A'}$
+  beliebig.
+
+  \begin{quote}
+    Satz
+
+    Aus einer Formel $\varphi$ kann man eine erfüllbarkeitsäquivalente
+    Formel $\varphi$ in Skolemform berechnen. Ist $\varphi$ gleichungsfrei,
+    so auch $\varphi$.
+  \end{quote}
+
+  Beweis: Es kann zu $\varphi$ äquivalente Formel
+  $\varphi_0 =Q_1 x_1 Q_2 x_2 ...Q_{\iota}  x_{\iota}   \psi$ in
+  Pränexform berechnet werden (mit $n\leq {\iota}  $ Existenzquantoren).
+  Durch wiederholte Anwendung des vorherigen Lemmas erhält man Formeln
+  $\varphi_1,\varphi_2,...\varphi_n$ mit - $\varphi_i$ und $\varphi_{i+1}$
+  sind erfüllbarkeitsäquivalent - $\varphi_{i+1}$ enthält einen
+  Existenzquantor weniger als $\varphi_i$ - $\varphi_{i+1}$ ist in
+  Pränexform - ist $\varphi_i$ gleichungsfrei, so auch $\varphi_{i+1}$
+
+  Dann ist $\bar{\varphi}=\varphi_n$ erfüllbarkeitsäquivalente (ggf.
+  gleichungsfreie) Formel in Skolemform.
+
+  \subsection{Herbrand-Strukturen und
+    Herbrand-Modelle}\label{herbrand-strukturen-und-herbrand-modelle}
+
+  Sei $\sum= (\Omega,Rel,ar)$ eine Signatur. Wir nehmen im folgenden an,
+  dass $\Omega$ wenigstens ein Konstantensymbol enthält.
+
+  Das Herbrand-Universum $D(\sum)$ ist die Menge aller variablenfreien
+  $\sum$-Terme.
+
+  Beispiel: $\Omega =\{b,f\}$ mit $ar(b) =0$ und $ar(f) =1$. Dann gilt
+  $D(\sum) =\{b,f(b),f(f(b)),f(f(f(b))),...\}$
+
+  Eine $\sum$-Struktur $A=(UA,(fA)f\in\Omega,(RA)R\in Rel)$ ist eine
+  Herbrand-Struktur, falls folgendes gilt: 1. $UA=D(\sum)$, 2. für alle
+  $f\in\Omega$ mit $ar(f)=k$ und alle $t_1,t_2,...,t_k\in D(\sum)$ ist
+  $f^A(t_1,t_2,...,t_k) =f(t_1,t_2,...,t_k)$.
+
+  Für jede Herbrand-Struktur $A$, alle Variableninterpretationen $\rho$
+  und alle variablenfreien Terme $t$ gilt dann $\rho(t) =t$.
+
+  Ein Herbrand-Modell von $\varphi$ ist eine Herbrand-Struktur, die
+  gleichzeitig ein Modell von $\varphi$ ist.
+
+  \begin{quote}
+    Satz
+
+    Sei $\varphi$ eine gleichungsfreie Aussage in Skolemform. $\varphi$ ist
+    genau dann erfüllbar, wenn $\varphi$ ein Herbrand-Modell besitzt.
+  \end{quote}
+
+  Beweis: - Falls $\varphi$ ein Herbrand-Modell hat, ist $\varphi$
+  natürlich erfüllbar. - Sei nun $\varphi=\forall y_1...\forall y_n\psi$
+  erfüllbar. Dann existieren eine $\sum$-Struktur
+  $A=(U_A,(f^A)_{f\in\Omega},(R^A)_{R\in Rel})$ und eine
+  Variableninterpretation $\rho$ mit $A\vdash_\rho \varphi$.
+
+  \paragraph{Plan des Beweises}\label{plan-des-beweises}
+
+  Wir definieren eine Herbrand-Struktur
+  $B=(D(\sum),(f^B)_{f\in\Omega},(R^B)_{R\in Rel})$: - Seien $f\in\Omega$
+  mit $ar(f)=k$ und $t_1,...,t_k\in D(\sum)$. Um eine Herbrand-Struktur
+  $B$ zu konstruieren setzen wir $f^B(t_1,...,t_k) =f(t_1,...,t_k)$ - Sei
+  $R\in Rel$ mit $ar(R)=k$ und seien $t_1,...,t_k\in D(\sum)$. Dann setze
+  $(t_1,...,t_k)\in RB:\Leftrightarrow (\rho(t_1),...,\rho(t_k))\in RA$.
+
+  Sei $\rho_B:Var \rightarrow D(\sum)$ beliebige Variableninterpretation.
+
+  \subparagraph{Behauptung 1:}\label{behauptung-1}
+
+  Ist $\psi$ eine quantoren- und gleichungsfreie Aussage, so gilt
+  $A\vdash_{\rho}\psi \Leftrightarrow B\vdash_{\rho B} \psi$. Diese
+  Behauptung wird induktiv über den Aufbau von $\psi$ gezeigt.
+
+  \subparagraph{Intermezzo}\label{intermezzo}
+
+  Behauptung 1 gilt nur für quantorenfreie Aussagen
+
+  $\sum = (\Omega,Rel,ar)$ mit $\Omega =\{a\},ar(a) =0$ und
+  $Rel=\{E\},ar(E) =2$. Betrachte die Formel
+  $\varphi=\forall x(E(x,x)\wedge E(a,a))$ in Skolemform.
+  $A\vdash_\rho \varphi$ mit $U^A=\{a^A,m\}$ und
+  $E^A=\{(m,m),(a^A,a^A)\}$. Die konstruierte Herbrand-Struktur
+  $B:U_B=D(\sum) =\{a\}$ und $E^B=\{(a,a)\}$.
+
+  Betrachte nun die Formel $\psi=\forall x,y E(x,y)$. Dann gilt
+  $B\vdash_{\rho B}\psi$ und $A\not\vdash_\rho \psi$.
+
+  Für allgemeine Formeln in Skolemform (also u.U. mit Quantoren) können
+  wir also Behauptung 1 nicht zeigen, sondern höchstens die folgende
+  Abschwächung.
+
+  \subparagraph{Behauptung 2:}\label{behauptung-2}
+
+  Ist $\psi$ eine gleichungsfreie Aussage in Skolemform, so gilt
+  $A\vdash_\rho \psi \Rightarrow B\vdash_{\rho B}\psi$. (hieraus folgt dann
+  $B\vdash_{\rho B}\varphi$ wegen $A\vdash_\rho \varphi$)
+
+  Diese Behauptung wird induktiv über die Anzahl $n$ der Quantoren in
+  $\psi$ bewiesen.
+
+  \subsection{Die Herbrand-Expansion}\label{die-herbrand-expansion}
+
+  verbleibende Frage: Wie erkennt man, ob eine gleichungsfreie Aussage in
+  Skolemform ein Herbrand-Modell hat?
+
+  Beispiel: Seien $\sum=(\{a,f\},\{P,R\},ar)$ und
+  $\varphi=\forall x\forall y (P(a,x)\wedge\lnot R(f(y)))$.  Jedes Herbrand-Modell A von $\varphi$ - hat als Universum das Herbrand-Universum $D(\sum)=\{a,f(a),f^2 (a),...\}=\{f^n(a)|n\geq 0\}$ - erfüllt $f^A(f^n(a))= f^{n+1} (a)$ für alle $n\geq 0$
+
+  Um ein Herbrand-Modell zu konstruieren, müssen (bzw. können) wir für
+  alle Elemente $s,t,u\in D(\sum)$ unabhängig und beliebig wählen, ob
+  $(s,t)\in P^A$ und $u\in R^A$ gilt. Wir fassen dies als
+  ``aussagenlogische B-Belegung'' B der ``aussagenlogischen atomaren
+  Formeln'' $P(s,t)$ bzw. $R(u)$ auf.
+
+  Jede solche aussagenlogische B-Belegung $B$ definiert dann eine
+  Herbrand-Struktur $A_B$: -
+  $P^{A_B} = \{(s,t)\in D(\sum)^2 |B(P(s,t))= 1\}$ -
+  $R^{A_B} = \{u\in D(\sum) |B(R(u))= 1\}$
+
+  Mit $\varphi=\forall x\forall y(P(a,x)\wedge\lnot R(f(y)))$ gilt dann
+  $A_B \Vdash_\rho \varphi$ -
+  $\Leftrightarrow A_B \Vdash_{\rho[x\rightarrow f^m(a)][y\rightarrow f^n(a)]} P(a,x)\wedge\lnot R(f(y))$
+  f.a. $m,n\geq 0$ - $\Leftrightarrow (a,fm(a))\in P^{A_B}$ und
+  $f^{n+1}(a)\not\in R^{A_B}$ f.a. $m,n\geq 0$ -
+  $\Leftrightarrow B(P(a,f^m(a)))= 1$ und $B(R(f^{n+1} (a)))= 0$ f.a.
+  $m,n\geq 0$ -
+  $\Leftrightarrow B(P(a,f^m(a))\wedge\lnot R(f^{n+1} (a)))= 1$ f.a.
+  $m,n\geq 0$
+
+  Also hat $\varphi$ genau dann ein Herbrand-Modell, wenn es eine
+  erfüllende B-Belegung $B$ der Menge aussagenlogischer Formeln
+  $E(\varphi)=\{P(a,f^m(a))\wedge\lnot R(f^{n+1}(a)) | m,n\geq 0\}$ gibt.
+
+  Beispiellösung: Setzt $B(P(s,t))= 1$ und $B(R(s))= 0$ für alle
+  $s,t\in D(\sum)$.
+
+  Diese B-Belegung erfüllt $E(\varphi)$ und ``erzeugt'' die
+  Herbrand-Struktur $A_B$ mit $P^{A_B}=D(\sum)^2$ und
+  $R^{A_B}=\varnothing$.
+
+  Nach obiger Überlegung gilt $A_B\Vdash\varphi$, wir haben also ein
+  Herbrand-Modell von $\varphi$ gefunden.
+
+  Sei $\varphi=\forall y_1\forall y_2...\forall y_n\psi$ gleichungsfreie
+  Aussage in Skolemform.
+
+  Ziel: Konstruktion einer Menge aussagenlogischer Formeln, die genau dann
+  erfüllbar ist, wenn $\varphi$ ein Herbrand-Modell hat.
+
+  Die Herbrand-Expansion von $\varphi$ ist die Menge der Aussagen
+  $E(\varphi)=\{\psi[y_1:=t_1][y_2:=t_2]...[y_n:=t_n]|t_1,t_2,...,t_n\in D(\sum)\}$
+
+  Die Formeln von $E(\varphi)$ entstehen also aus $\psi$, indem die
+  (variablenfreien) Terme aus $D(\sum)$ in jeder möglichen Weise in $\psi$
+  substituiert werden.
+
+  Wir betrachten die Herbrand-Expansion von $\varphi$ im folgenden als
+  eine Menge von aussagenlogischen Formeln.
+
+  Die atomaren Formeln sind hierbei von der Gestalt $P(t_1,...,t_k)$ für
+  $P\in Rel$ mit $ar(P)=k$ und $t_1,...,t_k\in D(\sum)$.
+
+  \begin{quote}
+    Konstruktion
+
+    Sei
+    $B:\{P(t_1,...,t_k)|P\in Rel,k=ar(P),t_1,...,t_k\in D(\sum)\}\rightarrow B$
+    eine B-Belegung. Die hiervon induzierte Herbrand-Struktur $A_B$ ist
+    gegeben durch
+    $P^{A_B} = \{(t_1,...,t_k)|t_1,...,t_k\in D(\sum),B(P(t_1,...,t_k))= 1\}$
+    für alle $P\in Rel$ mit $ar(P)=k$.
+  \end{quote}
+
+  \begin{quote}
+    Lemma
+
+    Für jede quantoren- und gleichungsfreie Aussage $\alpha$ und jede
+    Variableninterpretation $\rho$ in $A_B$ gilt
+    $A_B\Vdash_\rho\alpha \Leftrightarrow B(\alpha)= 1$.
+  \end{quote}
+
+  Beweis: - per Induktion über den Aufbau von $\alpha$ - I.A. $\alpha$ ist
+  atomar, d.h. $\alpha= P(t_1,...,t_k)$ mit $t_1,...,t_k$ variablenlos
+  $A_B\Vdash_\rho \alpha\Leftrightarrow (\rho(t_1),\rho(t_2),...,\rho(t_k))\in P^{A_B}\Leftarrow B(\alpha)= 1$
+  - I.S. -
+  $\alpha=\beta\wedge\gamma: A_B\Vdash_\rho \alpha\Leftrightarrow A_B \Vdash_\rho\beta$
+  und
+  $A_B\Vdash_\rho\gamma \Leftrightarrow B(\beta)=B(\gamma)= 1 \Leftrightarrow B(\alpha)= 1$
+  - $\alpha=\beta\vee\gamma$: analog - $\alpha=\beta\rightarrow\gamma$:
+  analog - $\alpha=\lnot\beta$: analog
+
+  \begin{quote}
+    Lemma
+
+    Sei $\varphi=\forall y_1 \forall y_2 ...\forall y_n\psi$ gleichungsfreie
+    Aussage in Skolemform. Sie hat genau dann ein Herbrand-Modell, wenn die
+    Formelmenge $E(\varphi)$ (im aussagenlogischen Sinn) erfüllbar ist.
+  \end{quote}
+
+  Beweis: Seien $A$ Herbrand-Struktur und $\rho$ Variableninterpretation.
+  Sei $B$ die B-Belegung mit
+  $B(P(t_1,...,t_k))= 1\Leftrightarrow(t_1,...,t_k)\in P^A$ für alle
+  $P\in Rel$ mit $k=ar(P)$ und $t_1,...,t_k\in D(\sum)$. Dann gilt
+  $A=A_B$.
+
+  \begin{quote}
+    Satz von Gödel-Herbrand-Skolem
+
+    Sei $\varphi$ gleichungsfreie Aussage in Skolemform. Sie ist genau dann
+    erfüllbar, wenn die Formelmenge $E(\varphi)$ (im aussagenlogischen Sinn)
+    erfüllbar ist.
+  \end{quote}
+
+  Beweis: $\varphi$ erfüllbar $\Leftrightarrow$ $\varphi$ hat ein
+  Herbrand-Modell $\Leftrightarrow$ $E(\varphi)$ ist im aussagenlogischen
+  Sinne erfüllbar.
+
+  \begin{quote}
+    Satz von Herbrand
+
+    Eine gleichungsfreie Aussage $\varphi$ in Skolemform ist genau dann
+    unerfüllbar, wenn es eine endliche Teilmenge von $E(\varphi)$ gibt, die
+    (im aussagenlogischen Sinn) unerfüllbar ist.
+
+    (Jacques Herbrand (1908-1931))
+  \end{quote}
+
+  Beweis: $\varphi$ unerfüllbar $\Leftrightarrow$ $E(\varphi)$ unerfüllbar
+  $\Leftrightarrow$ es gibt $M\subseteq E(\varphi)$ endlich und
+  unerfüllbar
+
+  \subsection{Algorithmus von Gilmore}\label{algorithmus-von-gilmore}
+
+  Sei $\varphi$ gleichungsfreie Aussage in Skolemform und sei
+  $\alpha_1,\alpha_2,\alpha_3,...$ eine Aufzählung von $E(\varphi)$.
+
+  Eingabe: $\varphi$
+
+  \begin{verbatim}
+n:=0;
+repeat n := n +1;
+until { alpha_1, alpha_2,..., alpha_n } ist unerfüllbar;
+  (dies kann mit Mitteln der Aussagenlogik, z.B. Wahrheitswertetabelle, getestet werden)
+Gib "unerfüllbar" aus und stoppe.
+\end{verbatim}
+
+  Folgerung: Sei $\varphi$ eine gleichungsfreie Aussage in Skolemform.
+  Dann gilt: - Wenn die Eingabeformel $\varphi$ unerfüllbar ist, dann
+  terminiert der Algorithmus von Gilmore und gibt ``unerfüllbar'' aus. -
+  Wenn die Eingabeformel $\varphi$ erfüllbar ist, dann terminiert der
+  Algorithmus von Gilmore nicht, d.h. er läuft unendlich lange.
+
+  Beweis: unmittelbar mit Satz von Herbrand
+
+  Zusammenfassung: alternative Semi-Entscheidungsverfahren für die Menge
+  der allgemeingültigen Formeln. - Berechne aus $\psi$ eine zu $\lnot\psi$
+  erfüllbarkeitsäquivalente gleichungsfreie Formel $\varphi$ in
+  Skolemform. - Suche mit dem Algorithmus von Gilmore nach einer endlichen
+  Teilmenge $E'$ von $E(\varphi)$, die unerfüllbar ist.
+
+  \subsection{Berechnung von Lösungen}\label{berechnung-von-luxf6sungen}
+
+  Beispiel - $\gamma = \forall x,y (R(x,f(y))\wedge R(g(x),y))$ -
+  $\varphi = \forall x,yR(x,y)$ - Gilt $\{\gamma\}\Vdash\varphi$? nein,
+  denn $A\Vdash\gamma\wedge\lnot\varphi$ mit - $A=(\mathbb{N},f^A,g^A,R)$
+  - f\textsuperscript{A(n)=g}A(n)=n+1\$ für alle $n\in\mathbb{N}$ -
+  $R^A = \mathbb{N}^2 \backslash\{( 0 , 0 )\}$ - Gibt es variablenfreie
+  Terme $s$ und $t$ mit $\{\gamma\}\Vdash R(s,t)$? - ja: z.B.
+  $(s,t)=(g(f(a)),g(a))$ oder $(s,t)=(g(a),g(a))$ oder $(s,t)=(a,f(b))$ -
+  Kann die Menge aller Termpaare $(s,t)$ (d.h. aller ``Lösungen'') mit
+  $\{\gamma\}\Vdash R(s,t)$ effektiv und übersichtlich angegeben werden? -
+  Wegen $\{\gamma\}\Vdash R(s,t) \Leftrightarrow\gamma\wedge\lnot R(s,t)$
+  unerfüllbar ist die gesuchte Menge der variablenfreien Terme $(s,t)$
+  semi-entscheidbar, d.h. durch eine Turing-Maschine beschrieben. - Im
+  Rest des Logikteils der Vorlesung ``Logik und Logikprogrammierung''
+  wollen wir diese Menge von Termpaaren ``besser'' beschreiben (zumindest
+  in einem Spezialfall, der die Grundlage der logischen Programmierung
+  bildet).
+
+  \begin{quote}
+    Erinnerung
+
+    Eine Horn-Klausel der Prädikatenlogik ist eine Aussage der Form
+    $\forall x_1\forall x_2...\forall x_n ((\lnot\bot\wedge\alpha_1 \wedge\alpha_2 \wedge...\wedge\alpha_m)\rightarrow\beta)$,
+    mit $m\geq 0$, atomaren Formeln $\alpha_1,...,\alpha_m$ und $\beta$
+    atomare Formel oder $\bot$.
+  \end{quote}
+
+  Aufgabe: $\varphi_1,...,\varphi_n$ gleichungsfreie Horn-Klauseln,
+  $\psi(x_1,x_2,...,x_{\iota} )=R(t_1,...,t_k)$ atomare Formel, keine
+  Gleichung. Bestimme die Menge der Tupel $(s_1,...,s_{\iota} )$ von
+  variablenfreien Termen mit
+  $\{\varphi_1,...,\varphi_n\}\Vdash\psi(s_1,...,s_{\iota} )=R(t_1,...,t_k)[x_1:=s_1]...[x_{\iota} :=s_{\iota} ]$,
+  d.h., für die die folgende Formel unerfüllbar ist:
+  $\bigwedge_{1\leq i\leq n} \varphi_i \wedge \lnot\psi(s_1,...,s_{\iota} ) \equiv \bigwedge_{1\leq i\leq n} \varphi_i\wedge(\psi(s_1,...,s_{\iota} )\rightarrow\bot)$
+
+  Erinnerung - Eine Horn-Formel der Prädikatenlogik ist eine Konjunktion
+  von Horn-Klauseln der Prädikatenlogik. - Eine Horn-Klausel der
+  Aussagenlogik ist eine Formel der Form
+  $(\lnot\bot\wedge q_1\wedge q_2 \wedge...\wedge q_m)\rightarrow r$ mit
+  $m\geq 0$, atomaren Formeln $q_1,q_2,...,q_m, r$ atomare Formel od.
+  $\bot$.
+
+  Beobachtung - Wir müssen die Unerfüllbarkeit einer gleichungsfreien
+  Horn-Formel der Prädikatenlogik testen. - Ist $\varphi$ gleichungsfreie
+  Horn-Klausel der Prädikatenlogik, so ist $E(\varphi)$ eine Menge von
+  Horn-Klauseln der Aussagenlogik.
+
+  Schreib- und Sprechweise -
+  $\{\alpha_1,\alpha_2,...,\alpha_n\}\rightarrow\beta$ für Horn-Klausel
+  der Prädikatenlogik
+  $(\lnot\bot\wedge\alpha_1 \wedge\alpha_2\wedge...\wedge\alpha_n)\rightarrow\beta$
+  insbes. $\varnothing\rightarrow\beta$ für $\lnot\bot\rightarrow\beta$ -
+  $\{(N_i\rightarrow\beta_i) | 1\leq i\leq m\}$ für Horn-Formel
+  $\bigwedge_{1\leq i\leq m} (N_i\rightarrow\beta_i)$
+
+  Folgerung: Sei $\varphi =\bigwedge_{1\leq i\leq n} \varphi_i$
+  gleichungsfreie Horn-Formel der Prädikatenlogik. Dann ist $\varphi$
+  genau dann unerfüllbar, wenn $\bigcup_{1\leq i\leq n} E(\varphi_i)$ im
+  aussagenlogischen Sinne unerfüllbar ist.
+
+  Beweis: Für $1\leq i\leq n$ sei
+  $\varphi_i=\forall x_1^i,x_2^i,...,x_{m_i}^i \psi_i$. Zur Vereinfachung
+  nehme wir an, daß die Variablen $x_j^i$ für $1\leq i\leq n$ und
+  $1\leq j\leq m_i$ paarweise verschieden sind.
+
+  Folgerung: Eine gleichungsfreie Horn-Formel der Prädikatenlogik
+  $\varphi=\bigwedge_{1\leq i\leq n} \varphi_i$ ist genau dann
+  unerfüllbar, wenn es eine SLD-Resolution
+  $(M_0\rightarrow\bot,M_1\rightarrow\bot,...,M_m\rightarrow\bot)$ aus
+  $\bigcup_{1\leq i\leq n} E(\varphi_i)$ mit $M_m =\varnothing$ gibt.
+
+  \subsection{Substitutionen}\label{substitutionen-1}
+
+  Eine verallgemeinerte Substitution $\sigma$ ist eine Abbildung der Menge
+  der Variablen in die Menge aller Terme, so daß nur endlich viele
+  Variable $x$ existieren mit $\sigma(x) \not=x$.
+
+  Sei $Def(\sigma)=\{x\ Variable|x\not =\sigma(x)\}$ der
+  Definitionsbereich der verallgemeinerten Substitution $\sigma$. Für
+  einen Term $t$ definieren wir den Term $t\sigma$ (Anwendung der
+  verallgemeinerten Substitution $\sigma$ auf den Term $t$) wie folgt
+  induktiv: - $x\sigma=\sigma(x)$ -
+  $[f(t_1 ,... ,t_k)]\sigma=f(t_1\sigma,... ,t_k\sigma)$ für Terme
+  $t_1,... ,t_k,f\in\Omega$ und $k=ar(f)$ Für eine atomare Formel
+  $\alpha=P(t_1 ,... ,t_k)$ (d.h. $P\in Rel,ar(P) =k,t_1 ,... ,t_k$ Terme)
+  sei $\alpha\sigma = P(t_1\sigma,... ,t_k\sigma)$
+
+  Verknüpfungvon verallgemeinerten Substitutionen: Sind $\sigma_1$ und
+  $\sigma_2$ verallgemeinerte Substitutionen, so definieren wir eine neue
+  verallgemeinerte Substitution $\sigma_1 \sigma_2$ durch
+  $(\sigma_1 \sigma_2)(x) = (x\sigma_1)\sigma_2$.
+
+  Beispiel: Sei $x$ Variable und $t$ Term. Dann ist $\sigma$ mit
+  $\sigma(y) =\begin{cases} t \quad\text{ falls } x=y \\ y \quad\text{ sonst }\end{cases}$
+  eine verallgemeinerte Substitution. Für alle Terme $s$ und alle atomaren
+  Formeln $\alpha$ gilt $s\sigma=s[x:=t]$ und $\alpha\sigma=\alpha[x:=t]$.
+  Substitutionen sind also ein Spezialfall der verallgemeinerten
+  Substitutionen.
+
+  Beispiel: Die verallgemeinerte Substitution $\sigma$ mit
+  $Def(\sigma)=\{x,y,z\}$ und
+  $\sigma(x) =f(h(x')), \sigma(y) =g(a,h(x')), \sigma(z) =h(x')$ ist
+  gleich der verallgemeinerten Substitution
+  $[x:=f(h(x'))] [y:=g(a,h(x'))] [z:=h(x')] = [x:=f(z)] [y:=g(a,z)] [z:=h(x')]$.
+  Es kann sogar jede verallgemeinerte Substitution $\sigma$ als
+  Verknüpfung von Substitutionen der Form $[x:=t]$ geschrieben werden.
+  Vereinbarung: Wir sprechen ab jetzt nur von ``Substitutionen'', auch
+  wenn wir ``verallgemeinerte Substitutionen'' meinen.
+
+  \begin{quote}
+    Lemma
+
+    Seien $\sigma$ Substitution, $x$ Variable und $t$ Term, so dass - (i)
+    $x\not\in Def(\sigma)$ und - (ii) $x$ in keinem der Terme $y\sigma$ mit
+    $y\in Def(\sigma)$ vorkommt. Dann gilt
+    $[x:=t]\sigma=\sigma[x:=t\sigma]$.
+  \end{quote}
+
+  Beispiele: Im folgenden sei $t=f(y)$. - Ist $\sigma=[x:=g(z)]$, so gilt
+  $x[x:=t]\sigma=t\sigma=t\not=g(z) =g(z)[x:=t\sigma] =x\sigma[x:=t\sigma]$.
+  - Ist $\sigma= [y:=g(x)]$, so gilt
+  $y[x:=t]\sigma=y\sigma=g(x) \not=g(f(g(x)))= g(x) [x:=t\sigma] =y\sigma[x:=t\sigma]$.
+  - Ist $\sigma= [y:=g(z)]$, so gelten
+  $Def([x:=t]\sigma) =\{x,y\}=Def(\sigma[x:=t\sigma]),[x:=t]\sigma(x) =f(g(z)) =\sigma[x:=t\sigma]$
+  und $[x:=t]\sigma(y) =\sigma(z) =\sigma[x:=t\sigma]$, also
+  $[x:=t]\sigma=\sigma[x:=t\sigma]$.
+
+  Beweis: Wir zeigen $y[x:=t]\sigma=y\sigma[x:=t\sigma]$ für alle
+  Variablen $y$. - $y=x$: Dann gilt $y[x:=t]\sigma=t\sigma$. Außerdem
+  $y\sigma=x$ wegen $y=x\not\in Def(\sigma)$ und damit
+  $y\sigma[x:=t\sigma]=x[x:=t\sigma]=t\sigma$. - $y\not =x$: Dann gilt
+  $y[x:=t]\sigma=y\sigma$ und ebenso $y\sigma[x:=t\sigma]=y\sigma$, da $x$
+  in $y\sigma$ nicht vorkommt.
+
+  \subsection{Unifikator/Allgemeinster
+    Unifikator}\label{unifikatorallgemeinster-unifikator}
+
+  Gegeben seien zwei gleichungsfreie Atomformeln $\alpha$ und $\beta$.
+  Eine Substitution $\sigma$ heißt Unifikator von $\alpha$ und $\beta$,
+  falls $\alpha\sigma=\beta\sigma$.
+
+  Ein Unifikator $\sigma$ von $\alpha$ und $\beta$ heißt allgemeinster
+  Unifikator von $\alpha$ und $\beta$, falls für jeden Unifikator
+  $\sigma'$ von $\alpha$ und $\beta$ eine Substitution $\tau$ mit
+  $\sigma'=\sigma \tau$ existiert.
+
+  Aufgabe: Welche der folgenden Paare $(\alpha,\beta)$ sind unifizierbar?
+  \textbar{} $\alpha$ \textbar{} $\beta$ \textbar{} Ja \textbar{} Nein
+  \textbar{} \textbar{} --- \textbar{} --- \textbar{} --- \textbar{} ---
+  \textbar{} \textbar{} $P(f(x))$ \textbar{} $P(g(y))$ \textbar{}
+  \textbar{} \textbar{} $P(x)$ \textbar{}$P(f(y))$\textbar{}\textbar{}
+  \textbar{}$Q(x,f(y))$\textbar{} $Q(f(u),z)$\textbar{}\textbar{}
+  \textbar{}$Q(x,f(y))$\textbar{} $Q(f(u),f(z))$\textbar{}\textbar{}
+  \textbar{}$Q(x,f(x))$\textbar{} $Q(f(y),y)$\textbar{}\textbar{}
+  \textbar{}$R(x,g(x),g^2 (x))$\textbar{} $R(f(z),w,g(w))$
+  \textbar{}\textbar{}
+
+  \subsubsection{Zum allgemeinsten
+    Unifikator}\label{zum-allgemeinsten-unifikator}
+
+  Eine Variablenumbenennung ist eine Substitution $\rho$, die $Def(\rho)$
+  injektiv in die Menge der Variablen abbildet.
+
+  \begin{quote}
+    Lemma
+
+    Sind $\sigma_1$ und $\sigma_2$ allgemeinste Unifikatoren von $\alpha$
+    und $\beta$, so existiert eine Variablenumbenennung $\rho$ mit
+    $\sigma_2=\sigma_1 \rho$.
+  \end{quote}
+
+  Beweis: $\sigma_1$ und $\sigma_2$ allgemeinste Unifikatoren
+  $\Rightarrow$ es gibt Substitutionen $\tau_1$ und $\tau_2$ mit
+  $\sigma_1\tau_1 =\sigma_2$ und $\sigma_2\tau_2 =\sigma_1$. Definiere
+  eine Substitution $\rho$ durch:
+  $\rho(y) =\begin{cases} y\tau_1 \quad\text{ falls es x gibt, so dass y in } x\sigma_1 \text{ vorkommt}\\ y \quad\text{ sonst }\end{cases}$
+  Wegen $Def(\rho)\subseteq Def(\tau_1)$ ist $Def(\rho)$ endlich, also
+  $\rho$ eine Substitution. - Für alle Variablen $x$ gilt dann
+  $x\sigma_1 \rho=x\sigma_1 \tau_1 =x\sigma_2$ und daher
+  $\sigma_2 =\sigma_1 \rho$. - Wir zeigen, dass $\rho(y)$ Variable und
+  $\rho$ auf $Def(\rho)$ injektiv ist: Sei $y\in Def(\rho)$. Dann
+  existiert Variable $x$, so dass $y$ in $x\sigma_1$ vorkommt. Es gilt
+  $x\sigma_1 =x\sigma_2\tau_2=x\sigma_1\tau_1\tau_2$, und damit
+  $y=y\tau_1 \tau_2 =y\rho \tau_2 =\rho(y)\tau_2$, d.h. $\rho(y)$ ist
+  Variable, die Abbildung $\rho:Def(\rho)\rightarrow\{z|z\ Variable\}$ ist
+  invertierbar (durch $\tau_2$) und damit injektiv.
+
+  \subsubsection{Unifikationsalgorithmus}\label{unifikationsalgorithmus}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Eingabe: Paar$(\alpha,\beta)$ gleichungsfreier Atomformeln $\sigma:=$
+          Substitution mit $Def(\sigma)=\varnothing$ (d.h. Identität)
+    \item
+          while $\alpha\sigma\not =\beta\sigma$ do
+    \item
+          Suche die erste Position, an der sich $\alpha\sigma$ und $\beta\sigma$
+          unterscheiden
+    \item
+          if keines der beiden Symbole an dieser Position ist eine Variable
+    \item
+          then stoppe mit ``nicht unifizierbar''
+    \item
+          else sei $x$ die Variable und $t$ der Term in der anderen Atomformel
+          (möglicherweise auch eine Variable)
+
+          \begin{itemize}
+            \item
+                  if $x$ kommt in $t$ vor
+            \item
+                  then stoppe mit ``nicht unifizierbar''
+            \item
+                  else $\sigma:=\sigma[x:=t]$
+          \end{itemize}
+    \item
+          endwhile
+    \item
+          Ausgabe: $\sigma$
+  \end{itemize}
+
+  \begin{quote}
+    Satz
+
+    \begin{itemize}
+      \item
+            \begin{enumerate}
+              \def\labelenumi{(\Alph{enumi})}
+              \item
+                    Der Unifikationsalgorithmus terminiert für jede Eingabe.
+            \end{enumerate}
+      \item
+            \begin{enumerate}
+              \def\labelenumi{(\Alph{enumi})}
+              \setcounter{enumi}{1}
+              \item
+                    Wenn die Eingabe nicht unifizierbar ist, so terminiert der
+                    Unifikationsalgorithmus mit der Ausgabe ``nicht unifizierbar''.
+            \end{enumerate}
+      \item
+            \begin{enumerate}
+              \def\labelenumi{(\Alph{enumi})}
+              \setcounter{enumi}{2}
+              \item
+                    Wenn die Eingabe $(\alpha,\beta)$ unifizierbar ist, dann findet der
+                    Unifikationsalgorithmus einen allgemeinsten Unifikator von $\alpha$
+                    und $\beta$.
+            \end{enumerate}
+    \end{itemize}
+  \end{quote}
+
+  \begin{enumerate}
+    \def\labelenumi{(\Alph{enumi})}
+    \setcounter{enumi}{2}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          besagt insbesondere, daß zwei unifizierbare gleichungsfreie
+          Atomformeln (wenigstens) einen allgemeinsten Unifikator haben. Nach
+          dem Lemma oben haben sie also genau einen allgemeinsten Unifikator
+          (bis auf Umbenennung der Variablen).
+  \end{enumerate}
+
+  Die drei Teilaussagen werden in getrennten Lemmata bewiesen werden.
+
+  \begin{quote}
+    Lemma (A) Der Unifikationsalgorithmus terminiert für jede
+    Eingabe($\alpha$, $\beta$).
+  \end{quote}
+
+  Beweis: Wir zeigen, daß die Anzahl der in $\alpha\sigma$ oder
+  $\beta\sigma$ vorkommenden Variablen in jedem Durchlauf der
+  while-Schleife kleiner wird. Betrachte hierzu einen Durchlauf durch die
+  while-Schleife. Falls der Algorithmus in diesem Durchlauf nicht
+  terminiert, so wird $\sigma$ auf $\sigma[x:=t]$ gesetzt. Hierbei kommt
+  $x$ in $\alpha\sigma$ oder in $\beta\sigma$ vor und der Term $t$ enthält
+  $x$ nicht. Also kommt $x$ weder in $\alpha\sigma[x:=t]$ noch in
+  $\beta\sigma[x:=t]$ vor.
+
+  \begin{quote}
+    Lemma (B) Wenn die Eingabe nicht unifizierbar ist, so terminiert der
+    Unifikationsalgorithmus mit der Ausgabe ``nicht unifizierbar''.
+  \end{quote}
+
+  Beweis: Sei die Eingabe $(\alpha,\beta)$ nicht unifizierbar. Falls die
+  Bedingung $\alpha\sigma\not=\beta\sigma$ der while-Schleife irgendwann
+  verletzt wäre, so wäre $(\alpha,\beta)$ doch unifizierbar (denn $\sigma$
+  wäre ja ein Unifikator). Da nach Lemma (A) der Algorithmus bei Eingabe
+  $(\alpha,\beta)$ terminiert, muss schließlich ``nicht unifizierbar''
+  ausgegeben werden.
+
+  \begin{quote}
+    Lemma (C1) Sei $\sigma'$ ein Unifikator der Eingabe $(\alpha,\beta)$, so
+    dass keine Variable aus $\alpha$ oder $\beta$ auch in einem Term aus
+    $\{y\sigma'|y\in Def(\sigma')\}$ vorkommt. Dann terminiert der
+    Unifikationsalgorithmus erfolgreich und gibt einen Unifikator $\sigma$
+    von $\alpha$ und $\beta$ aus. Außerdem gibt es eine Substitution $\tau$
+    mit $\sigma'=\sigma\tau$.
+  \end{quote}
+
+  Beweis: - Sei $N\in\mathbb{N}$ die Anzahl der Durchläufe der
+  while-Schleife (ein solches $N$ existiert, da der Algorithmus nach Lemma
+  (A) terminiert). - Sei $\sigma_0$ Substitution mit
+  $Def(\sigma_0) =\varnothing$, d.h. die Identität. Für $1\leq i\leq N$
+  sei $\sigma_i$ die nach dem $i$-ten Durchlauf der while-Schleife
+  berechnete Substitution $\sigma$. - Für $1\leq i\leq N$ sei $x_i$ die im
+  $i$-ten Durchlauf behandelte Variable $x$ und $t_i$ der entsprechende
+  Term $t$. - Für $0\leq i\leq N$ sei $\tau_i$ die Substitution mit
+  $\tau_i(x)=\sigma'(x)$ für alle
+  $x\in Def(\tau_i) =Def(\sigma')\backslash\{x_1,x_2,...,x_i\}$.
+
+  Behauptung: 1. Für alle $0\leq i\leq N$ gilt $\sigma'=\sigma_i\tau_i$.
+  2. Im $i$-ten Durchlauf durch die while-Schleife $(1\leq i\leq N)$
+  terminiert der Algorithmus entweder erfolgreich (und gibt die
+  Substitution $\sigma_N$ aus) oder der Algorithmus betritt die beiden
+  else-Zweige. 3. Für alle $0\leq i\leq N$ enthalten
+  $\{\alpha\sigma_i,\beta\sigma_i\}$ und $T_i=\{y\tau_i|y\in Def(\tau_i)\}$
+  keine gemeinsamen Variablen.
+
+  Aus dieser Behauptung folgt tatsächlich die Aussage des Lemmas: - Nach
+  (2) terminiert der Algorithmus erfolgreich mit der Substitution
+  $\sigma_N$. Daher gilt aber $\alpha\sigma_N=\beta\sigma_N$, d.h.
+  $\sigma_N$ ist ein Unifikator. - Nach (1) gibt es auch eine Substitution
+  $\tau_n$ mit $\sigma'=\sigma_N\tau_n$.
+
+  \begin{quote}
+    Lemma (C) Sei die Eingabe $(\alpha,\beta)$ unifizierbar. Dann terminiert
+    der Unifikationsalgorithmus erfolgreich und gibt einen allgemeinsten
+    Unifikator $\sigma$ von $\alpha$ und $\beta$ aus.
+  \end{quote}
+
+  Beweis: Sei $\sigma'$ ein beliebiger Unifikator von $\alpha$ und
+  $\beta$. Sei $Y=\{y_1,y_2,... ,y_n\}$ die Menge aller Variablen, die in
+  $\{y\sigma'|y\in Def(\sigma')\}$ vorkommen. Sei $Z=\{z_1,z_2,...,z_n\}$
+  eine Menge von Variablen, die weder in $\alpha$ noch in $\beta$
+  vorkommen. Sei $\rho$ die Variablenumbenennung mit
+  $Def(\rho)=Y\cup Z,\rho(y_i) =z_i$ und $\rho(z_i)=y_i$ für alle
+  $1\leq i\leq n$. Dann ist auch $\sigma'\rho$ ein Unifikator von $\alpha$
+  und $\beta$ und keine Variable aus $\alpha$ oder $\beta$ kommt in einem
+  der Terme aus $\{y\sigma'\rho|y\in Def(\sigma')\}$ vor. Nach Lemma (C1)
+  terminiert der Unifikationsalgorithmus erfolgreich mit einem Unifikator
+  $\sigma$ von $\alpha$ und $\beta$, so dass es eine Substitution $\tau$
+  gibt mit $\sigma'\rho=\sigma\tau$. Also gilt
+  $\sigma'=\sigma(\tau\rho^{-1})$. Da $\sigma'$ ein beliebiger Unifikator
+  von $\alpha$ und $\beta$ war und da die Ausgabe $\sigma$ des Algorithmus
+  nicht von $\sigma'$ abhängt, ist $\sigma$ also ein allgemeinster
+  Unifikator.
+
+  \begin{quote}
+    Satz
+
+    \begin{itemize}
+      \item
+            \begin{enumerate}
+              \def\labelenumi{(\Alph{enumi})}
+              \item
+                    Der Unifikationsalgorithmus terminiert für jede Eingabe.
+            \end{enumerate}
+      \item
+            \begin{enumerate}
+              \def\labelenumi{(\Alph{enumi})}
+              \setcounter{enumi}{1}
+              \item
+                    Wenn die Eingabe nicht unifizierbar ist, so terminiert der
+                    Unifikationsalgorithmus mit der Ausgabe ``nicht unifizierbar''.
+            \end{enumerate}
+      \item
+            \begin{enumerate}
+              \def\labelenumi{(\Alph{enumi})}
+              \setcounter{enumi}{2}
+              \item
+                    Wenn die Eingabe $(\alpha,\beta)$ unifizierbar ist, dann findet der
+                    Unifikationsalgorithmus immer einen allgemeinsten Unifikator von
+                    $\alpha$ und $\beta$.
+            \end{enumerate}
+    \end{itemize}
+  \end{quote}
+
+  \begin{enumerate}
+    \def\labelenumi{(\Alph{enumi})}
+    \setcounter{enumi}{2}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          besagt insbesondere, daß zwei unifizierbare gleichungsfreie
+          Atomformeln(wenigstens) einen allgemeinsten Unifikator haben. Damit
+          haben sie aber genau einen allgemeinsten Unifikator (bis auf
+          Umbenennung der Variablen).
+  \end{enumerate}
+
+  \subsection{Prädikatenlogische
+    SLD-Resolution}\label{pruxe4dikatenlogische-sld-resolution}
+
+  Erinnerung - Eine Horn-Klausel der Prädikatenlogik ist eine Aussage der
+  Form
+  $\forall x_1 \forall x_2... \forall x_n ((\lnot\bot \wedge\alpha_1 \wedge\alpha_2 \wedge...\wedge\alpha_m)\rightarrow\beta)=\Psi$
+  mit $m\geq 0$, atomaren Formeln $\alpha_1,...,\alpha_m$ und $\beta$
+  atomare Formel oder $\bot$. Sie ist definit, wenn $\beta\not =\bot$. -
+  $E(\varphi) =\{\Psi[x_1 :=t_1 ][x_2 :=t_2 ]...[x_n:=tn]|t_1 ,t_2 ,...,t_n\in D(\sigma)\}$
+  - Eine Horn-Klausel der Aussagenlogik ist eine Formel der Form
+  $(\lnot\bot\wedge q_1 \wedge q_2 \wedge... \wedge q_m)\rightarrow r$ mit
+  $m\geq 0$, atomaren Formeln $q_1,q_2,...,q_m,r$ atomare Formel oder
+  $\bot$.
+
+  Schreib- und Sprechweise: Für die Horn-Klausel der Prädikatenlogik
+  $\forall x_1...\forall x_n(\lnot\bot \wedge \alpha_1 \wedge \alpha_2 \wedge...\wedge \alpha_m)\rightarrow\beta$
+  schreiben wir kürzer
+  $\{\alpha_1,\alpha_2,...,\alpha_m\}\rightarrow\beta$. insbes.
+  $\varnothing\rightarrow\beta$ für
+  $\forall x_1...\forall x_n(\lnot\bot\rightarrow\beta)$
+
+  Erinnerung: Sei $\Gamma$ eine Menge von Horn-Klauseln der Aussagenlogik.
+  Eine aussagenlogische SLD-Resolution aus $\Gamma$ ist eine Folge
+  $(M_0 \rightarrow\bot,M_1 \rightarrow\bot,...,M_m\rightarrow\bot)$ von
+  Hornklauseln mit - $(M_0\rightarrow\bot)\in\Gamma$ und - für alle
+  $0\leq n<m$ existiert $(N\rightarrow q)\in\Gamma$ mit $q\in M_n$ und
+  $M_{n+1} =M_n\backslash\{q\}\cup N$
+
+  \begin{quote}
+    Definition
+
+    Sei $\Gamma$ eine Menge von gleichungsfreien Horn-Klauseln der
+    Prädikatenlogik. Eine SLD-Resolution aus $\Gamma$ ist eine Folge
+    $((M_0\rightarrow\bot,\sigma_0),(M_1\rightarrow\bot,\sigma_1),...,(M_m\rightarrow\bot,\sigma_m))$
+    von Horn-Klauseln und Substitutionen mit -
+    $(M_0\rightarrow\bot)\in\Gamma$ und $Def(\sigma_0)=\varnothing$ - für
+    alle $0\leq n<m$ existieren
+    $\varnothing\not=Q\subseteq M_n,(N\rightarrow\alpha)\in\Gamma$ und
+    Variablenumbenennung $\rho$, so dass - $(N\cup\{\alpha\})\rho$ und $M_n$
+    variablendisjunkt sind, - $\sigma_{n+1}$ ein allgemeinster Unifikator
+    von $\alpha\rho$ und $Q$ ist und -
+    $M_{n+1} = (M_n\backslash Q\cup N\rho)\sigma_{n+1}$.
+  \end{quote}
+
+  Ziel: Seien $\Gamma=\{\varphi_1,...,\varphi_n\}$ Menge gleichungsfreier
+  Horn-Klauseln, $\Psi(x_1,x_2 ,...,x_{\iota}) =R(t_1 ,...,t_k)$ atomare
+  Formel, keine Gleichung und $(s_1,...,s_{\iota})$ Tupel variablenloser
+  Terme. Dann sind äquivalent: 1.
+  $\Gamma\Vdash\Psi(s_1,...,s_{\iota}). 2.$ Es gibt eine SLD-Resolution $((M\_n\rightarrow\bot,\sigma\emph{n))}\{0\leq n\leq m\}$ aus $\Gamma\cup\{M_0\rightarrow\bot\}$ mit
+  $M_0=\{\Psi(x_1,...,x_{\iota})\}$ und $M_m=\varnothing$ und eine
+  Substitution $\tau$, so dass $s_i=x_i\sigma_0 \sigma_1 ...\sigma_m\tau$
+  für alle $1\leq i\leq \iota$ gilt.
+
+  \begin{quote}
+    Lemma
+
+    Sei $\Gamma$ Menge von gleichungsfreien Horn-Klauseln der
+    Prädikatenlogik und $(M_n \rightarrow\bot,\sigma_n))_{0\leq n\leq m}$
+    eine SLD-Resolution aus $\Gamma\cup\{M_0\rightarrow\bot\}$ mit
+    $M_m=\varnothing$. Dann gilt
+    $\Gamma\Vdash\Psi\sigma_0 \sigma_1\sigma_2...\sigma_m$ für alle
+    $\Psi\in M_0$.
+  \end{quote}
+
+  Konsequenz:
+  $\Gamma=\{\varphi_1,...,\varphi_n\},M_0 =\{\Psi(x_1,...,x_{\iota})\},\tau$
+  Substitution, so dass
+  $s_i=x_i \sigma_0 \sigma_1 \sigma_2 ...\sigma_m \tau$ variablenlos für
+  alle $1\leq i \leq \iota$. Nach dem Lemma gilt also
+  $\Gamma \Vdash\Psi(x_1,...,x_{\iota})\sigma_0 ...\sigma_m$ und damit
+  $\Gamma\Vdash\Psi(x_1 ,...,x_{\iota} )\sigma_0 ...\sigma_m\tau=\Psi(s_1,...,s_{\iota} )$.
+  Die Implikation $(2)\Rightarrow (1)$ des Ziels folgt also aus diesem
+  Lemma.
+
+  \begin{quote}
+    Lemma
+
+    Sei $\Gamma$ eine Menge von definiten gleichungsfreien Horn-Klauseln der
+    Prädikatenlogik, sei $M\rightarrow\bot$ eine gleichungsfreie
+    Horn-Klausel und sei $\nu$ Substitution, so dass $M\nu$ variablenlos ist
+    und $\Gamma\Vdash M\nu$ gilt. Dann existieren eine prädikatenlogische
+    SLD-Resolution $((M_n \rightarrow\bot,\sigma_n))_{0 \leq n\leq m}$ und
+    eine Substitution $\tau$ mit $M_0=M,M_m=\varnothing$ und
+    $M_0 \sigma_0 \sigma_1... \sigma_m \tau=M_{\nu}$.
+  \end{quote}
+
+  Konsequenz:
+  $\Gamma=\{\varphi_1,...,\varphi_n\},M=\{\psi(x_1 ,...,x_{\iota})\},s_1,...,s_\iota\}$
+  variablenlose Terme, so dass
+  $\{\varphi_1 ,...,\varphi_n\}\Vdash\psi(s_1,...,s_{\iota}) =\psi(x_1 ,...,x_{\iota})\nu$
+  mit $\nu(x_i)=s_i$. Dann existieren SLD-Resolution und Substitution
+  $\tau$ mit
+  $M_0\sigma 0...\sigma_m\tau=M\nu=\{\psi (s_1,...,s_{\iota} )\}$. Die
+  Implikation $(1)\Rightarrow (2)$ des Ziels folgt also aus diesem Lemma.
+
+  \begin{quote}
+    Satz
+
+    Sei $\Gamma$ eine Menge von definiten gleichungsfreien Horn-Klauseln der
+    Prädikatenlogik, sei $M\rightarrow\bot$ eine gleichungsfreie
+    Horn-Klausel und sei $\nu$ Substitution, so dass $M\nu$ variablenlos
+    ist. Dann sind äquivalent: - $\Gamma\Vdash M\nu$ - Es existieren eine
+    SLD-Resolution $((M_n\rightarrow\bot,\sigma_n))_{0\leq n\leq m}$ aus
+    $\Gamma\cup\{M\nu\rightarrow\bot\}$ und eine Substitution $\tau$ mit
+    $M_0=M,M_m=\varnothing$ und $M_0\sigma_0\sigma_1...\sigma_m\tau=M\nu$.
+  \end{quote}
+
+  Konsequenz:
+  $\Gamma =\{\varphi_1,...,\varphi_n\},M_0 =\{\psi(x_1,...,x_{\iota})\}=\{R(t_1,t_2,...,t_k)\}$.
+  Durch SLD-Resolutionen können genau die Tupel variablenloser Terme
+  gewonnen werden, für die gilt:
+  $\{\varphi_1,...,\varphi_n\}\Vdash\psi (s_1,...,s_{\iota})$
+
+  \subsection{Zusammenfassung
+    Prädikatenlogik}\label{zusammenfassung-pruxe4dikatenlogik}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Das natürliche Schließen formalisiert die ``üblichen'' Argumente in
+          mathematischen Beweisen.
+    \item
+          Das natürliche Schließen ist vollständig und korrekt.
+    \item
+          Die Menge der allgemeingültigen Formeln ist semi-entscheidbar, aber
+          nicht entscheidbar.
+    \item
+          Die Menge der Aussagen, die in $(\mathbb{N},+,*,0,1)$ gelten, ist
+          nicht semi-entscheidbar.
+    \item
+          Die SLD-Resolution ist ein praktikables Verfahren, um die Menge der
+          ``Lösungen'' $(s_1,...,s_{\iota})$ von
+          $\Gamma\Vdash\psi(s_1,...,s_{\iota})$ zu bestimmen (wobei $\Gamma$
+          Menge von gleichungsfreien Horn-Klauseln und $\psi$ Konjunktion von
+          gleichungsfreien Atomformeln sind.
+  \end{itemize}
+
+  \section{Logische Programmierung}\label{logische-programmierung}
+
+  \subsection{Einführung in die Künstliche Intelligenz
+    (KI)}\label{einfuxfchrung-in-die-kuxfcnstliche-intelligenz-ki}
+
+  Ziel: Mechanisierung von Denkprozessen
+
+  Grundidee (nach G.W. Leibniz) 1. lingua characteristica -
+  Wissensdarstellungssprache 2. calculus ratiocinator -
+  Wissensverarbeitungskalkül
+
+  \textbf{Teilgebiete der KI} - Wissensrepräsentation - maschinelles
+  Beweisen (Deduktion) - KI-Sprachen: Prolog, Lisp - Wissensbasierte
+  Systeme - Lernen (Induktion) - Wissensverarbeitungstechnologien
+  (Suchtechniken, fallbasiertes Schließen, Multiagenten-Systeme) - Sprach-
+  und Bildverarbeitung
+
+  \subsection{Logische Grundlagen}\label{logische-grundlagen}
+
+  \subsubsection{PROLOG - ein
+    ``Folgerungstool''}\label{prolog---ein-folgerungstool}
+
+  Sei $M$ eine Menge von Aussagen, $H$ eine Hypothese.
+
+  $H$ folgt aus $M(M \Vdash H)$, falls jede Interpretation, die zugleich
+  alle Elemente aus $M$ wahr macht (jedes Modell von M), auch $H$ wahr
+  macht. Für endliche Aussagenmengen $M=\{A_1, A_2, ... , A_n\}$ bedeutet
+  das: $M \Vdash H$, gdw. $ag(\bigwedge_{i=1}^n A_i\rightarrow A)$ bzw.
+  (was dasselbe ist) $kt(\bigwedge_{i=1}^n A_i \wedge \lnot A)$
+
+  \subsubsection{Aussagen in PROLOG: HORN-Klauseln des
+    PK1}\label{aussagen-in-prolog-horn-klauseln-des-pk1}
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-prolog-horn.png}
+
+  $A(X_1,...,X_n), A_i(X_1,...,X_n)$ quantorfreie Atomformeln, welche die
+  allquantifizierten Variablen $X_1,...,X_n$ enthalten können
+
+  Varianten / Spezialfälle 1. Regeln (vollständige HORN-Klauseln)
+  $\forall X_1... \forall X_n(A(X_1,...,X_n)\leftarrow \bigwedge_{i=1}^n A_i(X_1,...,X_n))$
+  2. Fakten (HORN-Klauseln mit leerem Klauselkörper)
+  $\forall X_1...\forall X_n(A(X_1,...,X_n)\leftarrow true)$ 3. Fragen
+  (HORN-Klauseln mit leerem Klauselkopf)
+  $\forall X_1...\forall X_n(false \leftarrow \bigwedge_{i=1}^n A_i(X_1,...,X_n))$
+  4. leere HORN-Klauseln (mit leeren Kopf \& leerem Körper)
+  $false\leftarrow true$
+
+  Effekte der Beschränkung auf HORN-Logik 1. Über HORN-Klauseln gibt es
+  ein korrektes und vollständiges Ableitungsverfahren. -
+  $\{K_1, ...,K_n\} \Vdash H$ , gdw. $\{K_1,...,K_n\} \vdash_{ROB} H$ 1.
+  Die Suche nach einer Folge von Resolutionsschritten ist
+  algorithmisierbar. - Das Verfahren ``Tiefensuche mit Backtrack'' sucht
+  systematisch eine Folge, die zur leeren Klausel führt. - Rekursive
+  und/oder metalogische Prädikate stellen dabei die Vollständigkeit in
+  Frage. 3. Eine Menge von HORN-Klauseln mit nichtleeren Klauselköpfen ist
+  stets erfüllbar; es lassen sich keine Widersprüche formulieren. -
+  $K_1 \wedge K_2...\wedge K_n \not= false$
+
+  Die systematische Erzeugung von (HORN-) Klauseln 1.
+  Verneinungstechnischen Normalform (VTNF): $\lnot$ steht nur vor
+  Atomformeln - $\lnot\lnot A\equiv A$ 2. Erzeugung der Pränexen
+  Normalform (PNF): $\forall, \exists$ stehen vor dem Gesamtausdruck -
+  $\forall X A(X) \rightarrow B \equiv \exists X(A(X)\rightarrow B)$ -
+  $\exists X A(X) \rightarrow B \equiv \forall X(A(X)\rightarrow B)$ 3.
+  Erzeugung der SKOLEM`schen Normalform (SNF): $\exists$ wird eliminiert
+  Notation aller existenzquantifizierten Variablen als Funktion derjenigen
+  allquantifizierten Variablen, in deren Wirkungsbereich ihr Quantor
+  steht. Dies ist keine äquivalente - , wohl aber eine die
+  Kontradiktorizität erhaltende Umformung. 4. Erzeugung der Konjunktiven
+  Normalform (KNF) Durch systematische Anwendung des Distributivgesetzes
+  $A\vee (B\wedge C)\equiv (A\vee B)\wedge(A\vee C)$ lässt sich aus der
+  SNF $\forall X_1...\forall X_n A(X_1,...,X_n)$ stets die äquivalente KNF
+  $\forall X_1...\forall X_n((L_1^1\vee...\vee L_1^{n_1})\wedge...\wedge(L_m^1\vee...\vee L_m^{n_m}))$
+  erzeugen. Die $L_i^k$ sind unnegierte oder negierte Atomformeln und
+  heißen positive bzw. negative Literale. 5. Erzeugung der Klauselform
+  (KF) Jede der Elementardisjunktionen $(L_1^j \vee...\vee L_1^{j_k})$ der
+  KNF kann man als äquivalente Implikation (Klausel)
+  $(L_i^j \vee...\vee L_i^{j_m})\leftarrow(L_1^{j_{m+1}}\wedge ...\wedge L_1^{j_k})$
+  notieren, indem man alle positiven Literale $L_i^j,...,L_i^{j_m}$
+  disjunktiv verknüpft in den DANN-Teil (Klauselkopf) und alle negativen
+  Literale $L_i^{j_{m+1}},...,L_i^{j_k}$ konjunktiv verknüpft in den
+  WENN-Teil (Klauselkörper) notiert. 6. Sind die Klauseln aus Schritt 5.
+  HORN? In dem Spezialfall, dass alle Klauselköpfe dabei aus genau einem
+  Literal bestehen, war die systematische Erzeugung von HORN-Klauseln
+  erfolgreich; anderenfalls gelingt sie auch nicht durch andere Verfahren.
+  Heißt das etwa, die HORN-Logik ist eine echte Beschränkung der
+  Ausdrucksfähigkeit? Richtig, das heißt es.
+
+  Im Logik-Teil dieser Vorlesung lernten Sie eine Resolutionsmethode für
+  Klauseln kennenlernen - \ldots{} deren Algorithmisierbarkeit allerdings
+  an der ``kombinatorischen Explosion'' der Resolutionsmöglichkeiten
+  scheitert, aber \ldots{} - \ldots{} in der LV ``Inferenzmethoden''
+  können Sie noch ein paar ``Tricks'' kennenlernen, die ``Explosion''
+  einzudämmen
+
+  \subsubsection{Inferenz in PROLOG: Resolution nach
+    ROBINSON}\label{inferenz-in-prolog-resolution-nach-robinson}
+
+  gegeben: - Menge von Regeln und Fakten M $M=\{K_1,...,K_n\}$ - negierte
+  Hypothese $\lnot H$
+  $\lnot H\equiv \bigwedge_{i=1}^m H_i \equiv false \rightarrow \bigwedge_{i=1}^m H_i$
+
+  Ziel: Beweis, dass $M \Vdash H$.
+  $kt(\bigwedge_{i=1}^n K_i \wedge \lnot H)$
+
+  Eine der Klauseln habe die Form $A\leftarrow \bigwedge_{k=1}^p B_k$.
+  ($A,B_k$- Atomformeln)
+
+  Es gebe eine Substitution (Variablenersetzung) $\nu$ für die $A$ und
+  eines der $H_i$ (etwa $H_l$) vorkommenden Variablen, welche $A$ und
+  $H_l$ syntaktisch identisch macht.
+
+  $M\equiv \bigwedge_{i=1}^n K_i\wedge \lnot(\bigwedge_{i=1}^m H_i)$ ist
+  kontradiktorisch ($kt\ M‘$), gdw. $M‘$ nach Ersetzen von $H$ durch
+  $\bigwedge_{i=1}^{l-1}\nu(H_i)\wedge\bigwedge_{k=1}^p\nu(B_k)\wedge\bigwedge_{i=l+1}^m \nu(H_i)$
+  noch immer kontradiktorisch ist.
+
+  Jetzt wissen wir also, wie man die zu zeigende Kontradiktorizität auf
+  eine andere - viel kompliziertere Kontradiktorizität zurückführen kann.
+  Für $p=0$ und $m=1$ wird es allerdings trivial. Die sukzessive Anwendung
+  von Resolutionen muss diesen Trivialfall systematisch herbeiführen:
+
+  \begin{quote}
+    Satz von ROBINSON
+
+    $M'\equiv\bigwedge_{i=1}^n K_i\wedge \lnot H$ ist kontradiktorisch
+    ($kt\ M‘$), gdw. durch wiederholte Resolutionen in endlich vielen
+    Schritten die negierte Hypothese $\lnot H\equiv false \leftarrow H$
+    durch die leere Klausel $false\leftarrow true$ ersetzt werden kann.
+  \end{quote}
+
+  Substitution - Eine (Variablen-) Substitution $\nu$ einer Atomformel $A$
+  ist eine Abbildung der Menge der in $A$ vorkommenden Variablen $X$ in
+  die Menge der Terme (aller Art: Konstanten, Variablen, strukturierte
+  Terme). - Sie kann als Menge von Paaren $[Variable,Ersetzung]$ notiert
+  werden: $\nu=\{[x,t]: x\in X, t=\nu(x)\}$ - Für strukturierte Terme wird
+  die Substitution auf deren Komponenten angewandt:
+  $\nu(f(t_1,...,t_n)) = f(\nu(t_1),...,\nu(t_n))$ - Verkettungsoperator
+  $\circ$ für Substitutionen drückt Hintereinander-anwendung aus:
+  $\sigma\circ\nu(t)=\sigma(\nu(t))$ - Substitutionen, die zwei Terme
+  syntaktisch identisch machen, heißen Unifikator: $\nu$unifiziert zwei
+  Atomformeln (oder Terme) $s$ und $t$ (oder: heißt Unifikator von $s$ und
+  $t$), falls dessen Einsetzung $s$ und $t$ syntaktisch identisch macht.
+
+  \paragraph{Unifikation}\label{unifikation}
+
+  Zwei Atomformeln $p(t_{11},...,t_{1n})$ und $p_2(t_{21},...,t_{2n})$
+  sind unifizierbar, gdw. - sie die gleichen Prädikatensymbole aufweisen
+  ($p_1= p_2$), - sie die gleichen Stelligkeiten aufweisen ($n = m$) und -
+  die Terme $t_{1i}$ und $t_{2i}$ jeweils miteinander unifizierbar sind.
+
+  Die Unifizierbarkeit zweier Terme richtet sich nach deren Sorte: 1. Zwei
+  Konstanten $t_1$ und $t_2$ sind unifizierbar, gdw. $t_1= t_2$ 2. Zwei
+  strukturierte Terme $f(t_{11},...,t_{1n})$ und $f(t_{21},...,t_{2n})$
+  sind unifizierbar, gdw. - sie die gleichen Funktionssymbole aufweisen
+  ($f_1= f_2$), - sie die gleichen Stelligkeiten aufweisen ($n=m$) und -
+  die Terme $t_{1i}$ und $t_{2i}$ jeweils miteinander unifizierbar sind.
+  3. Eine Variable $t_1$ ist mit einer Konstanten oder einem
+  strukturierten Term $t_2$ unifizierbar. $t_1$ wird durch $t_2$ ersetzt
+  (instanziert): $t_1:= t_2$ 4. Zwei Variablen $t_1$ und $_2$ sind
+  unifizierbar und werden gleichgesetzt: $t_1:=t_2$ bzw. $t_2:= t_1$
+
+  Genügt ``irgendein'' Unifikator? - ein Beispiel -
+  $K_1: p(A,B)\leftarrow q(A)\wedge r(B)$ - $K_2:q(c)\leftarrow true$ -
+  $K_3:r(d)\leftarrow true$ - $\lnot H: false\leftarrow p(X,Y)$ -
+  Unifikatoren für $H_1^0$ und Kopf von $K_1$: -
+  $\nu^1 = \{[X,a],[Y,b],[A,a],[B,b]\}$ - $\nu^2 =\{[X,a],[Y,B],[A,a]\}$ -
+  $\nu^3 =\{[X,A],[Y,b],[B,b]\}$ - $\nu^4=\{[A,X],[B,Y]\}$ \ldots{} -
+  Obwohl $\{K_1, K_2, K_3\}\Vdash H$, gibt es bei Einsetzung von $\nu^1$,
+  $\nu^2$ und $\nu^3$ keine Folge von Resolutionsschritten, die zur leeren
+  Klausel führt. - Bei Einsetzung von $\nu^4$ hingegen gibt es eine solche
+  Folge. - Die Vollständigkeit des Inferenzverfahrens hängt von der Wahl
+  des ``richtigen'' Unifikators ab. - Dieser Unifikator muss möglichst
+  viele Variablen variabel belassen. Unnötige Spezialisierungen versperren
+  zukünftige Inferenzschritte. - Ein solcher Unifikator heißt
+  \textbf{allgemeinster Unifikator} bzw. ``most general unifier''
+  (m.g.u.).
+
+  \begin{quote}
+    Allgemeinster Unifikator
+
+    Eine Substitution heißt allgemeinster Unifikator (most general unfier;
+    m.g.u.) zweier (Atomformeln oder) Terme $s$ und $t$
+    ($\nu= m.g.u.(s,t)$), gdw. 1. die Substitution $\nu$ ein Unifikator von
+    $s$ und $t$ ist und 2. für jeden anderen Unifikator $\sigma$ von $s$ und
+    $t$ eine nichtleere und nicht identische Substitution $\tau$ existiert,
+    so dass $\sigma=\tau\circ\nu$ ist. graphisch betrachtet:
+    \includegraphics[width=\linewidth]{Assets/Logik_allgemeinster-unifikator.png}
+  \end{quote}
+
+  Der Algorithmus zur Berechnung des m.g.u. zweier Terme $s$ und $t$
+  verwendet Unterscheidungsterme: Man lese $s$ und $t$ zeichenweise
+  simultan von links nach rechts. Am ersten Zeichen, bei welchem sich $s$
+  und $t$ unterscheiden, beginnen die Unterscheidungsterme $s*$ und $t*$
+  und umfassen die dort beginnenden (vollständigen) Teilterme.
+
+  Algorithmus zur Bestimmung des allgemeinsten Unifikators 2er Terme
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          input: s, t
+    \item
+          output: Unifizierbarkeitsaussage, ggf. $\nu= m.g.u.( s , t )$
+    \item
+          $i:=0;\nu_i:=\varnothing;s_i:=s; t_i=t$
+    \item
+          Start: $s_i$ und $t_i$ identisch?
+    \item
+          ja $\Rightarrow$ s und t sind unifizierbar, $\nu=\nu_i=m.g.u.(s,t)$
+          (fertig)
+    \item
+          nein $\Rightarrow$ Bilde die Unterscheidungsterme $s_i^*$ und $t_i^*$
+
+          \begin{itemize}
+            \item
+                  $s_i^*$ oder $t_i^*$ Variable?
+            \item
+                  nein $\Rightarrow$ $s$ und $t$ sind nicht unifizierbar (fertig)
+            \item
+                  ja $\Rightarrow$ sei (o.B.d.A.) $s_i^*$ eine Variable
+
+                  \begin{itemize}
+                    \itemsep1pt\parskip0pt\parsep0pt
+                    \item
+                          $s_i^* \subseteq t_i^*$? (enthält $t_i^*$ die Variable $s_i^*$?)
+                    \item
+                          ja $\Rightarrow$ $s$ und $t$ sind nicht unifizierbar (fertig)
+                    \item
+                          nein $\Rightarrow$
+
+                          \begin{itemize}
+                            \itemsep1pt\parskip0pt\parsep0pt
+                            \item
+                                  $\nu':=\{[s,t']: [s,t]\in\nu, t':=t|_{s*\rightarrow t*}\}\cup \{[s_i^*, t_i^*]\}$
+                            \item
+                                  $s':=\nu'(s_i); t':=\nu'(t_i); i:=i+1;$
+                            \item
+                                  $\nu_i:=\nu'; s_i:=s'; t_i:=t';$
+                            \item
+                                  gehe zu Start
+                          \end{itemize}
+                  \end{itemize}
+          \end{itemize}
+  \end{itemize}
+
+  \subsection{Logische Programmierung}\label{logische-programmierung-1}
+
+  \subsubsection{Einordnung des logischen
+    Paradigmas}\label{einordnung-des-logischen-paradigmas}
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-logische-programmierung-einordnung.png}
+
+  ``deskriptives'' Programmierparadigma = 1. Problembeschreibung - Die
+  Aussagenmenge $M =\{K_1,...,K_n\}$, über denen gefolgert wird, wird in
+  Form von Fakten und Regeln im PK1 notiert. - Eine mutmaßliche Folgerung
+  (Hypothese) $H$ wird in Form einer Frage als negierte Hypothese
+  hinzugefügt. 2. (+) Programmverarbeitung - Auf der Suche eines Beweises
+  für $M \Vdash H$ werden durch mustergesteuerte Prozedur-Aufrufe
+  Resolutions-Schritte zusammengestellt. - Dem ``Programmierer'' werden
+  (begrenzte) Möglichkeiten gegeben, die systematische Suche zu
+  beeinflussen.
+
+  \subsubsection{Syntax}\label{syntax}
+
+  Syntax von Klauseln \textbar{} \textbar{} Syntax \textbar{} Beispiel
+  \textbar{} --- \textbar{} --- \textbar{} --- \textbar{} Fakt \textbar{}
+  praedikatensymbol(term,\ldots{}term). \textbar{}
+  liefert(xy\_ag,motor,vw). Regel \textbar{}
+  praedikatensymbol(term,\ldots{}term) :-
+  praedikatensymbol(term,\ldots{}term) ,\ldots{} ,
+  praedikatensymbol(term,\ldots{}term). \textbar{} konkurrenten(Fa1,Fa2)
+  :- liefert(Fa1,Produkt,\emph{),liefert(Fa2,Produkt,}). Frage \textbar{}
+  ?- praedikatensymbol(term,\ldots{}term) , \ldots{}
+  ,praedikatensymbol(term,\ldots{}term). \textbar{} ?-
+  konkurrenten(ibm,X), liefert(ibm,\_,X).
+
+  Syntax von Termen \textbar{} \textbar{} \textbar{} Syntax \textbar{}
+  Beispiele \textbar{} \textbar{} --- \textbar{} --- \textbar{} ---
+  \textbar{} --- \textbar{} Konstante \textbar{} Name \textbar{}
+  Zeichenfolge, beginnend mit Kleinbuchstaben, die Buchstaben, Ziffern und
+  \_ enthalten kann. \textbar{} otto\_1 , tisch, hund \textbar{}\textbar{}
+  beliebige Zeichenfolge in ``\ldots{}'' geschlossen \textbar{} ``Otto'',
+  ``r@ho'' \textbar{}\textbar{} Sonderzeichenfolge \textbar{}
+  \%\&§$€ | Zahl | Ziffernfolge, ggf. mit Vorzeichen, Dezimalpunkt und Exponentendarstellung | 3, -5, 1001, 3.14E-12 Variable | allg. | Zeichenfolge, mit Großbuchstaben oder \_ beginnend | X, Was, _alter | anonym | Unterstrich | \_ strukturierter Term | allg. | funktionssymbol( term , ... , term ) | nachbar(chef(X)) | Liste | leere Liste | [ ] | | ${[}term\textbar{}restliste{]}\$
+  \textbar{} $[mueller|[mayer|[]]]$ \textbar{}\textbar{}
+  $[term , term , ... , term ]$ \textbar{} $[ mueller, mayer, schulze ]$
+
+  BACKUS-NAUR-Form - PROLOG-Programm ::= Wissensbasis Hypothese -
+  Wissensbasis ::= Klausel \textbar{} Klausel Wissensbasis - Klausel ::=
+  Fakt \textbar{} Regel - Fakt ::= Atomformel. - Atomformel ::=
+  Prädikatensymbol (Termfolge) - Prädikatensymbol ::= Name - Name ::=
+  Kleinbuchstabe \textbar{}Kleinbuchstabe Restname \textbar{}
+  ``Zeichenfolge'' \textbar{} Sonderzeichenfolge - RestName ::=
+  Kleinbuchstabe \textbar{} Ziffer \textbar{} \_ \textbar{} Kleinbuchstabe
+  RestName \textbar{} Ziffer RestName \textbar{} \_ RestName - \ldots{}
+
+  \subsubsection{PROLOG aus logischer
+    Sicht}\label{prolog-aus-logischer-sicht}
+
+  Was muss der Programmierer tun? - Formulierung einer Menge von Fakten
+  und Regeln (kurz: Klauseln), d.h. einer Wissensbasis
+  $M\equiv\{K_1,...,K_n\}$ - Formulierung einer negierten Hypothese
+  (Frage, Ziel)
+  $\lnot H\equiv\bigwedge_{i=1}^m H_i \equiv false\leftarrow \bigwedge_{i=1}^m H_i$
+
+  Was darf der Programmierer erwarten? - Dass das ``Deduktionstool''
+  PROLOG $M \Vdash H$ zu zeigen versucht, d.h.
+  $kt(\bigwedge_{i=1}^n K_i\wedge \lnot H)$ ) - \ldots{}, indem
+  systematisch die Resolutionsmethode auf $\lnot H$ und eine der Klauseln
+  aus $M$ angewandt wird, solange bis $\lnot H\equiv false\leftarrow true$
+  entsteht
+
+  \paragraph{Formulierung von
+    Wissensbasen}\label{formulierung-von-wissensbasen}
+
+  \begin{enumerate}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Beispiel 1 (BSP1.PRO)
+  \end{enumerate}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Yoshihito und Sadako sind die Eltern von Hirohito.
+    \item
+          vater\_von(yoshihito,hirohito).
+    \item
+          mutter\_von(sadako,hirohito).
+    \item
+          Kuniyoshi und Chikako sind die Eltern von Nagako.
+    \item
+          vater\_von(kunioshi,nagako).
+    \item
+          mutter\_von(chikako,nagako).
+    \item
+          Akihito`s und Hitachi`s Eltern sind Hirohito und Nagako.
+    \item
+          vater\_von(hirohito,akihito).
+    \item
+          vater\_von(hirohito,hitachi).
+    \item
+          mutter\_von(nagako,akihito).
+    \item
+          mutter\_von(nagako,hitachi).
+    \item
+          Der Großvater ist der Vater des Vaters oder der Vater der Mutter.
+    \item
+          grossvater\_von(G,E) :- vater\_von(G,V), vater\_von(V,E).
+    \item
+          grossvater\_von(G,E) :-vater\_von(G,M),mutter\_von(M,E).
+    \item
+          Geschwister haben den gleichen Vater und die gleiche Mutter.
+    \item
+          geschwister(X,Y) :- vater\_von(V,X), vater\_von(V,Y),
+          mutter\_von(M,X), mutter\_von(M,Y).
+  \end{itemize}
+
+  Visual Prolog benötigt aber einen Deklarationsteil für Datentypen und
+  Aritäten der Prädikate, der für die o.g. Wissensbasis so aussieht:
+
+  \begin{verbatim}
+domains
+  person = symbol
+predicates
+  vater_von(person,person)        % bei nichtdeterministischen Prädikaten
+  mutter_von(person,person).      % kann man "nondeterm" vor das
+  grossvater_von(person,person)   % Prädikat schreiben, um die
+  geschwister(person,person)      % Kompilation effizienter zu machen
+clauses
+  < Wissensbasis einfügen und Klauseln gleichen Kopfprädikates gruppieren>
+goal
+  < Frage ohne "?-" einfügen>
+\end{verbatim}
+
+  \begin{enumerate}
+    \setcounter{enumi}{1}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Beispiel 2 (BSP2.PRO)
+  \end{enumerate}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Kollegen Meier und Müller arbeiten im Raum 1, Kollege Otto im Raum 2
+          und Kollege Kraus im Raum 3.
+    \item
+          arbeitet\_in(meier,raum\_1).
+    \item
+          arbeitet\_in(mueller,raum\_1).
+    \item
+          arbeitet\_in(otto,raum\_2).
+    \item
+          arbeitet\_in(kraus,raum\_3).
+    \item
+          Netzanschlüsse gibt es in den Räumen 2 und 3.
+    \item
+          anschluss\_in(raum\_2).
+    \item
+          anschluss\_in(raum\_3).
+    \item
+          Ein Kollege ist erreichbar, wenn er in einem Raum mit Netzanschluss
+          arbeitet.
+    \item
+          erreichbar(K) :- arbeitet\_in(K,R),anschluss\_in(R).
+    \item
+          2 Kollegen können Daten austauschen, wenn sie im gleichen Raum
+          arbeiten oder beide erreichbar sind.
+    \item
+          koennen\_daten\_austauschen(K1,K2) :-
+          arbeitet\_in(K1,R),arbeitet\_in(K2,R).
+    \item
+          koennen\_daten\_austauschen(K1,K2) :- erreichbar(K1),erreichbar(K2).
+  \end{itemize}
+
+  Deklarationsteil
+
+  \begin{verbatim}
+domains
+  person, raum: symbol
+predicates
+  arbeitet_in(person,raum)
+  ansluss_in(raum)
+  erreichbar(person)
+  koennen_daten_austauschen(person,person)
+clauses
+  ...
+goal
+  ...
+\end{verbatim}
+
+  \paragraph{Verarbeitung Logischer
+    Programme}\label{verarbeitung-logischer-programme}
+
+  \subparagraph{Veranschaulichung ohne
+    Unifikation}\label{veranschaulichung-ohne-unifikation}
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-prolog-ohne-unifikation.png}
+
+  Tiefensuche mit Backtrack: Es werden anwendbare Klauseln für das erste
+  Teilziel gesucht. Gibt es \ldots{} - \ldots{} genau eine, so wird das 1.
+  Teilziel durch deren Körper ersetzt. - \ldots{} mehrere, so wird das
+  aktuelle Ziel inklusive alternativ anwendbarer Klauseln im
+  Backtrack-Keller abgelegt und die am weitesten oben stehende Klausel
+  angewandt. - \ldots{} keine (mehr), so wird mit dem auf dem
+  Backtrack-Keller liegendem Ziel die Bearbeitung fortgesetzt.
+
+  Dies geschieht solange, bis - das aktuelle Ziel leer ist oder - keine
+  Klausel (mehr) anwendbar ist und der Backtrack-Keller leer ist.
+
+  \subparagraph{Veranschaulichung mit
+    Unifikation}\label{veranschaulichung-mit-unifikation}
+
+
+  \includegraphics[width=\linewidth]{Assets/Logik-prolog-mit-unifikation.png}
+
+  Zusätzliche Markierung der Kanten mit der Variablenersetzung (dem
+  Unifikator).
+
+  \subsubsection{PROLOG aus prozeduraler
+    Sicht}\label{prolog-aus-prozeduraler-sicht}
+
+  Beispiel: die ``Hackordnung'' 1. chef\_von(mueller,mayer). 2.
+  chef\_von(mayer,otto). 3. chef\_von(otto,walter). 4.
+  chef\_von(walter,schulze). 5. weisungsrecht(X,Y) :- chef\_von(X,Y). 6.
+  weisungsrecht(X,Y) :- chef\_von(X,Z), weisungsrecht(Z,Y).
+
+  \begin{quote}
+    Deklarative Interpretation In einem Objektbereich
+    $I=\{mueller, mayer, schulze, ...\}$ bildet das Prädikat
+    $weisungsrecht(X,Y)$ $[X,Y]$ auf wahr ab, gdw. - das Prädikat
+    $chef_von(X,Y)$ das Paar $[X,Y]$ auf wahr abbildet oder - es ein
+    $Z\in I$ gibt, so dass - das Prädikat $chef_von(X,Z)$ das Paar $[X,Z]$
+    auf wahr abbildet und - das Prädikat $weisungsrecht(Z,Y)$ das Paar
+    $[Z,Y]$ auf wahr abbildet.
+  \end{quote}
+
+  \begin{quote}
+    Prozedurale Interpretation Die Prozedur $weisungsrecht(X,Y)$ wird
+    abgearbeitet, indem 1. die Unterprozedur $chef_von(X,Y)$ abgearbeitet
+    wird. Im Erfolgsfall ist die Abarbeitung beendet; anderenfalls werden 2.
+    die Unterprozeduren $chef_von(X,Z)$ und $weisungsrecht(Z,Y)$
+    abgearbeitet; indem systematisch Prozedurvarianten beider
+    Unterprozeduren aufgerufen werden.Dies geschieht bis zum Erfolgsfall
+    oder erfolgloser erschöpfender Suche.
+  \end{quote}
+
+  \begin{tabular}{ll}
+    deklarative Interpretation         & prozedurale Interpretation \\\hline
+    Prädikat                           & Prozedur                   \\
+    Ziel                               & Prozeduraufruf             \\
+    Teilziel                           & Unterprozedur              \\
+    Klauseln mit gleichem Kopfprädikat & Prozedur-varianten         \\
+    Klauselkopf                        & Prozedurkopf               \\
+    Klauselkörper                      & Prozedurrumpf              \\
+  \end{tabular}
+
+  Die Gratwanderung zwischen Wünschenswertem und technisch Machbarem
+  erfordert mitunter ``Prozedurales Mitdenken'', um 1. eine gewünschte
+  Reihenfolge konstruktiver Lösungen zu erzwingen, 2. nicht terminierende
+  (aber - deklarativ, d.h. logisch interpretiert - völlig korrekte)
+  Programme zu vermeiden, 3. seiteneffektbehaftete Prädikate sinnvoll
+  einzusetzen, 4. (laufzeit-) effizienter zu programmieren und 5. das
+  Suchverfahren gezielt zu manipulieren.
+
+  Programm inkl. Deklarationsteil (BSP3.PRO)
+
+  \begin{verbatim}
+domains
+  person = symbol
+predicates
+  chef_von(person,person)
+  weisungsrecht(person,person)
+clauses
+  chef_von(mueller,mayer).
+  chef_von(mayer,otto).
+  chef_von(otto,walter).
+  chef_von(walter,schulze).
+  weisungsrecht(X,Y) :- chef_von(X,Y).
+  weisungsrecht(X,Y) :- chef_von(X,Z), weisungsrecht(Z,Y).
+goal
+  weisungsrecht(Wer,Wem).
+\end{verbatim}
+
+  Prädikate zur Steuerung der Suche nach einer Folge von
+  Resolutionsschritten!
+
+  !(cut) Das Prädikat $!/0$ ist stets wahr. In Klauselkörpern eingefügt
+  verhindert es ein Backtrack der hinter $!/0$ stehenden Teilziele zu den
+  vor $!/0$ stehenden Teilzielen sowie zu alternativen Klauseln des
+  gleichen Kopfprädikats. Die Verarbeitung von $!/0$ schneidet demnach
+  alle vor der Verarbeitung verbliebenen Lösungswege betreffenden Prozedur
+  ab.
+
+  Prädikate zur Steuerung der Suche: $!/0$
+  \includegraphics[width=\linewidth]{Assets/Logik-prädikate-suche.png}
+
+  Prädikate zur Steuerung der Suche nach einer Folge von
+  Resolutionsschritten \textbf{fail} Das Prädikat $fail/0$ ist stets
+  falsch. In Klauselkörpern eingefügt löst es ein Backtrack aus bzw. führt
+  zum Misserfolg, falls der Backtrack-Keller leer ist, d.h. falls es keine
+  verbleibenden Lösungswege (mehr) gibt.
+  \includegraphics[width=\linewidth]{Assets/Logik-prädikate-suche-fail.png}
+  \includegraphics[width=\linewidth]{Assets/Logik-prädikate-suche-fail-2.png}
+
+  \subsubsection{Listen und rekursive
+    Problemlösungsstrategien}\label{listen-und-rekursive-problemluxf6sungsstrategien}
+
+  Listen 1. $[]$ ist eine Liste. 2. Wenn $T$ ein Term und $L$ eine Liste
+  ist, dann ist 1. $[T|L]$ eine Liste. 2. $T.L$ eine Liste.
+  (ungebräuchlich) 3. $.(T,L)$ eine Liste. (ungebräuchlich) Das erste
+  Element $T$ heißt Listenkopf, $L$ heißt Listenkörper oder Restliste. 3.
+  Wenn $t_1, ... ,t_n$ Terme sind, so ist $[t_1,...,t_n]$ eine Liste. 4.
+  Weitere Notationsformen von Listen gibt es nicht.
+
+  Listen als kompakte Wissensrepräsentation: ein bekanntes Beispiel
+  (BSP5.PRO) - arbeiten\_in({[}meier, mueller{]}, raum\_1). -
+  arbeitet\_in(meier, raum\_1). - arbeitet\_in(mueller, raum\_1). -
+  arbeitet\_in(otto, raum\_2). - arbeiten\_in({[}otto{]}, raum\_2 ). -
+  arbeitet\_in(kraus, raum\_3). - arbeiten\_in({[}kraus{]}, raum\_3 ). -
+  anschluesse\_in({[}raum\_2, raum\_3{]}). - anschluss\_in(raum\_2). -
+  anschluss\_in(raum\_3).
+
+  \textbf{Rekursion} in der Logischen Programmierung Eine Prozedur heißt
+  (direkt) rekursiv, wenn in mindestens einem der Klauselkörper ihrer
+  Klauseln ein erneuter Aufruf des Kopfprädikates erfolgt. Ist der
+  Selbstaufruf die letzte Atomformel des Klauselkörpers der letzten
+  Klausel dieser Prozedur - bzw. wird er es durch vorheriges
+  ``Abschneiden'' nachfolgender Klauseln mit dem Prädikat $!/0$ - , so
+  spricht man von Rechtsrekursion ; anderenfalls von Linksrekursion. Eine
+  Prozedur heißt indirekt rekursiv, wenn bei der Abarbeitung ihres
+  Aufrufes ein erneuter Aufruf derselben Prozedur erfolgt.
+
+  Wissensverarbeitung mit Listen: (BSP5.PRO) - erreichbar(K) :-
+  arbeitet\_in(K,R), anschluss\_in(R). - erreichbar(K) :-
+  anschluesse\_in(Rs), member(R, Rs), arbeiten\_in(Ks, R), member(K, Ks).
+  - koennen\_daten\_austauschen(K1,K2) :-
+  arbeitet\_in(K1,R),arbeitet\_in(K2,R). -
+  koennen\_daten\_austauschen(K1,K2) :-
+  arbeiten\_in(Ks,\emph{),member(K1,Ks), member(K2,Ks). -
+  koennen}daten\_austauschen(K1,K2) :- erreichbar(K1),erreichbar(K2). -
+  koennen\_daten\_austauschen(K1,K2) :- erreichbar(K1),erreichbar(K2).
+
+  BSP5.PRO
+
+  \begin{verbatim}
+domains
+  person, raum = symbol
+  raeume = raum*
+  personen = person*
+predicates
+  arbeiten_in(personen, raum)
+  anschluesse_in(raeume)
+  erreichbar(person)
+  koennen_daten_austauschen(person,person)
+  member(person,personen)
+  member(raum,raeume)
+clauses
+  ... (siehe oben)
+  member(E,[E|_]).
+  member(E,[_|R]) :- member(E,R).
+goal
+  erreichbar(Wer).
+\end{verbatim}
+
+  \textbf{Unifikation 2er Listen} 1. Zwei leere Listen sind (als
+  identische Konstanten aufzufassen und daher) miteinander unifizierbar.
+  1. Zwei nichtleere Listen $[K_1|R_1]$ und $[K_2|R_2]$ sind miteinander
+  unifizierbar, wenn ihre Köpfe ($K_1$ und $K_2$) und ihre Restlisten
+  ($R_1$ und $R_2$) jeweils miteinander unifizierbar sind. 3. Eine Liste
+  $L$ und eine Variable $X$ sind miteinander unifizierbar, wenn die
+  Variable selbst nicht in der Liste enthalten ist. Die Variable $X$ wird
+  bei erfolgreicher Unifikation mit der Liste $L$ instanziert: $X:=L$.
+
+  \textbf{Differenzlisten:} eine intuitive Erklärung Eine Differenzliste
+  $L_1 - L_2$ besteht aus zwei Listen $L_1$ und $L_2$ und wird im
+  allgemeinen als $[L_1,L_2]$ oder (bei vorheriger Definition eines pre-
+  bzw. infix notierten Funktionssymbols -/2) als $-(L_1,L_2)$ bzw.
+  $L_1-L_2$ notiert.
+
+  Sie wird (vom Programmierer, nicht vom PROLOG-System!) als eine Liste
+  interpretiert, deren Elemente sich aus denen von $L_1$ abzüglich derer
+  von $L_2$ ergeben. Differenzlisten verwendet man typischerweise, wenn
+  häufig Operationen am Ende von Listen vorzunehmen sind.
+
+  Eine Definition 1. Die Differenz aus einer leeren Liste und einer
+  (beliebigen) Liste ist die leere Liste: $[] - L = []$ 2. Die Differenz
+  aus einer Liste $[E|R]$ und der Liste $L$, welche $E$ enthält, ist die
+  Liste $D$, wenn die Differenz aus $R$ und $L$ (abzügl. $E$) die Liste
+  $D$ ist: $[E|R]-L = D$, wenn $E\in L$ und $R-(L-[E]) = D$ 3. Die
+  Differenz aus einer Liste $[E|R]$ und einer Liste $L$, welche $E$ nicht
+  enthält, ist die Liste $[E|D]$, wenn die Differenz aus $R$ und $L$ die
+  Liste $D$ ist: $[E|R] - L = [E|D]$, wenn $E\in L$ und $R-L=D$
+
+  Differenzlisten: Ein Interpreter
+  $interpret(Differenzliste,Interpretation)$ (BSP6.PRO) 1.
+  $interpret([[],_],[]).$ 2.
+  $interpret([[E|R],L] , D ) :- loesche(E , L, L1),! , interpret( [ R , L1 ] , D ).$
+  3. $interpret([[E|R],L] , [E|D] ) :- interpret([R,L],D).$ 4.
+  $loesche(E, [E|R], R) :-!.$ 5.
+  $loesche(E, [K|R], [K|L]) :- loesche(E,R,L).$
+
+  \subsubsection{Prolog-Fallen}\label{prolog-fallen}
+
+  \paragraph{Nicht terminierende
+    Programme}\label{nicht-terminierende-programme}
+
+  Ursache: ``ungeschickt'' formulierte (direkte oder indirekte) Rekursion
+
+  \subparagraph{Alternierende
+    Zielklauseln}\label{alternierende-zielklauseln}
+
+  Ein aktuelles Ziel wiederholt sich und die Suche nach einer Folge von
+  Resolutionsschritten endet nie: 1. $liegt_auf(X,Y) :- liegt_unter(Y,X).$
+  2. $liegt_unter(X,Y) :- liegt_auf(Y,X).$ -
+  $?- liegt_auf( skript, pult ).$ - logisch korrekte Antwort: nein -
+  tatsächliche Antwort: keine Logische Programmierung
+
+  oder die Suche nach Resolutionsschritten endet mit einem Überlauf des
+  Backtrack-Kellers: (BSP8.PRO) 1. $liegt_auf(X,Y) :- liegt_unter(Y,X).$
+  2. $liegt_auf( skript , pult ).$ 3.
+  $liegt_unter(X,Y) :- liegt_auf(Y,X).$ - $?- liegt_auf( skript , pult ).$
+  - logisch korrekte Antwort: ja - tatsächliche Antwort: keine
+
+  \subparagraph{Expandierende
+    Zielklauseln}\label{expandierende-zielklauseln}
+
+  Das erste Teilziel wird in jeden Resolutionsschritt durch mehrere neue
+  Teilziele ersetzt; die Suche endet mit einem Speicherüberlauf:
+  (BSP9.PRO) 1. $liegt_auf( notebook , pult ).$ 2.
+  $liegt_auf( skript , notebook ).$ 3.
+  $liegt_auf(X,Y) :- liegt_auf(X,Z), liegt_auf(Z,Y).$ -
+  $?- liegt_auf( handy , skript ).$ - logisch korrekte Antwort: nein -
+  tatsächliche Antwort: keine
+
+  Auch dieses Beispiel lässt sich so erweitern, dass die Hypothese
+  offensichtlich aus der Wissensbasis folgt, die Umsetzung der
+  Resolutionsmethode aber die Vollständigkeit zerstört: 1.
+  $liegt_auf( notebook , pult ).$ 2. $liegt_auf( skript , notebook ).$ 3.
+  $liegt_auf(X,Y) :- liegt_auf(X,Z), liegt_auf(Z,Y).$ 4.
+  $liegt_auf( handy , skript ).$ - $?- liegt_auf( handy , pult ).$ -
+  logisch korrekte Antwort: ja - tatsächliche Antwort: keine
+
+  Auch dieses Beispiel zeigt, dass das Suchverfahren ``Tiefensuche mit
+  Backtrack'' die Vollständigkeit des Inferenzverfahrens zerstört.
+
+  \paragraph{Metalogische Prädikate und konstruktive
+    Lösungen}\label{metalogische-pruxe4dikate-und-konstruktive-luxf6sungen}
+
+  Das Prädikat $not/1$ hat eine Aussage als Argument und ist somit eine
+  Aussage über eine Aussage, also metalogisch. I.allg. ist $not/1$
+  vordefiniert, kann aber mit Hilfe von $call/1$ definiert werden.
+  $call/1$ hat Erfolg, wenn sein Argument - als Ziel interpretiert -
+  Erfolg hat.
+
+  Beispiel (BSP10.PRO): 1. $fleissig(horst).$ 2. $fleissig(martin).$ 3.
+  $faul(X) :- not( fleissig(X) ).$ 4. $not(X) :- call(X), !, fail.$ 5.
+  $not( _ ).$ - $?- faul(horst).$ Antwort: nein - $?- faul(alex).$
+  Antwort: ja - $?- faul(Wer).$ Antwort: nein
+
+  Widerspruch \ldots{} und Beweis der Unvollständigkeit durch Metalogik
+
+  \subsubsection{Typische Problemklassen für die Anwendung der Logischen
+    Programmierung}\label{typische-problemklassen-fuxfcr-die-anwendung-der-logischen-programmierung}
+
+  \paragraph{Rekursive
+    Problemlösungsstrategien}\label{rekursive-problemluxf6sungsstrategien}
+
+  \begin{quote}
+    Botschaft 1 Man muss ein Problem nicht in allen Ebenen überblicken, um
+    eine Lösungsverfahren zu programmieren. Es genügt die Einsicht, 1. wie
+    man aus der Lösung eines einfacheren Problems die Lösung des präsenten
+    Problems macht und 2. wie es im Trivialfall zu lösen ist.
+  \end{quote}
+
+  \textbf{Türme von Hanoi} Es sind N Scheiben von der linken Säule auf die
+  mittlere Säule zu transportieren, wobei die rechte Säule als
+  Zwischenablage genutzt wird. Regeln: 1. Es darf jeweils nur eine Scheibe
+  transportiert werden. 2. Die Scheiben müssen mit fallendem Durchmesser
+  übereinander abgelegt werden.
+
+  (doppelt) rekursive Lösungsstrategie: - $N = 0:$ Das Problem ist gelöst.
+  - $N > 0:$ 1. Man löse das Problem für N-Scheiben, die von der
+  Start-Säule zur Hilfs-Säule zu transportieren sind. 2. Man lege eine
+  Scheibe von der Start- zur Ziel-Säule. 3. Man löse das Problem für
+  N-Scheiben, die von der Hilfs-Säule zur Ziel-Säule zu transportieren
+  sind.
+
+  Prädikate - $hanoi(N)$ löst das Problem für N Scheiben -
+  $verlege(N,Start,Ziel,Hilf)$ verlegt N Scheiben von Start nach Ziel
+  unter Nutzung von Hilf als Ablage
+
+  Die Regeln zur Kodierung der Strategie (BSP11.PRO) -
+  $hanoi( N ) :- verlege( N , s1 , s2 , s3 ).$ -
+  $verlege( 0 , \_ , \_ , \_ ).$ - $verlege( N , S , Z , H ) :-$ -
+  $N1 = N - 1,$ - $verlege( N1 , S , H , Z ),$ -
+  $write("Scheibe von ", S," nach ", Z),$ - $verlege( N1 , H , Z , S ).$
+
+  \paragraph{Sprachverarbeitung mit
+    PROLOG}\label{sprachverarbeitung-mit-prolog}
+
+  \begin{quote}
+    Botschaft 2 Wann immer man Objekte mit Mustern vergleicht, z.B. 1. eine
+    Struktur durch ``Auflegen von Schablonen'' identifiziert, 2.
+    Gemeinsamkeiten mehrerer Objekte identifiziert, d.h. ``eine Schablone
+    entwirft'' oder 3. ``gemeinsame Beispiele für mehrere Schablonen''
+    sucht, mache man sich den Unifikations-Mechanismus zu nutzen.
+  \end{quote}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Eine kontextfreie Grammatik (Chomsky-Typ 2) besteht aus
+  \end{itemize}
+
+  \begin{enumerate}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          einem Alphabet A, welches die terminalen (satzbildenden) Symbole
+          enthält
+    \item
+          einer Menge nichtterminaler (satzbeschreibender) Symbole N (=
+          Vokabular abzüglich des Alphabets: $N = V \backslash A$)
+    \item
+          einer Menge von Ableitungsregeln $R\subseteq N\times (N\cup A)^*$
+    \item
+          dem Satzsymbol $S\in N$
+  \end{enumerate}
+
+  \begin{itemize}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          \ldots{} und in PROLOG repräsentiert werden durch
+  \end{itemize}
+
+  \begin{enumerate}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          1-elementige Listen, welche zu satzbildenden Listen komponiert werden:
+          $[der],[tisch],[liegt],...$
+    \item
+          Namen, d.h. mit kleinem Buchstaben beginnende Zeichenfolgen:
+          $nebensatz,subjekt,attribut,...$
+    \item
+          PROLOG-Regeln mit $l\in N$ im Kopf und $r\in(N\cup A)^*$ im Körper
+    \item
+          einen reservierten Namen: $satz$
+  \end{enumerate}
+
+  Ein Ableitungsbaum beschreibt die grammatische Struktur eines Satzes.
+  Seine Wurzel ist das Satzsymbol, seine Blätter in Hauptreihenfolge
+  bilden den Satz.
+  \includegraphics[width=\linewidth]{Assets/Logik-ableitungsbaum-beispiel.png}
+
+  \begin{enumerate}
+    \itemsep1pt\parskip0pt\parsep0pt
+    \item
+          Alphabet
+          $ministerium, rektorat, problem, das, loest, ignoriert, verschaerft$
+    \item
+          nichtterminale Symbole
+          $satz, subjekt, substantiv, artikel, praedikat,objekt$
+    \item
+          Ableitungsregeln (in BACKUS-NAUR-Form)
+
+          \begin{itemize}
+            \item
+                  $satz ::= subjekt praedikat objekt$
+            \item
+                  $subjekt ::= artikel substantiv$
+            \item
+                  $objekt ::=  artikel substantiv$
+            \item
+                  $substantiv ::= ministerium | rektorat | problem$
+            \item
+                  $artikel ::= das$
+            \item
+                  $praedikat ::= loest | ignoriert | verschaerft$
+          \end{itemize}
+    \item
+          Satzsymbol $satz$
+  \end{enumerate}
+
+  Verketten einer Liste von Listen
+
+  \begin{verbatim}
+% die Liste ist leer
+verkette( [ ] , [ ] ) .
+
+% das erste Element ist eine leere Liste
+verkette( [ [ ] | Rest ] , L ) :-
+  verkette( Rest , L ).
+
+% das erste Element ist eine nichtleere Liste
+verkette([ [K | R ] | Rest ] , [ K | L ] ) :-
+  verkette( [ R | Rest ] , L ).4
+\end{verbatim}
+
+  \paragraph{Die ``Generate - and - Test''
+    Strategie}\label{die-generate---and---test-strategie}
+
+  \begin{quote}
+    Botschaft 3 Es ist mitunter leichter (oder überhaupt erst möglich), für
+    komplexe Probleme 1. eine potentielle Lösung zu ``erraten'' und dazu 2.
+    ein Verfahren zu entwickeln, welches diese Lösung auf Korrektheit
+    testet, als zielgerichtet die korrekte Lösung zu entwerfen. Hierbei kann
+    man den Backtrack-Mechanismus nutzen.
+  \end{quote}
+
+  Strategie: Ein Prädikat $moegliche_loesung(L)$ generiert eine
+  potentielle Lösung, welche von einem Prädikat $korrekte_loesung(L)$
+  geprüft wird: - Besteht $L$ diesen Korrektheitstest, ist eine Lösung
+  gefunden. - Fällt $L$ bei diesem Korrektheitstest durch, wird mit
+  Backtrack das Prädikat $moegliche_loesung(L)$ um eine alternative
+  potentielle Lösung ersucht. (vgl.: Lösen NP-vollständiger Probleme,
+  Entscheidung von Erfüllbarkeit)
+
+  ein Beispiel: konfliktfreie Anordnung von $N$ Damen auf einem
+  $N\times N$ Schachbrett (BSP13.PRO) - eine Variante: Liste
+  strukturierter Terme - $[dame(Zeile,Spalte),...,dame(Zeile,Spalte)]$ -
+  $[dame(1,2), dame(2,4), dame(3,1), dame(4,3) ]$ - noch eine Variante:
+  Liste von Listen - $[[Zeile, Spalte] , ... , [Zeile,Spalte] ]$ -
+  $[ [1,2] , [2,4] , [3,1] , [4,3] ]$ - \ldots{} und noch eine (in die
+  Wissensdarstellung etwas ``natürliche'' Intelligenz investierende, den
+  Problemraum enorm einschränkende) Variante: Liste der Spaltenindizes -
+  $[ Spalte_zu_Zeile_1, ..., Spalte_zu_Zeile_N ]$ - $[ 2, 4, 1, 3 ]$
+
+  \paragraph{Heuristische
+    Problemlösungsmethoden}\label{heuristische-problemluxf6sungsmethoden}
+
+  \begin{quote}
+    Botschaft 4 Heuristiken sind 1. eine Chance, auch solche Probleme einer
+    Lösung zuzuführen, für die man keinen (determinierten)
+    Lösungsalgorithmus kennt und 2. das klassische Einsatzgebiet zahlreicher
+    KI-Tools - auch der Logischen Programmierung.
+  \end{quote}
+
+  Was ist eine Heuristik? Worin unterscheidet sich eine heuristische
+  Problemlösungsmethode von einem Lösungsalgorithmus?
+
+  Heuristiken bewerten die Erfolgsaussichten alternativer
+  Problemlösungsschritte. Eine solche Bewertung kann sich z.B. ausdrücken
+  in - einer quantitativen Abschätzung der ``Entfernung'' zum gewünschten
+  Ziel oder der ``Kosten'' für das Erreichen des Ziels, - einer
+  quantitativen Abschätzung des Nutzens und/oder der Kosten der
+  alternativen nächsten Schritte, - eine Vorschrift zur Rangordnung der
+  Anwendung alternativer Schritte, z.B. durch Prioritäten oder gemäß einer
+  sequenziell abzuarbeitenden Checkliste.
+
+  Ein Beispiel: Das Milchgeschäft meiner Großeltern in den 40er Jahren -
+  Der Milchhof liefert Milch in großen Kannen. - Kunden können Milch nur
+  in kleinen Mengen kaufen. - Es gibt nur 2 Sorten geeichter Schöpfgefäße;
+  sie fassen 0.75 Liter bzw. 1.25 Liter. - Eine Kundin wünscht einen Liter
+  Milch. 1. Wenn das große Gefäß leer ist, dann fülle es. 2. Wenn das
+  kleine Gefäß voll ist, dann leere es. 3. Wenn beides nicht zutrifft,
+  dann schütte so viel wie möglich vom großen in das kleine Gefäß.
+
+  Prädikat $miss_ab(VolGr, VolKl, Ziel, InhGr, InhKl)$ mit - VolGr -
+  Volumen des großen Gefäßes - VolKl - Volumen des kleinen Gefäßes - Ziel
+  - die abzumessende (Ziel-) Menge - InhGr - der aktuelle Inhalt im großen
+  Gefäß - InhKl - der aktuelle Inhalt im kleinen Gefäß
+
+  Beispiel-Problem: $?- miss_ab( 1.25 , 0.75 , 1 , 0 , 0 )4$
+
+  \paragraph{Pfadsuche in gerichteten
+    Graphen}\label{pfadsuche-in-gerichteten-graphen}
+
+  \begin{quote}
+    Botschaft 5 1. Für die systematische Suche eines Pfades kann der
+    Suchprozess einer Folge von Resolutionsschritten genutzt werden. Man
+    muss den Suchprozess nicht selbst programmieren. 2. Für eine
+    heuristische Suche eines Pfades gilt Botschaft 4: Sie ist das klassische
+    Einsatzgebiet zahlreicher KI-Tools - auch der Logischen Programmierung.
+  \end{quote}
+
+  Anwendungen - Handlungsplanung, z.B. - Suche einer Folge von
+  Bearbeitungsschritten für ein Produkt, eine Dienstleistung, einen
+  ``Bürokratischen Vorgang'' - Suche eines optimalen Transportweges in
+  einem Netzwerk von Straßen-, Bahn-, Flugverbindungen - Programmsynthese
+  = Handlungsplanung mit \ldots{} - \ldots{} Schnittstellen für die
+  Datenübergabe zwischen ``Handlungsschritten'' (= Prozeduraufrufen) und -
+  \ldots{} einem hierarchischen Prozedurkonzept, welches die
+  Konfigurierung von ``Programmbausteinen'' auf mehreren Hierarchie-Ebenen
+
+  Ein Beispiel: Suche einer zeitoptimalen Flugverbindung (BSP15.PRO) -
+  Repräsentation als Faktenbasis $verbindung(Start,Zeit1,Ziel,Zeit2,Tag).$
+  - Start - Ort des Starts - Zeit1 - Zeit des Starts - Ziel - Ort der
+  Landung - Zeit2 - Zeit der Landung - Tag - 0, falls Zeit1 und Zeit2 am
+  gleichen Tag und 1 ansonsten - möglich: -
+  $verbindung(fra,z(11,45),ptb,z(21,0),0).$ -
+  $verbindung(fra,z(11,15),atl,z(21,25),0).$ -
+  $verbindung(ptb,z(24,0),orl,z(2,14),1).$ -
+  $verbindung(atl,z(23,30),orl,z(0,54),1).$
+
+  In einer dynamischen Wissensbasis wird die bislang günstigste Verbindung
+  in Form eines Faktes
+  $guenstigste([ v(Von,Zeit1,Nach,Zeit2,Tag), ... ], Ankunftszeit, Tag ).$
+  festgehalten und mit den eingebauten Prädikaten $assert(<Fakt>)$ - zum
+  Einfügen des Faktes - und $retract (<Fakt>)$ - zum Entfernen des Faktes
+  - bei Bedarf aktualisiert. Zum Beispiel
+  $guenstigste([v(fra,z(11,45),ptb,z(21,00),0),v(ptb,z(24,0),orl,z(2,14),1)],z(2,14),1).$
+  erklärt den Weg über Pittsburgh zum bislang günstigsten gefundenen Weg.
+
+  \paragraph{``Logeleien'' als
+    Prolog-Wissensbasen}\label{logeleien-als-prolog-wissensbasen}
+
+  \begin{quote}
+    Botschaft 6 1. ``Logeleien'' sind oft Aussagen über Belegungen von
+    Variablen mit endlichem Wertebereich, ergänzt um eine Frage zu einem
+    nicht explizit gegebenen Wert. 2. Dabei handelt es sich um Grunde um
+    eine Deduktionsaufgabe mit einer Hypothese zu einem mutmaßlichen Wert
+    der gesuchten Variablen. Deshalb ist es oft auch mit dem
+    ``Deduktionstool'' Prolog lösbar, denn Prolog tut im Grunde nichts
+    anderes als ein ziel-gerichtetes ``Durchprobieren'' legitimer
+    Deduktionsschritte im ``Generate - and - Test'' - Verfahren.
+  \end{quote}
+
+  Beispiel das ``Zebra-Rätsel''(BSP16.PRO): 1. Es gibt fünf Häuser. 2. Der
+  Engländer wohnt im roten Haus. 3. Der Spanier hat einen Hund. 4. Kaffee
+  wird im grünen Haus getrunken. 5. Der Ukrainer trinkt Tee. 6. Das grüne
+  Haus ist (vom Betrachter aus gesehen) direkt rechts vom weißen Haus. 7.
+  Der Raucher von Atem-Gold-Zigaretten hält Schnecken als Haustiere. 8.
+  Die Zigaretten der Marke Kools werden im gelben Haus geraucht. 9. Milch
+  wird im mittleren Haus getrunken. 10. Der Norweger wohnt im ersten Haus.
+  11. Der Mann, der Chesterfields raucht, wohnt neben dem Mann mit dem
+  Fuchs. 12. Die Marke Kools wird geraucht im Haus neben dem Haus mit dem
+  Pferd. 13. Der Lucky-Strike-Raucher trinkt am liebsten Orangensaft. 14.
+  Der Japaner raucht Zigaretten der Marke Parliament. 15. Der Norweger
+  wohnt neben dem blauen Haus.
+
+  Jedes Haus ist in einer anderen Farbe gestrichen und jeder Bewohner hat
+  eine andere Nationalität, besitzt ein anderes Haustier, trinkt ein von
+  den anderen Bewohnern verschiedenes Getränk und raucht eine von den
+  anderen Bewohnern verschiedene Zigarettensorte. Fragen: - Wer trinkt
+  Wasser? - Wem gehört das Zebra?
+
+  \begin{verbatim}
+loesung(WT, ZB) :-
+  haeuser(H), nationen(N), getraenke(G), tiere(T), zigaretten(Z),aussagentest(H,N,G,T,Z), !, wassertrinker(N,G,WT), zebrabesitzer(N,T,ZB).
+tiere(X) :- permutation([fuchs, hund, schnecke, pferd, zebra],X).
+nationen([norweger|R]) :- permmutation([englaender, spanier, ukrainer, japaner], R). % erfüllt damit Aussage 10
+getraenke(X) :- permutation([kaffee, tee, milch, osaft, wasser], X), a9(X). % erfüllt damit Aussage 9
+zigaretten(X) :- permmutation([atemgold,kools,chesterfield, luckystrike, parliament], X).
+haeuser(X) :- permutation([rot, gruen, weiss, gelb, blau], X), a6(X).
+% parmutation/2, fuege_ein/3: siehe BSP13-PRO
+wassertrinker([WT| _ ], [wasser| _ ], WT ) :- !.
+wassertrinker([ _ |R1], [ _ |R2],WT) :- wassertrinker(R1, R2, WT).
+zebrabesitzer([ZB| _ ], [zebra| _ ], ZB) :- !.
+zebrabesitzer([ _ |R1], [ _ |R2],ZB) :- zebrabesitzer(R1, R2, ZB).
+aussagentest(H,N,G,T,Z) :-
+  a2(H,N), a3(N,T), a4(H,G), a5(N,G), a7(Z,T), a8(H,Z), a11(Z,T), a12(Z,T),a13(Z,G), a14(N,Z), a15(H,N).
+  
+a2([rot| _ ], [englaender| _ ]) :- !
+.a2([ _ |R1], [ _ |R2]) :- a2(R1, R2).
+a3([spanier| _ ], [hund| _ ]) :- !.
+a3([ _ |R1], [ _ |R2]) :- a3(R1, R2).
+a4([gruen| _ ], [kaffee| _ ]) :- !.
+a4([ _ |R1], [ _ |R2]) :- a4(R1, R2).
+a5([ukrainer| _ ], [tee| _ ]) :- !.
+a5([ _ |R1], [ _ |R2]) :- a5(R1, R2).
+a6([weiss, gruen| _ ]) :- !.
+a6([weiss| _ ]) :- !, fail.a6([ _ |R]) :- a6(R).
+a7([atemgold| _ ], [schnecke| _ ]) :- !.
+a7([ _ |R1], [ _ |R2]) :- a7(R1, R2).
+a8([gelb| _ ], [kools| _ ]) :- !.
+a8([ _ |R1], [ _ |R2]) :- a8(R1, R2).
+a9([ _ , _ , milch, _ , _ ]).
+a11([chesterfield| _ ], [ _ ,fuchs| _ ]) :-!.
+a11([ _ , chesterfield| _ ], [fuchs| _ ]) :- !.
+a11([ _ |R1], [ _ |R2]) :- a11(R1, R2).
+a12([kools| _ ], [ _ , pferd| _ ]) :- !.
+a12([ _ , kools| _ ], [pferd| _ ] ) :- !.
+a12([ _ |R1], [ _ |R2]) :- a12(R1, R2).
+a13([luckystrike| _ ], [osaft| _ ]) :- !.
+a13([ _ |R1], [ _ |R2]) :- a13(R1, R2).
+a14([japaner| _ ], [parliament| _ ]) :- !.
+a14([ _ |R1], [ _ |R2]) :- a14(R1, R2).
+%a15([blau| _ ], [ _, norweger| _ ]) :- !.
+a15([ _ , blau| _ ], [norweger| _ ]) :- !.
+%a15([ _ |R1], [ _ |R2]) :- a15(R1, R2).
+
+?- loesung(Wassertrinker, Zebrabesitzer).
+\end{verbatim}
+
+  Beispiel SUDOKU (BSP17.PRO): - Liste 9-elementliger Listen, die (von
+  links nach rechts) die Zeilen (von oben nach unten)repräsentieren -
+  Elemente einer jeden eine Zeile repräsentierenden Liste: - Ziffer, falls
+  dort im gegebenen Sudoku eine Ziffer steht - Anonyme Variable
+  andernfalls
+
+  \paragraph{Tools für die formale
+    Logik}\label{tools-fuxfcr-die-formale-logik}
+
+  \begin{quote}
+    Botschaft 7 Auch in der formalen Logik gibt es Deduktionsaufgaben, bei
+    der Variablenbelegungen gesucht sind, welche eine Aussage wahr machen:
+    1. Meist geschieht das durch systematische Auswertung der Aussage, wozu
+    das Suchverfahren von Prolog genutzt werden kann. 2. Auch hier geht es
+    oft um gesuchte Werte für Variablen. Deshalb ist es oft auch mit dem
+    ``Deduktionstool'' Prolog lösbar, denn Prolog tut im Grunde nichts
+    anderes als ein ziel-gerichtetes ``Durchprobieren'' legitimer
+    Deduktionsschritte im ``Generate - and - Test'' - Verfahren.
+  \end{quote}
+
+  Repräsentation von Aussagen als PROLOG-Term: - true, false: atom(true),
+  atom(false) - $A1\wedge A2$: und(A1,A2) - $A1\vee A2$: oder(A1,A2) -
+  $\lnot A$: nicht(A) - $A1\rightarrow A2$: wenndann(A1,A2) -
+  $A1\leftarrow A2$: dannwenn(A1,A2) - $A1\leftrightarrow A2$: gdw(A1,A2)
+
+  Erfüllbarkeitstest: -
+  $?- erfuellbar(gdw(wenndann(nicht(oder(atom(false),atom(X))),atom(Y)), atom(Z))).$
+  - X=true, Y=\_, Z=true ; steht für 2 Modelle (eines mit Y = true und
+  eines mit Y = false) - X=false, Y=true, Z=true
+
+  Ketten von Konjunktionen und Disjunktionen als PROLOG-Listen: -
+  $A1\wedge A2\wedge ... \wedge An$: und verkettung({[}A1,A2, \ldots{},
+  An{]}) - $A1\vee A2\vee ...\vee An$: oder verkettung({[}A1,A2, \ldots{},
+  An{]})
+
+  Erfüllbarkeitstest: - $?- erfuellbar(undverkettung([true,X,Y,true]))$ -
+  X = true Y = true - $?- erfuellbar(oderverkettung([false,X,Y,false]))$ -
+  X = true Y = \emph{ - X = } Y = true
+
+  Repräsentation von Termen als PROLOG-Term: - Wert: atom() -
+  $A1\wedge A2$: und(A1,A2) - $A1\vee A2$: oder(A1,A2) - $\lnot A$:
+  nicht(A) - $A1\rightarrow A2$: wenndann(A1,A2) - $A1\leftarrow A2$:
+  dannwenn(A1,A2) - $A1\leftrightarrow A2$: gdw(A1,A2) -
+  $A1\wedge A2\wedge ... \wedge An$: und verkettung({[}A1,A2, \ldots{},
+  An{]}) - $A1\vee A2\vee ...\vee An$: oder verkettung({[}A1,A2, \ldots{},
+  An{]})
+
+  Termauswertung: -
+  $?- hat_wert(und(atom(0.5),oder(atom(0.7),atom(0.3))),X).$ - X=0.5
+
+  Termauswertung unter Vorgabe des Wertebereiches: -
+  $?- hat_wert([0,0.3, 0.5, 0.7, 1], und(atom(X),oder(atom(0.5),atom(0.7))),Wert).$
+  - X=0, Wert=0 - X=0.3, Wert=0.3 - X=0.5, Wert=0.5 - X=0.7, Wert=0.7 -
+  X=1, Wert=0.7
+
+\end{multicols}
+\end{document}
\ No newline at end of file