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zik$c|DXT6eTihpuKFzGwq?WLW>@pEJbL3YVv&M4N+vdH6z3TkgQUANcW|%uKbD9nF zx65hxmoKmm=@iroBs@GRV|5dSk#V8JLMnF6!YB6Zz8{wEAVjw!$L$YSF=&WSSiL2K z#CedPU5(gd?$%!fu>xLbvwW_GDBDrm z?@+omYIX}6Qz$3%TH}V3Bz6*p{(LSlAnnGAGo$9>yMD=|mse7z7Zo*t8Pfj?!+zW| diff --git a/Logik und Logikprogrammierung - Cheatsheet.tex b/Logik und Logikprogrammierung - Cheatsheet.tex index 64e41c6..a3a6682 100644 --- a/Logik und Logikprogrammierung - Cheatsheet.tex +++ b/Logik und Logikprogrammierung - Cheatsheet.tex @@ -14,6 +14,7 @@ \usepackage{verbatim} \usepackage[most]{tcolorbox} \usepackage[hidelinks,pdfencoding=auto]{hyperref} +\usepackage{bussproofs} \usepackage{fancyhdr} \usepackage{lastpage} \pagestyle{fancy} @@ -128,312 +129,142 @@ \subsubsection{Probleme mit natürlicher Sprache} - \begin{enumerate} + \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} + \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. - 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 zusammengesetzten Aussagen verwenden wir $\varphi,\psi$ usw. - Für solche zusammengesetzten Aussagen verwenden wir $\varphi,\psi$ usw. + Durch die Wahl der erlaubten Operatoren erhält man unterschiedliche ``Logiken''. - 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. - 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. + \subsection{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: + \begin{enumerate*} + \item Alle atomaren Formeln und $\bot$ sind Formeln. + \item Falls $\varphi$ und $\psi$ Formeln sind, sind auch $(\varphi\wedge\psi),(\varphi\wedge\psi)$($\varphi \rightarrow\psi$) und $\lnot\varphi$Formeln. + \item Nichts ist Formel, was sich nicht mittels der obigen Regeln erzeugen läßt. + \end{enumerate*} Beispielformel: $\lnot((\lnot p_4 \vee p_1)\wedge\bot)$ - Bezeichnungen: - Falsum: $\bot$ - Konjunktion: $\wedge$ - Disjunktion: - $\vee$ - Implikation: $\rightarrow$ - Negation: $\lnot$ + Präzedenz der Operatoren: + \begin{itemize*} + \item $\leftrightarrow$ bindet am schwächsten + \item $\rightarrow$\ldots{} + \item $\vee$\ldots{} + \item $\wedge$\ldots{} + \item $\lnot$ bindet am stärksten + \end{itemize*} - Abkürzungen $p,q,r...$ statt $p_0,p_1,p_2...$ + \subsection{Natürliches Schließen} + 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: + \begin{itemize*} + \item an der Wurzel steht die Behauptung (= Konklusion $\varphi$) + \item an den Blättern stehen Voraussetzungen (= Hypothesen oder Annahmen aus $\Gamma$) + \item an den inneren Knoten stehen ``Teilergebnisse'' und ``Begründungen'' + \end{itemize*} - $(\bigvee_{i=1}^n \varphi_i$ 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 + \subsection{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} - + \subsubsection{Konjunktionseinführung} + 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 $\Gamma$: +\begin{prooftree} + \AxiomC{$\varphi$} + \AxiomC{$\psi$} + \RightLabel{\scriptsize ($\wedge I$)} + \BinaryInfC{$\varphi\wedge\psi$} +\end{prooftree} + + \subsubsection{Konjunktionselimination} + 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 $\Gamma$: + + \begin{prooftree} + \AxiomC{$\varphi\wedge\psi$} + \RightLabel{\scriptsize ($\wedge E_1$)} + \UnaryInfC{$\varphi$} + \end{prooftree} + \begin{prooftree} + \AxiomC{$\varphi\wedge\psi$} + \RightLabel{\scriptsize ($\wedge E_2$)} + \UnaryInfC{$\psi$} + \end{prooftree} + + \subsubsection{Implikationseinführung} + 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 $\Gamma$: + \begin{prooftree} + \AxiomC{$\psi$} + \RightLabel(\scriptsize ($\rightarrow I)$) + \UnaryInfC{$\varphi\rightarrow\psi$} + \end{prooftree} + + \subsubsection{Implikationselimination oder modus ponens} + 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 $\Gamma$: + \begin{prooftree} + \AxiomC{$\varphi$} + \AxiomC{$\varphi\rightarrow\psi$} + \RightLabel(\scriptsize ($\rightarrow E)$) + \BinaryInfC{$\varphi$} + \end{prooftree} + + \subsubsection{Disjunktionselimination} + 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 $\Gamma$: + \begin{prooftree} + \AxiomC{$\varphi\vee\psi$} + \AxiomC{$\sigma$} + \AxiomC{$\sigma$} + \RightLabel(\scriptsize ($\vee E)$) + \TrinaryInfC{$\sigma$} + \end{prooftree} + + \subsubsection{Negationseinführung} + Ist D eine Deduktion von $\bot$ mit Hypothesen aus $\Gamma\cup\{\varphi\}$, so ergibt sich die folgende Deduktion von $\lnot\varphi$ mit Hypothesen aus $\Gamma$: + \begin{prooftree} + \AxiomC{$\bot$} + \RightLabel(\scriptsize ($\lnot I)$) + \UnaryInfC{$\varphi$} + \end{prooftree} + + \subsection{Negationselimination} + 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 $\Gamma$: + \begin{prooftree} + \AxiomC{$\lnot\varphi$} + \AxiomC{$\varphi$} + \RightLabel(\scriptsize ($\lnot E)$) + \BinaryInfC{$\bot$} + \end{prooftree} + + \subsubsection{Falsum} +Ist D eine Deduktion von $\bot$ mit Hypothesen aus $\Gamma$, so ergibt sich die folgende Deduktion von $\varphi$ mit Hypothesen aus $\Gamma$: +\begin{prooftree} + \AxiomC{$\bot$} + \RightLabel(\scriptsize ($\bot)$) + \UnaryInfC{$\varphi$} +\end{prooftree} + + \subsubsection{reductio ad absurdum} + Ist D eine Deduktion von $\bot$ mit Hypothesen aus $\Gamma\cup\{\lnot\varphi\}$, so ergibt sich die folgende Deduktion von $\varphi$ mit Hypothesen aus $\Gamma$: + \begin{prooftree} + \AxiomC{$\bot$} + \RightLabel(\scriptsize ($raa)$) + \UnaryInfC{$\varphi$} + \end{prooftree} + + \subsection{Regeln des natürlichen Schließens} \begin{quote} Definition @@ -610,7 +441,7 @@ \subsubsection{Beispiel}\label{beispiel-2} - \begin{itemize} + \begin{itemize*} \itemsep1pt\parskip0pt\parsep0pt \item Der Boolesche Wahrheitswertebereich B ist definiert durch die @@ -618,12 +449,12 @@ Funktionen $\lnot_B (a) = 1-a$, $\rightarrow_B(a,b) = max(b, 1 -a)$. Hier gelten: - \begin{itemize} + \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} + \end{itemize*} \item Der Kleenesche Wahrheitswertebereich $K_3$ ist definiert durch die Grundmenge $K_3=\{0,\frac{1}{2},1\}$ mit der natürlichen Ordnung @@ -670,7 +501,7 @@ \item Beispiele: $Inneres((0,1))=(0,1)=Inneres([0,1]),Inneres(N)=\varnothing,Inneres(\mathbb{R}_{\geq 0}) = \mathbb{R}_{> 0}$ - \end{itemize} + \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 @@ -748,19 +579,19 @@ $B(\lnot\lnot\phi\rightarrow\phi) = \rightarrow B(\lnot B\lnot B(B(\phi)),B(\phi Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den Wahrheitswertebereich $B_R$). \end{verbatim} - \begin{itemize} + \begin{itemize*} \item Sei $B:V\rightarrow H_R$ eine $H_R$-Belegung mit $B(\phi) =R\backslash\{0\}$. Dann gelten - \begin{itemize} + \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} + \end{itemize*} Also ist $\lnot\lnot\phi\rightarrow\phi$ keine $H_R$-Tautologie (gilt ebenso für die Wahrheitswertebereiche $K_3$ und $F$). @@ -788,7 +619,7 @@ Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den \item Ebenso erhält man: - \begin{itemize} + \begin{itemize*} \item $\{\lnot\phi\rightarrow\bot\}\Vdash_{K_3} \phi$ \item @@ -797,7 +628,7 @@ Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den $\{\lnot\phi\rightarrow\bot\}\Vdash_F\phi$ \item $\{\phi\}\Vdash F\lnot\phi\rightarrow\bot$ - \end{itemize} + \end{itemize*} \item Sei $B:D\rightarrow H_R$ eine $H_R$-Belegung mit $B(\phi) =\mathbb{R}\backslash\{0\}$. Dann gilt @@ -806,7 +637,7 @@ Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den also $\{\lnot\phi\rightarrow\bot\}\not\Vdash_{H_R} \phi$. Es gilt aber $\{\phi\}\Vdash_{H_R}\lnot \phi\rightarrow\bot$. - \end{itemize} + \end{itemize*} Zusammenfassung der Beispiele @@ -822,13 +653,13 @@ Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den $\{\phi\}\vdash\lnot\phi\rightarrow\bot$ \\ \end{tabular} - \begin{itemize} + \begin{itemize*} \itemsep1pt\parskip0pt\parsep0pt \item $Y$ in Spalte W:W-Folgerung gilt \item $-$ in Spalte W:W-Folgerung gilt nicht - \end{itemize} + \end{itemize*} \begin{quote} Überblick: Wir haben definiert - $\Gamma\vdash\phi$ syntaktische @@ -1019,7 +850,7 @@ Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den \subsubsection{Plan}\label{plan} - \begin{itemize} + \begin{itemize*} \itemsep1pt\parskip0pt\parsep0pt \item Sei $W$ einer der Wahrheitswertebereiche @@ -1046,7 +877,7 @@ Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den $\Leftrightarrow$ $\Gamma\not\Vdash_B \varphi$ \item $\Rightarrow$ $\Gamma\not\Vdash\varphi$ - \end{itemize} + \end{itemize*} \subsubsection{Konsistente Mengen}\label{konsistente-mengen} @@ -1140,14 +971,14 @@ Also ist $\lnot\lnot\phi\rightarrow\phi$ eine B-Tautologie (gilt ebenso für den Also $\Delta\vdash\bot$, ein Widerspruch zur Konsistenz von $\Delta$. Also ist $\Delta\cup\{\varphi\}$ konsistent. \end{verbatim} - \begin{enumerate} + \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} + \end{enumerate*} \begin{quote} Lemma 2 @@ -1197,7 +1028,7 @@ Also $\Delta\vdash\bot$, ein Widerspruch zur Konsistenz von $\Delta$. Also ist $ Der Beweis erfolgt per Induktion über die Länge von $\varphi$. - \begin{enumerate} + \begin{enumerate*} \itemsep1pt\parskip0pt\parsep0pt \item I.A.: hat $\varphi$ die Länge 1, so ist $\varphi$ atomare Formel. Hier @@ -1209,7 +1040,7 @@ Also $\Delta\vdash\bot$, ein Widerspruch zur Konsistenz von $\Delta$. Also ist $ Formeln $\alpha$ und $\beta$ der Länge$ \end{verbatim} - \begin{enumerate} + \begin{enumerate*} \setcounter{enumi}{1} \itemsep1pt\parskip0pt\parsep0pt \item Beispiel 2 (BSP2.PRO) - \end{enumerate} + \end{enumerate*} - \begin{itemize} + \begin{itemize*} \itemsep1pt\parskip0pt\parsep0pt \item Kollegen Meier und Müller arbeiten im Raum 1, Kollege Otto im Raum 2 @@ -4766,7 +4597,7 @@ goal arbeitet\_in(K1,R),arbeitet\_in(K2,R). \item koennen\_daten\_austauschen(K1,K2) :- erreichbar(K1),erreichbar(K2). - \end{itemize} + \end{itemize*} Deklarationsteil @@ -5103,13 +4934,13 @@ goal sucht, mache man sich den Unifikations-Mechanismus zu nutzen. \end{quote} - \begin{itemize} + \begin{itemize*} \itemsep1pt\parskip0pt\parsep0pt \item Eine kontextfreie Grammatik (Chomsky-Typ 2) besteht aus - \end{itemize} + \end{itemize*} - \begin{enumerate} + \begin{enumerate*} \itemsep1pt\parskip0pt\parsep0pt \item einem Alphabet A, welches die terminalen (satzbildenden) Symbole @@ -5121,15 +4952,15 @@ goal einer Menge von Ableitungsregeln $R\subseteq N\times (N\cup A)^*$ \item dem Satzsymbol $S\in N$ - \end{enumerate} + \end{enumerate*} - \begin{itemize} + \begin{itemize*} \itemsep1pt\parskip0pt\parsep0pt \item \ldots{} und in PROLOG repräsentiert werden durch - \end{itemize} + \end{itemize*} - \begin{enumerate} + \begin{enumerate*} \itemsep1pt\parskip0pt\parsep0pt \item 1-elementige Listen, welche zu satzbildenden Listen komponiert werden: @@ -5141,14 +4972,14 @@ goal PROLOG-Regeln mit $l\in N$ im Kopf und $r\in(N\cup A)^*$ im Körper \item einen reservierten Namen: $satz$ - \end{enumerate} + \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} + \begin{enumerate*} \itemsep1pt\parskip0pt\parsep0pt \item Alphabet @@ -5159,7 +4990,7 @@ goal \item Ableitungsregeln (in BACKUS-NAUR-Form) - \begin{itemize} + \begin{itemize*} \item $satz ::= subjekt praedikat objekt$ \item @@ -5172,10 +5003,10 @@ goal $artikel ::= das$ \item $praedikat ::= loest | ignoriert | verschaerft$ - \end{itemize} + \end{itemize*} \item Satzsymbol $satz$ - \end{enumerate} + \end{enumerate*} Verketten einer Liste von Listen