From f6e7a5ddd3c14e54bb550911089938da700e50c8 Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Tue, 2 Feb 2021 19:12:41 +0100 Subject: [PATCH] =?UTF-8?q?beweisverfahren=20hinzugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- Acronyms.tex | 1 + chapters/Logik/Beweisverfahren.tex | 51 +++++++++++++++++++++++++++++ images/folgerung.png | Bin 0 -> 70063 bytes images/inferenzalgorithmen.png | Bin 0 -> 89569 bytes images/inferenzregeln.png | Bin 0 -> 20340 bytes images/klauselform.png | Bin 0 -> 24287 bytes images/resolutionsalgorithmus1.png | Bin 0 -> 21863 bytes images/resolutionsalgorithmus2.png | Bin 0 -> 12856 bytes parts/Logik.tex | 3 +- 9 files changed, 54 insertions(+), 1 deletion(-) create mode 100644 chapters/Logik/Beweisverfahren.tex create mode 100644 images/folgerung.png create mode 100644 images/inferenzalgorithmen.png create mode 100644 images/inferenzregeln.png create mode 100644 images/klauselform.png create mode 100644 images/resolutionsalgorithmus1.png create mode 100644 images/resolutionsalgorithmus2.png diff --git a/Acronyms.tex b/Acronyms.tex index baf148c..dee41d4 100644 --- a/Acronyms.tex +++ b/Acronyms.tex @@ -9,4 +9,5 @@ \acro{PEAS}{Performance, Environment, Actuators, Sensors} \acro{TSP}{Traveling Salesman Problem} \acro{CSP}{Constraint Satisfaction Problem} + \acro{KNF}{Konjunktive Normalform} \end{acronym} \ No newline at end of file diff --git a/chapters/Logik/Beweisverfahren.tex b/chapters/Logik/Beweisverfahren.tex new file mode 100644 index 0000000..5b26052 --- /dev/null +++ b/chapters/Logik/Beweisverfahren.tex @@ -0,0 +1,51 @@ +\chapter{Beweisverfahren} +\label{logik: beweisverfahren} + \section{Logische Folgerung} + \label{logische folgerung} + $$\models \subseteq Formel(\Sigma)\times Formel(\Sigma)$$ + $$ F \models G: \text{Aus }F\text{ folgt logisch }G$$ + $$ F \models G \text{ gdw. } Mod(F)\subseteq Mod(G)$$ + \includegraphics[width = .8\textwidth]{folgerung.png}\\ + Hieraus lässt sich ein automatisiertes Beweisverfahren entwickeln, indem überprüft wird, ob in jeder Welt, ind der $F$ war ist auch $G$ war ist. + + \section{Widerspruchsbeweis} + \label{widerspruchsbeweis} + $$F\models G \text{ gdw. }F\wedge\neg G \text{ unerfüllbar}$$ + Hieraus lässt sich ein automatisiertes Beweisverfahren aufstellen, indem $\neg G$ zu $F$ hinzugefügt wird und daraus ein Widerspruch abgeleitet wird. + + \section{Inferenz} + \label{inferenz} + $$F\models G$$ + Dies ist auf 2 Arten beweisbar: + \begin{enumerate} + \item \textbf{Modellüberprüfung:}\\ + Überprüfung, ob in allen Welten, in denen $F$ war ist auch $G$ war ist + \item \textbf{Theoreme beweisen:}\\ + Abfolge von Beweisschritten (Inferenzregeln(\ref{inferenzregeln})) von $WB$ nach $F$ + \end{enumerate} + + \subsection{Inferenzalgorithmen} + \label{inferenzalgorithmen} + \includegraphics[width = \textwidth]{inferenzalgorithmen.png} + + \subsection{Inferenzregeln} + \label{inferenzregeln} + \includegraphics[width = .8\textwidth]{inferenzregeln.png} + + \subsection{Beweis durch Resolution} + \label{beweis durch resolution} + $$\frac{A\vee B, \neg B\vee C}{A\vee C}$$ + + \subsection{Allgemeine Resolutionsregel} + \label{allgemeine resolutionsregel} + $$\frac{(A_1\vee \dots \vee A_m\vee B), (\neg B \vee C_1 \vee\dots\vee C_n)}{(A_1\vee\dots\vee A_m\vee C_1\vee\dots\vee C_n)}$$ + + \subsection{Klauselform} + \label{klauselform} + Die Klauselform ist die Mengendarstellung der \ac{KNF}:\\ + \includegraphics[width = \textwidth]{klauselform.png} + + \subsection{Resolutionsalgorithmus} + \label{resolutionsalgorithmus} + \includegraphics[width = .8\textwidth]{resolutionsalgorithmus1.png}\\ + \includegraphics[width = .3\textwidth]{resolutionsalgorithmus2.png} \ No newline at end of file diff --git a/images/folgerung.png b/images/folgerung.png new file mode 100644 index 0000000000000000000000000000000000000000..a0b03a9b534ff2ca32c3e7184f218c26b9107122 GIT binary patch literal 70063 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