From 69ca84b57743c591d85b837391f04e43b661806e Mon Sep 17 00:00:00 2001 From: paul-loedige Date: Tue, 2 Feb 2021 20:43:14 +0100 Subject: [PATCH] =?UTF-8?q?Hornklauseln=20zu=20den=20Beweisverfahren=20hin?= =?UTF-8?q?zugef=C3=BCgt?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- chapters/Logik/Beweisverfahren.tex | 28 +++++++++++++++++++++++++++- images/backward_chaining.png | Bin 0 -> 44427 bytes images/forward_chaining.png | Bin 0 -> 44261 bytes 3 files changed, 27 insertions(+), 1 deletion(-) create mode 100644 images/backward_chaining.png create mode 100644 images/forward_chaining.png diff --git a/chapters/Logik/Beweisverfahren.tex b/chapters/Logik/Beweisverfahren.tex index 5b26052..10d97f1 100644 --- a/chapters/Logik/Beweisverfahren.tex +++ b/chapters/Logik/Beweisverfahren.tex @@ -48,4 +48,30 @@ \subsection{Resolutionsalgorithmus} \label{resolutionsalgorithmus} \includegraphics[width = .8\textwidth]{resolutionsalgorithmus1.png}\\ - \includegraphics[width = .3\textwidth]{resolutionsalgorithmus2.png} \ No newline at end of file + \includegraphics[width = .3\textwidth]{resolutionsalgorithmus2.png} + + \section{Hornklauseln} + \label{hornklauseln} + Klauseln in \ac{KNF} lassen sich nach positiven und negativen Literalen sortieren: + $$ (\neg A_1\vee\dots\vee\neg A_m\vee B_1\vee\dots\vee B_n)$$ + Hieraus ergibt sich durch umformen: + $$ A_1\wedge\dots\wedge A_m \Rightarrow B_1\vee\dots\vee B_n$$ + Dies ist besonders hilfreich, wenn die Klausel nur ein positives Literal enthält: + $$ \neg A_1\vee\dots\vee\neg A_m\vee B)$$ + $$A_1\wedge\dots\wedge A_m \Rightarrow B$$ + Diese Klauseln werden als \textbf{Hornklauseln} bezeichnet. + Der große Vorteil bei der Nutzung von Hornklauseln ist, + dass die Laufzeit einer Folgerung mit Hornklauseln \textbf{linear} in der Anzahl der Fakten der Wissensbasis ist. + + \subsection{Forward Chaining} + \label{forward chaining} + Der Modus Ponens (\ref{inferenzregeln}) ist für Hornklauseln eine vollständige Inferenzregel: + $$\frac{A_1\wedge\dots\wedge A_m, A_1\wedge\dots\wedge A_m \Rightarrow B}{B}$$ + \includegraphics[width = \textwidth]{forward_chaining.png} + + \subsection{Backward Chaining} + \label{backward chaining} + Während das Forward Chaining (\ref{forward chaining}) den einen datengetriebenen Ansatz verfolgt (Bei den Fakten starten und Anfrage herleiten) + beginnt das Backward Chaining bei der Anfrage und arbeitet die relevanten Teile des Und-Oder-Graphen rückwärts ab. + Dies hat den Vorteil, das anders als beim Forward Chaining keine potentiell unnötigen Formeln abgeleitet werden.\\ + \includegraphics[width = \textwidth]{backward_chaining.png} \ No newline at end of file diff --git a/images/backward_chaining.png b/images/backward_chaining.png new file mode 100644 index 0000000000000000000000000000000000000000..625f0dce2fd714f0f52ab99606245279ec98663f GIT binary patch literal 44427 zcma&OWmr|)7caa~1XNT&x|lGEd&B_NBWt#G6Hd}6@fs>!?*_D2^hrF 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