Mercurial > 13ws.ds
annotate notes/tex/logic.tex @ 28:f639b478a28b
typo
| author | Markus Kaiser <markus.kaiser@in.tum.de> |
|---|---|
| date | Mon, 02 Dec 2013 10:52:41 +0100 |
| parents | 4436f8006ebd |
| children | b5094b3e3111 |
| rev | line source |
|---|---|
| 20 | 1 \defineUnit{aussagenlogiksyntax}{% |
| 2 \begin{frame}[c] | |
| 3 \frametitle{Syntax der Aussagenlogik} | |
| 4 \setbeamercovered{dynamic} | |
| 5 | |
| 6 \begin{definition}[Syntax der Aussagenlogik] | |
| 7 Aussagenlogische \structure{Formeln} bestehen aus Konstanten, Variablen und Operatoren. Die Menge \structure{$\mathcal{F}$} aller Formeln ist induktiv definiert. | |
| 8 \begin{itemize} | |
|
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9 \item $\false = 0 = \bot \in \mathcal{F},\quad \true = 1 = \top \in \mathcal{F}$\hfill(\alert{Konstanten}) |
| 20 | 10 \item $V = \left\{ a, b, c,\ldots \right\} \subseteq \mathcal{F}$\hfill(\alert{Variablen})\\ |
| 11 \medskip | |
| 12 \item Ist $A \in \mathcal{F}$ eine aussagenlogische Formel, dann auch | |
| 13 \begin{align} | |
| 14 \neg A &\in \mathcal{F}\tag{\alert{Negation}}\\ | |
| 15 \intertext{\item Sind $A, B \in \mathcal{F}$ aussagenlogische Formeln, dann auch} | |
| 16 (A \wedge B)&\in \mathcal{F}\tag{\alert{Konjunktion}}\\ | |
| 17 (A \vee B)&\in \mathcal{F}\tag{\alert{Disjunktion}}\\ | |
| 18 (A \rightarrow B)&\in \mathcal{F}\tag{\alert{Implikation}} | |
| 19 \end{align} | |
| 20 \end{itemize} | |
| 21 Alle Formeln lassen sich so konstruieren. | |
| 22 \end{definition} | |
| 23 \end{frame} | |
| 24 | |
| 25 \begin{frame} | |
| 26 \frametitle{Operatorenbindung} | |
| 27 \setbeamercovered{dynamic} | |
| 28 | |
| 29 \begin{definition}[Bindungsregeln] | |
| 30 Die \structure{Bindungsstärke} der Operatoren in absteigender Reihenfolge ist | |
| 31 \[ \neg \quad \wedge \quad \vee \quad \rightarrow \quad \leftrightarrow \] | |
| 32 Die Implikation ist \structure{rechtsassoziativ} | |
| 33 \[ a \rightarrow b \rightarrow c \rightarrow d\text{\quad steht für\quad} \left( a \rightarrow \left( b \rightarrow \left( c \rightarrow d \right) \right) \right) \] | |
| 34 \end{definition} | |
| 35 \begin{itemize} | |
| 36 \item Üblicherweise klammert man $\wedge$ und $\vee$ | |
| 37 \[ (a \wedge b) \vee c \text{\quad statt\quad} a \wedge b \vee c \] | |
| 38 \end{itemize} | |
| 39 \begin{example}[] | |
| 40 \begin{itemize} | |
| 41 \item $\neg a \wedge b$\quad steht für \quad$ \left( \left( \neg a \right) \wedge b \right)$ | |
| 42 \item $a \wedge b \rightarrow c \vee \neg d$\quad steht für \quad$((a \wedge b) \rightarrow (c \vee \left( \neg d \right)))$ | |
| 43 \end{itemize} | |
| 44 \end{example} | |
| 45 \end{frame} | |
| 46 | |
| 47 \begin{frame} | |
| 48 \frametitle{Syntaxbaum} | |
| 49 \setbeamercovered{dynamic} | |
| 50 | |
| 51 \begin{block}{Syntaxbaum} | |
| 52 \structure{Syntaxbäume} visualisieren in welcher Reihenfolge die Regeln zur induktiven Definition angewandt werden müssen, um eine Formel zu erzeugen. | |
| 53 \end{block} | |
| 54 \begin{example}[] | |
| 55 Sei $F \defeq a \wedge b \rightarrow c \vee \neg d$ dann ist der dazu passende Syntaxbaum | |
| 56 \centering | |
| 57 \begin{tikzpicture}[grow=down, level distance = 33] | |
| 58 \tikzstyle{every node} = [] | |
| 59 \tikzstyle{op} = [pretty] | |
| 60 \tikzstyle{var} = [pretty, rectangle] | |
| 61 \tikzstyle{edge from parent} = [edge] | |
| 62 | |
| 63 \tikzstyle{level 1} = [sibling distance = 80] | |
| 64 \tikzstyle{level 2} = [sibling distance = 40] | |
| 65 \node[op] {$\rightarrow$} | |
| 66 child { | |
| 67 node[op] {$\wedge$} | |
| 68 edge from parent | |
| 69 child { | |
| 70 node[var] {$a$} | |
| 71 edge from parent | |
| 72 } | |
| 73 child { | |
| 74 node[var] {$b$} | |
| 75 edge from parent | |
| 76 } | |
| 77 } | |
| 78 child { | |
| 79 node[op] {$\vee$} | |
| 80 edge from parent | |
| 81 child { | |
| 82 node[var] {$c$} | |
| 83 edge from parent | |
| 84 } | |
| 85 child { | |
| 86 node[op] {$\neg$} | |
| 87 edge from parent | |
| 88 child { | |
| 89 node[var] {$d$} | |
| 90 edge from parent | |
| 91 } | |
| 92 } | |
| 93 }; | |
| 94 \end{tikzpicture} | |
| 95 \end{example} | |
| 96 \end{frame} | |
| 97 } | |
| 98 | |
| 99 \defineUnit{aussagenlogiksemantik}{% | |
| 100 \begin{frame} | |
| 101 \frametitle{Belegung} | |
| 102 \setbeamercovered{dynamic} | |
| 103 | |
| 104 \begin{definition}[Belegung] | |
| 105 Eine passende \structure{Belegung} $\beta$ zu einer Formel $F$ ordnet jeder Variable in $V$ einen Wahrheitswert aus $\left\{ 0, 1 \right\}$ zu. Es ist | |
| 106 \[ \beta : V \to \left\{ 0, 1 \right\} \] | |
| 107 \end{definition} | |
| 108 \begin{itemize} | |
| 109 \item Belegungen formalisieren Einsetzen | |
| 110 \item Für $n$ Variablen existieren $2^n$ Belegungen | |
| 111 \end{itemize} | |
| 112 \begin{example}[] | |
| 113 Sei $F \defeq \neg \left( a \wedge b \right)$ mit $V = \left\{ a, b \right\}$ und | |
| 114 \begin{align} | |
| 115 \beta : \left\{ a, b \right\} &\to \left\{ 0, 1 \right\}\\ | |
| 116 a &\mapsto 1\\ | |
| 117 b &\mapsto 0 | |
| 118 \end{align} | |
| 119 Dann ist $\beta$ eine zu $F$ passende \structure{Belegung}. | |
| 120 \end{example} | |
| 121 \end{frame} | |
| 122 | |
| 123 \begin{frame} | |
| 124 \frametitle{Semantik der Aussagenlogik} | |
| 125 \setbeamercovered{dynamic} | |
| 126 | |
| 127 \begin{definition}[Semantik einer Formel] | |
| 128 Die \structure{Semantik} $[F]$ einer aussagenlogischen Formel $F$ ist eine Funktion, die jeder passenden Belegung $\beta$ einen Wahrheitswert zuordnet.\\ | |
| 129 Sei $\mathcal{B} = \left\{ \beta_0, \beta_1, \ldots \right\}$ die Menge aller Belegungen zu $F$. Dann ist | |
| 130 \[[F] : \mathcal{B} \to \left\{ 0, 1 \right\}\] | |
| 131 \end{definition} | |
| 132 \begin{itemize} | |
| 133 \item Die Semantik löst eingesetzte Formeln auf | |
| 134 \item Wird anhand der induktiven Syntax definiert | |
| 135 \item Es gibt \alert{syntaktisch verschiedene} Formeln gleicher \structure{Semantik} | |
| 136 \end{itemize} | |
| 137 \begin{example}[] | |
| 138 Sei $F \defeq \left( G \rightarrow H \right)$ mit $G, H$ Formeln. Dann ist | |
| 139 \[ [F](\beta) = \begin{cases} | |
| 140 0 & \text{falls } [G](\beta) = 1 \text{ und } [H](\beta) = 0 \\ | |
| 141 1 & \text{sonst} | |
| 142 \end{cases}\] | |
| 143 \end{example} | |
| 144 \end{frame} | |
| 145 | |
| 146 \begin{frame} | |
| 147 \frametitle{Wahrheitstabelle} | |
| 148 \setbeamercovered{dynamic} | |
| 149 | |
| 150 \begin{block}{Wahrheitstabelle} | |
| 151 Die Semantik einer Formel kann mit Hilfe einer \structure{Wahrheitstabelle} visualisiert werden. Die Tabelle gibt den Wahrheitswert der Formel für jede mögliche Belegung an. | |
| 152 \end{block} | |
| 153 \begin{example}[] | |
| 154 Sei $F \defeq a \vee b \rightarrow \neg c \wedge b$. Die zu $[F]$ gehörige Wahrheitstabelle ist | |
| 155 \begin{center} | |
| 156 \begin{tabu} to .5\textwidth {cccX|[1.2pt]Xccccc} | |
| 157 a & b & c & & & $a \vee b$ & $\rightarrow$ & $\neg c$ & $\wedge$ & $b$ \\ \tabucline[1.2pt]{-} | |
| 158 0 & 0 & 0 & & & \onslide<2->{0} & \onslide<3->{\structure{1}} & \onslide<2->{1} & \onslide<2->{0} & \\ | |
| 159 0 & 0 & 1 & & & \onslide<2->{0} & \onslide<3->{\structure{1}} & \onslide<2->{0} & \onslide<2->{0} & \\ | |
| 160 0 & 1 & 0 & & & \onslide<2->{1} & \onslide<3->{\structure{1}} & \onslide<2->{1} & \onslide<2->{1} & \\ | |
| 161 0 & 1 & 1 & & & \onslide<2->{1} & \onslide<3->{\structure{0}} & \onslide<2->{0} & \onslide<2->{0} & \\ | |
| 162 1 & 0 & 0 & & & \onslide<2->{1} & \onslide<3->{\structure{0}} & \onslide<2->{1} & \onslide<2->{0} & \\ | |
| 163 1 & 0 & 1 & & & \onslide<2->{1} & \onslide<3->{\structure{0}} & \onslide<2->{0} & \onslide<2->{0} & \\ | |
| 164 1 & 1 & 0 & & & \onslide<2->{1} & \onslide<3->{\structure{1}} & \onslide<2->{1} & \onslide<2->{1} & \\ | |
| 165 1 & 1 & 1 & & & \onslide<2->{1} & \onslide<3->{\structure{0}} & \onslide<2->{0} & \onslide<2->{0} & \\ | |
| 166 \end{tabu} | |
| 167 \end{center} | |
| 168 \end{example} | |
| 169 \end{frame} | |
| 170 | |
| 171 \begin{frame} | |
| 172 \frametitle{Äquivalente Formeln} | |
| 173 \setbeamercovered{dynamic} | |
| 174 | |
| 175 \begin{definition}[Äquivalente Formeln] | |
| 176 Man nennt zwei Formeln \structure{äquivalent}, wenn sie dieselbe Semantik besitzen.\\ | |
| 177 Seien $F, G$ Formeln mit Belegungen $\mathcal{B} = \mathcal{B}_F = \mathcal{B}_G$. $F$ und $G$ sind äquivalent wenn | |
| 178 \[ \forall \beta \in \mathcal{B}. [F](\beta) = [G](\beta) \] | |
| 179 Man schreibt \structure{$F \equiv G$} oder \structure{$F \leftrightarrow G$}. | |
| 180 \end{definition} | |
| 181 | |
| 182 \begin{example}[] | |
| 183 Für $F \defeq a \rightarrow b$ und $G \defeq \neg a \vee b$ gilt $F \equiv G$. | |
| 184 \begin{center} | |
| 185 \begin{tabu} to .4\textwidth {cc|[1.2pt]XcX||Xccc} | |
| 186 a & b & & $a \Rightarrow b$ & & & $\neg a$ & $\vee$ & $b$ \\ \tabucline[1.2pt]{-} | |
| 187 0 & 0 & & \structure{1} & & & 1 & \structure{1} \\ | |
| 188 0 & 1 & & \structure{1} & & & 1 & \structure{1} \\ | |
| 189 1 & 0 & & \structure{0} & & & 0 & \structure{0} \\ | |
| 190 1 & 1 & & \structure{1} & & & 0 & \structure{1} \\ | |
| 191 \end{tabu} | |
| 192 \end{center} | |
| 193 \end{example} | |
| 194 \end{frame} | |
| 195 | |
| 196 \begin{frame} | |
| 197 \frametitle{Eigenschaften von Formeln} | |
| 198 \setbeamercovered{dynamic} | |
| 199 | |
| 200 \begin{block}{Eigenschaften aussagenlogischer Formeln} | |
| 201 Sei $F$ eine aussagenlogische Formel mit Variablen $V$ und der Menge der passenden Belegungen $\mathcal{B}$. Man nennt F | |
| 202 \begin{description}[\quad unerfüllbar] | |
| 203 \item[erfüllbar] $\exists \beta \in \mathcal{B}. [F](\beta) = 1$\hfill($F$ kann wahr sein) | |
| 204 \item[unerfüllbar] $\forall \beta \in \mathcal{B}. [F](\beta) = 0$\hfill($F$ ist nie wahr) | |
| 205 \item[gültig] $\forall \beta \in \mathcal{B}. [F](\beta) = 1$\hfill($F$ ist immer wahr) | |
| 206 \end{description} | |
| 207 \end{block} | |
| 208 \begin{itemize} | |
| 209 \item Eine unerfüllbare Formel nennt man \structure{Widerspruch} | |
| 210 \item Eine gültige Formel nennt man \structure{Tautologie} | |
| 211 \end{itemize} | |
| 212 | |
| 213 \end{frame} | |
| 214 } | |
| 215 | |
| 216 \defineUnit{aussagenlogikaequivalenzen}{% | |
| 217 { | |
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parents:
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changeset
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218 \newcommand{\spc}{\hfill} |
| 20 | 219 \newcommand{\F}{F} |
| 220 \newcommand{\G}{G} | |
| 221 \newcommand{\K}{H} | |
| 222 | |
| 223 \begin{frame} | |
| 224 \frametitle{Äquivalenzregeln} | |
| 225 \setbeamercovered{dynamic} | |
| 226 | |
| 227 \begin{description}[Triviale Kontradiktion\quad] | |
| 228 \item[Identität] $\F \wedge \true \equiv \F \spc \F \vee \false \equiv \F$ | |
| 229 \item[Dominanz] $\F \vee \true \equiv \true \spc \F \wedge \false \equiv \false$ | |
| 230 \item[Idempotenz] $\F \vee \F \equiv \F \spc \F \wedge \F \equiv \F$ | |
| 231 \item[Doppelte Negation] $\neg \neg \F \equiv \F$ | |
| 232 \item[Triviale Tautologie] $\F \vee \neg \F \equiv \true$ | |
| 233 \item[Triviale Kontradiktion] $\F \wedge \neg \F \equiv \false$ | |
| 234 \bigskip | |
| 235 \item[Kommutativität] $\F \vee \G \equiv \G \vee \F$\\ | |
| 236 $\F \wedge \G \equiv \G \wedge \F$ | |
| 237 \item[Assoziativität] $(\F \vee \G) \vee \K \equiv \F \vee (\G \vee \K)$\\ | |
| 238 $(\F \wedge \G) \wedge \K \equiv \F \wedge (\G \wedge \K)$ | |
| 239 \item[Distributivität] $\F \vee (\G \wedge \K) \equiv (\F \vee \G) \wedge (\F \vee \K)$\\ | |
| 240 $\F \wedge (\G \vee \K) \equiv (\F \wedge \G) \vee (\F \wedge \K)$ | |
| 241 \item[De Morgan] $\neg(\F \wedge \G) \equiv \neg \F \vee \neg \G$\\ | |
| 242 $\neg(\F \vee \G) \equiv \neg\F \wedge \neg\G$ | |
| 243 \bigskip | |
| 244 \item[Implikation] $\F \rightarrow \G \equiv \neg \F \vee \G$ | |
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245 \item[Bikonditional] $\F \leftrightarrow \G \equiv \neg(\F \otimes \G) \left[ \equiv (\F \rightarrow \G) \wedge (\G \rightarrow \F) \right]$ |
| 20 | 246 \end{description} |
| 247 \end{frame} | |
| 248 | |
| 249 %\begin{frame} | |
| 250 %\frametitle{Äquivalenzregeln} | |
| 251 %\setbeamercovered{dynamic} | |
| 252 | |
| 253 %\vspace{-2em} | |
| 254 %\begin{align} | |
| 255 %\F \wedge \true &\equiv \F \spc \F \vee \false \equiv \F \tag{\structure{Identität}}\\ | |
| 256 %\F \vee \true &\equiv \true \spc \F \wedge \false \equiv \false \tag{\structure{Dominanz}}\\ | |
| 257 %\F \vee \F &\equiv \F \spc \F \wedge \F \equiv \F \tag{\structure{Idempotenz}}\\ | |
| 258 %\neg \neg \F &\equiv \F \tag{\structure{Doppelte Negation}}\\ | |
| 259 %\F \vee \neg \F &\equiv \true \tag{\structure{Triviale Tautologie}}\\ | |
| 260 %\F \wedge \neg \F &\equiv \false \tag{\structure{Triviale Kontradiktion}}\\ | |
| 261 %\bigskip | |
| 262 %\F \vee \G &\equiv \G \vee \F \tag{\structure{Kommutativität}}\\ | |
| 263 %\F \wedge \G &\equiv \G \wedge \F\\ | |
| 264 %(\F \vee \G) \vee \K &\equiv \F \vee (\G \vee \K) \tag{\structure{Assoziativität}}\\ | |
| 265 %(\F \wedge \G) \wedge \K &\equiv \F \wedge (\G \wedge \K)\\ | |
| 266 %\F \vee (\G \wedge \K) &\equiv (\F \vee \G) \wedge (\F \vee \K) \tag{\structure{Distributivität}}\\ | |
| 267 %\F \wedge (\G \vee \K) &\equiv (\F \wedge \G) \vee (\F \wedge \K)\\ | |
| 268 %\neg(\F \wedge \G) &\equiv \neg \F \vee \neg \G \tag{\structure{De Morgan}}\\ | |
| 269 %\neg(\F \vee \G) &\equiv \neg\F \wedge \neg\G\\ | |
| 270 %\bigskip | |
| 271 %\F \rightarrow \G &\equiv \neg \F \vee \G \tag{\structure{Implikation}}\\ | |
| 272 %\F \leftrightarrow \G &\equiv (\F \rightarrow \G) \wedge (\G \rightarrow \F) \tag{\structure{Bikonditional}}\\ | |
| 273 %\end{align} | |
| 274 %\end{frame} | |
| 275 } | |
| 276 } | |
| 277 | |
| 278 \defineUnit{aussagenlogiknormalformen}{% | |
| 279 \begin{frame}[c] | |
| 280 \frametitle{Literale und Klauseln} | |
| 281 \setbeamercovered{dynamic} | |
| 282 | |
| 283 \begin{definition}[Literal] | |
| 284 Ein \structure{Literal} ist eine Variable $v \in V$ oder die Negation $\neg v$ einer Variable. | |
| 285 \end{definition} | |
| 286 | |
| 287 \begin{definition}[Klausel] | |
| 288 Eine \structure{Klausel} verknüpft mehrere Literale mit einem assoziativen Operator. | |
| 289 \end{definition} | |
| 290 | |
| 291 \vfill | |
| 292 | |
| 293 \begin{example}[] | |
| 294 Seien $a, \neg b, c$ Literale. Dann sind | |
| 295 \begin{itemize} | |
| 296 \item $a \wedge \neg b \wedge c$ | |
| 297 \item $a \vee \neg b \vee c$ | |
| 298 \end{itemize} | |
| 299 Klauseln. | |
| 300 \end{example} | |
| 301 \end{frame} | |
| 302 | |
| 303 { | |
| 304 \newcommand{\klausel}[2]{\underbracket{(##2)}_{\text{##1-Klausel}}} | |
| 305 \begin{frame} | |
| 306 \frametitle{DNF} | |
| 307 \setbeamercovered{dynamic} | |
| 308 | |
| 309 \begin{definition}[Disjunktive Normalform] | |
| 310 Eine \structure{DNF-Klausel} ist eine Konjunktion von Literalen $L_i$.\\ | |
| 311 Eine Formel $F$, ist in \structure{Disjunktiver Normalform}, wenn sie eine Disjunktion von DNF-Klauseln ist. | |
| 312 \[ F \defeq \bigvee \bigwedge_i L_i \] | |
| 313 \end{definition} | |
| 314 \begin{itemize} | |
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4436f8006ebd
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parents:
24
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changeset
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315 \item Ausnahme: $\false$ ist auch in DNF |
| 20 | 316 \end{itemize} |
| 317 | |
| 318 \begin{example}[] | |
| 319 $F$ ist in DNF. | |
| 320 \[ F \defeq \klausel{DNF}{a \structure{\wedge} b \structure{\wedge} \neg c} \alert{\vee} \klausel{DNF}{\neg b \structure{\wedge} c} \alert{\vee} \klausel{DNF}{\neg a \structure{\wedge} b \structure{\wedge} \neg c} \] | |
| 321 \end{example} | |
| 322 \end{frame} | |
| 323 | |
| 324 \begin{frame} | |
| 325 \frametitle{KNF} | |
| 326 \setbeamercovered{dynamic} | |
| 327 | |
| 328 \begin{definition}[Konjunktive Normalform] | |
| 329 Eine \structure{KNF-Klausel} ist eine Disjunktion von Literalen $L_i$.\\ | |
| 330 Eine Formel $F$, ist in \structure{Konjunktiver Normalform}, wenn sie eine Konjunktion von KNF-Klauseln ist. | |
| 331 \[ F \defeq \bigwedge \bigvee_i L_i \] | |
| 332 \end{definition} | |
| 333 \begin{itemize} | |
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4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
334 \item Ausnahme: $\true$ ist auch in KNF |
| 20 | 335 \end{itemize} |
| 336 | |
| 337 \begin{example}[] | |
| 338 $F$ ist in KNF. | |
| 339 \[ F \defeq \klausel{KNF}{\neg a \alert{\vee} b} \structure{\wedge} \klausel{KNF}{\neg b \alert{\vee} c} \structure{\wedge} \klausel{KNF}{a \alert{\vee} b \alert{\vee} \neg c} \] | |
| 340 \end{example} | |
| 341 \end{frame} | |
| 342 } | |
| 343 | |
| 344 \begin{frame} | |
| 345 \frametitle{Konstruktion der NF} | |
| 346 \setbeamercovered{dynamic} | |
| 347 | |
| 348 \begin{itemize} | |
| 349 \item \structure{Jede} nicht-triviale Formel ist in DNF und KNF umwandelbar | |
| 350 \item Durch Äquivalenzumformungen berechenbar (exponentiell groß!) | |
| 351 \item Oder: Konstruktion mit Wahrheitstabellen | |
| 352 \end{itemize} | |
| 353 | |
| 354 \vfill | |
| 355 | |
| 356 \begin{block}{Normalformen aus Wahrheitstabellen} | |
| 357 Gegeben eine Formel $F$ und ihre Wahrheitstabelle | |
| 358 \begin{itemize} | |
| 359 \item DNF | |
| 360 \begin{enumerate} | |
| 361 \item Betrachte Zeilen mit Eintrag \structure{$1$} | |
| 362 \item Bilde \structure{Konjunktion} aus der \structure{Belegung} | |
| 363 \item Bilde \structure{Disjunktion} aller erhaltenen Klauseln | |
| 364 \end{enumerate} | |
| 365 \bigskip | |
| 366 \item KNF | |
| 367 \begin{enumerate} | |
| 368 \item Betrachte Zeilen mit Eintrag \alert{$0$} | |
| 369 \item Bilde \alert{Disjunktion} aus der \alert{Negation} der Belegung | |
| 370 \item Bilde \alert{Konjunktion} aller erhaltenen Klauseln | |
| 371 \end{enumerate} | |
| 372 \end{itemize} | |
| 373 \end{block} | |
| 374 \end{frame} | |
| 375 | |
| 376 { | |
| 377 \newcommand{\klausel}[4]{(##2 a ##1 ##3 b ##1 ##4 c)} | |
| 378 \begin{frame} | |
| 379 \frametitle{Konstruktion der NF} | |
| 380 \setbeamercovered{dynamic} | |
| 381 | |
| 382 \begin{example}[] | |
| 383 Gegeben eine Formel $F$ mit folgender Semantik | |
| 384 \begin{center} | |
| 385 \begin{tabu} to .4\textwidth {ccc|[1.2pt]c} | |
| 386 a & b & c & $F$ \\ \tabucline[1.2pt]{-} | |
| 387 0 & 0 & 0 & 0 \\ | |
| 388 0 & 0 & 1 & 0 \\ | |
| 389 0 & 1 & 0 & 0 \\ | |
| 390 0 & 1 & 1 & 1 \\ | |
| 391 1 & 0 & 0 & 1 \\ | |
| 392 1 & 0 & 1 & 1 \\ | |
| 393 1 & 1 & 0 & 1 \\ | |
| 394 1 & 1 & 1 & 0 | |
| 395 \end{tabu} | |
| 396 \end{center} | |
| 397 $F$ dargestellt in | |
| 398 \begin{itemize} | |
| 399 \item DNF $\hfill\klausel{\wedge}{\neg}{}{} \vee (a \wedge \neg b) \vee \klausel{\wedge}{}{}{\neg}\hfill$ | |
| 400 \medskip | |
| 401 \item KNF $\hfill(a \vee b) \wedge \klausel{\vee}{\neg}{}{\neg} \wedge \klausel{\vee}{\neg}{\neg}{\neg}\hfill$ | |
| 402 \end{itemize} | |
| 403 \end{example} | |
| 404 \end{frame} | |
| 405 | |
| 406 \begin{frame} | |
| 407 \frametitle{Mengendarstellung der KNF} | |
| 408 \setbeamercovered{dynamic} | |
| 409 | |
| 410 \begin{block}{Mengendarstellung der KNF} | |
| 411 Eine Formel $F = \bigwedge \bigvee L_i$ in \structure{KNF} kann in einer \structure{Mengendarstellung} repräsentiert werden. | |
| 412 \begin{itemize} | |
| 413 \item Klauseln werden durch Mengen von Literalen dargestellt | |
| 414 \[\left\{ a, \neg b, c \right\} \text{ steht für } (a \vee \neg b \vee c)\] | |
| 415 \item KNF-Formeln sind Mengen von Klauseln | |
| 416 \[ \left\{ \left\{ \neg a \right\}, \left\{ a, \neg b, c \right\} \right\} \text{ steht für } \neg a \wedge (a \vee \neg b \vee c) \] | |
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417 \item $\emptyset$ steht für $\true$, $\left\{ \emptyset \right\}$ für $\false$ |
| 20 | 418 \end{itemize} |
| 419 \end{block} | |
| 420 \begin{example}[] | |
| 421 Gegeben $F \defeq (a \vee b) \wedge \klausel{\vee}{\neg}{}{\neg} \wedge \klausel{\vee}{\neg}{\neg}{\neg}$ in KNF. | |
| 422 \[ \left\{ \left\{ a, b \right\}, \left\{ \neg a, b, \neg c \right\}, \left\{ \neg a, \neg b, \neg c \right\} \right\}\] | |
| 423 \end{example} | |
| 424 \end{frame} | |
| 425 } | |
| 426 } | |
| 427 | |
| 428 \defineUnit{DPLL}{% | |
| 429 \begin{frame} | |
| 430 \frametitle{KNF aus Syntaxbaum} | |
| 431 \setbeamercovered{dynamic} | |
| 432 | |
| 433 \begin{block}{Idee} | |
| 434 Erzeuge die KNF aus dem Syntaxbaum | |
| 435 \begin{enumerate} | |
| 436 \item<1-> Weise jedem \structure{inneren Knoten} eine Variable zu | |
| 437 \item<1,3-> Variablen sind \structure{abhängig} von ihren Kindern | |
| 438 \item<1,4-> Berechne \structure{kleine} KNFs und führe diese \structure{zusammen} | |
| 439 \end{enumerate} | |
| 440 \end{block} | |
| 441 \begin{columns}[T] | |
| 442 \begin{column}{.7\textwidth} | |
| 443 % FIXME: onlys around A_\vee needed for "correct" fadein | |
| 444 \begin{align} | |
| 445 (x \wedge y) \vee z | |
| 446 \uncover<3->{\equiv &\hphantom{{}\wedge {}}\only<3->{A_\vee}\\ | |
| 447 &\structure{\wedge (A_\vee \leftrightarrow A_\wedge \vee z)}\\ | |
| 448 &\alert{\wedge (A_\wedge \leftrightarrow x \wedge y)}\\} | |
| 449 \uncover<4->{\equiv &\hphantom{{}\wedge {}}\only<4->{A_\vee}\\ | |
| 450 &\structure{\wedge (A_\vee \vee \neg A_\wedge) \wedge (A_\vee \vee \neg z)}\\ | |
| 451 &\qquad\structure{\wedge (\neg A_\vee \vee A_\wedge \vee z)}\\ | |
| 452 &\alert{\wedge (\neg A_\wedge \vee x) \wedge (\neg A_\wedge \vee y)} \\ | |
| 453 &\qquad\alert{\wedge(A_\wedge \vee \neg x \vee \neg y)}} | |
| 454 \end{align} | |
| 455 \end{column} | |
| 456 \begin{column}{.3\textwidth} | |
| 457 \begin{tikzpicture}[grow=down, level distance = 33] | |
| 458 \tikzstyle{op} = [pretty] | |
| 459 \tikzstyle{var} = [pretty, rectangle] | |
| 460 \tikzstyle{edge from parent} = [edge] | |
| 461 | |
| 462 \tikzstyle{level 1} = [sibling distance = 50] | |
| 463 \tikzstyle{level 2} = [sibling distance = 30] | |
| 464 \node at (0, 0) {}; | |
| 465 \node[op] at (0, -1) {\alt<1>{$\vee$}{$A_\vee$}} | |
| 466 child { | |
| 467 node[op] {\alt<1>{$\wedge$}{$A_\wedge$}} | |
| 468 edge from parent | |
| 469 child { | |
| 470 node[var] {$x$} | |
| 471 edge from parent | |
| 472 } | |
| 473 child { | |
| 474 node[var] {$y$} | |
| 475 edge from parent | |
| 476 } | |
| 477 } | |
| 478 child { | |
| 479 node[var] {$z$} | |
| 480 }; | |
| 481 \end{tikzpicture} | |
| 482 \end{column} | |
| 483 \end{columns} | |
| 484 \end{frame} | |
| 485 | |
| 486 \begin{frame} | |
| 487 \frametitle{DPLL} | |
| 488 \setbeamercovered{dynamic} | |
| 489 | |
| 490 \begin{definition}[DPLL-Belegung] | |
| 491 Sei $F$ eine Formel in KNF und $p$ eine Variable von $F$.\\ | |
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492 Dann bezeichnet \structure{$F[p\backslash\true]$} die Formel, die entsteht, wenn jedes Vorkommnis von $p$ in F durch $\true$ ersetzt und vereinfacht wird. |
| 20 | 493 \end{definition} |
| 494 | |
| 495 \begin{block}{DPLL} | |
| 496 Gegeben eine Formel $F$ in KNF | |
| 497 \begin{itemize} | |
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498 \item Wenn $F = \true$ dann antworte \structure{erfüllbar} |
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499 \item Wenn $F = \false$ dann antworte \structure{unerfüllbar} |
| 20 | 500 \item Sonst |
| 501 \begin{enumerate} | |
| 502 \item Wähle eine Variable $p$ in $F$ | |
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503 \item Prüfe ob $F[p\backslash\true]$ oder $F[p\backslash\false]$ erfüllbar |
| 20 | 504 \end{enumerate} |
| 505 \end{itemize} | |
| 506 \end{block} | |
| 507 \begin{itemize} | |
| 508 \item Schlaue Wahl der Variable beschleunigt Ausführung | |
| 509 \item Wähle Variablen die einzeln stehen (\structure{One-Literal-Rule}) | |
| 510 \end{itemize} | |
| 511 \end{frame} | |
| 512 } | |
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513 |
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514 \defineUnit{resolution}{% |
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515 \begin{frame} |
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516 \frametitle{Resolution} |
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517 \setbeamercovered{dynamic} |
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518 |
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519 \begin{definition}[Resolvent] |
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520 Seien $K_1$, $K_2$ und $R$ Klauseln in Mengendarstellung. Dann heißt $R$ \structure{Resolvent} von $K_1$ und $K_2$ wenn $L \in K_1$, $\neg L \in K_2$ und |
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521 \begin{align} |
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522 R &= \left( K_1 \setminus \left\{ L \right\} \right) \cup \left( K_2 \setminus \left\{ \neg L \right\} \right) |
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523 \end{align} |
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524 \end{definition} |
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525 |
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526 \begin{block}{Resolution} |
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527 Gegeben eine Formel $F$ in KNF in Mengendarstellung. |
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528 \begin{algorithmic} |
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529 \While{$\square = \emptyset \not \in F$} |
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530 \State{$R \gets \text{Resolvent aus } F \text{ mit } R \not\in F$} |
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531 \If{$R$ existiert} |
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532 \State{$F \gets F \cup R$} |
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533 \Else |
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534 \State{\textbf{return} \structure{erfüllbar}} |
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535 \EndIf |
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536 \EndWhile |
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537 \State{\textbf{return} \structure{unerfüllbar}} |
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538 \end{algorithmic} |
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539 \end{block} |
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540 \end{frame} |
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541 } |
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542 |
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543 \defineUnit{natuerlichesschliessen}{% |
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544 \begin{frame} |
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545 \frametitle{Kalküle} |
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546 \setbeamercovered{dynamic} |
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547 |
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548 \begin{definition}[Kalkül] |
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549 Ein \structure{Logikkalkül} stellt \structure{Inferenzregeln} bereit, mit denen Formeln \alert{syntaktisch} umgeformt werden können. |
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550 \end{definition} |
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551 |
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552 \begin{definition}[Folgerung] |
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553 $F$ \structure{folgt aus} $A$, wenn mit Hilfe der \alert{Semantik} der \alert{Aussagenlogik} $F$ unter der Annahme dass $A$ gilt zu $\true$ ausgewertet wird. Wir schreiben |
|
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554 \[ A \vDash F \] |
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555 \end{definition} |
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556 |
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557 \begin{definition}[Ableitung] |
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558 $F$ kann aus $A$ \structure{abgeleitet} werden, wenn mit Hilfe \alert{syntaktischer} Umformungen in einem \alert{Logikkalkül} $F$ unter der Annahme $A$ bewiesen werden kann. Wir schreiben |
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559 \[ A \vdash F \] |
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560 \end{definition} |
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561 \end{frame} |
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562 |
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563 \begin{frame} |
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564 \frametitle{Eigenschaften von Kalkülen} |
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565 \setbeamercovered{dynamic} |
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566 |
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567 \begin{block}{Eigenschaften von Kalkülen} |
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568 \begin{description}[\quad vollständig (complete)] |
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569 \item[korrekt (sound)] Es können \alert{nur} semantisch gültige Formeln abgeleitet werden. |
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570 \[ \text{Aus } A \vdash F \text{ folgt } A \vDash F \] |
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571 \item[vollständig (complete)] \alert{Alle} semantisch gültigen Formeln können abgeleitet werden. |
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572 \[ \text{Aus } A \vDash F \text{ folgt } A \vdash F \] |
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573 \end{description} |
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574 \end{block} |
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575 |
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576 \begin{itemize} |
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577 \item Für uns nur korrekte vollständige Kalküle |
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578 \item Beispiel für die Aussagenlogik: \structure{Natürliches Schließen} |
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579 \medskip |
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580 \item Es gibt keine solchen Kalküle für die |
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581 \begin{itemize} |
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582 \item Prädikatenlogik |
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583 \item Arithmetik |
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584 \end{itemize} |
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585 \item Deshalb sind nicht alle Sätze der Mathematik beweisbar |
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586 \end{itemize} |
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587 \end{frame} |
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588 |
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589 { |
| 24 | 590 \newcommand{\F}{\tau} |
| 591 \newcommand{\G}{\varphi} | |
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592 \newcommand{\K}{\chi} |
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593 \newcommand{\subproof}[2]{% |
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594 \begin{tikzpicture}[y=.9em] |
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595 \path |
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596 (0, 0) node (a) {##1} |
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597 ++(0, -1) node (b) {$\vdots$} |
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598 ++(0, -1) node[below] (c) {##2}; |
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599 \node[fit=(a)(b)(c), draw, inner sep=1pt] {}; |
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600 \end{tikzpicture}% |
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601 } |
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602 \newcommand{\topproof}[1]{% |
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603 ##1 |
|
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604 \bottomAlignProof |
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605 \DisplayProof |
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606 } |
|
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607 |
|
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608 \begin{frame} |
|
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609 \frametitle{Natürliches Schließen} |
|
dc6b569c57c8
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|
610 \setbeamercovered{dynamic} |
|
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|
611 |
|
dc6b569c57c8
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|
612 \vspace{-4pt} |
|
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|
613 \tabulinesep=4pt |
|
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614 \begin{tabu} to \textwidth {X[c,m,.5]|[1pt]X[c,b,5]X[c,b,5]} |
|
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diff
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|
615 & \structure{Introduktion} & \structure{Elimination}\\\tabucline[1pt]{-} |
|
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616 $\wedge$ & |
|
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617 \topproof{ |
|
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618 \AxiomC{$\F$} |
|
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fifth slides and sheet
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|
619 \AxiomC{$\G$} |
|
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620 \RightLabel{\scriptsize $+\wedge$} |
|
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621 \BinaryInfC{$\F \wedge \G$} |
|
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622 } |
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623 & |
|
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624 \topproof{ |
|
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625 \AxiomC{$\F \wedge \G$} |
|
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|
626 \RightLabel{\scriptsize $-\wedge_1$} |
|
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627 \UnaryInfC{$\F$} |
|
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|
628 } |
|
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|
629 \quad |
|
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|
630 \topproof{ |
|
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|
631 \AxiomC{$\F \wedge \G$} |
|
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632 \RightLabel{\scriptsize $-\wedge_2$} |
|
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633 \UnaryInfC{$\G$} |
|
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|
634 } |
|
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|
635 \\ |
|
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|
636 $\vee$ & |
|
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|
637 \topproof{ |
|
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|
638 \AxiomC{$\F$} |
|
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|
639 \RightLabel{\scriptsize $+\vee_1$} |
|
dc6b569c57c8
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|
640 \UnaryInfC{$\F \vee \G$} |
|
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|
641 } |
|
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|
642 \quad |
|
dc6b569c57c8
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|
643 \topproof{ |
|
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|
644 \AxiomC{$\G$} |
|
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|
645 \RightLabel{\scriptsize $+\vee_2$} |
|
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|
646 \UnaryInfC{$\F \vee \G$} |
|
dc6b569c57c8
fifth slides and sheet
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|
647 } |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
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diff
changeset
|
648 & |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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|
649 \topproof{ |
|
dc6b569c57c8
fifth slides and sheet
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diff
changeset
|
650 \AxiomC{$\F \vee \G$} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
651 \AxiomC{\subproof{$\F$}{$\K$}} |
|
dc6b569c57c8
fifth slides and sheet
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parents:
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diff
changeset
|
652 \AxiomC{\subproof{$\G$}{$\K$}} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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diff
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|
653 \RightLabel{\scriptsize $-\vee$} |
| 23 | 654 \TrinaryInfC{$\K$\vphantom{$\F\G$}} |
|
22
dc6b569c57c8
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|
655 } |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
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changeset
|
656 \\ |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
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changeset
|
657 $\rightarrow$ & |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
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diff
changeset
|
658 \topproof{ |
|
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fifth slides and sheet
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|
659 \AxiomC{\subproof{$\F$}{$\G$}} |
|
dc6b569c57c8
fifth slides and sheet
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|
660 \RightLabel{\scriptsize $+\rightarrow$} |
|
dc6b569c57c8
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|
661 \UnaryInfC{$\F \rightarrow \G$} |
|
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fifth slides and sheet
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|
662 } |
|
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changeset
|
663 & |
|
dc6b569c57c8
fifth slides and sheet
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|
664 \topproof{ |
|
dc6b569c57c8
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|
665 \AxiomC{$\F$} |
|
dc6b569c57c8
fifth slides and sheet
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|
666 \AxiomC{$\F \rightarrow \G$} |
|
dc6b569c57c8
fifth slides and sheet
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|
667 \RightLabel{\scriptsize $-\rightarrow$, MP} |
|
dc6b569c57c8
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|
668 \BinaryInfC{$\G$} |
|
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|
669 } |
|
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|
670 \\ |
|
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|
671 $\neg$ & |
|
dc6b569c57c8
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parents:
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changeset
|
672 \topproof{ |
|
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|
673 \AxiomC{\subproof{$\F$}{$\bot$}} |
|
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fifth slides and sheet
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|
674 \RightLabel{\scriptsize $+\neg$} |
|
dc6b569c57c8
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|
675 \UnaryInfC{$\neg \F$} |
|
dc6b569c57c8
fifth slides and sheet
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parents:
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changeset
|
676 } |
|
dc6b569c57c8
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parents:
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diff
changeset
|
677 & |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
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changeset
|
678 \topproof{ |
|
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changeset
|
679 \AxiomC{$\F$} |
|
dc6b569c57c8
fifth slides and sheet
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parents:
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changeset
|
680 \AxiomC{$\neg\F$} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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changeset
|
681 \RightLabel{\scriptsize $-\neg$} |
| 24 | 682 \BinaryInfC{$\bot$} |
|
22
dc6b569c57c8
fifth slides and sheet
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diff
changeset
|
683 } |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
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diff
changeset
|
684 \end{tabu} |
|
dc6b569c57c8
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|
685 \end{frame} |
|
dc6b569c57c8
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diff
changeset
|
686 |
|
dc6b569c57c8
fifth slides and sheet
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parents:
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diff
changeset
|
687 \begin{frame} |
|
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|
688 \frametitle{Natürliches Schließen} |
|
dc6b569c57c8
fifth slides and sheet
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changeset
|
689 \setbeamercovered{dynamic} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
690 |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
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diff
changeset
|
691 \vspace{-4pt} |
|
dc6b569c57c8
fifth slides and sheet
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changeset
|
692 \tabulinesep=4pt |
|
dc6b569c57c8
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20
diff
changeset
|
693 \begin{tabu} to \textwidth {X[c,m,.5]|[1pt]X[c,b,5]X[c,b,5]} |
|
dc6b569c57c8
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diff
changeset
|
694 & \structure{Introduktion} & \structure{Elimination}\\\tabucline[1pt]{-} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
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changeset
|
695 $\bot$ & |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
696 & |
|
dc6b569c57c8
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parents:
20
diff
changeset
|
697 \topproof{ |
|
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diff
changeset
|
698 \AxiomC{$\bot$} |
|
dc6b569c57c8
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diff
changeset
|
699 \RightLabel{\scriptsize $-\bot$} |
|
dc6b569c57c8
fifth slides and sheet
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20
diff
changeset
|
700 \UnaryInfC{$\F$} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
701 } |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
702 \\ |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
703 $\neg\neg$ & |
|
dc6b569c57c8
fifth slides and sheet
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parents:
20
diff
changeset
|
704 \topproof{ |
|
dc6b569c57c8
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parents:
20
diff
changeset
|
705 \AxiomC{$\F$} |
|
dc6b569c57c8
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20
diff
changeset
|
706 \RightLabel{\scriptsize $+\neg\neg$} |
|
dc6b569c57c8
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parents:
20
diff
changeset
|
707 \UnaryInfC{$\neg\neg\F$} |
|
dc6b569c57c8
fifth slides and sheet
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parents:
20
diff
changeset
|
708 } |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
709 & |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
710 \topproof{ |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
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20
diff
changeset
|
711 \AxiomC{$\neg\neg\F$} |
|
dc6b569c57c8
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20
diff
changeset
|
712 \RightLabel{\scriptsize $-\neg\neg$} |
|
dc6b569c57c8
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parents:
20
diff
changeset
|
713 \UnaryInfC{$\F$} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
714 } |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
715 \end{tabu} |
|
dc6b569c57c8
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parents:
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diff
changeset
|
716 \begin{itemize} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
717 \vfill |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
718 \item Praktische abgeleitete Regeln |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
719 \end{itemize} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
720 \tabulinesep=10pt |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
721 \begin{tabu} to \textwidth {X[c,m,.5]X[c,b,5]X[c,b,5]} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
722 & |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
723 \topproof{ |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
724 \AxiomC{ } |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
725 \RightLabel{\scriptsize LEM} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
726 \UnaryInfC{$\F \vee \neg\F$} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
727 } |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
728 & |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
729 \topproof{ |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
730 \AxiomC{\subproof{$\neg\F$}{$\bot$}} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
731 \RightLabel{\scriptsize $-\neg$, PBC} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
732 \UnaryInfC{$\F$} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
733 } |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
734 \\ |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
735 & |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
736 & |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
737 \topproof{ |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
738 \AxiomC{$\neg\G$} |
|
dc6b569c57c8
fifth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
739 \AxiomC{$\F \rightarrow \G$} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
740 \RightLabel{\scriptsize MT} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
741 \BinaryInfC{$\neg\F$} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
742 } |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
changeset
|
743 \end{tabu} |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
20
diff
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|
744 \end{frame} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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20
diff
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|
745 } |
|
dc6b569c57c8
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Markus Kaiser <markus.kaiser@in.tum.de>
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20
diff
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|
746 } |
|
26
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24
diff
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|
747 |
|
4436f8006ebd
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
748 \defineUnit{praedikatenlogiksyntax}{% |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
749 { |
|
4436f8006ebd
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
750 \newcommand{\terms}{\mathcal{T}} |
|
4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
751 \newcommand{\preds}{\mathcal{P}} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
752 \newcommand{\logic}{\mathcal{L}} |
|
4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
753 \begin{frame}[c] |
|
4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
754 \frametitle{Syntax der Prädikatenlogik} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
755 \setbeamercovered{dynamic} |
|
4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
756 |
|
4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
757 \begin{definition}[Term] |
|
4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
758 Die Menge $\terms$ aller \structure{Terme} ist induktiv definiert. |
|
4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
759 \begin{itemize} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
760 \item Jede Konstante ist in $\terms$ |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
761 \item Jede Variable ist in $\terms$ |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
762 \item Sind $f$ eine Funktion und $t_1, \dots, t_n$ Terme, dann auch |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
763 \[ f(t_1, \dots, t_n) \] |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
764 \end{itemize} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
765 Funktionen wandeln Terme in \alert{Terme} um. Wir beschreiben sie mit Kleinbuchstaben. |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
766 \end{definition} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
767 |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
768 \begin{definition}[Prädikat] |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
769 \structure{Prädikate} $P$ wandeln Terme in \alert{Wahrheitswerte} um. |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
770 Wir beschreiben sie mit Großbuchstaben.\\ |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
771 Die Menge $\preds$ enthält alle \structure{Prädikate}. |
|
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fix minor errors; sixth slides and sheet
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24
diff
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|
772 \end{definition} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
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|
773 \end{frame} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
774 |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
775 \begin{frame}[c] |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
776 \frametitle{Syntax der Prädikatenlogik} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
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|
777 \setbeamercovered{dynamic} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
778 |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
779 \begin{definition}[Syntax der Prädikatenlogik] |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
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|
780 Die Menge \structure{$\logic$} aller \structure{prädikatenlogischen Formeln} ist induktiv definiert. |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
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|
781 Seien $A, B\in \logic$, $t_i \in \terms$ und $P \in \preds$. Dann sind alle Formeln |
|
4436f8006ebd
fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
782 { |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
783 \setlength{\belowdisplayskip}{0pt} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
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|
784 \setlength{\abovedisplayskip}{0pt} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
785 \begin{itemize} |
| 28 | 786 \item Grundbausteine |
|
26
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fix minor errors; sixth slides and sheet
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24
diff
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|
787 \smallskip |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
788 \begin{align} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
789 V = \left\{ a, b, \dots \right\} &\subseteq \logic\tag{\alert{Variablen}}\\ |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
790 P(t_1, \dots, t_n) &\in \logic\tag{\alert{Prädikate, Konstanten}}\\ |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
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24
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|
791 t_i = t_j &\in\logic\tag{\alert{Gleichheit}}\\ |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
792 \shortintertext{\item Verknüpfungen der Aussagenlogik} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
793 \neg A &\in \logic\tag{\alert{Negation}}\\ |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
794 (A \wedge B), (A \vee B)&\in \logic\tag{\alert{Konjunktion, Disjunktion}}\\ |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
795 (A \rightarrow B)&\in \logic\tag{\alert{Implikation}}\\ |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
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|
796 \shortintertext{\item \structure{Quantoren}} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
797 \exists x. A &\in \logic\tag{\alert{Existenzquantor}}\\ |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
798 \forall x. A &\in \logic\tag{\alert{Allquantor}}\\ |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
799 \end{align} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
800 \end{itemize} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
801 } |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
802 \vspace{-1.5em} |
|
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fix minor errors; sixth slides and sheet
Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
803 \end{definition} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
804 \end{frame} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
805 |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
806 \begin{frame} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
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|
807 \frametitle{Operatorenbindung} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
808 \setbeamercovered{dynamic} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
809 |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
810 \begin{definition}[Bindungsregeln] |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
811 Die \structure{Bindungsstärke} der Operatoren in absteigender Reihenfolge ist |
|
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24
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|
812 \[ \forall \quad \exists \quad \neg \quad \wedge \quad \vee \quad \rightarrow \quad \leftrightarrow \] |
|
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24
diff
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|
813 Die Implikation ist \structure{rechtsassoziativ}. |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
814 \end{definition} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
815 \begin{itemize} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
816 \item Üblicherweise klammert man wieder $\wedge$ und $\vee$ |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
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|
817 \item Genauso klammert man Quantoren |
|
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24
diff
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|
818 \[ \left( \forall x. F \right) \rightarrow G \text{\quad statt\quad} \forall x. F \rightarrow G \] |
|
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24
diff
changeset
|
819 \vfill |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
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changeset
|
820 \item \alert{Achtung!} Äußere Quantoren werden öfter anders interpretiert |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
821 \[ \forall x \forall y. F \wedge G \leftrightarrow H \] |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
822 Bindet formal \alert{nur an das F}! |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
823 \end{itemize} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
824 \end{frame} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
825 } |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
826 } |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
827 |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
828 \defineUnit{praedikatenlogikstruktur}{% |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
829 \begin{frame}[c] |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
830 \frametitle{Struktur} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
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24
diff
changeset
|
831 \setbeamercovered{dynamic} |
|
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Markus Kaiser <markus.kaiser@in.tum.de>
parents:
24
diff
changeset
|
832 |
|
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833 \begin{definition}[Struktur] |
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834 Eine passende \structure{Struktur} $S = \left( U_s, I_s \right)$ zu einer Formel $F$ besteht aus einem \structure{Universum} $U_s$ und einer \structure{Interpretation} $I_s$. |
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835 \begin{itemize} |
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836 \item Alle Terme werten zu einem Wert im \structure{Universum} $U_s$ aus |
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837 \item Die \structure{Interpretation} $I_s$ weist den Atomen der Formel Werte zu.\\ |
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838 Sie spezifiziert |
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839 \begin{itemize} |
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840 \item \alert{Variablen} $x$ mit |
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841 { |
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842 \setlength{\belowdisplayskip}{0pt} |
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843 \setlength{\abovedisplayskip}{0pt} |
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844 \begin{align} |
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845 x_s \in {} &U_s\\ |
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846 \shortintertext{\item \alert{Konstanten} $a$ mit} |
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847 a_s \in {} &U_s\\ |
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848 \shortintertext{\item \alert{k-stellige Prädikate} $P$ mit} |
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849 P_s \subseteq {} &U_s^k\\ |
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850 \shortintertext{\item \alert{Funktionen} $f$ mit} |
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851 f_s : {} &U_s^k \to U_s |
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852 \end{align} |
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853 } |
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854 \end{itemize} |
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855 \end{itemize} |
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856 \end{definition} |
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857 \end{frame} |
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858 } |