Mercurial > 13ws.ds
annotate notes/tex/logic.tex @ 22:dc6b569c57c8
fifth slides and sheet
| author | Markus Kaiser <markus.kaiser@in.tum.de> |
|---|---|
| date | Mon, 18 Nov 2013 23:49:37 +0100 |
| parents | d9b98c2ba5ac |
| children | 41623ba498a9 |
| 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} | |
| 9 \item $\mathrm{false} = 0 \in \mathcal{F},\quad \mathrm{true} = 1 \in \mathcal{F}$\hfill(\alert{Konstanten}) | |
| 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 { | |
| 218 \newcommand{\true}{1} | |
| 219 \newcommand{\false}{0} | |
| 220 \newcommand{\spc}{\hspace{3em}} | |
| 221 \newcommand{\F}{F} | |
| 222 \newcommand{\G}{G} | |
| 223 \newcommand{\K}{H} | |
| 224 | |
| 225 \begin{frame} | |
| 226 \frametitle{Äquivalenzregeln} | |
| 227 \setbeamercovered{dynamic} | |
| 228 | |
| 229 \begin{description}[Triviale Kontradiktion\quad] | |
| 230 \item[Identität] $\F \wedge \true \equiv \F \spc \F \vee \false \equiv \F$ | |
| 231 \item[Dominanz] $\F \vee \true \equiv \true \spc \F \wedge \false \equiv \false$ | |
| 232 \item[Idempotenz] $\F \vee \F \equiv \F \spc \F \wedge \F \equiv \F$ | |
| 233 \item[Doppelte Negation] $\neg \neg \F \equiv \F$ | |
| 234 \item[Triviale Tautologie] $\F \vee \neg \F \equiv \true$ | |
| 235 \item[Triviale Kontradiktion] $\F \wedge \neg \F \equiv \false$ | |
| 236 \bigskip | |
| 237 \item[Kommutativität] $\F \vee \G \equiv \G \vee \F$\\ | |
| 238 $\F \wedge \G \equiv \G \wedge \F$ | |
| 239 \item[Assoziativität] $(\F \vee \G) \vee \K \equiv \F \vee (\G \vee \K)$\\ | |
| 240 $(\F \wedge \G) \wedge \K \equiv \F \wedge (\G \wedge \K)$ | |
| 241 \item[Distributivität] $\F \vee (\G \wedge \K) \equiv (\F \vee \G) \wedge (\F \vee \K)$\\ | |
| 242 $\F \wedge (\G \vee \K) \equiv (\F \wedge \G) \vee (\F \wedge \K)$ | |
| 243 \item[De Morgan] $\neg(\F \wedge \G) \equiv \neg \F \vee \neg \G$\\ | |
| 244 $\neg(\F \vee \G) \equiv \neg\F \wedge \neg\G$ | |
| 245 \bigskip | |
| 246 \item[Implikation] $\F \rightarrow \G \equiv \neg \F \vee \G$ | |
| 247 \item[Bikonditional] $\F \leftrightarrow \G \equiv (\F \rightarrow \G) \wedge (\G \rightarrow \F)$ | |
| 248 \end{description} | |
| 249 \end{frame} | |
| 250 | |
| 251 %\begin{frame} | |
| 252 %\frametitle{Äquivalenzregeln} | |
| 253 %\setbeamercovered{dynamic} | |
| 254 | |
| 255 %\vspace{-2em} | |
| 256 %\begin{align} | |
| 257 %\F \wedge \true &\equiv \F \spc \F \vee \false \equiv \F \tag{\structure{Identität}}\\ | |
| 258 %\F \vee \true &\equiv \true \spc \F \wedge \false \equiv \false \tag{\structure{Dominanz}}\\ | |
| 259 %\F \vee \F &\equiv \F \spc \F \wedge \F \equiv \F \tag{\structure{Idempotenz}}\\ | |
| 260 %\neg \neg \F &\equiv \F \tag{\structure{Doppelte Negation}}\\ | |
| 261 %\F \vee \neg \F &\equiv \true \tag{\structure{Triviale Tautologie}}\\ | |
| 262 %\F \wedge \neg \F &\equiv \false \tag{\structure{Triviale Kontradiktion}}\\ | |
| 263 %\bigskip | |
| 264 %\F \vee \G &\equiv \G \vee \F \tag{\structure{Kommutativität}}\\ | |
| 265 %\F \wedge \G &\equiv \G \wedge \F\\ | |
| 266 %(\F \vee \G) \vee \K &\equiv \F \vee (\G \vee \K) \tag{\structure{Assoziativität}}\\ | |
| 267 %(\F \wedge \G) \wedge \K &\equiv \F \wedge (\G \wedge \K)\\ | |
| 268 %\F \vee (\G \wedge \K) &\equiv (\F \vee \G) \wedge (\F \vee \K) \tag{\structure{Distributivität}}\\ | |
| 269 %\F \wedge (\G \vee \K) &\equiv (\F \wedge \G) \vee (\F \wedge \K)\\ | |
| 270 %\neg(\F \wedge \G) &\equiv \neg \F \vee \neg \G \tag{\structure{De Morgan}}\\ | |
| 271 %\neg(\F \vee \G) &\equiv \neg\F \wedge \neg\G\\ | |
| 272 %\bigskip | |
| 273 %\F \rightarrow \G &\equiv \neg \F \vee \G \tag{\structure{Implikation}}\\ | |
| 274 %\F \leftrightarrow \G &\equiv (\F \rightarrow \G) \wedge (\G \rightarrow \F) \tag{\structure{Bikonditional}}\\ | |
| 275 %\end{align} | |
| 276 %\end{frame} | |
| 277 } | |
| 278 } | |
| 279 | |
| 280 \defineUnit{aussagenlogiknormalformen}{% | |
| 281 \begin{frame}[c] | |
| 282 \frametitle{Literale und Klauseln} | |
| 283 \setbeamercovered{dynamic} | |
| 284 | |
| 285 \begin{definition}[Literal] | |
| 286 Ein \structure{Literal} ist eine Variable $v \in V$ oder die Negation $\neg v$ einer Variable. | |
| 287 \end{definition} | |
| 288 | |
| 289 \begin{definition}[Klausel] | |
| 290 Eine \structure{Klausel} verknüpft mehrere Literale mit einem assoziativen Operator. | |
| 291 \end{definition} | |
| 292 | |
| 293 \vfill | |
| 294 | |
| 295 \begin{example}[] | |
| 296 Seien $a, \neg b, c$ Literale. Dann sind | |
| 297 \begin{itemize} | |
| 298 \item $a \wedge \neg b \wedge c$ | |
| 299 \item $a \vee \neg b \vee c$ | |
| 300 \end{itemize} | |
| 301 Klauseln. | |
| 302 \end{example} | |
| 303 \end{frame} | |
| 304 | |
| 305 { | |
| 306 \newcommand{\klausel}[2]{\underbracket{(##2)}_{\text{##1-Klausel}}} | |
| 307 \begin{frame} | |
| 308 \frametitle{DNF} | |
| 309 \setbeamercovered{dynamic} | |
| 310 | |
| 311 \begin{definition}[Disjunktive Normalform] | |
| 312 Eine \structure{DNF-Klausel} ist eine Konjunktion von Literalen $L_i$.\\ | |
| 313 Eine Formel $F$, ist in \structure{Disjunktiver Normalform}, wenn sie eine Disjunktion von DNF-Klauseln ist. | |
| 314 \[ F \defeq \bigvee \bigwedge_i L_i \] | |
| 315 \end{definition} | |
| 316 \begin{itemize} | |
| 317 \item Ausnahme: $\textrm{false}$ ist auch in DNF | |
| 318 \end{itemize} | |
| 319 | |
| 320 \begin{example}[] | |
| 321 $F$ ist in DNF. | |
| 322 \[ F \defeq \klausel{DNF}{a \structure{\wedge} b \structure{\wedge} \neg c} \alert{\vee} \klausel{DNF}{\neg b \structure{\wedge} c} \alert{\vee} \klausel{DNF}{\neg a \structure{\wedge} b \structure{\wedge} \neg c} \] | |
| 323 \end{example} | |
| 324 \end{frame} | |
| 325 | |
| 326 \begin{frame} | |
| 327 \frametitle{KNF} | |
| 328 \setbeamercovered{dynamic} | |
| 329 | |
| 330 \begin{definition}[Konjunktive Normalform] | |
| 331 Eine \structure{KNF-Klausel} ist eine Disjunktion von Literalen $L_i$.\\ | |
| 332 Eine Formel $F$, ist in \structure{Konjunktiver Normalform}, wenn sie eine Konjunktion von KNF-Klauseln ist. | |
| 333 \[ F \defeq \bigwedge \bigvee_i L_i \] | |
| 334 \end{definition} | |
| 335 \begin{itemize} | |
| 336 \item Ausnahme: $\textrm{true}$ ist auch in KNF | |
| 337 \end{itemize} | |
| 338 | |
| 339 \begin{example}[] | |
| 340 $F$ ist in KNF. | |
| 341 \[ F \defeq \klausel{KNF}{\neg a \alert{\vee} b} \structure{\wedge} \klausel{KNF}{\neg b \alert{\vee} c} \structure{\wedge} \klausel{KNF}{a \alert{\vee} b \alert{\vee} \neg c} \] | |
| 342 \end{example} | |
| 343 \end{frame} | |
| 344 } | |
| 345 | |
| 346 \begin{frame} | |
| 347 \frametitle{Konstruktion der NF} | |
| 348 \setbeamercovered{dynamic} | |
| 349 | |
| 350 \begin{itemize} | |
| 351 \item \structure{Jede} nicht-triviale Formel ist in DNF und KNF umwandelbar | |
| 352 \item Durch Äquivalenzumformungen berechenbar (exponentiell groß!) | |
| 353 \item Oder: Konstruktion mit Wahrheitstabellen | |
| 354 \end{itemize} | |
| 355 | |
| 356 \vfill | |
| 357 | |
| 358 \begin{block}{Normalformen aus Wahrheitstabellen} | |
| 359 Gegeben eine Formel $F$ und ihre Wahrheitstabelle | |
| 360 \begin{itemize} | |
| 361 \item DNF | |
| 362 \begin{enumerate} | |
| 363 \item Betrachte Zeilen mit Eintrag \structure{$1$} | |
| 364 \item Bilde \structure{Konjunktion} aus der \structure{Belegung} | |
| 365 \item Bilde \structure{Disjunktion} aller erhaltenen Klauseln | |
| 366 \end{enumerate} | |
| 367 \bigskip | |
| 368 \item KNF | |
| 369 \begin{enumerate} | |
| 370 \item Betrachte Zeilen mit Eintrag \alert{$0$} | |
| 371 \item Bilde \alert{Disjunktion} aus der \alert{Negation} der Belegung | |
| 372 \item Bilde \alert{Konjunktion} aller erhaltenen Klauseln | |
| 373 \end{enumerate} | |
| 374 \end{itemize} | |
| 375 \end{block} | |
| 376 \end{frame} | |
| 377 | |
| 378 { | |
| 379 \newcommand{\klausel}[4]{(##2 a ##1 ##3 b ##1 ##4 c)} | |
| 380 \begin{frame} | |
| 381 \frametitle{Konstruktion der NF} | |
| 382 \setbeamercovered{dynamic} | |
| 383 | |
| 384 \begin{example}[] | |
| 385 Gegeben eine Formel $F$ mit folgender Semantik | |
| 386 \begin{center} | |
| 387 \begin{tabu} to .4\textwidth {ccc|[1.2pt]c} | |
| 388 a & b & c & $F$ \\ \tabucline[1.2pt]{-} | |
| 389 0 & 0 & 0 & 0 \\ | |
| 390 0 & 0 & 1 & 0 \\ | |
| 391 0 & 1 & 0 & 0 \\ | |
| 392 0 & 1 & 1 & 1 \\ | |
| 393 1 & 0 & 0 & 1 \\ | |
| 394 1 & 0 & 1 & 1 \\ | |
| 395 1 & 1 & 0 & 1 \\ | |
| 396 1 & 1 & 1 & 0 | |
| 397 \end{tabu} | |
| 398 \end{center} | |
| 399 $F$ dargestellt in | |
| 400 \begin{itemize} | |
| 401 \item DNF $\hfill\klausel{\wedge}{\neg}{}{} \vee (a \wedge \neg b) \vee \klausel{\wedge}{}{}{\neg}\hfill$ | |
| 402 \medskip | |
| 403 \item KNF $\hfill(a \vee b) \wedge \klausel{\vee}{\neg}{}{\neg} \wedge \klausel{\vee}{\neg}{\neg}{\neg}\hfill$ | |
| 404 \end{itemize} | |
| 405 \end{example} | |
| 406 \end{frame} | |
| 407 | |
| 408 \begin{frame} | |
| 409 \frametitle{Mengendarstellung der KNF} | |
| 410 \setbeamercovered{dynamic} | |
| 411 | |
| 412 \begin{block}{Mengendarstellung der KNF} | |
| 413 Eine Formel $F = \bigwedge \bigvee L_i$ in \structure{KNF} kann in einer \structure{Mengendarstellung} repräsentiert werden. | |
| 414 \begin{itemize} | |
| 415 \item Klauseln werden durch Mengen von Literalen dargestellt | |
| 416 \[\left\{ a, \neg b, c \right\} \text{ steht für } (a \vee \neg b \vee c)\] | |
| 417 \item KNF-Formeln sind Mengen von Klauseln | |
| 418 \[ \left\{ \left\{ \neg a \right\}, \left\{ a, \neg b, c \right\} \right\} \text{ steht für } \neg a \wedge (a \vee \neg b \vee c) \] | |
| 419 \item $\emptyset$ steht für \textrm{true}, $\left\{ \emptyset \right\}$ für \textrm{false} | |
| 420 \end{itemize} | |
| 421 \end{block} | |
| 422 \begin{example}[] | |
| 423 Gegeben $F \defeq (a \vee b) \wedge \klausel{\vee}{\neg}{}{\neg} \wedge \klausel{\vee}{\neg}{\neg}{\neg}$ in KNF. | |
| 424 \[ \left\{ \left\{ a, b \right\}, \left\{ \neg a, b, \neg c \right\}, \left\{ \neg a, \neg b, \neg c \right\} \right\}\] | |
| 425 \end{example} | |
| 426 \end{frame} | |
| 427 } | |
| 428 } | |
| 429 | |
| 430 \defineUnit{DPLL}{% | |
| 431 \begin{frame} | |
| 432 \frametitle{KNF aus Syntaxbaum} | |
| 433 \setbeamercovered{dynamic} | |
| 434 | |
| 435 \begin{block}{Idee} | |
| 436 Erzeuge die KNF aus dem Syntaxbaum | |
| 437 \begin{enumerate} | |
| 438 \item<1-> Weise jedem \structure{inneren Knoten} eine Variable zu | |
| 439 \item<1,3-> Variablen sind \structure{abhängig} von ihren Kindern | |
| 440 \item<1,4-> Berechne \structure{kleine} KNFs und führe diese \structure{zusammen} | |
| 441 \end{enumerate} | |
| 442 \end{block} | |
| 443 \begin{columns}[T] | |
| 444 \begin{column}{.7\textwidth} | |
| 445 % FIXME: onlys around A_\vee needed for "correct" fadein | |
| 446 \begin{align} | |
| 447 (x \wedge y) \vee z | |
| 448 \uncover<3->{\equiv &\hphantom{{}\wedge {}}\only<3->{A_\vee}\\ | |
| 449 &\structure{\wedge (A_\vee \leftrightarrow A_\wedge \vee z)}\\ | |
| 450 &\alert{\wedge (A_\wedge \leftrightarrow x \wedge y)}\\} | |
| 451 \uncover<4->{\equiv &\hphantom{{}\wedge {}}\only<4->{A_\vee}\\ | |
| 452 &\structure{\wedge (A_\vee \vee \neg A_\wedge) \wedge (A_\vee \vee \neg z)}\\ | |
| 453 &\qquad\structure{\wedge (\neg A_\vee \vee A_\wedge \vee z)}\\ | |
| 454 &\alert{\wedge (\neg A_\wedge \vee x) \wedge (\neg A_\wedge \vee y)} \\ | |
| 455 &\qquad\alert{\wedge(A_\wedge \vee \neg x \vee \neg y)}} | |
| 456 \end{align} | |
| 457 \end{column} | |
| 458 \begin{column}{.3\textwidth} | |
| 459 \begin{tikzpicture}[grow=down, level distance = 33] | |
| 460 \tikzstyle{op} = [pretty] | |
| 461 \tikzstyle{var} = [pretty, rectangle] | |
| 462 \tikzstyle{edge from parent} = [edge] | |
| 463 | |
| 464 \tikzstyle{level 1} = [sibling distance = 50] | |
| 465 \tikzstyle{level 2} = [sibling distance = 30] | |
| 466 \node at (0, 0) {}; | |
| 467 \node[op] at (0, -1) {\alt<1>{$\vee$}{$A_\vee$}} | |
| 468 child { | |
| 469 node[op] {\alt<1>{$\wedge$}{$A_\wedge$}} | |
| 470 edge from parent | |
| 471 child { | |
| 472 node[var] {$x$} | |
| 473 edge from parent | |
| 474 } | |
| 475 child { | |
| 476 node[var] {$y$} | |
| 477 edge from parent | |
| 478 } | |
| 479 } | |
| 480 child { | |
| 481 node[var] {$z$} | |
| 482 }; | |
| 483 \end{tikzpicture} | |
| 484 \end{column} | |
| 485 \end{columns} | |
| 486 \end{frame} | |
| 487 | |
| 488 \begin{frame} | |
| 489 \frametitle{DPLL} | |
| 490 \setbeamercovered{dynamic} | |
| 491 | |
| 492 \begin{definition}[DPLL-Belegung] | |
| 493 Sei $F$ eine Formel in KNF und $p$ eine Variable von $F$.\\ | |
| 494 Dann bezeichnet \structure{$F[p\backslash\mathrm{true}]$} die Formel, die entsteht, wenn jedes Vorkommnis von $p$ in F durch $\mathrm{true}$ ersetzt und vereinfacht wird. | |
| 495 \end{definition} | |
| 496 | |
| 497 \begin{block}{DPLL} | |
| 498 Gegeben eine Formel $F$ in KNF | |
| 499 \begin{itemize} | |
|
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500 \item Wenn $F = \mathrm{true}$ dann antworte \structure{erfüllbar} |
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501 \item Wenn $F = \mathrm{false}$ dann antworte \structure{unerfüllbar} |
| 20 | 502 \item Sonst |
| 503 \begin{enumerate} | |
| 504 \item Wähle eine Variable $p$ in $F$ | |
| 505 \item Prüfe ob $F[p\backslash\mathrm{true}]$ oder $F[p\backslash\mathrm{false}]$ erfüllbar | |
| 506 \end{enumerate} | |
| 507 \end{itemize} | |
| 508 \end{block} | |
| 509 \begin{itemize} | |
| 510 \item Schlaue Wahl der Variable beschleunigt Ausführung | |
| 511 \item Wähle Variablen die einzeln stehen (\structure{One-Literal-Rule}) | |
| 512 \end{itemize} | |
| 513 \end{frame} | |
| 514 } | |
|
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515 |
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516 \defineUnit{resolution}{% |
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517 \begin{frame} |
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518 \frametitle{Resolution} |
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519 \setbeamercovered{dynamic} |
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520 |
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521 \begin{definition}[Resolvent] |
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522 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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523 \begin{align} |
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524 R &= \left( K_1 \setminus \left\{ L \right\} \right) \cup \left( K_2 \setminus \left\{ \neg L \right\} \right) |
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525 \end{align} |
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526 \end{definition} |
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527 |
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528 \begin{block}{Resolution} |
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529 Gegeben eine Formel $F$ in KNF in Mengendarstellung. |
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530 \begin{algorithmic} |
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531 \While{$\square = \emptyset \not \in F$} |
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532 \State{$R \gets \text{Resolvent aus } F \text{ mit } R \not\in F$} |
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533 \If{$R$ existiert} |
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534 \State{$F \gets F \cup R$} |
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535 \Else |
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536 \State{\textbf{return} \structure{erfüllbar}} |
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537 \EndIf |
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538 \EndWhile |
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539 \State{\textbf{return} \structure{unerfüllbar}} |
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540 \end{algorithmic} |
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541 \end{block} |
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542 \end{frame} |
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543 } |
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544 |
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545 \defineUnit{natuerlichesschliessen}{% |
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546 \begin{frame} |
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547 \frametitle{Kalküle} |
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548 \setbeamercovered{dynamic} |
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549 |
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550 \begin{definition}[Kalkül] |
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551 Ein \structure{Logikkalkül} stellt \structure{Inferenzregeln} bereit, mit denen Formeln \alert{syntaktisch} umgeformt werden können. |
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552 \end{definition} |
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553 |
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554 \begin{definition}[Folgerung] |
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555 $F$ \structure{folgt aus} $A$, wenn mit Hilfe der \alert{Semantik} der \alert{Aussagenlogik} $F$ unter der Annahme dass $A$ gilt zu $\mathrm{true}$ ausgewertet wird. Wir schreiben |
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556 \[ A \vDash F \] |
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557 \end{definition} |
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558 |
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559 \begin{definition}[Ableitung] |
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560 $F$ 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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561 \[ A \vdash F \] |
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562 \end{definition} |
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563 \end{frame} |
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564 |
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565 \begin{frame} |
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566 \frametitle{Eigenschaften von Kalkülen} |
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567 \setbeamercovered{dynamic} |
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568 |
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569 \begin{block}{Eigenschaften von Kalkülen} |
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570 \begin{description}[\quad vollständig (complete)] |
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571 \item[korrekt (sound)] Es können \alert{nur} semantisch gültige Formeln abgeleitet werden. |
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572 \[ \text{Aus } A \vdash F \text{ folgt } A \vDash F \] |
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573 \item[vollständig (complete)] \alert{Alle} semantisch gültigen Formeln können abgeleitet werden. |
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574 \[ \text{Aus } A \vDash F \text{ folgt } A \vdash F \] |
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575 \end{description} |
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576 \end{block} |
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577 |
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578 \begin{itemize} |
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579 \item Für uns nur korrekte vollständige Kalküle |
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580 \item Beispiel für die Aussagenlogik: \structure{Natürliches Schließen} |
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581 \medskip |
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582 \item Es gibt keine solchen Kalküle für die |
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583 \begin{itemize} |
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584 \item Prädikatenlogik |
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585 \item Arithmetik |
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586 \end{itemize} |
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587 \item Deshalb sind nicht alle Sätze der Mathematik beweisbar |
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588 \end{itemize} |
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589 \end{frame} |
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590 |
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591 { |
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592 \newcommand{\F}{\phi} |
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593 \newcommand{\G}{\psi} |
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594 \newcommand{\K}{\chi} |
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595 \newcommand{\subproof}[2]{% |
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596 \begin{tikzpicture}[y=.9em] |
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597 \path |
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598 (0, 0) node (a) {##1} |
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599 ++(0, -1) node (b) {$\vdots$} |
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600 ++(0, -1) node[below] (c) {##2}; |
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601 \node[fit=(a)(b)(c), draw, inner sep=1pt] {}; |
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602 \end{tikzpicture}% |
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603 } |
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604 \newcommand{\topproof}[1]{% |
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605 ##1 |
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606 \bottomAlignProof |
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607 \DisplayProof |
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608 } |
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609 |
|
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610 \begin{frame} |
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611 \frametitle{Natürliches Schließen} |
|
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612 \setbeamercovered{dynamic} |
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613 |
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614 \vspace{-4pt} |
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615 \tabulinesep=4pt |
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616 \begin{tabu} to \textwidth {X[c,m,.5]|[1pt]X[c,b,5]X[c,b,5]} |
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617 & \structure{Introduktion} & \structure{Elimination}\\\tabucline[1pt]{-} |
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618 $\wedge$ & |
|
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619 \topproof{ |
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620 \AxiomC{$\F$} |
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621 \AxiomC{$\G$} |
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622 \RightLabel{\scriptsize $+\wedge$} |
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623 \BinaryInfC{$\F \wedge \G$} |
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624 } |
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625 & |
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626 \topproof{ |
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627 \AxiomC{$\F \wedge \G$} |
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628 \RightLabel{\scriptsize $-\wedge_1$} |
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629 \UnaryInfC{$\F$} |
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630 } |
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631 \quad |
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632 \topproof{ |
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633 \AxiomC{$\F \wedge \G$} |
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634 \RightLabel{\scriptsize $-\wedge_2$} |
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635 \UnaryInfC{$\G$} |
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636 } |
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637 \\ |
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638 $\vee$ & |
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639 \topproof{ |
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640 \AxiomC{$\F$} |
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641 \RightLabel{\scriptsize $+\vee_1$} |
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642 \UnaryInfC{$\F \vee \G$} |
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643 } |
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644 \quad |
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645 \topproof{ |
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646 \AxiomC{$\G$} |
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647 \RightLabel{\scriptsize $+\vee_2$} |
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648 \UnaryInfC{$\F \vee \G$} |
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649 } |
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650 & |
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651 \topproof{ |
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parents:
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652 \AxiomC{$\F \vee \G$} |
|
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diff
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|
653 \AxiomC{\subproof{$\F$}{$\K$}} |
|
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|
654 \AxiomC{\subproof{$\G$}{$\K$}} |
|
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655 \RightLabel{\scriptsize $-\vee$} |
|
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656 \TrinaryInfC{$\K$} |
|
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657 } |
|
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658 \\ |
|
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659 $\rightarrow$ & |
|
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|
660 \topproof{ |
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661 \AxiomC{\subproof{$\F$}{$\G$}} |
|
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662 \RightLabel{\scriptsize $+\rightarrow$} |
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663 \UnaryInfC{$\F \rightarrow \G$} |
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664 } |
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665 & |
|
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666 \topproof{ |
|
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667 \AxiomC{$\F$} |
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668 \AxiomC{$\F \rightarrow \G$} |
|
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669 \RightLabel{\scriptsize $-\rightarrow$, MP} |
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670 \BinaryInfC{$\G$} |
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671 } |
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672 \\ |
|
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673 $\neg$ & |
|
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674 \topproof{ |
|
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675 \AxiomC{\subproof{$\F$}{$\bot$}} |
|
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676 \RightLabel{\scriptsize $+\neg$} |
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677 \UnaryInfC{$\neg \F$} |
|
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678 } |
|
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679 & |
|
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680 \topproof{ |
|
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681 \AxiomC{$\F$} |
|
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682 \AxiomC{$\neg\F$} |
|
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683 \RightLabel{\scriptsize $-\neg$} |
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684 \UnaryInfC{$\bot$} |
|
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685 } |
|
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|
686 \end{tabu} |
|
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687 \end{frame} |
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688 |
|
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689 \begin{frame} |
|
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690 \frametitle{Natürliches Schließen} |
|
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691 \setbeamercovered{dynamic} |
|
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|
692 |
|
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|
693 \vspace{-4pt} |
|
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694 \tabulinesep=4pt |
|
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695 \begin{tabu} to \textwidth {X[c,m,.5]|[1pt]X[c,b,5]X[c,b,5]} |
|
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696 & \structure{Introduktion} & \structure{Elimination}\\\tabucline[1pt]{-} |
|
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697 $\bot$ & |
|
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|
698 & |
|
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|
699 \topproof{ |
|
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|
700 \AxiomC{$\bot$} |
|
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701 \RightLabel{\scriptsize $-\bot$} |
|
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|
702 \UnaryInfC{$\F$} |
|
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|
703 } |
|
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|
704 \\ |
|
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|
705 $\neg\neg$ & |
|
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|
706 \topproof{ |
|
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707 \AxiomC{$\F$} |
|
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|
708 \RightLabel{\scriptsize $+\neg\neg$} |
|
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|
709 \UnaryInfC{$\neg\neg\F$} |
|
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|
710 } |
|
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|
711 & |
|
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|
712 \topproof{ |
|
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|
713 \AxiomC{$\neg\neg\F$} |
|
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|
714 \RightLabel{\scriptsize $-\neg\neg$} |
|
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715 \UnaryInfC{$\F$} |
|
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|
716 } |
|
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diff
changeset
|
717 \end{tabu} |
|
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|
718 \begin{itemize} |
|
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|
719 \vfill |
|
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|
720 \item Praktische abgeleitete Regeln |
|
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|
721 \end{itemize} |
|
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|
722 \tabulinesep=10pt |
|
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723 \begin{tabu} to \textwidth {X[c,m,.5]X[c,b,5]X[c,b,5]} |
|
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724 & |
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725 \topproof{ |
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726 \AxiomC{ } |
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727 \RightLabel{\scriptsize LEM} |
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728 \UnaryInfC{$\F \vee \neg\F$} |
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729 } |
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730 & |
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731 \topproof{ |
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732 \AxiomC{\subproof{$\neg\F$}{$\bot$}} |
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733 \RightLabel{\scriptsize $-\neg$, PBC} |
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734 \UnaryInfC{$\F$} |
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735 } |
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736 \\ |
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737 & |
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Markus Kaiser <markus.kaiser@in.tum.de>
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738 & |
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739 \topproof{ |
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740 \AxiomC{$\neg\G$} |
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741 \AxiomC{$\F \rightarrow \G$} |
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742 \RightLabel{\scriptsize MT} |
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743 \BinaryInfC{$\neg\F$} |
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744 } |
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745 \end{tabu} |
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746 \end{frame} |
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747 } |
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Markus Kaiser <markus.kaiser@in.tum.de>
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748 } |