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move frame-definitions out of presentations
author Markus Kaiser <markus.kaiser@in.tum.de>
date Thu, 11 Jul 2013 20:42:36 +0200
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children c14b92bfa07f
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5d10471f5585 move frame-definitions out of presentations
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1 \defineUnit{grammatik}{%
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2 \begin{frame}
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3 \frametitle{Grammatiken}
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4 \setbeamercovered{dynamic}
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5
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6 \begin{definition}[Kontextfreie Grammatik]
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7 Eine \alert{kontextfreie Grammatik} $G = (V, \Sigma, P, S)$ ist ein 4-Tupel:
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8 \begin{description}
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9 \item[V] endlich viele \alert{Nichtterminale} (Variablen)
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10 \item[$\Sigma$] ein Alphabet von \alert{Terminalen}
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11 \item[P] endlich viele \alert{Produktionen} $\subseteq V \times \left( V \cup \Sigma \right)^*$
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12 \item[S] ein \alert{Startsymbol}
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13 \end{description}
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14 \end{definition}
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15
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16 \begin{example}[Vorbereitung 3]
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17 $\Sigma = \left\{ 0, 1 \right\}$. Grammatik für alle Wörter ungerader Länge, bei denen alle Nullen vor der ersten Eins stehen und weniger Nullen als Einsen vorhanden sind.
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18 \pause
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19 \[
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20 S \rightarrow 0S1 \mid S11 \mid 1
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21 \]
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22 \end{example}
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23 \end{frame}
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24 }
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25
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26 \defineUnit{ableitung}{%
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27 \begin{frame}
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28 \frametitle{Ableitungsrelation}
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29 \setbeamercovered{dynamic}
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30
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31 \begin{definition}[Ableitungsrelation]
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32 Eine CFG $G$ induziert eine \alert{Ableitungsrelation} $\rightarrow_G$ auf Wörtern über $V \cup \Sigma$:
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33 \[
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34 \alpha \rightarrow_G \beta
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35 \]
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36 gdw es eine Regel $A \rightarrow \gamma$ in $P$ mit Wörtern $\alpha_1, \alpha_2$ gibt, so dass
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37 \[
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38 \alpha = \alpha_1\alert{A}\alpha_2 \quad \text{und} \quad \beta = \alpha_1 \alert{\gamma} \alpha_2
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39 \]
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40 \end{definition}
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41
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42 \begin{example}[Vorbereitung 3]
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43 Mit den Produktionen $S \rightarrow 0S1 \mid S11 \mid 1$:
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44 \begin{align*}
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45 S &\rightarrow_G 0S1 \rightarrow_G 00S11 \rightarrow_G 00S1111 \rightarrow_G 0011111 \\
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46 \Rightarrow S &\rightarrow_G^* 0011111
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47 \end{align*}
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48 \end{example}
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49 \end{frame}
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50 }
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51
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52 \defineUnit{cfl}{%
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53 \begin{frame}[c]
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54 \frametitle{Kontextfreie Sprache}
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55 \setbeamercovered{dynamic}
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56
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57 \begin{definition}[Kontextfreie Sprache]
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58 Eine kontextfreie Grammatik $G = (V, \Sigma, P, S)$ \alert{erzeugt} die Sprache
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59 \[
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60 L(G) := \left\{ w \in \Sigma^* \mid S \rightarrow_G^* w \right\}
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61 \]
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62 Eine Sprache $L \subseteq \Sigma^*$ heißt \alert{kontextfrei} gdw es eine kontextfreie Grammatik $G$ gibt mit $L = L(G)$.
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63 \end{definition}
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64 \end{frame}
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65 }
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66
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67 \defineUnit{cnf}{%
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68 \begin{frame}
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69 \frametitle{CNF}
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70 \setbeamercovered{dynamic}
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71
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72 \begin{definition}[Chomsky-Normalform]
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73 Eine kontextfreie Grammatik ist in \alert{Chomsky-Normalform} (CNF) genau dann wenn alle Produktionen die Form
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74 \[
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75 A \rightarrow \alert{a} \quad \text{oder} \quad A \rightarrow \alert{BC}
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76 \]
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77 haben.
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78 \end{definition}
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79
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80 \vfill
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81
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82 \begin{theorem}
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83 Zu \alert{jeder} CFG $G$ existiert eine CFG $G'$ in Chomsky-Normalform mit
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84 \[
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85 L(G') = L(G) \alert{\setminus \left\{ \epsilon \right\}}
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86 \]
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87 \end{theorem}
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88 \end{frame}
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89 }
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90
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91 \defineUnit{cnfkonstruktion}{%
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92 \begin{frame}
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93 \frametitle{CNF Konstruktion}
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94 \setbeamercovered{dynamic}
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95
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96 \begin{block}{Idee}
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97 Sei $G = (V, \Sigma, P, S)$ eine CFG.
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98 \begin{enumerate}
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99 \item<1,2-> Eliminiere \alert{$\epsilon$-Produktionen}
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100 \item<1,3-> Eliminiere \alert{Kettenproduktionen}
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101 \item<1,4-> \alert{Ersetze Terminale} durch Nichtterminale
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102 \item<1,5-> \alert{Verkürze Ketten} von Nichtterminalen der Länge $\geq 3$
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103 \end{enumerate}
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104 \end{block}
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105
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106 \vspace{1em}
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107
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108 \only<2> {
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109 Sind \alert{$B \rightarrow \epsilon$} und \alert{$A \rightarrow \alpha B \beta$} in $P$, dann füge \alert{$A \rightarrow \alpha \beta$} hinzu. Entferne danach alle $\epsilon$-Produktionen.
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110 \begin{align*}
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111 S &\rightarrow Ab, \quad A \rightarrow aAA \mid \epsilon \\
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112 \intertext{neu:}
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113 S &\rightarrow \alert{b} \\
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114 A &\rightarrow \alert{aA \mid a}
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115 \end{align*}
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116 }
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117
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118 \only<3> {
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119 Sind \alert{$A \rightarrow B$} und \alert{$B \rightarrow \alpha$} in $P$, dann füge \alert{$A \rightarrow \alpha$} hinzu. Entferne danach alle Kettenproduktionen und unerreichbaren Symbole.
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120 \begin{align*}
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121 S &\rightarrow A, \quad A \rightarrow a \mid B, \quad B \rightarrow bS \\
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122 \intertext{neu:}
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123 A &\rightarrow \alert{a \mid bS} \\
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124 S &\rightarrow \alert{a \mid bS}
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125 \end{align*}
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126 }
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127
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128 \only<4> {
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129 Ersetze jedes \alert{$a \in \Sigma$} in einer rechten Seite \alert{länger als $1$} durch ein neues Nichtterminal.
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130 \begin{align*}
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131 S &\rightarrow aa \mid Bb \mid b, \quad B \rightarrow \ldots \\
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132 \intertext{neu:}
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133 S &\rightarrow \alert{X_aX_a \mid BX_b \mid b} \\
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134 X_a &\rightarrow \alert{a}, \quad X_b \rightarrow \alert{b}
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135 \end{align*}
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136 }
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137
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138 \only<5> {
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139 Ersetze jede Produktion der Form $A \rightarrow B_1B_2\ldots B_k$ durch neue Nichtterminale mit Produktionen der Länge $2$.
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140 \begin{align*}
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141 S &\rightarrow X_aX_bBX_a, \quad X_a \rightarrow a, \quad X_b \rightarrow b, \quad B \rightarrow \ldots \\
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142 \intertext{neu:}
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143 S &\rightarrow \alert{X_aT_1} \\
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144 T_1 &\rightarrow \alert{X_bT_2}, \quad T_2 \rightarrow \alert{BX_a} \\
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145 \end{align*}
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146 }
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147 \end{frame}
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148 }
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149
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150 \defineUnit{nuetzlichessymbol}{%
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151 \begin{frame}
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152 \frametitle{Eigenschaften von Symbolen}
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153 \setbeamercovered{dynamic}
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154
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155 \begin{definition}
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156 Sei $G = (V, \Sigma, P, S)$ eine CFG. \\
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157 Ein Symbol $X \in V \cup \Sigma$ ist
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158 \begin{description}
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159 \item[nützlich] es gibt $S \rightarrow_G^* w \in \Sigma^*$ in der X \alert{vorkommt}
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160 \item[erzeugend] es gibt $\alert{X} \rightarrow_G^* w \in \Sigma^*$
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161 \item[erreichbar] es gibt $S \rightarrow_G^* \alpha \alert{X} \beta$
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162 \end{description}
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163 \end{definition}
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164
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165 \vfill
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166
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167 \begin{theorem}
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168 Nützliche Symbole \alert{sind} erzeugend und erreichbar. Aber \alert{nicht} notwendigerweise umgekehrt.
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169 \[
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170 S \rightarrow AB \mid a, \quad A \rightarrow b
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171 \]
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172 \end{theorem}
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173 \end{frame}
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174 }
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175
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176 \defineUnit{cyk}{%
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177 \begin{frame}
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178 \frametitle{CYK}
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179 \setbeamercovered{dynamic}
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180
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181 \begin{definition}[Cocke-Younger-Kasami-Algorithmus]
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182 Der \alert{CYK-Algorithmus} entscheidet das Wortproblem für kontextfreie Grammatiken in Chomsky-Normalform in $\Oh(n^3)$. \\
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183 Gegeben eine \alert{Grammatik} $G = (V, \Sigma, P, S)$ in CNF und ein \alert{Wort} $w = a_1 \ldots a_n \in \Sigma^*$.
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184 Mit \[ V_{ij} := \left\{ A \in V \mid A \rightarrow_G^* \alert{a_i \ldots a_j} \right\}\]
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185 ist \[ w \in L(G) \Leftrightarrow S \in V_{\alert{1n}} \]
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186 \end{definition}
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187
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188 \begin{align*}
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189 V_{ii} &= \left\{ A \in V \mid (A \rightarrow a_i) \in P \right\} \\
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190 V_{ij} &= \left\{ A \in V \mid \exists k, B \in V_{ik}, C \in V_{k+1,j} \;.\; (A \rightarrow BC) \in P \right\}
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191 \end{align*}
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192 \end{frame}
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193 }
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194
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195 \defineUnit{cykbeispiel}{%
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196 \begin{frame}
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197 \frametitle{CYK}
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198 \setbeamercovered{dynamic}
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199
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200 \begin{block}{Idee}
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201 Kombiniere \alert{Teilwörter} zum ganzen Wort, wenn möglich.
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202 \begin{enumerate}
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203 \item Initialisiere mit den \alert{$V_{ii}$}.
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204 \item<3-5> Befülle die Tabelle von unten nach oben.
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205 \end{enumerate}
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206 \end{block}
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207
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208 \[ S \rightarrow AB \mid BC, \quad A \rightarrow BA \mid a, \quad B \rightarrow CC \mid b, \quad C \rightarrow AB \mid a \]
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209 \begin{center}
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210 \extrarowsep=5pt
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211 \begin{tabu}to .8\textwidth{r|X[c]|X[c]|X[c]|X[c]|}
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212 \tabucline{2-2}
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213 4 & \alt<-4>{}{$S,\ldots$} \\ \tabucline{2-3}
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214 3 & \alt<-3>{}{$\emptyset$} & \alt<-3>{}{$S, A, C$} \\ \tabucline{2-4}
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215 2 & \alt<-2>{}{$A$} & \alt<-2>{}{$B$} & \alt<-2>{}{$B$} \\ \tabucline{2-5}
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216 1 & \alt<-1>{}{$B$} & \alt<1>{}{$A,C$} & \alt<1>{}{$A,C$} & \alt<1>{}{$A,C$} \\ \tabucline{2-5}
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217 \multicolumn{1}{r}{} & \multicolumn{1}{c}{\alert{b}} & \multicolumn{1}{c}{\alert{a}} & \multicolumn{1}{c}{\alert{a}} & \multicolumn{1}{c}{\alert{a}} \\
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218 \end{tabu}
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219 \end{center}
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220 \end{frame}
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221 }
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222
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223 \defineUnit{pda}{%
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224 \begin{frame}
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225 \frametitle{Kellerautomaten}
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226 \setbeamercovered{dynamic}
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227
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228 \begin{definition}[Kellerautomat]
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229 Ein \alert{PDA} (Push-Down-Automaton) ist ein Tupel $P = (Q, \Sigma, \Gamma, \delta, q_0, Z_0, F)$ aus einer/einem
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230 \begin{itemize}
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231 \item endlichen Menge von \alert{Zuständen} $Q$
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232 \item endlichen \alert{Eingabealphabet} $\Sigma$
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233 \item endlichen \alert{Kelleralphabet} $\Gamma$
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234 \item \alert{Übergangsfunktion} $\delta : Q \times \left( \Sigma \cup \{\epsilon\} \right) \times \Gamma \to P(Q \times \Gamma^*)$
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235 \item \alert{Startzustand} $q_0 \in Q$
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236 \item \alert{Kellerinitialisierung} $Z_0 \in \Gamma$
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237 \item Menge von \alert{Endzuständen} $F \subseteq Q$
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238 \end{itemize}
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239 \end{definition}
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240
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241 \begin{center}
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242 \begin{tikzpicture}[automaton, node distance=4cm]
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243 \node[state] (q0) {$q_i$};
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244 \node[state] (q1) [right of = q0] {$q_j$};
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245
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246 \draw[every edge] (q0) edge node {$a, X/\gamma$} (q1);
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247 \end{tikzpicture}
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248 \end{center}
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249 \end{frame}
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250 }
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251
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252 \defineUnit{pdaakzeptanz}{%
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253 \begin{frame}
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254 \frametitle{Kellerautomaten}
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255 \setbeamercovered{dynamic}
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256
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257 \begin{definition}[Kellerautomat]
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258 Ein \alert{PDA} (Push-Down-Automaton) ist ein Tupel $P = (Q, \Sigma, \Gamma, \delta, q_0, Z_0, F)$ aus einer/einem
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259 \begin{itemize}
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260 \item \alert{Übergangsfunktion} $\delta : Q \times \left( \Sigma \cup \{\epsilon\} \right) \times \Gamma \to P(Q \times \Gamma^*)$
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261 \end{itemize}
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262 \end{definition}
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263
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264 \vfill
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265
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266 \begin{definition}[Akzeptanz]
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267 Ein PDA $P$ akzeptiert $w \in \Sigma^*$ \alert{mit Endzustand} gdw
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268 \[ \exists \alert{f \in F}, \gamma \in \Gamma^*.(q_0, w, Z_0) \rightarrow_P^* (\alert{f}, \epsilon, \gamma) \]
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269 Ein PDA $P$ akzeptiert $w \in \Sigma^*$ \alert{mit leerem Keller} gdw
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270 \[ \exists q \in Q.(q_0, w, Z_0) \rightarrow_P^* (q, \epsilon, \alert{\epsilon}) \]
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271 \end{definition}
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272 \end{frame}
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273 }
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274
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275 \defineUnit{pdabeispiel}{%
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276 \begin{frame}
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277 \frametitle{Kellerautomaten}
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278 \setbeamercovered{dynamic}
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279
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280 \begin{definition}[Kellerautomat]
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281 Ein \alert{PDA} (Push-Down-Automaton) ist ein Tupel $P = (Q, \Sigma, \Gamma, \delta, q_0, Z_0, F)$ aus einer/einem
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282 \begin{itemize}
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283
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284 \item \alert{Übergangsfunktion} $\delta : Q \times \left( \Sigma \cup \{\epsilon\} \right) \times \Gamma \to P(Q \times \Gamma^*)$
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285 \end{itemize}
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286 \end{definition}
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287
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288 \vfill
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289
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290 \begin{example}[]
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291 PDA akzeptierend \alert{mit leerem Keller} zu $L = \left\{ a^nb^n \mid n \in \N \right\}$.
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292
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293 \centering
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294 \begin{tikzpicture}[automaton]
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295
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296 \node[state, initial] (q0) {$q_0$};
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297 \node[state] (q1) [right of = q0] {$q_1$};
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298
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299 \draw[->] (q0) edge [bend left] node {$\epsilon, A/A$} (q1);
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300 \draw[->] (q0) edge [bend right] node [below] {$\epsilon, Z_0/Z_0$} (q1);
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301
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302 \draw[->] (q0) edge [loop above] node {$a, Z_0/AZ_0$} (q0);
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303 \draw[->] (q0) edge [loop below] node {$a, A/AA$} (q0);
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304
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305 \draw[->] (q1) edge [loop above] node {$b, A/\epsilon$} (q1);
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306 \draw[->] (q1) edge [loop below] node {$\epsilon, Z_0/\epsilon$} (q1);
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307 \end{tikzpicture}
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308 \end{example}
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309 \end{frame}
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310 }