1 (*<*) |
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2 theory Slides8 |
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3 imports "~~/src/HOL/Library/LaTeXsugar" "Nominal" |
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4 begin |
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5 |
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6 declare [[show_question_marks = false]] |
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7 |
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8 notation (latex output) |
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9 set ("_") and |
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10 Cons ("_::/_" [66,65] 65) |
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11 |
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12 (*>*) |
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13 |
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14 text_raw {* |
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15 \renewcommand{\slidecaption}{Copenhagen, 23rd~May 2011} |
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16 |
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17 \newcommand{\abst}[2]{#1.#2}% atom-abstraction |
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18 \newcommand{\pair}[2]{\langle #1,#2\rangle} % pairing |
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19 \newcommand{\susp}{{\boldsymbol{\cdot}}}% for suspensions |
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20 \newcommand{\unit}{\langle\rangle}% unit |
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21 \newcommand{\app}[2]{#1\,#2}% application |
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22 \newcommand{\eqprob}{\mathrel{{\approx}?}} |
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23 \newcommand{\freshprob}{\mathrel{\#?}} |
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24 \newcommand{\redu}[1]{\stackrel{#1}{\Longrightarrow}}% reduction |
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25 \newcommand{\id}{\varepsilon}% identity substitution |
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26 |
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27 \newcommand{\bl}[1]{\textcolor{blue}{#1}} |
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30 |
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31 \newcommand{\ok}{\includegraphics[scale=0.07]{ok.png}} |
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32 \newcommand{\notok}{\includegraphics[scale=0.07]{notok.png}} |
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34 |
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37 \newcommand{\VeryHuge}{\fontsize{89.16}{112}\selectfont} |
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38 \newcommand{\VERYHuge}{\fontsize{107}{134}\selectfont} |
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39 |
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40 \newcommand{\LL}{$\mathbb{L}\,$} |
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41 |
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42 |
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43 \pgfdeclareradialshading{smallbluesphere}{\pgfpoint{0.5mm}{0.5mm}}% |
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48 |
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51 {\pgftransformscale{0.8}\pgftext{\normalsize\pgfuseshading{bigsphere}}} |
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52 \pgftext{% |
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53 \usebeamerfont*{subitem projected}} |
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54 \end{pgfpicture}} |
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55 |
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56 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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57 \mode<presentation>{ |
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58 \begin{frame}<1>[t] |
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59 \frametitle{% |
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60 \begin{tabular}{@ {\hspace{-3mm}}c@ {}} |
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61 \\ |
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62 \LARGE Verifying a Regular Expression\\[-1mm] |
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63 \LARGE Matcher and Formal Language\\[-1mm] |
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64 \LARGE Theory\\[5mm] |
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65 \end{tabular}} |
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66 \begin{center} |
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67 Christian Urban\\ |
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68 \small Technical University of Munich, Germany |
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69 \end{center} |
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70 |
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71 |
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72 \begin{center} |
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73 \small joint work with Chunhan Wu and Xingyuan Zhang from the PLA |
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74 University of Science and Technology in Nanjing |
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75 \end{center} |
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76 \end{frame}} |
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77 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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78 |
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79 *} |
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80 |
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81 |
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82 text_raw {* |
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83 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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84 \mode<presentation>{ |
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85 \begin{frame}[c] |
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86 \frametitle{This Talk: 4 Points} |
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87 \large |
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88 \begin{itemize} |
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89 \item It is easy to make mistakes.\medskip |
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90 \item Theorem provers can prevent mistakes, {\bf if} the problem |
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91 is formulated so that it is suitable for theorem provers.\medskip |
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92 \item This re-formulation can be done, even in domains where |
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93 we least expect it.\medskip |
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94 \item Where theorem provers are superior to the {\color{gray}{(best)}} human reasoners. ;o) |
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95 \end{itemize} |
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96 |
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97 \end{frame}} |
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98 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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99 *} |
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100 |
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101 |
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102 text_raw {* |
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103 |
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104 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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105 \mode<presentation>{ |
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106 \begin{frame}[c] |
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107 \frametitle{} |
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108 |
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109 \begin{tabular}{c@ {\hspace{2mm}}c} |
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110 \\[6mm] |
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111 \begin{tabular}{c} |
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112 \includegraphics[scale=0.12]{harper.jpg}\\[-2mm] |
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113 {\footnotesize Bob Harper}\\[-2.5mm] |
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114 {\footnotesize (CMU)} |
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115 \end{tabular} |
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116 \begin{tabular}{c} |
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117 \includegraphics[scale=0.36]{pfenning.jpg}\\[-2mm] |
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118 {\footnotesize Frank Pfenning}\\[-2.5mm] |
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119 {\footnotesize (CMU)} |
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120 \end{tabular} & |
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121 |
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122 \begin{tabular}{p{6cm}} |
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123 \raggedright |
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124 \color{gray}{published a proof in\\ {\bf ACM Transactions on Computational Logic} (2005), |
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125 $\sim$31pp} |
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126 \end{tabular}\\ |
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127 |
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128 \pause |
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129 \\[0mm] |
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130 |
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131 \begin{tabular}{c} |
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132 \includegraphics[scale=0.36]{appel.jpg}\\[-2mm] |
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133 {\footnotesize Andrew Appel}\\[-2.5mm] |
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134 {\footnotesize (Princeton)} |
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135 \end{tabular} & |
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136 |
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137 \begin{tabular}{p{6cm}} |
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138 \raggedright |
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139 \color{gray}{relied on their proof in a\\ {\bf security} critical application} |
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140 \end{tabular} |
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141 \end{tabular} |
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142 |
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143 |
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144 |
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145 |
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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 *} |
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151 |
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152 text_raw {* |
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153 |
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154 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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155 \mode<presentation>{ |
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156 \begin{frame} |
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157 \frametitle{Proof-Carrying Code} |
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158 |
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159 \begin{textblock}{10}(2.5,2.2) |
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160 \begin{block}{Idea:} |
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172 |
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176 {\small\begin{tabular}{@ {}p{1.9cm}@ {}}\bf\centering proof- checker\end{tabular}};} |
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177 |
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178 \node at (3.8,3.0) [single arrow, fill=red,text=white, minimum height=3cm]{\bf code}; |
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179 \onslide<2->{ |
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180 \node at (3.8,1.3) [single arrow, fill=red,text=white, minimum height=3cm]{\bf certificate}; |
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181 \node at (3.8,1.9) {\small\color{gray}{\mbox{}\hspace{-1mm}a proof in LF}}; |
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182 } |
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183 |
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184 |
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185 \end{tikzpicture} |
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186 \end{center} |
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187 \end{block} |
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188 \end{textblock} |
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189 |
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190 %\begin{textblock}{15}(2,12) |
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191 %\small |
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192 %\begin{itemize} |
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193 %\item<4-> Appel's checker is $\sim$2700 lines of code (1865 loc of\\ LF definitions; |
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194 %803 loc in C including 2 library functions)\\[-3mm] |
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195 %\item<5-> 167 loc in C implement a type-checker |
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196 %\end{itemize} |
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197 %\end{textblock} |
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198 |
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199 \end{frame}} |
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200 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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201 |
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202 *} |
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203 |
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204 text {* |
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205 \tikzstyle{every node}=[node distance=25mm,text height=1.5ex, text depth=.25ex] |
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222 { \&[-10mm] |
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223 \node (def1) [node1] {\large\hspace{1mm}Spec\hspace{1mm}\mbox{}}; \& |
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224 \node (proof1) [node1] {\large Proof}; \& |
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225 \node (alg1) [node1] {\large\hspace{1mm}Alg\hspace{1mm}\mbox{}}; \\ |
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226 |
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235 \onslide<5->{\node (alg3) [node2] {\large Alg$^\text{-ex}$};} \\ |
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241 }; |
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254 |
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256 \end{tikzpicture} |
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257 |
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258 \end{textblock} |
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259 \end{column} |
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260 \end{columns} |
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261 |
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262 |
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263 \begin{textblock}{3}(12,3.6) |
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264 \onslide<4->{ |
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265 \begin{tikzpicture} |
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266 \node at (0,0) [single arrow, shape border rotate=270, fill=red,text=white]{2h}; |
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267 \end{tikzpicture}} |
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268 \end{textblock} |
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269 |
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270 \end{frame}} |
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271 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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272 |
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273 *} |
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274 |
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275 |
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276 (*<*) |
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277 atom_decl name |
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278 |
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279 nominal_datatype lam = |
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280 Var "name" |
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281 | App "lam" "lam" |
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282 | Lam "\<guillemotleft>name\<guillemotright>lam" ("Lam [_]._" [100,100] 100) |
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283 |
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284 nominal_primrec |
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285 subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_[_::=_]") |
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286 where |
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287 "(Var x)[y::=s] = (if x=y then s else (Var x))" |
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288 | "(App t\<^isub>1 t\<^isub>2)[y::=s] = App (t\<^isub>1[y::=s]) (t\<^isub>2[y::=s])" |
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289 | "x\<sharp>(y,s) \<Longrightarrow> (Lam [x].t)[y::=s] = Lam [x].(t[y::=s])" |
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290 apply(finite_guess)+ |
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291 apply(rule TrueI)+ |
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292 apply(simp add: abs_fresh) |
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293 apply(fresh_guess)+ |
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294 done |
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295 |
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296 lemma subst_eqvt[eqvt]: |
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297 fixes pi::"name prm" |
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298 shows "pi\<bullet>(t1[x::=t2]) = (pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)]" |
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299 by (nominal_induct t1 avoiding: x t2 rule: lam.strong_induct) |
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300 (auto simp add: perm_bij fresh_atm fresh_bij) |
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301 |
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302 lemma fresh_fact: |
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303 fixes z::"name" |
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304 shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]" |
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305 by (nominal_induct t avoiding: z y s rule: lam.strong_induct) |
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306 (auto simp add: abs_fresh fresh_prod fresh_atm) |
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307 |
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308 lemma forget: |
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309 assumes asm: "x\<sharp>L" |
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310 shows "L[x::=P] = L" |
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311 using asm |
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312 by (nominal_induct L avoiding: x P rule: lam.strong_induct) |
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313 (auto simp add: abs_fresh fresh_atm) |
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314 (*>*) |
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315 |
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316 text_raw {* |
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317 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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318 \mode<presentation>{ |
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319 \begin{frame} |
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320 |
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321 \begin{textblock}{16}(1,1) |
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322 \renewcommand{\isasymbullet}{$\cdot$} |
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323 \tiny\color{black} |
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324 *} |
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325 lemma substitution_lemma_not_to_be_tried_at_home: |
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326 assumes asm: "x\<noteq>y" "x\<sharp>L" |
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327 shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]" |
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328 using asm |
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329 proof (induct M arbitrary: x y N L rule: lam.induct) |
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330 case (Lam z M1) |
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331 have ih: "\<And>x y N L. \<lbrakk>x\<noteq>y; x\<sharp>L\<rbrakk> \<Longrightarrow> M1[x::=N][y::=L] = M1[y::=L][x::=N[y::=L]]" by fact |
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332 have "x\<noteq>y" by fact |
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333 have "x\<sharp>L" by fact |
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334 obtain z'::"name" where fc: "z'\<sharp>(x,y,z,M1,N,L)" by (rule exists_fresh) (auto simp add: fs_name1) |
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335 have eq: "Lam [z'].([(z',z)]\<bullet>M1) = Lam [z].M1" using fc |
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336 by (auto simp add: lam.inject alpha fresh_prod fresh_atm) |
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337 have fc': "z'\<sharp>N[y::=L]" using fc by (simp add: fresh_fact fresh_prod) |
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338 have "([(z',z)]\<bullet>x) \<noteq> ([(z',z)]\<bullet>y)" using `x\<noteq>y` by (auto simp add: calc_atm) |
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339 moreover |
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340 have "([(z',z)]\<bullet>x)\<sharp>([(z',z)]\<bullet>L)" using `x\<sharp>L` by (simp add: fresh_bij) |
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341 ultimately |
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342 have "M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)] |
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343 = M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]]" |
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344 using ih by simp |
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345 then have "[(z',z)]\<bullet>(M1[([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)][([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)] |
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346 = M1[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)][([(z',z)]\<bullet>x)::=([(z',z)]\<bullet>N)[([(z',z)]\<bullet>y)::=([(z',z)]\<bullet>L)]])" |
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347 by (simp add: perm_bool) |
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348 then have ih': "([(z',z)]\<bullet>M1)[x::=N][y::=L] = ([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]]" |
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349 by (simp add: eqvts perm_swap) |
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350 show "(Lam [z].M1)[x::=N][y::=L] = (Lam [z].M1)[y::=L][x::=N[y::=L]]" (is "?LHS=?RHS") |
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351 proof - |
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352 have "?LHS = (Lam [z'].([(z',z)]\<bullet>M1))[x::=N][y::=L]" using eq by simp |
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353 also have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[x::=N][y::=L])" using fc by (simp add: fresh_prod) |
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354 also from ih have "\<dots> = Lam [z'].(([(z',z)]\<bullet>M1)[y::=L][x::=N[y::=L]])" sorry |
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355 also have "\<dots> = (Lam [z'].([(z',z)]\<bullet>M1))[y::=L][x::=N[y::=L]]" using fc fc' by (simp add: fresh_prod) |
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356 also have "\<dots> = ?RHS" using eq by simp |
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357 finally show "?LHS = ?RHS" . |
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358 qed |
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359 qed (auto simp add: forget) |
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360 text_raw {* |
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361 \end{textblock} |
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362 \mbox{} |
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363 |
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364 \only<2->{ |
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365 \begin{textblock}{11.5}(4,2.3) |
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366 \begin{minipage}{9.3cm} |
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367 \begin{block}{}\footnotesize |
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368 *} |
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369 lemma substitution_lemma\<iota>: |
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370 assumes asm: "x \<noteq> y" "x \<sharp> L" |
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371 shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]" |
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372 using asm |
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373 by (nominal_induct M avoiding: x y N L rule: lam.strong_induct) |
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374 (auto simp add: forget fresh_fact) |
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375 text_raw {* |
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376 \end{block} |
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377 \end{minipage} |
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378 \end{textblock}} |
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379 \end{frame}} |
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380 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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381 *} |
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382 |
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383 |
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384 text_raw {* |
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385 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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386 \mode<presentation>{ |
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387 \begin{frame}<1->[c] |
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388 \frametitle{Lesson Learned} |
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389 |
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390 \begin{textblock}{11.5}(1.2,5) |
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391 \begin{minipage}{10.5cm} |
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392 \begin{block}{} |
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393 Theorem provers can keep large proofs and definitions consistent and |
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394 make them modifiable. |
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395 \end{block} |
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396 \end{minipage} |
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397 \end{textblock} |
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398 |
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399 |
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400 \end{frame}} |
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401 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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402 *} |
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403 |
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404 text_raw {* |
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405 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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406 \mode<presentation>{ |
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407 \begin{frame} |
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408 \frametitle{} |
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409 |
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410 \begin{textblock}{11.5}(0.8,2.3) |
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411 \begin{minipage}{11.2cm} |
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412 In most papers/books: |
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413 \begin{block}{} |
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414 \color{darkgray} |
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415 ``\ldots this necessary hygienic discipline is somewhat swept under the carpet via |
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416 the so-called `{\bf variable convention}' \ldots |
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417 The {\color{black}{\bf belief}} that this is {\bf sound} came from the calculus |
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418 with nameless binders in de Bruijn'' |
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419 \end{block}\medskip |
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420 \end{minipage} |
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421 \end{textblock} |
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422 |
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423 \begin{textblock}{11.5}(0.8,10) |
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424 \includegraphics[scale=0.25]{LambdaBook.jpg}\hspace{-3mm}\includegraphics[scale=0.3]{barendregt.jpg} |
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425 \end{textblock} |
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426 \end{frame}} |
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427 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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428 *} |
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429 |
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430 |
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431 text_raw {* |
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432 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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433 \mode<presentation>{ |
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434 \begin{frame}<1->[t] |
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435 \frametitle{Regular Expressions} |
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436 |
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437 \begin{textblock}{6}(2,4) |
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438 \begin{tabular}{@ {}rrl} |
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439 \bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\ |
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440 & \bl{$\mid$} & \bl{[]}\\ |
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441 & \bl{$\mid$} & \bl{c}\\ |
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442 & \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\ |
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443 & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\ |
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444 & \bl{$\mid$} & \bl{r$^*$}\\ |
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445 \end{tabular} |
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446 \end{textblock} |
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447 |
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448 \begin{textblock}{6}(8,3.5) |
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449 \includegraphics[scale=0.35]{Screen1.png} |
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450 \end{textblock} |
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451 |
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452 \begin{textblock}{6}(10.2,2.8) |
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453 \footnotesize Isabelle: |
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454 \end{textblock} |
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455 |
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456 \only<2>{ |
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457 \begin{textblock}{9}(3.6,11.8) |
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458 \bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm] |
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459 |
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460 \hspace{10mm}\begin{tikzpicture} |
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461 \coordinate (m1) at (0.4,1); |
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462 \draw (0,0.3) node (m2) {\small\color{gray}rexp}; |
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463 \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1); |
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464 |
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465 \coordinate (s1) at (0.81,1); |
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466 \draw (1.3,0.3) node (s2) {\small\color{gray} string}; |
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467 \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1); |
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468 \end{tikzpicture} |
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469 \end{textblock}} |
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470 |
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471 |
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472 |
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473 \end{frame}} |
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474 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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475 *} |
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476 |
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477 text_raw {* |
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478 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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479 \mode<presentation>{ |
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480 \begin{frame}<1->[t] |
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481 \frametitle{Specification} |
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482 |
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483 \small |
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484 \begin{textblock}{6}(0,3.5) |
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485 \begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l} |
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486 \multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\ |
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487 &\bl{\LL ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\ |
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488 &\bl{\LL ([])} & \bl{$\dn$} & \bl{\{[]\}}\\ |
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489 &\bl{\LL (c)} & \bl{$\dn$} & \bl{\{c\}}\\ |
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490 &\bl{\LL (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\ |
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491 \rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\ |
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492 \rd{$\Rightarrow$} &\bl{\LL (r$^*$)} & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\ |
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493 \end{tabular} |
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494 \end{textblock} |
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495 |
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496 \begin{textblock}{9}(7.3,3) |
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497 {\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip} |
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498 \includegraphics[scale=0.325]{Screen3.png} |
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499 \end{textblock} |
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500 |
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501 \end{frame}} |
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502 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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503 *} |
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504 |
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505 |
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506 text_raw {* |
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507 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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508 \mode<presentation>{ |
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509 \begin{frame}<1->[t] |
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510 \frametitle{Version 1} |
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511 \small |
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512 \mbox{}\\[-8mm]\mbox{} |
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513 |
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514 \begin{center}\def\arraystretch{1.05} |
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515 \begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}} |
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516 \bl{match [] []} & \bl{$=$} & \bl{true}\\ |
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517 \bl{match [] (c::s)} & \bl{$=$} & \bl{false}\\ |
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518 \bl{match ($\varnothing$::rs) s} & \bl{$=$} & \bl{false}\\ |
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519 \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\ |
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520 \bl{match (c::rs) []} & \bl{$=$} & \bl{false}\\ |
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521 \bl{match (c::rs) (d::s)} & \bl{$=$} & \bl{if c = d then match rs s else false}\\ |
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522 \bl{match (r$_1$ + r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::rs) s $\vee$ match (r$_2$::rs) s}\\ |
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523 \bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\ |
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524 \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\ |
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525 \end{tabular} |
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526 \end{center} |
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527 |
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528 \begin{textblock}{9}(0.2,1.6) |
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529 \hspace{10mm}\begin{tikzpicture} |
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530 \coordinate (m1) at (0.44,-0.5); |
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531 \draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps}; |
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532 \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1); |
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533 |
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534 \coordinate (s1) at (0.86,-0.5); |
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535 \draw (1.5,0.3) node (s2) {\small\color{gray} string}; |
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536 \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1); |
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537 \end{tikzpicture} |
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538 \end{textblock} |
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539 |
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540 \begin{textblock}{9}(2.8,11.8) |
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541 \bl{matches$_1$ r s $\;=\;$ match [r] s} |
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542 \end{textblock} |
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543 |
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544 \end{frame}} |
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545 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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546 *} |
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547 |
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548 text_raw {* |
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549 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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550 \mode<presentation>{ |
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551 \begin{frame}<1->[c] |
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552 \frametitle{Testing} |
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553 |
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554 \small |
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555 Every good programmer should do thourough tests: |
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556 |
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557 \begin{center} |
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558 \begin{tabular}{@ {\hspace{-20mm}}lcl} |
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559 \bl{matches$_1$ (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\ |
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560 \bl{matches$_1$ (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\ |
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561 \bl{matches$_1$ (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\ |
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562 \bl{matches$_1$ (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\ |
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563 \bl{matches$_1$ (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\ |
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564 \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\ |
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565 \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\ |
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566 \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false}} |
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567 \end{tabular} |
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568 \end{center} |
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569 |
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570 \onslide<3-> |
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571 {Looks OK \ldots let's ship it to customers\hspace{5mm} |
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572 \raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}} |
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573 |
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574 \end{frame}} |
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575 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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576 *} |
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577 |
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578 text_raw {* |
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579 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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580 \mode<presentation>{ |
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581 \begin{frame}<1->[c] |
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582 \frametitle{Version 1} |
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583 |
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584 \only<1->{Several hours later\ldots}\pause |
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585 |
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586 |
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587 \begin{center} |
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588 \begin{tabular}{@ {\hspace{0mm}}lcl} |
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589 \bl{matches$_1$ []$^*$ s} & \bl{$\mapsto$} & loops\\ |
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590 \onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s} & \bl{$\mapsto$} & loops\\} |
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591 \end{tabular} |
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592 \end{center} |
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593 |
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594 \small |
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595 \onslide<3->{ |
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596 \begin{center} |
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597 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} |
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598 \ldots\\ |
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599 \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\ |
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600 \ldots\\ |
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601 \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\ |
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602 \end{tabular} |
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603 \end{center}} |
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604 |
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605 |
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606 \end{frame}} |
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607 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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608 *} |
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609 |
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610 |
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611 text_raw {* |
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612 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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613 \mode<presentation>{ |
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614 \begin{frame}<1->[c] |
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615 \frametitle{Testing} |
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616 |
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617 \begin{itemize} |
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618 \item We can only test a {\bf finite} amount of examples:\bigskip |
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619 |
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620 \begin{center} |
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621 \colorbox{cream} |
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622 {\gr{\begin{minipage}{10cm} |
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623 ``Testing can only show the presence of errors, never their |
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624 absence.'' (Edsger W.~Dijkstra) |
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625 \end{minipage}}} |
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626 \end{center}\bigskip\pause |
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627 |
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628 \item In a theorem prover we can establish properties that apply to |
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629 {\bf all} input and {\bf all} output. |
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630 |
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631 \end{itemize} |
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632 |
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633 \end{frame}} |
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634 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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635 *} |
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636 |
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637 |
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638 text_raw {* |
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639 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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640 \mode<presentation>{ |
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641 \begin{frame}<1->[t] |
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642 \frametitle{Version 2} |
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643 \mbox{}\\[-14mm]\mbox{} |
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644 |
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645 \small |
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646 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} |
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647 \bl{nullable ($\varnothing$)} & \bl{$=$} & \bl{false} &\\ |
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648 \bl{nullable ([])} & \bl{$=$} & \bl{true} &\\ |
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649 \bl{nullable (c)} & \bl{$=$} & \bl{false} &\\ |
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650 \bl{nullable (r$_1$ + r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\vee$ nullable r$_2$} & \\ |
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651 \bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\wedge$ nullable r$_2$} & \\ |
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652 \bl{nullable (r$^*$)} & \bl{$=$} & \bl{true} & \\ |
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653 \end{tabular}\medskip |
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654 |
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655 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} |
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656 \bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\ |
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657 \bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\ |
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658 \bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\ |
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659 \bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\ |
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660 \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\ |
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661 & & \bl{\;\;\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\ |
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662 \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\ |
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663 |
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664 \bl{derivative r []} & \bl{$=$} & \bl{r} & \\ |
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665 \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\ |
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666 \end{tabular}\medskip |
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667 |
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668 \bl{matches$_2$ r s $=$ nullable (derivative r s)} |
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669 |
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670 \begin{textblock}{6}(9.5,0.9) |
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671 \begin{flushright} |
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672 \color{gray}``if r matches []'' |
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673 \end{flushright} |
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674 \end{textblock} |
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675 |
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676 \begin{textblock}{6}(9.5,6.18) |
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677 \begin{flushright} |
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678 \color{gray}``derivative w.r.t.~a char'' |
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679 \end{flushright} |
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680 \end{textblock} |
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681 |
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682 \begin{textblock}{6}(9.5,12.1) |
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683 \begin{flushright} |
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684 \color{gray}``deriv.~w.r.t.~a string'' |
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685 \end{flushright} |
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686 \end{textblock} |
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687 |
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688 \begin{textblock}{6}(9.5,13.98) |
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689 \begin{flushright} |
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690 \color{gray}``main'' |
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691 \end{flushright} |
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692 \end{textblock} |
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693 |
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694 \end{frame}} |
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695 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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696 *} |
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697 |
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698 text_raw {* |
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700 \mode<presentation>{ |
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701 \begin{frame}<1->[t] |
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702 \frametitle{Is the Matcher Error-Free?} |
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703 |
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704 We expect that |
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705 |
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706 \begin{center} |
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707 \begin{tabular}{lcl} |
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708 \bl{matches$_2$ r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% |
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709 \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\ |
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710 \bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% |
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711 \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\ |
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712 \end{tabular} |
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713 \end{center} |
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714 \pause\pause\bigskip |
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715 By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip |
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716 |
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717 \begin{tabular}{lrcl} |
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718 Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\ |
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719 & \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\ |
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720 \end{tabular} |
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721 |
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722 \only<4->{ |
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723 \begin{textblock}{3}(0.9,4.5) |
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724 \rd{\huge$\forall$\large{}r s.} |
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725 \end{textblock}} |
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726 \end{frame}} |
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727 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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728 *} |
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729 |
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730 text_raw {* |
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731 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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732 \mode<presentation>{ |
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733 \begin{frame}<1>[c] |
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734 \frametitle{ |
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735 \begin{tabular}{c} |
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736 \mbox{}\\[23mm] |
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737 \LARGE Demo |
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738 \end{tabular}} |
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739 |
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740 \end{frame}} |
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741 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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742 *} |
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743 |
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744 |
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745 text_raw {* |
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746 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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747 \mode<presentation>{ |
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748 \begin{frame}<1->[t] |
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749 |
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750 \mbox{}\\[-2mm] |
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751 |
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752 \small |
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753 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} |
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754 \bl{nullable (NULL)} & \bl{$=$} & \bl{false} &\\ |
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755 \bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} &\\ |
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756 \bl{nullable (CHR c)} & \bl{$=$} & \bl{false} &\\ |
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757 \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\ |
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758 \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\ |
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759 \bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \\ |
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760 \end{tabular}\medskip |
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761 |
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762 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} |
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763 \bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \\ |
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764 \bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \\ |
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765 \bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\ |
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766 \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\ |
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767 \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\ |
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768 & & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\ |
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769 \bl{der c (STAR r)} & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\ |
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770 |
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771 \bl{derivative r []} & \bl{$=$} & \bl{r} & \\ |
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772 \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\ |
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773 \end{tabular}\medskip |
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774 |
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775 \bl{matches r s $=$ nullable (derivative r s)} |
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776 |
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777 \only<2>{ |
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778 \begin{textblock}{8}(1.5,4) |
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779 \includegraphics[scale=0.3]{approved.png} |
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780 \end{textblock}} |
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781 |
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782 \end{frame}} |
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783 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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784 *} |
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785 |
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786 |
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787 text_raw {* |
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788 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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789 \mode<presentation>{ |
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790 \begin{frame}[c] |
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791 \frametitle{No Automata?} |
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792 |
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793 You might be wondering why I did not use any automata? |
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794 |
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795 \begin{itemize} |
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796 \item {\bf Def.:} A \alert{regular language} is one where there is a DFA that |
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797 recognises it.\bigskip\pause |
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798 \end{itemize} |
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799 |
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800 |
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801 There are many reasons why this is a good definition:\medskip |
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802 \begin{itemize} |
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803 \item pumping lemma |
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804 \item closure properties of regular languages\\ (e.g.~closure under complement) |
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805 \end{itemize} |
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806 |
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807 \end{frame}} |
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808 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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809 |
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810 *} |
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811 |
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812 text_raw {* |
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813 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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814 \mode<presentation>{ |
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815 \begin{frame}[t] |
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816 \frametitle{Really Bad News!} |
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817 |
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818 DFAs are bad news for formalisations in theorem provers. They might |
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819 be represented as: |
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820 |
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821 \begin{itemize} |
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822 \item graphs |
|
823 \item matrices |
|
824 \item partial functions |
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825 \end{itemize} |
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826 |
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827 All constructions are messy to reason about.\bigskip\bigskip |
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828 \pause |
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829 |
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830 \small |
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831 \only<2>{ |
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832 Constable et al needed (on and off) 18 months for a 3-person team |
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833 to formalise automata theory in Nuprl including Myhill-Nerode. There is |
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834 only very little other formalised work on regular languages I know of |
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835 in Coq, Isabelle and HOL.} |
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836 \only<3>{Typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two |
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837 automata with no inaccessible states \ldots'' |
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838 } |
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839 |
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840 \end{frame}} |
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841 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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842 |
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843 *} |
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844 |
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845 text_raw {* |
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846 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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847 \mode<presentation>{ |
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848 \begin{frame}[c] |
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849 \frametitle{} |
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850 \large |
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851 \begin{center} |
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852 \begin{tabular}{p{9cm}} |
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853 My point:\bigskip\\ |
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854 |
|
855 The theory about regular languages can be reformulated |
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856 to be more\\ suitable for theorem proving. |
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857 \end{tabular} |
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858 \end{center} |
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859 \end{frame}} |
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860 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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861 *} |
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862 |
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863 text_raw {* |
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864 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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865 \mode<presentation>{ |
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866 \begin{frame}[c] |
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867 \frametitle{\LARGE The Myhill-Nerode Theorem} |
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868 |
|
869 \begin{itemize} |
|
870 \item provides necessary and suf\!ficient conditions for a language |
|
871 being regular (pumping lemma only necessary)\medskip |
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872 |
|
873 \item will help with closure properties of regular languages\bigskip\pause |
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874 |
|
875 \item key is the equivalence relation:\smallskip |
|
876 \begin{center} |
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877 \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L} |
|
878 \end{center} |
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879 \end{itemize} |
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880 |
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881 \end{frame}} |
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882 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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883 *} |
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884 |
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885 text_raw {* |
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886 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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887 \mode<presentation>{ |
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888 \begin{frame}[c] |
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889 \frametitle{\LARGE The Myhill-Nerode Theorem} |
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890 |
|
891 \mbox{}\\[5cm] |
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892 |
|
893 \begin{itemize} |
|
894 \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} |
|
895 \end{itemize} |
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896 |
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897 \end{frame}} |
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898 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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899 |
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900 *} |
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901 |
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902 text_raw {* |
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903 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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904 \mode<presentation>{ |
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905 \begin{frame}[c] |
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906 \frametitle{\LARGE Equivalence Classes} |
|
907 |
|
908 \begin{itemize} |
|
909 \item \smath{L = []} |
|
910 \begin{center} |
|
911 \smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}} |
|
912 \end{center}\bigskip\bigskip |
|
913 |
|
914 \item \smath{L = [c]} |
|
915 \begin{center} |
|
916 \smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}} |
|
917 \end{center}\bigskip\bigskip |
|
918 |
|
919 \item \smath{L = \varnothing} |
|
920 \begin{center} |
|
921 \smath{\Big\{U\!N\!IV\Big\}} |
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922 \end{center} |
|
923 |
|
924 \end{itemize} |
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925 |
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926 \end{frame}} |
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927 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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928 |
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929 *} |
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930 |
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931 text_raw {* |
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932 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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933 \mode<presentation>{ |
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934 \begin{frame}[c] |
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935 \frametitle{\LARGE Regular Languages} |
|
936 |
|
937 \begin{itemize} |
|
938 \item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M} |
|
939 such that \smath{\mathbb{L}(M) = L}\\[1.5cm] |
|
940 |
|
941 \item Myhill-Nerode: |
|
942 |
|
943 \begin{center} |
|
944 \begin{tabular}{l} |
|
945 finite $\Rightarrow$ regular\\ |
|
946 \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r.\; L = \mathbb{L}(r)}\\[3mm] |
|
947 regular $\Rightarrow$ finite\\ |
|
948 \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})} |
|
949 \end{tabular} |
|
950 \end{center} |
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951 |
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952 \end{itemize} |
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953 |
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954 \end{frame}} |
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955 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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956 |
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957 *} |
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958 |
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959 text_raw {* |
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960 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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961 \mode<presentation>{ |
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962 \begin{frame}[c] |
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963 \frametitle{\LARGE Final Equiv.~Classes} |
|
964 |
|
965 \mbox{}\\[3cm] |
|
966 |
|
967 \begin{itemize} |
|
968 \item \smath{\text{finals}\,L \dn |
|
969 \{{\lbrack\mkern-2mu\lbrack{s}\rbrack\mkern-2mu\rbrack}_\approx\;|\; s \in L\}}\\ |
|
970 \medskip |
|
971 |
|
972 \item we can prove: \smath{L = \bigcup (\text{finals}\,L)} |
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973 |
|
974 \end{itemize} |
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975 |
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976 \end{frame}} |
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977 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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978 *} |
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979 |
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980 text_raw {* |
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981 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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982 \mode<presentation>{ |
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983 \begin{frame}[c] |
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984 \frametitle{\LARGE Transitions between ECs} |
|
985 |
|
986 \smath{L = \{[c]\}} |
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987 |
|
988 \begin{tabular}{@ {\hspace{-7mm}}cc} |
|
989 \begin{tabular}{c} |
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990 \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] |
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991 \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] |
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992 |
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993 %\draw[help lines] (0,0) grid (3,2); |
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994 |
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995 \node[state,initial] (q_0) {$R_1$}; |
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996 \node[state,accepting] (q_1) [above right of=q_0] {$R_2$}; |
|
997 \node[state] (q_2) [below right of=q_0] {$R_3$}; |
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998 |
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999 \path[->] (q_0) edge node {c} (q_1) |
|
1000 edge node [swap] {$\Sigma-{c}$} (q_2) |
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1001 (q_2) edge [loop below] node {$\Sigma$} () |
|
1002 (q_1) edge node {$\Sigma$} (q_2); |
|
1003 \end{tikzpicture} |
|
1004 \end{tabular} |
|
1005 & |
|
1006 \begin{tabular}[t]{ll} |
|
1007 \\[-20mm] |
|
1008 \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm] |
|
1009 |
|
1010 \smath{R_1}: & \smath{\{[]\}}\\ |
|
1011 \smath{R_2}: & \smath{\{[c]\}}\\ |
|
1012 \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm] |
|
1013 \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ;; [c] \subseteq Y}}} |
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1014 \end{tabular} |
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1015 |
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1016 \end{tabular} |
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1017 |
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1018 \end{frame}} |
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1019 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1020 *} |
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1021 |
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1022 |
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1023 text_raw {* |
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1024 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1025 \mode<presentation>{ |
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1026 \begin{frame}[c] |
|
1027 \frametitle{\LARGE Systems of Equations} |
|
1028 |
|
1029 Inspired by a method of Brzozowski\;'64, we can build an equational system |
|
1030 characterising the equivalence classes: |
|
1031 |
|
1032 \begin{center} |
|
1033 \begin{tabular}{@ {\hspace{-20mm}}c} |
|
1034 \\[-13mm] |
|
1035 \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] |
|
1036 \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] |
|
1037 |
|
1038 %\draw[help lines] (0,0) grid (3,2); |
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1039 |
|
1040 \node[state,initial] (p_0) {$R_1$}; |
|
1041 \node[state,accepting] (p_1) [right of=q_0] {$R_2$}; |
|
1042 |
|
1043 \path[->] (p_0) edge [bend left] node {a} (p_1) |
|
1044 edge [loop above] node {b} () |
|
1045 (p_1) edge [loop above] node {a} () |
|
1046 edge [bend left] node {b} (p_0); |
|
1047 \end{tikzpicture}\\ |
|
1048 \\[-13mm] |
|
1049 \end{tabular} |
|
1050 \end{center} |
|
1051 |
|
1052 \begin{center} |
|
1053 \begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l} |
|
1054 & \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\ |
|
1055 & \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\ |
|
1056 \onslide<3->{we can prove} |
|
1057 & \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}} |
|
1058 & \onslide<3->{\smath{R_1;; \mathbb{L}(b) \,\cup\, R_2;;\mathbb{L}(b) \,\cup\, \{[]\}}}\\ |
|
1059 & \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}} |
|
1060 & \onslide<3->{\smath{R_1;; \mathbb{L}(a) \,\cup\, R_2;;\mathbb{L}(a)}}\\ |
|
1061 \end{tabular} |
|
1062 \end{center} |
|
1063 |
|
1064 \end{frame}} |
|
1065 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1066 *} |
|
1067 |
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1068 |
|
1069 text_raw {* |
|
1070 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1071 \mode<presentation>{ |
|
1072 \begin{frame}<1>[t] |
|
1073 \small |
|
1074 |
|
1075 \begin{center} |
|
1076 \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} |
|
1077 \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}} |
|
1078 & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ |
|
1079 \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}} |
|
1080 & \onslide<1->{\smath{R_1; a + R_2; a}}\\ |
|
1081 |
|
1082 & & & \onslide<2->{by Arden}\\ |
|
1083 |
|
1084 \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}} |
|
1085 & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ |
|
1086 \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}} |
|
1087 & \only<2>{\smath{R_1; a + R_2; a}}% |
|
1088 \only<3->{\smath{R_1; a\cdot a^\star}}\\ |
|
1089 |
|
1090 & & & \onslide<4->{by Arden}\\ |
|
1091 |
|
1092 \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} |
|
1093 & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\ |
|
1094 \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}} |
|
1095 & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\ |
|
1096 |
|
1097 & & & \onslide<5->{by substitution}\\ |
|
1098 |
|
1099 \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} |
|
1100 & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\ |
|
1101 \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}} |
|
1102 & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\ |
|
1103 |
|
1104 & & & \onslide<6->{by Arden}\\ |
|
1105 |
|
1106 \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}} |
|
1107 & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ |
|
1108 \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}} |
|
1109 & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\ |
|
1110 |
|
1111 & & & \onslide<7->{by substitution}\\ |
|
1112 |
|
1113 \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}} |
|
1114 & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ |
|
1115 \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}} |
|
1116 & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star |
|
1117 \cdot a\cdot a^\star}}\\ |
|
1118 \end{tabular} |
|
1119 \end{center} |
|
1120 |
|
1121 \end{frame}} |
|
1122 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1123 *} |
|
1124 |
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1125 text_raw {* |
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1126 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1127 \mode<presentation>{ |
|
1128 \begin{frame}[c] |
|
1129 \frametitle{\LARGE A Variant of Arden's Lemma} |
|
1130 |
|
1131 {\bf Arden's Lemma:}\smallskip |
|
1132 |
|
1133 If \smath{[] \not\in A} then |
|
1134 \begin{center} |
|
1135 \smath{X = X; A + \text{something}} |
|
1136 \end{center} |
|
1137 has the (unique) solution |
|
1138 \begin{center} |
|
1139 \smath{X = \text{something} ; A^\star} |
|
1140 \end{center} |
|
1141 |
|
1142 |
|
1143 \end{frame}} |
|
1144 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1145 *} |
|
1146 |
|
1147 |
|
1148 text_raw {* |
|
1149 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1150 \mode<presentation>{ |
|
1151 \begin{frame}<1->[t] |
|
1152 \small |
|
1153 |
|
1154 \begin{center} |
|
1155 \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} |
|
1156 \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}} |
|
1157 & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ |
|
1158 \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}} |
|
1159 & \onslide<1->{\smath{R_1; a + R_2; a}}\\ |
|
1160 |
|
1161 & & & \onslide<2->{by Arden}\\ |
|
1162 |
|
1163 \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}} |
|
1164 & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ |
|
1165 \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}} |
|
1166 & \only<2>{\smath{R_1; a + R_2; a}}% |
|
1167 \only<3->{\smath{R_1; a\cdot a^\star}}\\ |
|
1168 |
|
1169 & & & \onslide<4->{by Arden}\\ |
|
1170 |
|
1171 \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} |
|
1172 & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\ |
|
1173 \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}} |
|
1174 & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\ |
|
1175 |
|
1176 & & & \onslide<5->{by substitution}\\ |
|
1177 |
|
1178 \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} |
|
1179 & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\ |
|
1180 \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}} |
|
1181 & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\ |
|
1182 |
|
1183 & & & \onslide<6->{by Arden}\\ |
|
1184 |
|
1185 \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}} |
|
1186 & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ |
|
1187 \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}} |
|
1188 & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\ |
|
1189 |
|
1190 & & & \onslide<7->{by substitution}\\ |
|
1191 |
|
1192 \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}} |
|
1193 & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ |
|
1194 \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}} |
|
1195 & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star |
|
1196 \cdot a\cdot a^\star}}\\ |
|
1197 \end{tabular} |
|
1198 \end{center} |
|
1199 |
|
1200 \only<8->{ |
|
1201 \begin{textblock}{6}(2.5,4) |
|
1202 \begin{block}{} |
|
1203 \begin{minipage}{8cm}\raggedright |
|
1204 |
|
1205 \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm] |
|
1206 \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] |
|
1207 |
|
1208 %\draw[help lines] (0,0) grid (3,2); |
|
1209 |
|
1210 \node[state,initial] (p_0) {$R_1$}; |
|
1211 \node[state,accepting] (p_1) [right of=q_0] {$R_2$}; |
|
1212 |
|
1213 \path[->] (p_0) edge [bend left] node {a} (p_1) |
|
1214 edge [loop above] node {b} () |
|
1215 (p_1) edge [loop above] node {a} () |
|
1216 edge [bend left] node {b} (p_0); |
|
1217 \end{tikzpicture} |
|
1218 |
|
1219 \end{minipage} |
|
1220 \end{block} |
|
1221 \end{textblock}} |
|
1222 |
|
1223 \end{frame}} |
|
1224 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1225 *} |
|
1226 |
|
1227 |
|
1228 text_raw {* |
|
1229 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1230 \mode<presentation>{ |
|
1231 \begin{frame}[c] |
|
1232 \frametitle{\LARGE The Equ's Solving Algorithm} |
|
1233 |
|
1234 \begin{itemize} |
|
1235 \item The algorithm must terminate: Arden makes one equation smaller; |
|
1236 substitution deletes one variable from the right-hand sides.\bigskip |
|
1237 |
|
1238 \item We need to maintain the invariant that Arden is applicable |
|
1239 (if \smath{[] \not\in A} then \ldots):\medskip |
|
1240 |
|
1241 \begin{center}\small |
|
1242 \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} |
|
1243 \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\ |
|
1244 \smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\ |
|
1245 |
|
1246 & & & by Arden\\ |
|
1247 |
|
1248 \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\ |
|
1249 \smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\ |
|
1250 \end{tabular} |
|
1251 \end{center} |
|
1252 |
|
1253 \end{itemize} |
|
1254 |
|
1255 |
|
1256 \end{frame}} |
|
1257 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1258 *} |
|
1259 |
|
1260 |
|
1261 |
|
1262 |
|
1263 text_raw {* |
|
1264 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1265 \mode<presentation>{ |
|
1266 \begin{frame}[c] |
|
1267 \frametitle{\LARGE Other Direction} |
|
1268 |
|
1269 One has to prove |
|
1270 |
|
1271 \begin{center} |
|
1272 \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})} |
|
1273 \end{center} |
|
1274 |
|
1275 by induction on \smath{r}. Not trivial, but after a bit |
|
1276 of thinking, one can prove that if |
|
1277 |
|
1278 \begin{center} |
|
1279 \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm} |
|
1280 \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})} |
|
1281 \end{center} |
|
1282 |
|
1283 then |
|
1284 |
|
1285 \begin{center} |
|
1286 \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})} |
|
1287 \end{center} |
|
1288 |
|
1289 |
|
1290 |
|
1291 \end{frame}} |
|
1292 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1293 *} |
|
1294 |
|
1295 |
|
1296 |
|
1297 text_raw {* |
|
1298 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1299 \mode<presentation>{ |
|
1300 \begin{frame}[c] |
|
1301 \frametitle{\LARGE What Have We Achieved?} |
|
1302 |
|
1303 \begin{itemize} |
|
1304 \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} |
|
1305 \bigskip\pause |
|
1306 \item regular languages are closed under complementation; this is now easy\medskip |
|
1307 \begin{center} |
|
1308 \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}} |
|
1309 \end{center} |
|
1310 \end{itemize} |
|
1311 |
|
1312 |
|
1313 \end{frame}} |
|
1314 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1315 *} |
|
1316 |
|
1317 text_raw {* |
|
1318 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
1319 \mode<presentation>{ |
|
1320 \begin{frame}[c] |
|
1321 \frametitle{\LARGE Examples} |
|
1322 |
|
1323 \begin{itemize} |
|
1324 \item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular |
|
1325 \begin{quote}\small |
|
1326 \begin{tabular}{lcl} |
|
1327 \smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\ |
|
1328 \smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\ |
|
1329 \smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\ |
|
1330 \smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\ |
|
1331 \end{tabular} |
|
1332 \end{quote} |
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1333 |
|
1334 \item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular |
|
1335 \begin{quote}\small |
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1336 \begin{tabular}{lcl} |
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1337 \smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\ |
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1338 \smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\ |
|
1339 \smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\ |
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1340 \smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\ |
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1341 & \smath{\vdots} &\\ |
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1342 \end{tabular} |
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1343 \end{quote} |
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1344 \end{itemize} |
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1345 |
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1346 \end{frame}} |
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1347 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1348 *} |
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1349 |
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1350 |
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1351 text_raw {* |
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1352 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1353 \mode<presentation>{ |
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1354 \begin{frame}[c] |
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1355 \frametitle{\LARGE What We Have Not Achieved} |
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1356 |
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1357 \begin{itemize} |
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1358 \item regular expressions are not good if you look for a minimal |
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1359 one for a language (DFAs have this notion)\pause\bigskip |
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1360 |
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1361 \item Is there anything to be said about context free languages:\medskip |
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1362 |
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1363 \begin{quote} |
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1364 A context free language is where every string can be recognised by |
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1365 a pushdown automaton.\bigskip |
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1366 \end{quote} |
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1367 \end{itemize} |
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1368 |
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1369 \textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.} |
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1370 |
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1371 \end{frame}} |
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1372 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1373 *} |
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1374 |
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1375 |
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1376 text_raw {* |
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1377 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1378 \mode<presentation>{ |
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1379 \begin{frame}[c] |
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1380 \frametitle{\LARGE Conclusion} |
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1381 |
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1382 \begin{itemize} |
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1383 \item We formalised the Myhill-Nerode theorem based on |
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1384 regular expressions only (DFAs are difficult to deal with in a theorem prover).\smallskip |
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1385 |
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1386 \item Seems to be a common theme: algorithms need to be reformulated |
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1387 to better suit formal treatment.\smallskip |
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1388 |
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1389 \item The most interesting aspect is that we are able to |
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1390 implement the matcher directly inside the theorem prover |
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1391 (ongoing work).\smallskip |
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1392 |
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1393 \item Parsing is a vast field which seem to offer new results. |
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1394 \end{itemize} |
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1395 |
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1396 \end{frame}} |
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1397 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1398 *} |
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1399 |
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1400 text_raw {* |
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1401 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1402 \mode<presentation>{ |
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1403 \begin{frame}<1>[b] |
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1404 \frametitle{ |
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1405 \begin{tabular}{c} |
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1406 \mbox{}\\[13mm] |
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1407 \alert{\LARGE Thank you very much!}\\ |
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1408 \alert{\Large Questions?} |
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1409 \end{tabular}} |
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1410 |
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1411 \end{frame}} |
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1412 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1413 *} |
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1414 |
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1415 |
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1416 |
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1417 (*<*) |
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1418 end |
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1419 (*>*) |
|