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1 (*<*) |
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2 theory Slides7 |
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3 imports "~~/src/HOL/Library/LaTeXsugar" "Main" |
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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}{Hefei, 15.~April 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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28 \newcommand{\gr}[1]{\textcolor{gray}{#1}} |
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29 \newcommand{\rd}[1]{\textcolor{red}{#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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33 \newcommand{\largenotok}{\includegraphics[scale=1]{notok.png}} |
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34 |
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35 \renewcommand{\Huge}{\fontsize{61.92}{77}\selectfont} |
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36 \newcommand{\veryHuge}{\fontsize{74.3}{93}\selectfont} |
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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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44 {rgb(0mm)=(0,0,0.9); |
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46 rgb(1.3mm)=(0,0,0.5); |
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47 rgb(1.4mm)=(1,1,1)} |
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48 |
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49 \def\myitemi{\begin{pgfpicture}{-1ex}{-0.55ex}{1ex}{1ex} |
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50 \usebeamercolor[fg]{subitem projected} |
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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: 3 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.\bigskip |
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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.\bigskip |
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92 \item This re-formulation can be done, even in domains where |
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93 we do not expect it. |
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94 \end{itemize} |
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95 |
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96 \end{frame}} |
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97 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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98 *} |
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99 |
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100 text_raw {* |
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101 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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102 \mode<presentation>{ |
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103 \begin{frame}<1->[t] |
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104 \frametitle{Regular Expressions} |
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105 |
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106 \begin{textblock}{6}(2,4) |
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107 \begin{tabular}{@ {}rrl} |
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108 \bl{r} & \bl{$::=$} & \bl{$\varnothing$}\\ |
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109 & \bl{$\mid$} & \bl{[]}\\ |
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110 & \bl{$\mid$} & \bl{c}\\ |
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111 & \bl{$\mid$} & \bl{r$_1$ + r$_2$}\\ |
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112 & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$}\\ |
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113 & \bl{$\mid$} & \bl{r$^*$}\\ |
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114 \end{tabular} |
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115 \end{textblock} |
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116 |
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117 \begin{textblock}{6}(8,3.5) |
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118 \includegraphics[scale=0.35]{Screen1.png} |
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119 \end{textblock} |
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120 |
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121 \begin{textblock}{6}(10.2,2.8) |
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122 \footnotesize Isabelle: |
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123 \end{textblock} |
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124 |
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125 \only<2>{ |
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126 \begin{textblock}{9}(3.6,11.8) |
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127 \bl{matches r s $\;\Longrightarrow\;$ true $\vee$ false}\\[3.5mm] |
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128 |
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129 \hspace{10mm}\begin{tikzpicture} |
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130 \coordinate (m1) at (0.4,1); |
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131 \draw (0,0.3) node (m2) {\small\color{gray}rexp}; |
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132 \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1); |
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133 |
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134 \coordinate (s1) at (0.81,1); |
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135 \draw (1.3,0.3) node (s2) {\small\color{gray} string}; |
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136 \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1); |
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137 \end{tikzpicture} |
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138 \end{textblock}} |
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139 |
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140 |
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141 |
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142 \end{frame}} |
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143 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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144 *} |
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145 |
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146 text_raw {* |
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147 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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148 \mode<presentation>{ |
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149 \begin{frame}<1->[t] |
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150 \frametitle{Specification} |
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151 |
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152 \small |
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153 \begin{textblock}{6}(0,3.5) |
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154 \begin{tabular}{r@ {\hspace{0.5mm}}r@ {\hspace{1.5mm}}c@ {\hspace{1.5mm}}l} |
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155 \multicolumn{4}{c}{rexp $\Rightarrow$ set of strings}\bigskip\\ |
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156 &\bl{\LL ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$}\\ |
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157 &\bl{\LL ([])} & \bl{$\dn$} & \bl{\{[]\}}\\ |
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158 &\bl{\LL (c)} & \bl{$\dn$} & \bl{\{c\}}\\ |
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159 &\bl{\LL (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) $\cup$ \LL (r$_2$)}\\ |
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160 \rd{$\Rightarrow$} &\bl{\LL (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{\LL (r$_1$) ;; \LL (r$_2$)}\\ |
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161 \rd{$\Rightarrow$} &\bl{\LL (r$^*$)} & \bl{$\dn$} & \bl{(\LL (r))$^\star$}\\ |
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162 \end{tabular} |
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163 \end{textblock} |
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164 |
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165 \begin{textblock}{9}(7.3,3) |
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166 {\mbox{}\hspace{2cm}\footnotesize Isabelle:\smallskip} |
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167 \includegraphics[scale=0.325]{Screen3.png} |
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168 \end{textblock} |
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169 |
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170 \end{frame}} |
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171 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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172 *} |
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173 |
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174 |
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175 text_raw {* |
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176 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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177 \mode<presentation>{ |
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178 \begin{frame}<1->[t] |
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179 \frametitle{Version 1} |
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180 \small |
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181 \mbox{}\\[-8mm]\mbox{} |
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182 |
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183 \begin{center}\def\arraystretch{1.05} |
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184 \begin{tabular}{@ {\hspace{-5mm}}l@ {\hspace{2.5mm}}c@ {\hspace{2.5mm}}l@ {}} |
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185 \bl{match [] []} & \bl{$=$} & \bl{true}\\ |
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186 \bl{match [] (c::s)} & \bl{$=$} & \bl{false}\\ |
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187 \bl{match ($\varnothing$::rs) s} & \bl{$=$} & \bl{false}\\ |
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188 \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\ |
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189 \bl{match (c::rs) []} & \bl{$=$} & \bl{false}\\ |
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190 \bl{match (c::rs) (d::s)} & \bl{$=$} & \bl{if c = d then match rs s else false}\\ |
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191 \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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192 \bl{match (r$_1$ $\cdot$ r$_2$::rs) s} & \bl{$=$} & \bl{match (r$_1$::r$_2$::rs) s}\\ |
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193 \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\ |
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194 \end{tabular} |
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195 \end{center} |
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196 |
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197 \begin{textblock}{9}(0.2,1.6) |
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198 \hspace{10mm}\begin{tikzpicture} |
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199 \coordinate (m1) at (0.44,-0.5); |
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200 \draw (0,0.3) node (m2) {\small\color{gray}\mbox{}\hspace{-9mm}list of rexps}; |
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201 \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (m2) edge (m1); |
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202 |
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203 \coordinate (s1) at (0.86,-0.5); |
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204 \draw (1.5,0.3) node (s2) {\small\color{gray} string}; |
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205 \path[overlay, ->, line width = 0.5mm, shorten <=-1mm, draw = gray] (s2) edge (s1); |
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206 \end{tikzpicture} |
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207 \end{textblock} |
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208 |
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209 \begin{textblock}{9}(2.8,11.8) |
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210 \bl{matches$_1$ r s $\;=\;$ match [r] s} |
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211 \end{textblock} |
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212 |
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213 \end{frame}} |
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214 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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215 *} |
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216 |
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217 text_raw {* |
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218 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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219 \mode<presentation>{ |
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220 \begin{frame}<1->[c] |
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221 \frametitle{Testing} |
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222 |
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223 \small |
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224 Every good programmer should do thourough tests: |
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225 |
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226 \begin{center} |
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227 \begin{tabular}{@ {\hspace{-20mm}}lcl} |
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228 \bl{matches$_1$ (a$\cdot$b)$^*\;$ []} & \bl{$\mapsto$} & \bl{true}\\ |
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229 \bl{matches$_1$ (a$\cdot$b)$^*\;$ ab} & \bl{$\mapsto$} & \bl{true}\\ |
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230 \bl{matches$_1$ (a$\cdot$b)$^*\;$ aba} & \bl{$\mapsto$} & \bl{false}\\ |
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231 \bl{matches$_1$ (a$\cdot$b)$^*\;$ abab} & \bl{$\mapsto$} & \bl{true}\\ |
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232 \bl{matches$_1$ (a$\cdot$b)$^*\;$ abaa} & \bl{$\mapsto$} & \bl{false}\medskip\\ |
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233 \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x} & \bl{$\mapsto$} & \bl{true}}\\ |
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234 \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x0} & \bl{$\mapsto$} & \bl{true}}\\ |
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235 \onslide<2->{\bl{matches$_1$ x$\cdot$(0$|$1)$^*\;$ x3} & \bl{$\mapsto$} & \bl{false}} |
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236 \end{tabular} |
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237 \end{center} |
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238 |
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239 \onslide<3-> |
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240 {looks OK \ldots let's ship it to customers\hspace{5mm} |
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241 \raisebox{-5mm}{\includegraphics[scale=0.05]{sun.png}}} |
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242 |
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243 \end{frame}} |
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244 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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245 *} |
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246 |
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247 text_raw {* |
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248 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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249 \mode<presentation>{ |
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250 \begin{frame}<1->[c] |
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251 \frametitle{Version 1} |
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252 |
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253 \only<1->{Several hours later\ldots}\pause |
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254 |
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255 |
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256 \begin{center} |
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257 \begin{tabular}{@ {\hspace{0mm}}lcl} |
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258 \bl{matches$_1$ []$^*$ s} & \bl{$\mapsto$} & loops\\ |
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259 \onslide<4->{\bl{matches$_1$ ([] + \ldots)$^*$ s} & \bl{$\mapsto$} & loops\\} |
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260 \end{tabular} |
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261 \end{center} |
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262 |
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263 \small |
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264 \onslide<3->{ |
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265 \begin{center} |
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266 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {}} |
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267 \ldots\\ |
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268 \bl{match ([]::rs) s} & \bl{$=$} & \bl{match rs s}\\ |
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269 \ldots\\ |
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270 \bl{match (r$^*$::rs) s} & \bl{$=$} & \bl{match rs s $\vee$ match (r::r$^*$::rs) s}\\ |
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271 \end{tabular} |
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272 \end{center}} |
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273 |
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274 |
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275 \end{frame}} |
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276 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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277 *} |
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278 |
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279 |
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280 text_raw {* |
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281 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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282 \mode<presentation>{ |
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283 \begin{frame}<1->[t] |
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284 \frametitle{Testing} |
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285 |
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286 \begin{itemize} |
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287 \item While testing is an important part in the process of programming development\pause\ldots |
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288 |
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289 \item we can only test a {\bf finite} amount of examples.\bigskip\pause |
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290 |
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291 \begin{center} |
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292 \colorbox{cream} |
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293 {\gr{\begin{minipage}{10cm} |
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294 ``Testing can only show the presence of errors, never their |
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295 absence.'' (Edsger W.~Dijkstra) |
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296 \end{minipage}}} |
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297 \end{center}\bigskip\pause |
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298 |
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299 \item In a theorem prover we can establish properties that apply to |
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300 {\bf all} input and {\bf all} output. |
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301 |
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302 \end{itemize} |
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303 |
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304 \end{frame}} |
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305 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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306 *} |
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307 |
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308 |
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309 text_raw {* |
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310 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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311 \mode<presentation>{ |
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312 \begin{frame}<1->[t] |
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313 \frametitle{Version 2} |
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314 \mbox{}\\[-14mm]\mbox{} |
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315 |
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316 \small |
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317 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} |
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318 \bl{nullable ($\varnothing$)} & \bl{$=$} & \bl{false} &\\ |
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319 \bl{nullable ([])} & \bl{$=$} & \bl{true} &\\ |
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320 \bl{nullable (c)} & \bl{$=$} & \bl{false} &\\ |
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321 \bl{nullable (r$_1$ + r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\vee$ nullable r$_2$} & \\ |
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322 \bl{nullable (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{nullable r$_1$ $\wedge$ nullable r$_2$} & \\ |
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323 \bl{nullable (r$^*$)} & \bl{$=$} & \bl{true} & \\ |
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324 \end{tabular}\medskip |
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325 |
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326 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} |
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327 \bl{der c ($\varnothing$)} & \bl{$=$} & \bl{$\varnothing$} & \\ |
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328 \bl{der c ([])} & \bl{$=$} & \bl{$\varnothing$} & \\ |
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329 \bl{der c (d)} & \bl{$=$} & \bl{if c = d then [] else $\varnothing$} & \\ |
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330 \bl{der c (r$_1$ + r$_2$)} & \bl{$=$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\ |
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331 \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$=$} & \bl{((der c r$_1$) $\cdot$ r$_2$)} & \\ |
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332 & & \bl{\;\;+ (if nullable r$_1$ then der c r$_2$ else $\varnothing$)}\\ |
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333 \bl{der c (r$^*$)} & \bl{$=$} & \bl{(der c r) $\cdot$ r$^*$} &\smallskip\\ |
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334 |
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335 \bl{derivative r []} & \bl{$=$} & \bl{r} & \\ |
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336 \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\ |
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337 \end{tabular}\medskip |
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338 |
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339 \bl{matches$_2$ r s $=$ nullable (derivative r s)} |
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340 |
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341 \begin{textblock}{6}(9.5,0.9) |
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342 \begin{flushright} |
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343 \color{gray}``if r matches []'' |
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344 \end{flushright} |
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345 \end{textblock} |
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346 |
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347 \begin{textblock}{6}(9.5,6.18) |
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348 \begin{flushright} |
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349 \color{gray}``derivative w.r.t.~a char'' |
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350 \end{flushright} |
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351 \end{textblock} |
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352 |
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353 \begin{textblock}{6}(9.5,12.1) |
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354 \begin{flushright} |
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355 \color{gray}``deriv.~w.r.t.~a string'' |
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356 \end{flushright} |
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357 \end{textblock} |
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358 |
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359 \begin{textblock}{6}(9.5,13.98) |
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360 \begin{flushright} |
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361 \color{gray}``main'' |
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362 \end{flushright} |
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363 \end{textblock} |
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364 |
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365 \end{frame}} |
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366 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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367 *} |
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368 |
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369 text_raw {* |
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370 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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371 \mode<presentation>{ |
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372 \begin{frame}<1->[t] |
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373 \frametitle{Is the Matcher Error-Free?} |
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374 |
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375 We expect that |
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376 |
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377 \begin{center} |
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378 \begin{tabular}{lcl} |
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379 \bl{matches$_2$ r s = true} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% |
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380 \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\in$ \LL(r)}\\ |
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381 \bl{matches$_2$ r s = false} & \only<1>{\rd{$\Longrightarrow\,\,$}}\only<2>{\rd{$\Longleftarrow\,\,$}}% |
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382 \only<3->{\rd{$\Longleftrightarrow$}} & \bl{s $\notin$ \LL(r)}\\ |
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383 \end{tabular} |
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384 \end{center} |
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385 \pause\pause\bigskip |
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386 ??? By \alert<4->{induction}, we can {\bf prove} these properties.\bigskip |
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387 |
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388 \begin{tabular}{lrcl} |
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389 Lemmas: & \bl{nullable (r)} & \bl{$\Longleftrightarrow$} & \bl{[] $\in$ \LL (r)}\\ |
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390 & \bl{s $\in$ \LL (der c r)} & \bl{$\Longleftrightarrow$} & \bl{(c::s) $\in$ \LL (r)}\\ |
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391 \end{tabular} |
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392 |
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393 \only<4->{ |
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394 \begin{textblock}{3}(0.9,4.5) |
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395 \rd{\huge$\forall$\large{}r s.} |
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396 \end{textblock}} |
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397 \end{frame}} |
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398 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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399 *} |
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400 |
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401 text_raw {* |
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402 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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403 \mode<presentation>{ |
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404 \begin{frame}<1>[c] |
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405 \frametitle{ |
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406 \begin{tabular}{c} |
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407 \mbox{}\\[23mm] |
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408 \LARGE Demo |
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409 \end{tabular}} |
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410 |
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411 \end{frame}} |
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412 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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413 *} |
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414 |
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415 |
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416 text_raw {* |
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417 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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418 \mode<presentation>{ |
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419 \begin{frame}<1->[t] |
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420 |
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421 \mbox{}\\[-2mm] |
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422 |
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423 \small |
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424 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}ll@ {}} |
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425 \bl{nullable (NULL)} & \bl{$=$} & \bl{false} &\\ |
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426 \bl{nullable (EMPTY)} & \bl{$=$} & \bl{true} &\\ |
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427 \bl{nullable (CHR c)} & \bl{$=$} & \bl{false} &\\ |
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428 \bl{nullable (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) orelse (nullable r$_2$)} & \\ |
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429 \bl{nullable (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{(nullable r$_1$) andalso (nullable r$_2$)} & \\ |
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430 \bl{nullable (STAR r)} & \bl{$=$} & \bl{true} & \\ |
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431 \end{tabular}\medskip |
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432 |
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433 \begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} |
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434 \bl{der c (NULL)} & \bl{$=$} & \bl{NULL} & \\ |
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435 \bl{der c (EMPTY)} & \bl{$=$} & \bl{NULL} & \\ |
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436 \bl{der c (CHR d)} & \bl{$=$} & \bl{if c=d then EMPTY else NULL} & \\ |
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437 \bl{der c (ALT r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (der c r$_1$) (der c r$_2$)} & \\ |
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438 \bl{der c (SEQ r$_1$ r$_2$)} & \bl{$=$} & \bl{ALT (SEQ (der c r$_1$) r$_2$)} & \\ |
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439 & & \bl{\phantom{ALT} (if nullable r$_1$ then der c r$_2$ else NULL)}\\ |
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440 \bl{der c (STAR r)} & \bl{$=$} & \bl{SEQ (der c r) (STAR r)} &\smallskip\\ |
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441 |
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442 \bl{derivative r []} & \bl{$=$} & \bl{r} & \\ |
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443 \bl{derivative r (c::s)} & \bl{$=$} & \bl{derivative (der c r) s} & \\ |
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444 \end{tabular}\medskip |
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445 |
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446 \bl{matches r s $=$ nullable (derivative r s)} |
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447 |
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448 \only<2>{ |
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449 \begin{textblock}{8}(1.5,4) |
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450 \includegraphics[scale=0.3]{approved.png} |
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451 \end{textblock}} |
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452 |
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453 \end{frame}} |
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454 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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455 *} |
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456 |
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457 |
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458 text_raw {* |
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459 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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460 \mode<presentation>{ |
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461 \begin{frame}[c] |
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462 \frametitle{No Automata?} |
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463 |
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464 You might be wondering why I did not use any automata? |
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465 |
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466 \begin{itemize} |
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467 \item {\bf Def.:} A \alert{regular language} is one where there is a DFA that |
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468 recognises it.\bigskip\pause |
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469 \end{itemize} |
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470 |
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471 |
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472 There are many reasons why this is a good definition:\medskip |
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473 \begin{itemize} |
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474 \item pumping lemma |
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475 \item closure properties of regular languages\\ (e.g.~closure under complement) |
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476 \end{itemize} |
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477 |
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478 \end{frame}} |
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479 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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480 |
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481 *} |
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482 |
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483 text_raw {* |
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484 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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485 \mode<presentation>{ |
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486 \begin{frame}[t] |
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487 \frametitle{Really Bad News!} |
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488 |
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489 DFAs are bad news for formalisations in theorem provers. They might |
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490 be represented as: |
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491 |
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492 \begin{itemize} |
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493 \item graphs |
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494 \item matrices |
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495 \item partial functions |
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496 \end{itemize} |
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497 |
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498 All constructions are messy to reason about.\bigskip\bigskip |
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499 \pause |
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500 |
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501 \small |
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502 \only<2>{ |
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503 Constable et al needed (on and off) 18 months for a 3-person team |
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504 to formalise automata theory in Nuprl including Myhill-Nerode. There is |
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505 only very little other formalised work on regular languages I know of |
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506 in Coq, Isabelle and HOL.} |
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507 \only<3>{Typical textbook reasoning goes like: ``\ldots if \smath{M} and \smath{N} are any two |
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508 automata with no inaccessible states \ldots'' |
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509 } |
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510 |
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511 \end{frame}} |
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512 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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513 |
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514 *} |
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515 |
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516 text_raw {* |
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517 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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518 \mode<presentation>{ |
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519 \begin{frame}[c] |
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520 \frametitle{} |
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521 \large |
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522 \begin{center} |
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523 \begin{tabular}{p{9cm}} |
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524 My point:\bigskip\\ |
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525 |
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526 The theory about regular languages can be reformulated |
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527 to be more suitable for theorem proving. |
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528 \end{tabular} |
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529 \end{center} |
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530 \end{frame}} |
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531 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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532 *} |
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533 |
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534 text_raw {* |
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535 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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536 \mode<presentation>{ |
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537 \begin{frame}[c] |
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538 \frametitle{\LARGE The Myhill-Nerode Theorem} |
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539 |
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540 \begin{itemize} |
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541 \item provides necessary and suf\!ficient conditions for a language |
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542 being regular (pumping lemma only necessary)\medskip |
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543 |
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544 \item will help with closure properties of regular languages\bigskip\pause |
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545 |
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546 \item key is the equivalence relation:\smallskip |
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547 \begin{center} |
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548 \smath{x \approx_{L} y \,\dn\, \forall z.\; x @ z \in L \Leftrightarrow y @ z \in L} |
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549 \end{center} |
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550 \end{itemize} |
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551 |
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552 \end{frame}} |
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553 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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554 *} |
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555 |
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556 text_raw {* |
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557 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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558 \mode<presentation>{ |
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559 \begin{frame}[c] |
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560 \frametitle{\LARGE The Myhill-Nerode Theorem} |
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561 |
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562 \mbox{}\\[5cm] |
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563 |
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564 \begin{itemize} |
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565 \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} |
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566 \end{itemize} |
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567 |
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568 \end{frame}} |
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569 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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570 |
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571 *} |
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572 |
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573 text_raw {* |
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574 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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575 \mode<presentation>{ |
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576 \begin{frame}[c] |
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577 \frametitle{\LARGE Equivalence Classes} |
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578 |
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579 \begin{itemize} |
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580 \item \smath{L = []} |
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581 \begin{center} |
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582 \smath{\Big\{\{[]\},\; U\!N\!IV - \{[]\}\Big\}} |
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583 \end{center}\bigskip\bigskip |
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584 |
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585 \item \smath{L = [c]} |
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586 \begin{center} |
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587 \smath{\Big\{\{[]\},\; \{[c]\},\; U\!N\!IV - \{[], [c]\}\Big\}} |
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588 \end{center}\bigskip\bigskip |
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589 |
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590 \item \smath{L = \varnothing} |
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591 \begin{center} |
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592 \smath{\Big\{U\!N\!IV\Big\}} |
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593 \end{center} |
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594 |
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595 \end{itemize} |
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596 |
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597 \end{frame}} |
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598 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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599 |
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600 *} |
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601 |
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602 text_raw {* |
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603 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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604 \mode<presentation>{ |
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605 \begin{frame}[c] |
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606 \frametitle{\LARGE Regular Languages} |
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607 |
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608 \begin{itemize} |
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609 \item \smath{L} is regular \smath{\dn} if there is an automaton \smath{M} |
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610 such that \smath{\mathbb{L}(M) = L}\\[1.5cm] |
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611 |
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612 \item Myhill-Nerode: |
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613 |
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614 \begin{center} |
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615 \begin{tabular}{l} |
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616 finite $\Rightarrow$ regular\\ |
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617 \;\;\;\smath{\text{finite}\,(U\!N\!IV /\!/ \approx_L) \Rightarrow \exists r. L = \mathbb{L}(r)}\\[3mm] |
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618 regular $\Rightarrow$ finite\\ |
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619 \;\;\;\smath{\text{finite}\, (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})} |
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620 \end{tabular} |
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621 \end{center} |
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622 |
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623 \end{itemize} |
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624 |
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625 \end{frame}} |
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626 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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627 |
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628 *} |
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629 |
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630 text_raw {* |
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631 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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632 \mode<presentation>{ |
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633 \begin{frame}[c] |
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634 \frametitle{\LARGE Final States} |
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635 |
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636 \mbox{}\\[3cm] |
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637 |
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638 \begin{itemize} |
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639 \item ??? \smath{\text{final}_L\,X \dn \{[|s|]_\approx\;|\; s \in X\}}\\ |
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640 \medskip |
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641 |
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642 \item we can prove: \smath{L = \bigcup \{X\;|\;\text{final}_L\,X\}} |
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643 |
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644 \end{itemize} |
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645 |
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646 \end{frame}} |
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647 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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648 *} |
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649 |
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650 text_raw {* |
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651 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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652 \mode<presentation>{ |
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653 \begin{frame}[c] |
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654 \frametitle{\LARGE Transitions between\\[-3mm] Equivalence Classes} |
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655 |
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656 \smath{L = \{[c]\}} |
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657 |
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658 \begin{tabular}{@ {\hspace{-7mm}}cc} |
|
659 \begin{tabular}{c} |
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660 \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] |
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661 \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] |
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662 |
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663 %\draw[help lines] (0,0) grid (3,2); |
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664 |
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665 \node[state,initial] (q_0) {$R_1$}; |
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666 \node[state,accepting] (q_1) [above right of=q_0] {$R_2$}; |
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667 \node[state] (q_2) [below right of=q_0] {$R_3$}; |
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668 |
|
669 \path[->] (q_0) edge node {c} (q_1) |
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670 edge node [swap] {$\Sigma-{c}$} (q_2) |
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671 (q_2) edge [loop below] node {$\Sigma$} () |
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672 (q_1) edge node {$\Sigma$} (q_2); |
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673 \end{tikzpicture} |
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674 \end{tabular} |
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675 & |
|
676 \begin{tabular}[t]{ll} |
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677 \\[-20mm] |
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678 \multicolumn{2}{l}{\smath{U\!N\!IV /\!/\approx_L} produces}\\[4mm] |
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679 |
|
680 \smath{R_1}: & \smath{\{[]\}}\\ |
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681 \smath{R_2}: & \smath{\{[c]\}}\\ |
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682 \smath{R_3}: & \smath{U\!N\!IV - \{[], [c]\}}\\[6mm] |
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683 \multicolumn{2}{l}{\onslide<2->{\smath{X \stackrel{c}{\longrightarrow} Y \dn X ;; [c] \subseteq Y}}} |
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684 \end{tabular} |
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685 |
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686 \end{tabular} |
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687 |
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688 \end{frame}} |
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689 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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690 *} |
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691 |
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692 |
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693 text_raw {* |
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694 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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695 \mode<presentation>{ |
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696 \begin{frame}[c] |
|
697 \frametitle{\LARGE Systems of Equations} |
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698 |
|
699 Inspired by a method of Brzozowski\;'64, we can build an equational system |
|
700 characterising the equivalence classes: |
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701 |
|
702 \begin{center} |
|
703 \begin{tabular}{@ {\hspace{-20mm}}c} |
|
704 \\[-13mm] |
|
705 \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick] |
|
706 \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] |
|
707 |
|
708 %\draw[help lines] (0,0) grid (3,2); |
|
709 |
|
710 \node[state,initial] (p_0) {$R_1$}; |
|
711 \node[state,accepting] (p_1) [right of=q_0] {$R_2$}; |
|
712 |
|
713 \path[->] (p_0) edge [bend left] node {a} (p_1) |
|
714 edge [loop above] node {b} () |
|
715 (p_1) edge [loop above] node {a} () |
|
716 edge [bend left] node {b} (p_0); |
|
717 \end{tikzpicture}\\ |
|
718 \\[-13mm] |
|
719 \end{tabular} |
|
720 \end{center} |
|
721 |
|
722 \begin{center} |
|
723 \begin{tabular}{@ {\hspace{-6mm}}ll@ {\hspace{1mm}}c@ {\hspace{1mm}}l} |
|
724 & \smath{R_1} & \smath{\equiv} & \smath{R_1;b + R_2;b \onslide<2->{\alert<2>{+ \lambda;[]}}}\\ |
|
725 & \smath{R_2} & \smath{\equiv} & \smath{R_1;a + R_2;a}\medskip\\ |
|
726 \onslide<3->{we can prove} |
|
727 & \onslide<3->{\smath{R_1}} & \onslide<3->{\smath{=}} |
|
728 & \onslide<3->{\smath{R_1; \mathbb{L}(b) \,\cup\, R_2;\mathbb{L}(b) \,\cup\, \{[]\};\{[]\}}}\\ |
|
729 & \onslide<3->{\smath{R_2}} & \onslide<3->{\smath{=}} |
|
730 & \onslide<3->{\smath{R_1; \mathbb{L}(a) \,\cup\, R_2;\mathbb{L}(a)}}\\ |
|
731 \end{tabular} |
|
732 \end{center} |
|
733 |
|
734 \end{frame}} |
|
735 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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736 *} |
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737 |
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738 |
|
739 text_raw {* |
|
740 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
741 \mode<presentation>{ |
|
742 \begin{frame}<1>[t] |
|
743 \small |
|
744 |
|
745 \begin{center} |
|
746 \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} |
|
747 \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}} |
|
748 & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ |
|
749 \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}} |
|
750 & \onslide<1->{\smath{R_1; a + R_2; a}}\\ |
|
751 |
|
752 & & & \onslide<2->{by Arden}\\ |
|
753 |
|
754 \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}} |
|
755 & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ |
|
756 \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}} |
|
757 & \only<2>{\smath{R_1; a + R_2; a}}% |
|
758 \only<3->{\smath{R_1; a\cdot a^\star}}\\ |
|
759 |
|
760 & & & \onslide<4->{by Arden}\\ |
|
761 |
|
762 \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} |
|
763 & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\ |
|
764 \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}} |
|
765 & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\ |
|
766 |
|
767 & & & \onslide<5->{by substitution}\\ |
|
768 |
|
769 \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} |
|
770 & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\ |
|
771 \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}} |
|
772 & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\ |
|
773 |
|
774 & & & \onslide<6->{by Arden}\\ |
|
775 |
|
776 \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}} |
|
777 & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ |
|
778 \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}} |
|
779 & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\ |
|
780 |
|
781 & & & \onslide<7->{by substitution}\\ |
|
782 |
|
783 \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}} |
|
784 & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ |
|
785 \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}} |
|
786 & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star |
|
787 \cdot a\cdot a^\star}}\\ |
|
788 \end{tabular} |
|
789 \end{center} |
|
790 |
|
791 \end{frame}} |
|
792 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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793 *} |
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794 |
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795 text_raw {* |
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796 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
797 \mode<presentation>{ |
|
798 \begin{frame}[c] |
|
799 \frametitle{\LARGE A Variant of Arden's Lemma} |
|
800 |
|
801 {\bf Arden's Lemma:}\smallskip |
|
802 |
|
803 If \smath{[] \not\in A} then |
|
804 \begin{center} |
|
805 \smath{X = X; A + \text{something}} |
|
806 \end{center} |
|
807 has the (unique) solution |
|
808 \begin{center} |
|
809 \smath{X = \text{something} ; A^\star} |
|
810 \end{center} |
|
811 |
|
812 |
|
813 \end{frame}} |
|
814 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
815 *} |
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816 |
|
817 |
|
818 text_raw {* |
|
819 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
|
820 \mode<presentation>{ |
|
821 \begin{frame}<1->[t] |
|
822 \small |
|
823 |
|
824 \begin{center} |
|
825 \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} |
|
826 \onslide<1->{\smath{R_1}} & \onslide<1->{\smath{=}} |
|
827 & \onslide<1->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ |
|
828 \onslide<1->{\smath{R_2}} & \onslide<1->{\smath{=}} |
|
829 & \onslide<1->{\smath{R_1; a + R_2; a}}\\ |
|
830 |
|
831 & & & \onslide<2->{by Arden}\\ |
|
832 |
|
833 \onslide<2->{\smath{R_1}} & \onslide<2->{\smath{=}} |
|
834 & \onslide<2->{\smath{R_1; b + R_2; b + \lambda;[]}}\\ |
|
835 \onslide<2->{\smath{R_2}} & \onslide<2->{\smath{=}} |
|
836 & \only<2>{\smath{R_1; a + R_2; a}}% |
|
837 \only<3->{\smath{R_1; a\cdot a^\star}}\\ |
|
838 |
|
839 & & & \onslide<4->{by Arden}\\ |
|
840 |
|
841 \onslide<4->{\smath{R_1}} & \onslide<4->{\smath{=}} |
|
842 & \onslide<4->{\smath{R_2; b \cdot b^\star+ \lambda;b^\star}}\\ |
|
843 \onslide<4->{\smath{R_2}} & \onslide<4->{\smath{=}} |
|
844 & \onslide<4->{\smath{R_1; a\cdot a^\star}}\\ |
|
845 |
|
846 & & & \onslide<5->{by substitution}\\ |
|
847 |
|
848 \onslide<5->{\smath{R_1}} & \onslide<5->{\smath{=}} |
|
849 & \onslide<5->{\smath{R_1; a\cdot a^\star \cdot b \cdot b^\star+ \lambda;b^\star}}\\ |
|
850 \onslide<5->{\smath{R_2}} & \onslide<5->{\smath{=}} |
|
851 & \onslide<5->{\smath{R_1; a\cdot a^\star}}\\ |
|
852 |
|
853 & & & \onslide<6->{by Arden}\\ |
|
854 |
|
855 \onslide<6->{\smath{R_1}} & \onslide<6->{\smath{=}} |
|
856 & \onslide<6->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ |
|
857 \onslide<6->{\smath{R_2}} & \onslide<6->{\smath{=}} |
|
858 & \onslide<6->{\smath{R_1; a\cdot a^\star}}\\ |
|
859 |
|
860 & & & \onslide<7->{by substitution}\\ |
|
861 |
|
862 \onslide<7->{\smath{R_1}} & \onslide<7->{\smath{=}} |
|
863 & \onslide<7->{\smath{\lambda;b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star}}\\ |
|
864 \onslide<7->{\smath{R_2}} & \onslide<7->{\smath{=}} |
|
865 & \onslide<7->{\smath{\lambda; b^\star\cdot (a\cdot a^\star \cdot b \cdot b^\star)^\star |
|
866 \cdot a\cdot a^\star}}\\ |
|
867 \end{tabular} |
|
868 \end{center} |
|
869 |
|
870 \only<8->{ |
|
871 \begin{textblock}{6}(2.5,4) |
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872 \begin{block}{} |
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873 \begin{minipage}{8cm}\raggedright |
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874 |
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875 \begin{tikzpicture}[shorten >=1pt,node distance=2cm,auto, ultra thick, inner sep=1mm] |
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876 \tikzstyle{state}=[circle,thick,draw=blue!75,fill=blue!20,minimum size=0mm] |
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877 |
|
878 %\draw[help lines] (0,0) grid (3,2); |
|
879 |
|
880 \node[state,initial] (p_0) {$R_1$}; |
|
881 \node[state,accepting] (p_1) [right of=q_0] {$R_2$}; |
|
882 |
|
883 \path[->] (p_0) edge [bend left] node {a} (p_1) |
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884 edge [loop above] node {b} () |
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885 (p_1) edge [loop above] node {a} () |
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886 edge [bend left] node {b} (p_0); |
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887 \end{tikzpicture} |
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888 |
|
889 \end{minipage} |
|
890 \end{block} |
|
891 \end{textblock}} |
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892 |
|
893 \end{frame}} |
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894 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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895 *} |
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896 |
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897 |
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898 text_raw {* |
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899 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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900 \mode<presentation>{ |
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901 \begin{frame}[c] |
|
902 \frametitle{\LARGE The Equ's Solving Algorithm} |
|
903 |
|
904 \begin{itemize} |
|
905 \item The algorithm must terminate: Arden makes one equation smaller; |
|
906 substitution deletes one variable from the right-hand sides.\bigskip |
|
907 |
|
908 \item We need to maintain the invariant that Arden is applicable |
|
909 (if \smath{[] \not\in A} then \ldots):\medskip |
|
910 |
|
911 \begin{center}\small |
|
912 \begin{tabular}{l@ {\hspace{1mm}}c@ {\hspace{1mm}}ll} |
|
913 \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\ |
|
914 \smath{R_2} & \smath{=} & \smath{R_1; a + R_2; a}\\ |
|
915 |
|
916 & & & by Arden\\ |
|
917 |
|
918 \smath{R_1} & \smath{=} & \smath{R_1; b + R_2; b + \lambda;[]}\\ |
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919 \smath{R_2} & \smath{=} & \smath{R_1; a\cdot a^\star}\\ |
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920 \end{tabular} |
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921 \end{center} |
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922 |
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923 \end{itemize} |
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924 |
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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 Other Direction} |
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936 |
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937 One has to prove |
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938 |
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939 \begin{center} |
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940 \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r)})} |
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941 \end{center} |
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942 |
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943 by induction on \smath{r}. Not trivial, but after a bit |
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944 of thinking, one can prove that if |
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945 |
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946 \begin{center} |
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947 \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1)})}\hspace{5mm} |
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948 \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_2)})} |
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949 \end{center} |
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950 |
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951 then |
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952 |
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953 \begin{center} |
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954 \smath{\text{finite} (U\!N\!IV /\!/ \approx_{\mathbb{L}(r_1) \,\cup\, \mathbb{L}(r_2)})} |
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955 \end{center} |
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956 |
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957 |
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958 |
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959 \end{frame}} |
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960 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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961 *} |
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962 |
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963 |
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964 text_raw {* |
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965 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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966 \mode<presentation>{ |
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967 \begin{frame}[c] |
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968 \frametitle{\LARGE What Have We Achieved?} |
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969 |
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970 \begin{itemize} |
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971 \item \smath{\text{finite}\, (U\!N\!IV /\!/ \approx_L) \;\Leftrightarrow\; L\; \text{is regular}} |
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972 \bigskip\pause |
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973 \item regular languages are closed under complementation; this is now easy\medskip |
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974 \begin{center} |
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975 \smath{U\!N\!IV /\!/ \approx_L \;\;=\;\; U\!N\!IV /\!/ \approx_{-L}} |
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976 \end{center} |
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977 \end{itemize} |
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978 |
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979 |
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980 \end{frame}} |
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981 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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982 *} |
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983 |
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984 text_raw {* |
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985 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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986 \mode<presentation>{ |
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987 \begin{frame}[c] |
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988 \frametitle{\LARGE Examples} |
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989 |
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990 \begin{itemize} |
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991 \item \smath{L \equiv \Sigma^\star 0 \Sigma} is regular |
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992 \begin{quote}\small |
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993 \begin{tabular}{lcl} |
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994 \smath{A_1} & \smath{=} & \smath{\Sigma^\star 00}\\ |
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995 \smath{A_2} & \smath{=} & \smath{\Sigma^\star 01}\\ |
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996 \smath{A_3} & \smath{=} & \smath{\Sigma^\star 10 \cup \{0\}}\\ |
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997 \smath{A_4} & \smath{=} & \smath{\Sigma^\star 11 \cup \{1\} \cup \{[]\}}\\ |
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998 \end{tabular} |
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999 \end{quote} |
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1000 |
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1001 \item \smath{L \equiv \{ 0^n 1^n \,|\, n \ge 0\}} is not regular |
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1002 \begin{quote}\small |
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1003 \begin{tabular}{lcl} |
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1004 \smath{B_0} & \smath{=} & \smath{\{0^n 1^n \,|\, n \ge 0\}}\\ |
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1005 \smath{B_1} & \smath{=} & \smath{\{0^n 1^{(n-1)} \,|\, n \ge 1\}}\\ |
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1006 \smath{B_2} & \smath{=} & \smath{\{0^n 1^{(n-2)} \,|\, n \ge 2\}}\\ |
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1007 \smath{B_3} & \smath{=} & \smath{\{0^n 1^{(n-3)} \,|\, n \ge 3\}}\\ |
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1008 & \smath{\vdots} &\\ |
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1009 \end{tabular} |
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1010 \end{quote} |
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1011 \end{itemize} |
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1012 |
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1013 \end{frame}} |
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1014 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1015 *} |
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1016 |
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1017 |
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1018 text_raw {* |
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1019 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1020 \mode<presentation>{ |
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1021 \begin{frame}[c] |
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1022 \frametitle{\LARGE What We Have Not Achieved} |
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1023 |
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1024 \begin{itemize} |
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1025 \item regular expressions are not good if you look for a minimal |
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1026 one for a language (DFAs have this notion)\pause\bigskip |
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1027 |
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1028 \item Is there anything to be said about context free languages:\medskip |
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1029 |
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1030 \begin{quote} |
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1031 A context free language is where every string can be recognised by |
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1032 a pushdown automaton.\bigskip |
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1033 \end{quote} |
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1034 \end{itemize} |
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1035 |
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1036 \textcolor{gray}{\footnotesize Yes. Derivatives also work for c-f grammars. Ongoing work.} |
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1037 |
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1038 \end{frame}} |
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1039 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1040 *} |
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1041 |
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1042 |
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1043 text_raw {* |
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1044 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1045 \mode<presentation>{ |
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1046 \begin{frame}[c] |
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1047 \frametitle{\LARGE Conclusion} |
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1048 |
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1049 \begin{itemize} |
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1050 \item We formalised the Myhill-Nerode theorem based on |
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1051 regular expressions only (DFAs are difficult to deal with in a theorem prover).\smallskip |
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1052 |
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1053 \item Seems to be a common theme: algorithms need to be reformulated |
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1054 to better suit formal treatment.\smallskip |
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1055 |
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1056 \item The most interesting aspect is that we are able to |
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1057 implement the matcher directly inside the theorem prover |
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1058 (ongoing work).\smallskip |
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1059 |
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1060 \item Parsing is a vast field which seem to offer new results. |
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1061 \end{itemize} |
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1062 |
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1063 \end{frame}} |
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1064 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1065 *} |
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1066 |
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1067 text_raw {* |
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1068 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1069 \mode<presentation>{ |
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1070 \begin{frame}<1>[b] |
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1071 \frametitle{ |
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1072 \begin{tabular}{c} |
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1073 \mbox{}\\[13mm] |
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1074 \alert{\LARGE Thank you very much!}\\ |
|
1075 \alert{\Large Questions?} |
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1076 \end{tabular}} |
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1077 |
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1078 \end{frame}} |
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1079 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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1080 *} |
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1081 |
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1082 |
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1083 |
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1084 (*<*) |
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1085 end |
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1086 (*>*) |