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1 \documentclass[dvipsnames,14pt,t]{beamer} |
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2 \usepackage{proof} |
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3 \usepackage{beamerthemeplainculight} |
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4 \usepackage[T1]{fontenc} |
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5 \usepackage[latin1]{inputenc} |
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6 \usepackage{mathpartir} |
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7 \usepackage{isabelle} |
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8 \usepackage{isabellesym} |
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9 \usepackage[absolute,overlay]{textpos} |
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10 \usepackage{ifthen} |
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11 \usepackage{tikz} |
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12 \usepackage{courier} |
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13 \usepackage{listings} |
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14 \usetikzlibrary{arrows} |
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15 \usetikzlibrary{positioning} |
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16 \usetikzlibrary{calc} |
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17 \usepackage{graphicx} |
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18 |
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19 \isabellestyle{rm} |
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20 \renewcommand{\isastyle}{\rm}% |
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21 \renewcommand{\isastyleminor}{\rm}% |
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22 \renewcommand{\isastylescript}{\footnotesize\rm\slshape}% |
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23 \renewcommand{\isatagproof}{} |
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24 \renewcommand{\endisatagproof}{} |
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25 \renewcommand{\isamarkupcmt}[1]{#1} |
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26 |
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27 % Isabelle characters |
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28 \renewcommand{\isacharunderscore}{\_} |
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29 \renewcommand{\isacharbar}{\isamath{\mid}} |
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30 \renewcommand{\isasymiota}{} |
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31 \renewcommand{\isacharbraceleft}{\{} |
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32 \renewcommand{\isacharbraceright}{\}} |
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33 \renewcommand{\isacharless}{$\langle$} |
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34 \renewcommand{\isachargreater}{$\rangle$} |
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35 \renewcommand{\isasymsharp}{\isamath{\#}} |
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36 \renewcommand{\isasymdots}{\isamath{...}} |
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37 \renewcommand{\isasymbullet}{\act} |
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38 |
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39 |
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40 |
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41 \definecolor{javared}{rgb}{0.6,0,0} % for strings |
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42 \definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments |
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43 \definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords |
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44 \definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc |
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45 |
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46 \lstset{language=Java, |
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47 basicstyle=\ttfamily, |
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48 keywordstyle=\color{javapurple}\bfseries, |
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49 stringstyle=\color{javagreen}, |
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50 commentstyle=\color{javagreen}, |
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51 morecomment=[s][\color{javadocblue}]{/**}{*/}, |
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52 numbers=left, |
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53 numberstyle=\tiny\color{black}, |
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54 stepnumber=1, |
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55 numbersep=10pt, |
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56 tabsize=2, |
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57 showspaces=false, |
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58 showstringspaces=false} |
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59 |
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60 \lstdefinelanguage{scala}{ |
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61 morekeywords={abstract,case,catch,class,def,% |
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62 do,else,extends,false,final,finally,% |
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63 for,if,implicit,import,match,mixin,% |
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64 new,null,object,override,package,% |
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65 private,protected,requires,return,sealed,% |
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66 super,this,throw,trait,true,try,% |
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67 type,val,var,while,with,yield}, |
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68 otherkeywords={=>,<-,<\%,<:,>:,\#,@}, |
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69 sensitive=true, |
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70 morecomment=[l]{//}, |
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71 morecomment=[n]{/*}{*/}, |
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72 morestring=[b]", |
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73 morestring=[b]', |
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74 morestring=[b]""" |
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75 } |
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76 |
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77 \lstset{language=Scala, |
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78 basicstyle=\ttfamily, |
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79 keywordstyle=\color{javapurple}\bfseries, |
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80 stringstyle=\color{javagreen}, |
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81 commentstyle=\color{javagreen}, |
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82 morecomment=[s][\color{javadocblue}]{/**}{*/}, |
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83 numbers=left, |
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84 numberstyle=\tiny\color{black}, |
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85 stepnumber=1, |
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86 numbersep=10pt, |
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87 tabsize=2, |
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88 showspaces=false, |
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89 showstringspaces=false} |
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90 |
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91 % beamer stuff |
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92 \renewcommand{\slidecaption}{APP 06, King's College London, 29 October 2012} |
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93 |
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94 \newcommand{\bl}[1]{\textcolor{blue}{#1}} |
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95 \begin{document} |
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96 |
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97 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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98 \mode<presentation>{ |
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99 \begin{frame}<1>[t] |
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100 \frametitle{% |
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101 \begin{tabular}{@ {}c@ {}} |
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102 \\ |
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103 \LARGE Access Control and \\[-3mm] |
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104 \LARGE Privacy Policies (6)\\[-6mm] |
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105 \end{tabular}}\bigskip\bigskip\bigskip |
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106 |
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107 %\begin{center} |
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108 %\includegraphics[scale=1.3]{pics/barrier.jpg} |
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109 %\end{center} |
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110 |
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111 \normalsize |
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112 \begin{center} |
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113 \begin{tabular}{ll} |
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114 Email: & christian.urban at kcl.ac.uk\\ |
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115 Of$\!$fice: & S1.27 (1st floor Strand Building)\\ |
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116 Slides: & KEATS (also homework is there)\\ |
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117 \end{tabular} |
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118 \end{center} |
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119 |
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120 |
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121 \end{frame}} |
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122 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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123 |
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124 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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125 \mode<presentation>{ |
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126 \begin{frame}[c] |
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127 \frametitle{1st Week} |
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128 |
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129 \begin{itemize} |
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130 \item What are hashes and salts?\bigskip\pause |
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131 \item \ldots can be use to store securely data on a client, but |
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132 you cannot make your protocol dependent on the |
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133 presence of the data\bigskip\pause |
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134 \item \ldots can be used to store and verify passwords |
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135 |
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136 \end{itemize} |
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137 |
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138 \end{frame}} |
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139 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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140 |
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141 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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142 \mode<presentation>{ |
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143 \begin{frame}[c] |
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144 \frametitle{2nd Week} |
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145 |
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146 \begin{itemize} |
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147 \item Buffer overflows\bigskip |
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148 \item choice of programming language can mitigate or even eliminate this problem |
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149 \end{itemize} |
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150 |
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151 \end{frame}} |
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152 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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153 |
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154 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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155 \mode<presentation>{ |
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156 \begin{frame}[c] |
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157 \frametitle{3rd Week} |
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158 |
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159 \begin{itemize} |
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160 \item defence in depth\bigskip |
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161 \item privilege separation afforded by the OS |
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162 \end{itemize} |
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163 |
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164 \begin{center} |
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165 \begin{tikzpicture}[scale=1] |
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166 |
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167 \draw[line width=1mm] (0, 1.1) rectangle (1.2,2); |
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168 \draw (4.7,1) node {Internet}; |
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169 \draw (0.6,1.7) node {\footnotesize Slave}; |
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170 \draw[line width=1mm] (0, 0) rectangle (1.2,0.9); |
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171 \draw (0.6,1.7) node {\footnotesize Slave}; |
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172 \draw (0.6,0.6) node {\footnotesize Slave}; |
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173 \draw (0.6,-0.5) node {\footnotesize \begin{tabular}{c}unprivileged\\[-1mm] processes\end{tabular}}; |
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174 \draw (-2.7,-0.4) node {\footnotesize \begin{tabular}{c}privileged\\[-1mm] process\end{tabular}}; |
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175 |
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176 \draw[line width=1mm] (-1.8, 0) rectangle (-3.6,2); |
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177 \draw (-2.9,1.7) node {\footnotesize Monitor}; |
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178 |
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179 \draw[white] (1.7,1) node (X) {}; |
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180 \draw[white] (3.7,1) node (Y) {}; |
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181 \draw[red, <->, line width = 2mm] (X) -- (Y); |
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182 |
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183 \draw[red, <->, line width = 1mm] (-0.4,1.4) -- (-1.4,1.1); |
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184 \draw[red, <->, line width = 1mm] (-0.4,0.6) -- (-1.4,0.9); |
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185 |
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186 \end{tikzpicture} |
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187 \end{center} |
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188 |
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189 \end{frame}} |
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190 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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191 |
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192 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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193 \mode<presentation>{ |
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194 \begin{frame}[c] |
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195 \frametitle{4th Week} |
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196 |
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197 \begin{itemize} |
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198 \item voting\ldots has security requirements that are in tension with each other |
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199 \begin{center} |
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200 integrity vs ballot secrecy\\ |
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201 authentication vs enfranchisment |
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202 \end{center}\bigskip |
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203 |
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204 \item electronic voting makes `whole sale' fraud easier as opposed to `retail attacks' |
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205 \end{itemize} |
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206 |
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207 \end{frame}} |
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208 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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209 |
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210 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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211 \mode<presentation>{ |
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212 \begin{frame}[c] |
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213 \frametitle{5th Week} |
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214 |
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215 \begin{itemize} |
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216 \item access control logic\bigskip |
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217 |
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218 \item formulas |
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219 \item judgements |
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220 \item inference rules |
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221 \end{itemize} |
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222 |
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223 \end{frame}} |
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224 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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225 |
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226 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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227 \mode<presentation>{ |
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228 \begin{frame}[t] |
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229 \frametitle{\Large\begin{tabular}{@ {}c@ {}}Access Control Logic\end{tabular}} |
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230 |
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231 Formulas |
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232 |
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233 \begin{itemize} |
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234 \item[] |
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235 |
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236 \begin{center}\color{blue} |
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237 \begin{tabular}[t]{rcl@ {\hspace{10mm}}l} |
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238 \isa{F} & \isa{{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3A}{\isacharcolon}}{\isaliteral{3D}{\isacharequal}}} & \isa{true} \\ |
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239 & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{false} \\ |
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240 & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{F\ {\isaliteral{5C3C616E643E}{\isasymand}}\ F} \\ |
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241 & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{F\ {\isaliteral{5C3C6F723E}{\isasymor}}\ F} \\ |
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242 & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{F\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ F}\\ |
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243 & \isa{{\isaliteral{7C}{\isacharbar}}} & \isa{p\ {\isaliteral{28}{\isacharparenleft}}t\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{2C}{\isacharcomma}}{\isaliteral{5C3C646F74733E}{\isasymdots}}{\isaliteral{2C}{\isacharcomma}}t\isaliteral{5C3C5E697375623E}{}\isactrlisub n{\isaliteral{29}{\isacharparenright}}} \\ |
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244 & \isa{{\isaliteral{7C}{\isacharbar}}} & \alert{\isa{P\ says\ F}} & \textcolor{black}{``saying predicate''}\\ |
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245 \end{tabular} |
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246 \end{center} |
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247 |
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248 \end{itemize} |
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249 |
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250 Judgements |
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251 |
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252 \begin{itemize} |
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253 \item[] \mbox{\hspace{9mm}}\bl{$\Gamma \vdash \text{F}$} |
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254 \end{itemize} |
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255 |
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256 \end{frame}} |
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257 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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258 |
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259 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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260 \mode<presentation>{ |
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261 \begin{frame}[c] |
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262 \frametitle{Inference Rules} |
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263 |
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264 \begin{center} |
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265 \bl{\infer{\Gamma, F\vdash F}{}}\bigskip\\ |
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266 |
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267 \bl{\infer{\Gamma \vdash F_2}{\Gamma \vdash F_1 \Rightarrow F_2 \quad \Gamma \vdash F_2}} |
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268 \qquad |
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269 \bl{\infer{\Gamma \vdash F_1 \Rightarrow F_2}{F_1, \Gamma \vdash F_2}}\bigskip\\ |
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270 |
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271 \bl{\infer{\Gamma \vdash P\,\text{says}\, F}{\Gamma \vdash F}}\medskip\\ |
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272 |
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273 \bl{\infer{\Gamma \vdash P \,\text{says}\, F_2} |
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274 {\Gamma \vdash P \,\text{says}\, (F_1\Rightarrow F_2) \quad |
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275 \Gamma \vdash P \,\text{says}\, F_1}} |
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276 \end{center} |
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277 |
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278 \end{frame}} |
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279 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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280 |
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281 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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282 \mode<presentation>{ |
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283 \begin{frame}[c] |
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284 \frametitle{Proofs} |
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285 |
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286 |
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287 |
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288 \end{frame}} |
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289 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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290 |
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291 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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292 \mode<presentation>{ |
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293 \begin{frame}[c] |
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294 \frametitle{The Access Control Problem} |
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295 |
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296 |
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297 \begin{center} |
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298 \begin{tikzpicture}[scale=1] |
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299 |
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300 \draw[line width=1mm] (-.3, -0.5) rectangle (1.5,2); |
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301 \draw (-2.7,1) node {\begin{tabular}{l}access\\request\\ (\bl{$F$})\end{tabular}}; |
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302 \draw (4.2,1) node {\begin{tabular}{l}provable/\\not provable\end{tabular}}; |
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303 \draw (0.6,0.8) node {\footnotesize \begin{tabular}{l}AC-\\ Checker:\\ applies\\ inference\\ rules\end{tabular}}; |
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304 |
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305 \draw[red, ->, line width = 2mm] (1.7,1) -- (2.7,1); |
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306 \draw[red,<-, line width = 2mm] (-0.6,1) -- (-1.6,1); |
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307 \draw[red, <-, line width = 3mm] (0.6,2.2) -- (0.6,3.2); |
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308 |
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309 \draw (0.6,4) node {\begin{tabular}{l}\large Access Policy (\bl{$\Gamma$})\end{tabular}}; |
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310 |
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311 \end{tikzpicture} |
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312 \end{center} |
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313 |
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314 \end{frame}} |
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315 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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316 |
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317 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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318 \mode<presentation>{ |
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319 \begin{frame}[c] |
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320 \frametitle{} |
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321 |
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322 Recall the following scenario: |
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323 |
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324 \begin{itemize} |
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325 \item If \textcolor{blue}{Admin} says that \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}} |
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326 should be deleted, then this file must be deleted. |
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327 \item \textcolor{blue}{Admin} trusts \textcolor{blue}{Bob} to decide whether |
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328 \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}} should be deleted. |
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329 \item \textcolor{blue}{Bob} wants to delete \textcolor{blue}{\isa{file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}}. |
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330 \end{itemize}\bigskip |
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331 |
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332 \small |
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333 \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{3D}{\isacharequal}}}\small\begin{tabular}{l} |
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334 \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}},\\ |
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335 \isa{{\isaliteral{28}{\isacharparenleft}}Admin\ says\ {\isaliteral{28}{\isacharparenleft}}{\isaliteral{28}{\isacharparenleft}}Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}\ {\isaliteral{5C3C52696768746172726F773E}{\isasymRightarrow}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}{\isaliteral{29}{\isacharparenright}}{\isaliteral{29}{\isacharparenright}}},\\ |
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336 \isa{Bob\ says\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}\\ |
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337 \end{tabular}}\medskip |
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338 |
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339 \textcolor{blue}{\isa{{\isaliteral{5C3C47616D6D613E}{\isasymGamma}}\ {\isaliteral{5C3C7475726E7374696C653E}{\isasymturnstile}}\ del{\isaliteral{5F}{\isacharunderscore}}file\isaliteral{5C3C5E697375623E}{}\isactrlisub {\isadigit{1}}}} |
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340 \end{frame}} |
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341 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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342 |
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343 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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344 \mode<presentation>{ |
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345 \begin{frame}[c] |
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346 |
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347 How to prove \bl{$\Gamma \vdash F$}?\bigskip\bigskip |
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348 |
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349 \begin{center} |
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350 \Large \bl{\infer{\Gamma, F\vdash F}{}} |
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351 \end{center} |
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352 |
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353 \end{frame}} |
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354 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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355 |
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356 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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357 \mode<presentation>{ |
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358 \begin{frame}[c] |
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359 |
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360 \begin{center} |
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361 \Large |
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362 \bl{\infer{\Gamma \vdash F_1 \Rightarrow F_2}{F_1, \Gamma \vdash F_2}} |
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363 \end{center} |
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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 \mode<presentation>{ |
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370 \begin{frame}[c] |
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371 |
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372 \begin{center} |
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373 \Large |
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374 \bl{\infer{\Gamma \vdash P \,\text{says}\, F}{\Gamma \vdash F}} |
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375 \end{center} |
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376 |
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377 \end{frame}} |
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378 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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379 |
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380 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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381 \mode<presentation>{ |
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382 \begin{frame}[c] |
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383 |
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384 \begin{center} |
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385 \Large |
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386 \bl{\infer{\Gamma \vdash F_1 \vee F_2}{\Gamma \vdash F_1}}\qquad |
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387 \bl{\infer{\Gamma \vdash F_1 \vee F_2}{\Gamma \vdash F_2}}\ |
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388 \end{center} |
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389 |
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390 \end{frame}} |
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391 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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392 |
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393 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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394 \mode<presentation>{ |
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395 \begin{frame}[c] |
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396 |
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397 \begin{center} |
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398 \Large |
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399 \bl{\infer{\Gamma \vdash F_1 \wedge F_2}{\Gamma \vdash F_1 \quad \Gamma \vdash F_2}} |
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400 \end{center} |
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401 |
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402 \end{frame}} |
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403 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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404 |
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405 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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406 \mode<presentation>{ |
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407 \begin{frame}[t] |
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408 |
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409 I want to prove \bl{$\Gamma \vdash \text{Pred}$}\bigskip\bigskip\pause |
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410 |
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411 \begin{enumerate} |
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412 \item I found that \bl{$\Gamma$} contains the assumption \bl{$F_1 \Rightarrow F_2$}\bigskip\pause |
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413 \item If I can prove \bl{$\Gamma \vdash F_1$},\pause{} then I can prove |
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414 \begin{center} |
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415 \bl{$\Gamma \vdash F_2$} |
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416 \end{center}\bigskip\pause |
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417 |
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418 \item So better I try to prove \bl{$\Gamma \vdash \text{Pred}$} with the additional assumption |
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419 \bl{$F_2$}.\bigskip |
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420 \begin{center} |
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421 \bl{$F_2, \Gamma \vdash \text{Pred}$} |
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422 \end{center} |
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423 |
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424 \end{enumerate} |
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425 |
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426 \end{frame}} |
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427 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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428 |
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429 \end{document} |
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430 |
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431 %%% Local Variables: |
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432 %%% mode: latex |
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433 %%% TeX-master: t |
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434 %%% End: |
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435 |