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1 \documentclass[dvipsnames,14pt,t]{beamer} |
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2 \usepackage{beamerthemeplainculight} |
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3 \usepackage[T1]{fontenc} |
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4 \usepackage[latin1]{inputenc} |
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5 \usepackage{mathpartir} |
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6 \usepackage[absolute,overlay]{textpos} |
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7 \usepackage{ifthen} |
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8 \usepackage{tikz} |
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9 \usepackage{pgf} |
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10 \usepackage{calc} |
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11 \usepackage{ulem} |
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12 \usepackage{courier} |
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13 \usepackage{listings} |
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14 \renewcommand{\uline}[1]{#1} |
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15 \usetikzlibrary{arrows} |
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16 \usetikzlibrary{automata} |
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17 \usetikzlibrary{shapes} |
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18 \usetikzlibrary{shadows} |
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19 \usetikzlibrary{positioning} |
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20 \usetikzlibrary{calc} |
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21 \usetikzlibrary{plotmarks} |
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22 \usepackage{graphicx} |
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23 \usepackage{pgfplots} |
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24 |
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25 |
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26 \definecolor{javared}{rgb}{0.6,0,0} % for strings |
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27 \definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments |
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28 \definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords |
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29 \definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc |
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30 |
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31 \lstset{language=Java, |
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32 basicstyle=\ttfamily, |
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33 keywordstyle=\color{javapurple}\bfseries, |
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34 stringstyle=\color{javagreen}, |
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35 commentstyle=\color{javagreen}, |
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36 morecomment=[s][\color{javadocblue}]{/**}{*/}, |
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37 numbers=left, |
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38 numberstyle=\tiny\color{black}, |
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39 stepnumber=1, |
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40 numbersep=10pt, |
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41 tabsize=2, |
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42 showspaces=false, |
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43 showstringspaces=false} |
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44 |
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45 \lstdefinelanguage{scala}{ |
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46 morekeywords={abstract,case,catch,class,def,% |
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47 do,else,extends,false,final,finally,% |
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48 for,if,implicit,import,match,mixin,% |
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49 new,null,object,override,package,% |
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50 private,protected,requires,return,sealed,% |
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51 super,this,throw,trait,true,try,% |
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52 type,val,var,while,with,yield}, |
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53 otherkeywords={=>,<-,<\%,<:,>:,\#,@}, |
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54 sensitive=true, |
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55 morecomment=[l]{//}, |
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56 morecomment=[n]{/*}{*/}, |
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57 morestring=[b]", |
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58 morestring=[b]', |
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59 morestring=[b]""" |
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60 } |
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61 |
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62 |
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63 \lstset{language=Scala, |
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64 basicstyle=\ttfamily, |
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65 keywordstyle=\color{javapurple}\bfseries, |
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66 stringstyle=\color{javagreen}, |
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67 commentstyle=\color{javagreen}, |
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68 morecomment=[s][\color{javadocblue}]{/**}{*/}, |
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69 numbers=left, |
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70 numberstyle=\tiny\color{black}, |
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71 stepnumber=1, |
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72 numbersep=10pt, |
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73 tabsize=2, |
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74 showspaces=false, |
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75 showstringspaces=false} |
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76 |
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77 \lstdefinelanguage{while}{ |
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78 morekeywords={if,then,else,while,do,true,false,write}, |
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79 otherkeywords={=,!=,:=,<,>,;}, |
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80 sensitive=true, |
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81 morecomment=[n]{/*}{*/}, |
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82 } |
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83 |
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84 |
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85 \lstset{language=While, |
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86 basicstyle=\ttfamily, |
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87 keywordstyle=\color{javapurple}\bfseries, |
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88 stringstyle=\color{javagreen}, |
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89 commentstyle=\color{javagreen}, |
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90 morecomment=[s][\color{javadocblue}]{/**}{*/}, |
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91 numbers=left, |
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92 numberstyle=\tiny\color{black}, |
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93 stepnumber=1, |
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94 numbersep=10pt, |
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95 tabsize=2, |
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96 showspaces=false, |
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97 showstringspaces=false} |
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98 |
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99 |
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100 % beamer stuff |
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101 \renewcommand{\slidecaption}{AFL 10, King's College London, 5.~December 2012} |
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102 \newcommand{\bl}[1]{\textcolor{blue}{#1}} |
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103 \newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions |
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104 |
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105 |
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106 % The data files, written on the first run. |
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107 \begin{filecontents}{compiled.data} |
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108 %1 0.234146 |
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112 100000 3.713165 |
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113 250000 21.6624545 |
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114 500000 85.872613 |
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115 750000 203.6408015 |
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116 1000000 345.736574 |
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117 \end{filecontents} |
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118 |
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119 \begin{filecontents}{interpreted.data} |
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120 %1 0.00503 |
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129 \end{filecontents} |
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130 |
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131 \begin{filecontents}{interpreted2.data} |
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132 %1 0.00503 |
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133 200 1.005863 |
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134 400 7.8296765 |
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135 600 27.2321885 |
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136 800 65.249271 |
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137 1000 135.4493445 |
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138 1200 232.134097 |
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139 1400 382.527227 |
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140 \end{filecontents} |
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141 |
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142 \begin{filecontents}{compiled2.data} |
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143 200 0.222058 |
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150 \end{filecontents} |
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151 |
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152 \begin{document} |
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153 |
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154 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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155 \mode<presentation>{ |
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156 \begin{frame}<1>[t] |
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157 \frametitle{% |
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158 \begin{tabular}{@ {}c@ {}} |
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159 \\[-3mm] |
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160 \LARGE Automata and \\[-2mm] |
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161 \LARGE Formal Languages (10)\\[3mm] |
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162 \end{tabular}} |
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163 |
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164 \normalsize |
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165 \begin{center} |
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166 \begin{tabular}{ll} |
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167 Email: & christian.urban at kcl.ac.uk\\ |
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168 Of$\!$fice: & S1.27 (1st floor Strand Building)\\ |
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169 Slides: & KEATS (also home work is there)\\ |
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170 \end{tabular} |
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171 \end{center} |
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172 |
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173 \end{frame}} |
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174 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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175 |
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176 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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177 \mode<presentation>{ |
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178 \begin{frame}[c] |
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179 |
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180 \Large\bf |
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181 There are more problems, than there are programs.\bigskip\bigskip\pause\\ |
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182 |
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183 There must be a problem for which there is no program. |
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184 \end{frame}} |
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185 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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186 |
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187 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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188 \mode<presentation>{ |
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189 \begin{frame}[c] |
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190 \frametitle{Revision: Proofs} |
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191 |
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192 \begin{center} |
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193 \includegraphics[scale=0.4]{river-stones.jpg} |
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194 \end{center} |
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195 |
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196 \end{frame}} |
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197 |
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198 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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199 \mode<presentation>{ |
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200 \begin{frame}[c] |
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201 \frametitle{Subsets} |
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202 |
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203 \Large |
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204 \bl{$A \subseteq B$}\bigskip\bigskip\\ |
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205 |
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206 \bl{$\forall e.\; e \in A \Rightarrow e \in B$} |
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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{Subsets} |
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214 |
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215 \Large |
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216 \bl{$A \subseteq B$} and \bl{$B \subseteq A$}\bigskip |
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217 |
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218 then \bl{$A = B$} |
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219 \end{frame}} |
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220 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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221 |
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222 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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223 \mode<presentation>{ |
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224 \begin{frame}[c] |
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225 \frametitle{Injective Function} |
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226 |
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227 \Large |
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228 \bl{f} is an injective function iff \bigskip |
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229 |
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230 \bl{$\forall x y.\; f(x) = f(y) \Rightarrow x = y$} |
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231 |
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232 \end{frame}} |
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233 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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234 |
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235 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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236 \mode<presentation>{ |
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237 \begin{frame}[c] |
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238 \frametitle{Cardinality} |
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239 |
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240 \Large |
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241 \bl{$|A|$} $\dn$ ``how many elements''\bigskip\\ |
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242 |
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243 \bl{$A \subseteq B \Rightarrow |A| \leq |B|$}\bigskip\\\pause |
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244 |
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245 if there is an injective function \bl{$f: A \rightarrow B$} then \bl{$|A| \leq |B|$}\ |
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246 |
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247 \end{frame}} |
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248 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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249 |
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250 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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251 \mode<presentation>{ |
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252 \begin{frame}[c] |
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253 \frametitle{Natural Numbers} |
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254 |
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255 \Large |
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256 \bl{$\mathbb{N}$} \bl{$\dn$} \bl{$\{0, 1, 2, 3, .......\}$}\bigskip\pause |
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257 |
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258 \bl{$A$} is \alert{countable} iff \bl{$|A| \leq |\mathbb{N}|$} |
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259 |
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260 \end{frame}} |
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261 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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262 |
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263 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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264 \mode<presentation>{ |
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265 \begin{frame}[c] |
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266 \frametitle{First Question} |
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267 |
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268 \Large |
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269 \bl{$|\mathbb{N} - \{0\}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip |
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270 |
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271 \normalsize |
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272 \bl{$\geq$} or \bl{$\leq$} or \bl{$=$} |
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273 \end{frame}} |
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274 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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275 |
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276 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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277 \mode<presentation>{ |
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278 \begin{frame}[c] |
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279 |
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280 \Large |
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281 \bl{$|\mathbb{N} - \{0, 1\}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\pause |
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282 |
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283 \bl{$|\mathbb{N} - \mathbb{O}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip |
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284 |
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285 \normalsize |
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286 \bl{$\mathbb{O}$} $\dn$ odd numbers\quad \bl{$\{1,3,5......\}$}\\ \pause |
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287 \bl{$\mathbb{E}$} $\dn$ even numbers\quad \bl{$\{0,2,4......\}$}\\ |
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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 |
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295 \Large |
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296 \bl{$|\mathbb{N} \cup \mathbb{-N}| \;\;\;\alert{?}\;\;\; |\mathbb{N}| $}\bigskip\bigskip |
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297 |
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298 |
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299 \normalsize |
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300 \bl{$\mathbb{\phantom{-}N}$} $\dn$ positive numbers\quad \bl{$\{0,1,2,3,......\}$}\\ |
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301 \bl{$\mathbb{-N}$} $\dn$ negative numbers\quad \bl{$\{0,-1,-2,-3,......\}$}\\ |
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302 \end{frame}} |
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303 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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304 |
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305 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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306 \mode<presentation>{ |
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307 \begin{frame}[c] |
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308 |
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309 \Large |
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310 \bl{$A$} is \alert{countable} if there exists an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip |
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311 |
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312 \bl{$A$} is \alert{uncountable} if there does not exist an injective \bl{$f : A \rightarrow \mathbb{N}$}\bigskip\bigskip |
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313 |
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314 |
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315 countable: \bl{$|A| \leq |\mathbb{N}|$}\\ |
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316 uncountable: \bl{$|A| > |\mathbb{N}|$}\pause\bigskip |
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317 |
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318 |
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319 Does there exist such an \bl{$A$} ? |
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320 |
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321 \end{frame}} |
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322 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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323 |
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324 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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325 \mode<presentation>{ |
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326 \begin{frame}[c] |
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327 \frametitle{Halting Problem} |
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328 |
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329 \large |
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330 Assume a program \bl{$H$} that decides for all programs \bl{$A$} and all |
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331 input data \bl{$D$} whether\bigskip |
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332 |
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333 \begin{itemize} |
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334 \item \bl{$H(A, D) \dn 1$} iff \bl{$A(D)$} terminates |
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335 \item \bl{$H(A, D) \dn 0$} otherwise |
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336 \end{itemize} |
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337 |
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338 \end{frame}} |
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339 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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340 |
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341 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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342 \mode<presentation>{ |
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343 \begin{frame}[c] |
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344 \frametitle{Halting Problem} |
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345 |
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346 \large |
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347 Given such a program \bl{$H$} define the following program \bl{$C$}: |
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348 for all programs \bl{$A$}\bigskip |
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349 |
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350 \begin{itemize} |
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351 \item \bl{$C(A) \dn 0$} iff \bl{$H(A, A) = 0$} |
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352 \item \bl{$C(A) \dn 1$} otherwise |
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353 \end{itemize} |
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354 |
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355 \end{frame}} |
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356 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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357 |
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358 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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359 \mode<presentation>{ |
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360 \begin{frame}[c] |
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361 \frametitle{Halting Problem (2)} |
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362 |
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363 \large |
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364 Given such a program \bl{$H$} define the following program \bl{$C$}: |
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365 for all programs \bl{$A$}\bigskip |
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366 |
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367 \begin{itemize} |
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368 \item \bl{$C(A) \dn 0$} iff \bl{$H(A, A) = 0$} |
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369 \item \bl{$C(A) \dn 1$} otherwise |
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370 \end{itemize} |
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371 |
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372 \end{frame}} |
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373 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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374 |
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375 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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376 \mode<presentation>{ |
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377 \begin{frame}[c] |
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378 \frametitle{Contradiction} |
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379 |
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380 |
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381 \bl{$H(C, C)$} is either \bl{$0$} or \bl{$1$}. |
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382 |
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383 \begin{itemize} |
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384 \item \bl{$H(C, C) = 1$} $\stackrel{def H}{\Rightarrow}$ \bl{$C(C)\downarrow$} $\stackrel{def C}{\Rightarrow}$ \bl{$H(C, C)=0$} |
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385 \item \bl{$H(C, C) = 0$} $\stackrel{def H}{\Rightarrow}$ \bl{$C(C)$} loops $\stackrel{def C}{\Rightarrow}$\\ |
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386 \hspace{7cm}\bl{$H(C, C)=1$} |
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387 \end{itemize} |
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388 |
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389 Contradiction in both cases. So \bl{$H$} cannot exist. |
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390 |
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391 \end{frame}} |
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392 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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393 |
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394 \end{document} |
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395 |
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396 %%% Local Variables: |
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397 %%% mode: latex |
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398 %%% TeX-master: t |
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399 %%% End: |
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400 |