1 | \documentclass[nopanel]{beamer} |
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2 | \usepackage{graphicx,color} |
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3 | |
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4 | \usetheme{Frankfurt} |
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5 | \logo{\includegraphics[height=1.0cm]{fetopen.png}} |
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6 | |
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7 | \usepackage[english]{babel} |
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8 | %\usepackage{inputenc} |
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9 | |
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10 | |
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11 | %\documentclass[nopanel,ucs]{beamer} |
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12 | \usepackage[utf8x]{inputenc} |
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13 | %\usepackage[T1,OT1]{fontenc} |
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14 | %\usepackage{pifont} |
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15 | %\usepackage{aeguill} |
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16 | % using aeguill forces us to T1 everywhere, resulting in horribleness |
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17 | % \def\selectguillfont{\fontencoding{T1}\fontfamily{cmtt}\selectfont}% |
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18 | %\def\guillemotleft{{\selectguillfont\symbol{19}}}% |
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19 | %\def\guillemotright{{\selectguillfont\symbol{20}}}% |
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20 | %\SetUnicodeOption{postscript} |
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21 | %\usepackage{hyperref} |
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22 | \usepackage{latexsym} |
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23 | \usepackage{listings} |
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24 | |
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25 | \def\cerco{CerCo} |
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26 | \def\ocaml{OCaml} |
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27 | |
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28 | \newcommand{\red}[1]{\textcolor{red}{#1}} |
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29 | \newcommand{\blue}[1]{\textcolor{blue}{#1}} |
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30 | \definecolor{darkgreen}{rgb}{0,0.4,0} |
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31 | \newcommand{\green}[1]{\textcolor{darkgreen}{#1}} |
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32 | \newcommand{\gray}[1]{\textcolor{gray}{#1}} |
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33 | |
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34 | %\setbeamertemplate{footline}[page number] |
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35 | %\setbeamertemplate{sidebar right}{} |
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36 | \setbeamertemplate{navigation symbols}{} |
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37 | |
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38 | \lstdefinelanguage{coq} |
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39 | {keywords={Definition,Lemma,Theorem,Remark,Qed,Save,Inductive,Record}, |
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40 | morekeywords={[2]if,then,else}, |
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41 | } |
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42 | |
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43 | \lstdefinelanguage{matita} |
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44 | {keywords={definition,lemma,theorem,remark,inductive,record,qed,let,rec,match,with,Type,and,on,return}, |
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45 | morekeywords={[2]whd,normalize,elim,cases,destruct}, |
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46 | mathescape=true, |
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47 | morecomment=[n]{(*}{*)}, |
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48 | } |
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49 | |
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50 | \lstset{language=matita,basicstyle=\small\tt,columns=flexible,breaklines=false, |
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51 | keywordstyle=\color{red}\bfseries, |
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52 | keywordstyle=[2]\color{blue}, |
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53 | commentstyle=\color{darkgreen}, |
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54 | stringstyle=\color{blue}, |
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55 | showspaces=false,showstringspaces=false} |
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56 | |
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57 | \lstset{extendedchars=false} |
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58 | \lstset{inputencoding=utf8x} |
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59 | \DeclareUnicodeCharacter{8797}{:=} |
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60 | \DeclareUnicodeCharacter{10746}{++} |
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61 | \DeclareUnicodeCharacter{9001}{\ensuremath{\langle}} |
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62 | \DeclareUnicodeCharacter{9002}{\ensuremath{\rangle}} |
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63 | |
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64 | |
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65 | \begin{document} |
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66 | |
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67 | \title{CerCo Work Package 3\\ |
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68 | Verified Compiler --- Front-end} |
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69 | \author{Brian Campbell\\ \scriptsize\smallskip |
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70 | \href{mailto:Brian.Campbell@ed.ac.uk}{\texttt{Brian.Campbell@ed.ac.uk}}} |
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71 | \institute{Laboratory for Foundations of Computer Science\\ |
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72 | The University of Edinburgh} |
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73 | \date{March 16, 2011} |
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74 | \frame{\titlepage} |
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75 | |
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76 | |
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77 | \frame{ |
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78 | \frametitle{Achievements in period 2} |
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79 | |
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80 | \begin{description}[T3.2] |
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81 | \item[T3.2] \alert{Completed}: Matita encoding of compiler front-end |
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82 | \item[T3.3] \alert{Completed}: Executable semantics for intermediate languages |
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83 | \item[T3.4] \alert{In progress}: Correctness proofs |
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84 | \end{description} |
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85 | |
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86 | \bigskip |
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87 | Deliverables D3.2 and D3.3 submitted. |
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88 | |
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89 | \bigskip |
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90 | This talk is an interleaved presentation of these results. |
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91 | |
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92 | |
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93 | } |
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94 | |
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95 | |
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96 | \frame{ |
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97 | \frametitle{Outline} |
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98 | |
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99 | \begin{itemize} |
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100 | \item Shared definitions |
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101 | %\item The front-end: |
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102 | %\end{itemize} |
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103 | %\vspace{-3ex} |
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104 | \item \begin{tabbing} |
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105 | \quad \= $\downarrow$ \quad \= \kill |
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106 | \gray{\textsf{C} (unformalized)}\\ |
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107 | \> \gray{$\downarrow$} \> \gray{CIL parser (unformalized \ocaml)}\\ |
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108 | \textsf{Clight}\\ |
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109 | \> $\downarrow$ \> cast removal\\ |
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110 | \> \gray{$\downarrow$} \> \gray{add runtime functions}\\ |
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111 | \> $\downarrow$ \> cost labelling\\ |
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112 | \> $\downarrow$ \> stack variable allocation and control structure |
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113 | simplification\\ |
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114 | \textsf{Cminor}\\ |
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115 | \> $\downarrow$ \> generate global variable initialisation code\\ |
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116 | \> $\downarrow$ \> transform to RTL graph\\ |
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117 | \textsf{RTLabs}\\ |
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118 | \> \,\gray{\vdots} \> \gray{start of target specific back-end} |
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119 | \end{tabbing} |
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120 | %\vspace{-3ex} |
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121 | %\begin{itemize} |
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122 | \item Structured traces |
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123 | \end{itemize} |
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124 | |
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125 | } |
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126 | |
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127 | \section{Shared} |
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128 | |
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129 | \begin{frame}[fragile] |
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130 | \frametitle{Shared definitions} |
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131 | |
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132 | Identifiers: variables, cost labels, CFG labels, \dots |
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133 | |
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134 | \begin{itemize} |
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135 | \item Represented by arbitrarily large binary numbers and trees |
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136 | \begin{itemize} |
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137 | \item D3.3 `lazy failure' approach reusing existing structures |
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138 | \item Invariant's types may depend on success of name generation |
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139 | %\item Use this CompCert-like system instead |
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140 | \end{itemize} |
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141 | \item Tags for some type safety: |
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142 | \begin{lstlisting}[language=matita] |
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143 | inductive identifier (tag:String) : Type[0] ≝ |
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144 | an_identifier : Word $\rightarrow$ identifier tag. |
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145 | \end{lstlisting} |
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146 | \end{itemize} |
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147 | |
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148 | Memory, global environments from D3.1\\ |
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149 | Bit vector based arithmetic from D4.1 |
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150 | \begin{itemize} |
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151 | \item added extra operations, increased efficiency |
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152 | \end{itemize} |
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153 | %Common record type for small-step executable semantics |
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154 | % \begin{itemize} |
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155 | % \item used for animation |
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156 | % \item intended to be used when defining simulations |
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157 | % \end{itemize} |
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158 | |
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159 | |
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160 | |
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161 | \textsf{Cminor} and \textsf{RTLabs} share operations on values. |
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162 | |
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163 | |
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164 | \end{frame} |
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165 | |
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166 | \section{Clight} |
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167 | \frame{ |
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168 | \frametitle{Clight: syntax and semantics} |
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169 | |
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170 | Modest evolution from D3.1: |
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171 | |
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172 | \medskip |
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173 | \begin{itemize} |
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174 | \item use precise bit vectors instead of integer and range proof |
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175 | \item adapting to changes in common definitions |
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176 | \item added function to produce fresh name generator |
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177 | \begin{itemize} |
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178 | \item used for adding temporary variables |
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179 | \item works by finding largest existing identifier |
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180 | \item have shown necessary freshness proof |
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181 | \end{itemize} |
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182 | \end{itemize} |
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183 | } |
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184 | |
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185 | \begin{frame}[fragile] |
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186 | \frametitle{Clight: cast simplification} |
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187 | |
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188 | C insists on arithmetic promotion |
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189 | \begin{itemize} |
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190 | \item CIL-based \ocaml{} parser adds suitable casts |
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191 | \item bad for our target (32-bit ops instead of 8-bit) |
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192 | \end{itemize} |
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193 | |
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194 | \medskip |
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195 | Prototype recognises fixed patterns to simplify: |
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196 | \[ (t)((t_1)e_1\ op\ (t_2)e_2) \] |
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197 | \vspace{-4ex} |
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198 | \begin{itemize} |
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199 | \item Deep pattern matching is awkward in Matita |
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200 | \item Misses (e.g.) \lstinline[language=C]'(char)((int)a+(int)b+(int)c)' |
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201 | \end{itemize} |
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202 | |
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203 | \medskip |
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204 | \red{Instead}: recursive `coerce to desired type' approach. |
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205 | |
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206 | Have done some preliminary proofs that this |
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207 | works. |
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208 | % (Doesn't get some things, |
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209 | % e.g., \lstinline'==') |
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210 | \end{frame} |
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211 | |
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212 | \frame{ |
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213 | \frametitle{Clight: cost labelling} |
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214 | |
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215 | Adds cost labels to Clight program. |
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216 | |
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217 | \begin{itemize} |
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218 | \item a simple recursive function, like prototype |
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219 | \item uses common identifiers definition to produce fresh cost labels |
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220 | \end{itemize} |
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221 | |
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222 | The cost labelling is sound and precise, so we will be able to prove |
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223 | some syntactic properties required later: |
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224 | \begin{enumerate} |
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225 | \item every function starts with a cost label |
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226 | \item every branching instruction is followed by a cost label |
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227 | \item the head of every loop (including \lstinline'goto's) is a cost label |
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228 | \end{enumerate} |
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229 | |
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230 | } |
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231 | |
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232 | \section{Cminor} |
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233 | \begin{frame}[fragile] |
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234 | \frametitle{Cminor syntax and semantics} |
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235 | |
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236 | Similar to CompCert's --- but developed from prototype. |
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237 | |
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238 | \begin{itemize} |
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239 | \item Some type enforcement for expressions: |
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240 | \end{itemize} |
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241 | |
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242 | \begin{lstlisting}[language=matita,basicstyle=\footnotesize\tt] |
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243 | inductive expr : typ $\rightarrow$ Type[0] ≝ |
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244 | | Id : $\forall$t. ident $\rightarrow$ expr t |
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245 | | Cst : $\forall$t. constant $\rightarrow$ expr t |
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246 | | Op1 : $\forall$t,t'. unary_operation t t' $\rightarrow$ expr t $\rightarrow$ expr t' |
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247 | | Op2 : $\forall$t1,t2,t'. binary_operation $\rightarrow$ expr t1 $\rightarrow$ expr t2 $\rightarrow$ expr t' |
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248 | | Mem : $\forall$t,r. memory_chunk $\rightarrow$ expr (ASTptr r) $\rightarrow$ expr t |
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249 | | Cond : $\forall$sz,sg,t. expr (ASTint sz sg) $\rightarrow$ expr t $\rightarrow$ expr t $\rightarrow$ expr t |
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250 | | Ecost : $\forall$t. costlabel $\rightarrow$ expr t $\rightarrow$ expr t. |
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251 | \end{lstlisting} |
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252 | \begin{itemize} |
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253 | %\item enforces some checking during production |
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254 | \item indirectly forces checking that temporaries are fresh |
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255 | \item[$\star$] reduces failure cases in translation to RTLabs |
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256 | \end{itemize} |
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257 | |
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258 | Semantics unexciting small-step function. |
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259 | \end{frame} |
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260 | |
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261 | \begin{frame} |
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262 | \frametitle{Adding invariants} |
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263 | |
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264 | Lots of sources of failure in front-end due to badly formed programs: |
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265 | \begin{itemize} |
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266 | \item missing variables, graph nodes, \dots |
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267 | \item badly structured programs |
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268 | \end{itemize} |
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269 | |
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270 | \bigskip |
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271 | Have been refining the intermediate languages to rule these out: |
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272 | \begin{enumerate} |
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273 | \item \gray{Indexing Cminor syntax by nesting depth} |
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274 | \item adding type constraints |
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275 | \item adding invariants asserting presence of variables, etc. |
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276 | \end{enumerate} |
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277 | |
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278 | \end{frame} |
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279 | |
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280 | \begin{frame}[fragile] |
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281 | \frametitle{Invariants for identifiers} |
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282 | |
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283 | |
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284 | Invariants change between languages and compilation: |
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285 | |
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286 | \begin{description} |
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287 | \item[syntax] constraints between parts of function body and other information |
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288 | about the function |
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289 | \item[compilation] constraints between statements and expressions and data structures in the translation |
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290 | \end{description} |
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291 | |
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292 | When using or creating function definitions have proved that we can switch between |
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293 | invariants: |
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294 | \begin{lstlisting}[language=matita,basicstyle=\footnotesize\tt] |
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295 | lemma populates_env : $\forall$l,e,u,l',e',u'. |
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296 | distinct_env ?? l $\rightarrow$ (* distinct names in l *) |
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297 | populate_env e u l = $\langle$l',e',u'$\rangle$ $\rightarrow$ (* build register mapping *) |
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298 | $\forall$i,t. Exists ? ($\lambda$x.x = $\langle$i,t$\rangle$) l $\rightarrow$ (* Anything in the environment... *) |
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299 | Env_has e' i t. (* maps to something of the correct type *) |
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300 | \end{lstlisting} |
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301 | |
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302 | \vfill |
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303 | \end{frame} |
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304 | |
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305 | \begin{frame}[fragile] |
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306 | \frametitle{Invariants in Cminor} |
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307 | |
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308 | Embedded invariant in the function record: |
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309 | \begin{lstlisting}[language=matita,basicstyle=\scriptsize\tt,morekeywords=f_inv] |
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310 | record internal_function : Type[0] ≝ |
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311 | { f_return : option typ |
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312 | ; f_params : list (ident $\times$ typ) |
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313 | ; f_vars : list (ident $\times$ typ) |
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314 | ; f_stacksize : nat |
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315 | ; f_body : stmt |
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316 | ; f_inv : |
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317 | stmt_P ($\lambda$s. |
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318 | stmt_vars ($\lambda$i,t.Exists ? ($\lambda$x. x = $\langle$i,t$\rangle$) (f_params @ f_vars)) s $\wedge$ |
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319 | stmt_labels ($\lambda$l.Exists ? ($\lambda$l'.l' = l) (labels_of f_body)) s) |
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320 | f_body |
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321 | }. |
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322 | \end{lstlisting} |
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323 | \begin{enumerate} |
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324 | \item All variables are in the parameters or locals lists\\ |
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325 | \quad --- with the correct type |
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326 | \item All \lstinline'goto' labels mentioned are defined in the body |
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327 | \end{enumerate} |
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328 | |
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329 | \end{frame} |
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330 | |
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331 | |
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332 | \section{Clight to Cminor} |
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333 | |
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334 | \begin{frame} |
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335 | \frametitle{Clight to Cminor} |
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336 | |
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337 | Two main jobs: |
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338 | \begin{enumerate} |
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339 | \item make memory allocation of variables explicit |
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340 | \item use simpler control structures |
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341 | \end{enumerate} |
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342 | Again, based on prototype rather than CompCert. |
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343 | |
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344 | \bigskip |
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345 | Added type checking: |
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346 | \begin{itemize} |
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347 | \item satisfies the restrictions for Cminor expressions |
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348 | \item could separate out, have a `nice Clight' language |
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349 | \end{itemize} |
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350 | Similarly, code checks |
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351 | \begin{itemize} |
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352 | \item variable environments are well-formed |
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353 | \item all \lstinline'goto' labels are translated |
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354 | \end{itemize} |
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355 | \end{frame} |
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356 | |
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357 | \begin{frame} |
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358 | \frametitle{Clight to Cminor proofs} |
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359 | |
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360 | Beyond the invariants already shown, we will prove: |
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361 | |
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362 | \medskip |
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363 | \begin{enumerate} |
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364 | \item a simulation relation |
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365 | \begin{itemize} |
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366 | \item between statement and continuation pairs |
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367 | \item using memory injection (similar to CompCert) |
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368 | \end{itemize} |
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369 | \item syntactic cost labelling properties are preserved |
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370 | \begin{itemize} |
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371 | \item structural induction on function body |
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372 | \end{itemize} |
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373 | \end{enumerate} |
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374 | \end{frame} |
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375 | |
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376 | \section{Initialisation} |
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377 | \begin{frame} |
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378 | \frametitle{Cminor: initialization} |
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379 | |
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380 | Replace initialization data by code in the initial function. |
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381 | |
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382 | \begin{itemize} |
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383 | \item straightforward to define |
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384 | \item instantiate a slightly different version of the semantics for it |
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385 | \begin{itemize} |
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386 | \item Can't accidentally forget the pass |
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387 | \end{itemize} |
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388 | \end{itemize} |
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389 | |
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390 | \bigskip |
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391 | Correctness should follow because the state after the initialisation will |
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392 | be the same as the initial state of the original. |
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393 | \end{frame} |
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394 | |
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395 | \section{RTLabs} |
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396 | \begin{frame}[fragile] |
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397 | \frametitle{RTLabs: syntax and semantics} |
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398 | |
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399 | Register Transfer Language with front-end operations. |
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400 | |
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401 | Control flow graph implemented by generic identifiers map: |
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402 | |
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403 | \begin{lstlisting}[language=matita,basicstyle=\footnotesize\tt] |
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404 | definition label := identifier LabelTag. |
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405 | definition graph : Type[0] $\rightarrow$ Type[0] := identifier_map LabelTag. |
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406 | |
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407 | inductive statement : Type[0] ≝ |
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408 | | St_skip : label $\rightarrow$ statement |
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409 | | St_const : register $\rightarrow$ constant $\rightarrow$ label $\rightarrow$ statement |
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410 | | St_op1 : $\forall$t,t'. unary_operation t' t $\rightarrow$ register |
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411 | $\rightarrow$ register $\rightarrow$ label $\rightarrow$ statement |
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412 | ... |
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413 | \end{lstlisting} |
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414 | \vspace{-1.5ex} |
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415 | \begin{itemize} |
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416 | \item Shares basic operations (including semantics) with Cminor. |
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417 | \item Tags prevent confusion of labels (graph vs.~goto vs.~cost). |
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418 | \end{itemize} |
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419 | |
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420 | Semantics straightforward interpretation of statements. |
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421 | \end{frame} |
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422 | |
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423 | \begin{frame}[fragile] |
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424 | \frametitle{RTLabs: syntax and semantics} |
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425 | |
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426 | \begin{lstlisting}[language=matita] |
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427 | record internal_function : Type[0] ≝ |
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428 | { f_labgen : universe LabelTag |
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429 | ; f_reggen : universe RegisterTag |
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430 | ... |
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431 | ; f_graph : graph statement |
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432 | ; f_closed : graph_closed f_graph |
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433 | ; f_typed : graph_typed (f_locals @ f_params) f_graph |
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434 | ... |
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435 | \end{lstlisting} |
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436 | |
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437 | Enforce that |
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438 | \begin{itemize} |
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439 | \item every statement's successor labels are present in the CFG |
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440 | \item every statement should be well-typed (for limited type system) |
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441 | \end{itemize} |
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442 | |
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443 | \end{frame} |
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444 | |
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445 | \section{Cminor to RTLabs} |
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446 | \begin{frame} |
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447 | \frametitle{Cminor to RTLabs} |
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448 | |
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449 | Break down statements and expressions into RTL graph. |
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450 | |
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451 | \begin{itemize} |
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452 | \item Incrementally build function backwards |
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453 | \item all state is in function record, like prototype |
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454 | \end{itemize} |
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455 | |
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456 | %\bigskip |
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457 | %May investigate whether a state monad makes invariant management easier. |
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458 | |
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459 | \bigskip |
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460 | Showing graph closure requires |
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461 | \begin{itemize} |
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462 | \item monotonicity of graph construction |
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463 | \item eventual insertion of all \lstinline'goto' destinations |
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464 | \end{itemize} |
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465 | |
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466 | Carry these along in the partially built function record. |
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467 | |
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468 | \end{frame} |
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469 | |
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470 | % \begin{frame}[fragile] |
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471 | % \frametitle{Cminor nesting depth} |
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472 | |
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473 | % Want to prevent |
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474 | % \begin{lstlisting}[language=C] |
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475 | % void f() { |
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476 | % block { loop { exit 5; } } |
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477 | % } |
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478 | % \end{lstlisting} |
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479 | |
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480 | % Index statements by block depth |
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481 | % \begin{lstlisting}[language=matita] |
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482 | % inductive stmt : $\forall$blockdepth:nat. Type[0] ≝ |
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483 | % | St_skip : $\forall$n. stmt n |
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484 | % ... |
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485 | % | St_block : $\forall$n. stmt (S n) $\rightarrow$ stmt n |
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486 | % | St_exit : $\forall$n. Fin n $\rightarrow$ stmt n |
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487 | % ... |
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488 | % \end{lstlisting} |
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489 | |
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490 | % \begin{itemize} |
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491 | % \item In semantics index continuations too |
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492 | % \item Use \lstinline[language=matita]'Fin n' to make \lstinline'k_exit' |
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493 | % a total function |
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494 | % \end{itemize} |
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495 | |
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496 | % (May be able to remove entirely if we redo \lstinline[language=C]{switch}?) |
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497 | % \end{frame} |
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498 | |
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499 | \begin{frame}[fragile] |
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500 | \frametitle{Establishing RTLabs invariants} |
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501 | |
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502 | Use dependent pairs to show invariant along with results. |
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503 | |
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504 | \begin{lstlisting}[language=matita,basicstyle=\footnotesize\tt,escapechar=\%] |
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505 | let rec add_expr ... (e:expr ty) |
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506 | (Env:expr_vars ty e (present ?? env)) %\quad% (* invariant *) |
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507 | (f:partial_fn le) |
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508 | on e: $\Sigma$f':partial_fn le. fn_graph_included le f f' ≝ |
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509 | |
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510 | match e return |
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511 | $\lambda$ty,e.expr_vars ty e (present ?? env) $\rightarrow$ ... with |
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512 | [ ... |
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513 | | Cst _ c $\Rightarrow$ $\lambda$_. %\guillemotleft%add_fresh_to_graph ? (St_const dst c) f ?, ?%\guillemotright% |
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514 | ... |
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515 | \end{lstlisting} |
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516 | |
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517 | Continually appeal to monotonicity (\lstinline'fn_graph_included').\\ |
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518 | Use unification hints to simplify stepping back. |
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519 | |
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520 | \end{frame} |
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521 | |
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522 | \frame{ |
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523 | \frametitle{Cminor to RTLabs: cost labels} |
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524 | |
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525 | Two cost label properties are the same, the third will require more |
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526 | work: |
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527 | \begin{enumerate} |
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528 | \item cost label at head of function |
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529 | \item cost label after branching instructions |
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530 | \item cost labels at the head of each loop / \lstinline'goto' destination |
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531 | \end{enumerate} |
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532 | |
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533 | No simple notion of the head of a loop or \lstinline'goto' any more. |
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534 | |
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535 | \medskip |
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536 | Instead: will prove in \alert{Cminor} that after following a finite |
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537 | number of instructions we reach either |
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538 | \begin{itemize} |
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539 | \item a cost label, or |
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540 | \item the end of the function |
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541 | \end{itemize} |
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542 | |
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543 | } |
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544 | |
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545 | \section{Structured traces} |
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546 | \frame{ |
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547 | \frametitle{RTLabs structured traces} |
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548 | |
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549 | Front-end only uses flat traces consisting of single steps. |
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550 | |
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551 | \bigskip |
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552 | The back-end will need the function call structure and the labelling |
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553 | properties in order to show that the generated costs are correct. |
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554 | |
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555 | \begin{itemize} |
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556 | \item Traces are structured in sections from cost label to cost label, |
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557 | \item the entire execution of function calls nested as a single `step', |
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558 | \item a coinductive definition presents non-terminating traces, using the |
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559 | inductive definition for all terminating function calls |
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560 | \end{itemize} |
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561 | |
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562 | \bigskip |
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563 | RTLabs chosen because it is the first languages where statements: |
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564 | \begin{itemize} |
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565 | \item take one step each (modulo function calls) |
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566 | \item have individual `addresses' |
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567 | \end{itemize} |
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568 | } |
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569 | |
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570 | \frame{ |
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571 | \frametitle{RTLabs structured traces} |
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572 | Have already established the existence of such traces |
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573 | \begin{itemize} |
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574 | \item termination decided classically |
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575 | \item syntactic labelling properties used to build semantic structure |
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576 | \item show stack preservation to ensure that function calls \texttt{return} |
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577 | to the correct location |
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578 | \item tricky to establish guardedness of definitions |
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579 | \end{itemize} |
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580 | |
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581 | \bigskip |
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582 | Next, prove that flattening these traces yields the original. |
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583 | } |
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584 | |
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585 | \frame{ |
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586 | \frametitle{Conclusion} |
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587 | |
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588 | Syntax, semantics and translations of prototype now implemented in Matita. |
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589 | |
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590 | \bigskip |
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591 | Have defined and established |
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592 | \begin{itemize} |
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593 | \item invariants regarding variables, typing and program structure |
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594 | \item a rich form of execution trace to pass to the back-end |
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595 | \end{itemize} |
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596 | |
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597 | \medskip |
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598 | Work in progress: |
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599 | \begin{itemize} |
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600 | \item showing functional correctness of the front-end |
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601 | \item proving that cost labelling is appropriate, and preserved |
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602 | \end{itemize} |
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603 | |
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604 | Finally, \alert{end-to-end} functional and cost correctness results. |
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605 | |
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606 | } |
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607 | |
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608 | \frame{ |
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609 | \frametitle{Extra detail beyond here...} |
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610 | } |
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611 | |
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612 | \begin{frame}[fragile] |
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613 | \frametitle{Invariants in Cminor} |
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614 | |
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615 | \begin{lstlisting}[language=matita,basicstyle=\footnotesize\tt] |
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616 | let rec stmt_P (P:stmt $\rightarrow$ Prop) (s:stmt) on s : Prop ≝ |
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617 | match s with |
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618 | [ St_seq s1 s2 $\Rightarrow$ P s $\wedge$ stmt_P P s1 $\wedge$ stmt_P P s2 |
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619 | | St_ifthenelse _ _ _ s1 s2 $\Rightarrow$ P s $\wedge$ stmt_P P s1 $\wedge$ stmt_P P s2 |
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620 | | St_loop s' $\Rightarrow$ P s $\wedge$ stmt_P P s' |
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621 | | St_block s' $\Rightarrow$ P s $\wedge$ stmt_P P s' |
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622 | | St_label _ s' $\Rightarrow$ P s $\wedge$ stmt_P P s' |
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623 | | St_cost _ s' $\Rightarrow$ P s $\wedge$ stmt_P P s' |
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624 | | _ $\Rightarrow$ P s |
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625 | ]. |
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626 | \end{lstlisting} |
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627 | |
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628 | Higher order predicate |
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629 | \begin{itemize} |
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630 | \item When we pattern match on the statement the predicate can unfold |
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631 | \item Easy to discharge proof obligations for substatements |
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632 | \item similar definition for expressions |
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633 | \end{itemize} |
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634 | |
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635 | \end{frame} |
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636 | \end{document} |
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