1 | \documentclass[11pt,epsf,a4wide]{article} |
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2 | \usepackage[mathletters]{ucs} |
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3 | \usepackage[utf8x]{inputenc} |
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4 | \usepackage{listings} |
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5 | \usepackage{../../style/cerco} |
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6 | \newcommand{\ocaml}{OCaml} |
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7 | \newcommand{\clight}{Clight} |
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8 | \newcommand{\matita}{Matita} |
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9 | \newcommand{\sdcc}{\texttt{sdcc}} |
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10 | |
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11 | \newcommand{\textSigma}{\ensuremath{\Sigma}} |
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12 | |
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13 | % LaTeX Companion, p 74 |
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14 | \newcommand{\todo}[1]{\marginpar{\raggedright - #1}} |
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15 | |
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16 | \lstdefinelanguage{coq} |
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17 | {keywords={Definition,Lemma,Theorem,Remark,Qed,Save,Inductive,Record}, |
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18 | morekeywords={[2]if,then,else}, |
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19 | } |
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20 | |
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21 | \lstdefinelanguage{matita} |
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22 | {keywords={definition,lemma,theorem,remark,inductive,record,qed,let,rec,match,with,Type,and,on}, |
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23 | morekeywords={[2]whd,normalize,elim,cases,destruct}, |
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24 | mathescape=true, |
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25 | morecomment=[n]{(*}{*)}, |
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26 | } |
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27 | |
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28 | \lstset{language=matita,basicstyle=\small\tt,columns=flexible,breaklines=false, |
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29 | keywordstyle=\color{red}\bfseries, |
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30 | keywordstyle=[2]\color{blue}, |
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31 | commentstyle=\color{green}, |
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32 | stringstyle=\color{blue}, |
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33 | showspaces=false,showstringspaces=false} |
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34 | |
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35 | \lstset{extendedchars=false} |
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36 | \lstset{inputencoding=utf8x} |
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37 | \DeclareUnicodeCharacter{8797}{:=} |
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38 | \DeclareUnicodeCharacter{10746}{++} |
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39 | \DeclareUnicodeCharacter{9001}{\ensuremath{\langle}} |
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40 | \DeclareUnicodeCharacter{9002}{\ensuremath{\rangle}} |
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41 | |
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42 | |
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43 | \title{ |
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44 | INFORMATION AND COMMUNICATION TECHNOLOGIES\\ |
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45 | (ICT)\\ |
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46 | PROGRAMME\\ |
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47 | \vspace*{1cm}Project FP7-ICT-2009-C-243881 \cerco{}} |
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48 | |
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49 | \date{ } |
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50 | \author{} |
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51 | |
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52 | \begin{document} |
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53 | \thispagestyle{empty} |
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54 | |
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55 | \vspace*{-1cm} |
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56 | \begin{center} |
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57 | \includegraphics[width=0.6\textwidth]{../../style/cerco_logo.png} |
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58 | \end{center} |
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59 | |
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60 | \begin{minipage}{\textwidth} |
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61 | \maketitle |
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62 | \end{minipage} |
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63 | |
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64 | |
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65 | \vspace*{0.5cm} |
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66 | \begin{center} |
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67 | \begin{LARGE} |
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68 | \bf |
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69 | Report n. D3.4\\ |
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70 | Front-end Correctness Proofs\\ |
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71 | \end{LARGE} |
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72 | \end{center} |
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73 | |
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74 | \vspace*{2cm} |
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75 | \begin{center} |
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76 | \begin{large} |
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77 | Version 1.0 |
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78 | \end{large} |
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79 | \end{center} |
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80 | |
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81 | \vspace*{0.5cm} |
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82 | \begin{center} |
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83 | \begin{large} |
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84 | Authors:\\ |
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85 | Brian~Campbell, Ilias~Garnier, James~McKinna, Ian~Stark |
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86 | \end{large} |
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87 | \end{center} |
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88 | |
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89 | \vspace*{\fill} |
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90 | \noindent |
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91 | Project Acronym: \cerco{}\\ |
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92 | Project full title: Certified Complexity\\ |
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93 | Proposal/Contract no.: FP7-ICT-2009-C-243881 \cerco{}\\ |
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94 | |
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95 | \clearpage \pagestyle{myheadings} \markright{\cerco{}, FP7-ICT-2009-C-243881} |
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96 | |
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97 | \newpage |
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98 | |
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99 | \vspace*{7cm} |
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100 | \paragraph{Abstract} |
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101 | We report on the correctness proofs for the front-end of the \cerco{} cost |
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102 | lifting compiler, considering three distinct parts of the task: showing that |
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103 | the \emph{annotated source code} output by the compiler has equivalent |
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104 | behaviour to the original input (up to the annotations); showing that a |
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105 | \emph{measurable} subtrace of the annotated source code corresponds to an |
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106 | equivalent measurable subtrace in the code produced by the front-end, including |
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107 | costs; and finally showing that the enriched \emph{structured} execution traces |
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108 | required for cost correctness in the back-end can be constructed from the |
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109 | properties of the code produced by the front-end. |
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110 | |
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111 | A key part of our work is that the intensional correctness results that show |
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112 | that we get consistent cost measurements throughout the intermediate languages |
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113 | of the compiler can be layered on top of normal forward simulation results, |
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114 | if we split them into local call-structure preserving results. |
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115 | |
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116 | This deliverable shows correctness results about the formalised compiler |
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117 | described in D3.2, using the source language semantics from D3.1 and |
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118 | intermediate language semantics from D3.3. Together with the companion |
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119 | deliverable about the correctness of the back-end, D4.4, we obtain results |
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120 | about the whole formalised compiler. |
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121 | |
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122 | % TODO: mention the deliverable about the extracted compiler et al? |
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123 | |
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124 | \newpage |
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125 | |
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126 | \tableofcontents |
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127 | |
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128 | % CHECK: clear up any -ize vs -ise |
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129 | % CHECK: clear up any "front end" vs "front-end" |
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130 | % CHECK: clear up any mentions of languages that aren't textsf'd. |
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131 | % CHECK: fix unicode in listings |
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132 | |
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133 | \section{Introduction} |
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134 | |
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135 | TODO: briefly, cerco compiler's purpose, prototype, formalisation, whole |
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136 | compiler. Full functional correctness too ambitious; fortunately get nice |
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137 | layering of intensional correctness on top of mildly refined normal simulations. |
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138 | Hence, concentrate on new intensional correctness results; add stack space for |
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139 | more precise statement. Also do some |
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140 | translation validation on sound, precise labelling properties. Fluffly version |
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141 | of correctness statement. |
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142 | |
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143 | The \cerco{} compiler compiles C code targeting microcontrollers implementing |
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144 | the Intel 8051 architecture, which produces both the object code and source |
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145 | code containing annotations describing the timing behavior of the object code. |
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146 | There are two versions: first, an initial prototype implemented in |
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147 | \ocaml{}~\cite{d2.2}, and a version formalised in the \matita{} proof |
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148 | assistant~\cite{d3.2,d4.2} and then extracted to \ocaml{} code to produce an |
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149 | executable compiler. In this document we present results formalised in |
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150 | \matita{} about the front-end of that version of the compiler, and how that fits |
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151 | into the verification of the whole compiler. |
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152 | |
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153 | % TODO: maybe mention stack space here? other additions? refer to "layering"? |
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154 | A key part of this work was to separate the proofs about the compiled code's |
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155 | extensional behaviour (that is, the functional correctness of the compiler) |
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156 | from the intensional correctness that the costs given are correct. |
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157 | Unfortunately, the ambitious goal of completely verifying the compiler was not |
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158 | feasible within the time available, but thanks to this separation of extensional |
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159 | and intensional proofs we are able to axiomatize simulation results similar to |
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160 | those in other compiler verification projects and concentrate on the novel |
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161 | intensional proofs. The proofs were also made more tractable by introducing |
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162 | compile-time checks for the `sound and precise' cost labelling properties |
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163 | rather than proving that they are preserved throughout. |
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164 | |
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165 | The overall statement of correctness says that the annotated program has the |
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166 | same behaviour as the input, and that for any suitably well-structured part of |
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167 | the execution (which we call \emph{measurable}), the object code will execute |
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168 | the same behaviour taking precisely the time given by the cost annotations in |
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169 | the annotated source program. |
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170 | |
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171 | In the next section we recall the structure of the compiler and make the overall |
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172 | statement more precise. Following that, in Section~\ref{sec:fegoals} we |
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173 | describe the statements we need to prove about the intermediate \textsf{RTLabs} |
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174 | programs sufficient for the back-end proofs. TODO: rest of document structure |
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175 | |
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176 | \section{The compiler and main goals} |
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177 | |
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178 | TODO: outline compiler, maybe figure from talk, maybe something like the figure |
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179 | below. |
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180 | |
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181 | TODO: might want a version of this figure |
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182 | \begin{figure} |
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183 | \begin{center} |
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184 | \begin{minipage}{.8\linewidth} |
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185 | \begin{tabbing} |
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186 | \quad \= $\downarrow$ \quad \= \kill |
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187 | \textsf{C} (unformalized)\\ |
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188 | \> $\downarrow$ \> CIL parser (unformalized \ocaml)\\ |
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189 | \textsf{Clight}\\ |
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190 | \> $\downarrow$ \> cast removal\\ |
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191 | \> $\downarrow$ \> add runtime functions\\ |
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192 | \> $\downarrow$ \> labelling\\ |
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193 | \> $\downarrow$ \> stack variable allocation and control structure |
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194 | simplification\\ |
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195 | \textsf{Cminor}\\ |
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196 | \> $\downarrow$ \> generate global variable initialization code\\ |
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197 | \> $\downarrow$ \> transform to RTL graph\\ |
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198 | \textsf{RTLabs}\\ |
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199 | \> $\downarrow$ \> start of target specific back-end\\ |
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200 | \>\quad \vdots |
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201 | \end{tabbing} |
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202 | \end{minipage} |
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203 | \end{center} |
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204 | \caption{Front-end languages and transformations} |
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205 | \label{fig:summary} |
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206 | \end{figure} |
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207 | |
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208 | The compiler function returns the following record on success: |
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209 | \begin{lstlisting}[language=matita] |
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210 | record compiler_output : Type[0] := |
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211 | { c_labelled_object_code: labelled_object_code |
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212 | ; c_stack_cost: stack_cost_model |
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213 | ; c_max_stack: nat |
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214 | ; c_init_costlabel: costlabel |
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215 | ; c_labelled_clight: clight_program |
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216 | ; c_clight_cost_map: clight_cost_map |
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217 | }. |
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218 | \end{lstlisting} |
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219 | It consists of annotated 8051 object code, a mapping from function |
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220 | identifiers to the function's stack space usage\footnote{The compiled |
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221 | code's only stack usage is to allocate a fixed-size frame on each |
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222 | function entry and discard it on exit.}, a cost label covering the |
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223 | initialisation of global variables and calling the |
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224 | \lstinline[language=C]'main' function, the annotated source code, and |
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225 | finally a mapping from cost labels to actual execution time costs. |
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226 | |
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227 | An \ocaml{} pretty printer is used to provide a concrete version of the output |
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228 | code. In the case of the annotated source code, it also inserts the actual |
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229 | costs alongside the cost labels, and optionally adds a global cost variable |
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230 | and instrumentation to support further reasoning. \todo{Cross-ref case study |
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231 | deliverables} |
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232 | |
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233 | \subsection{Revisions to the prototype compiler} |
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234 | |
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235 | TODO: could be a good idea to have this again |
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236 | |
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237 | \subsection{Main goals} |
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238 | |
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239 | TODO: need an example for this |
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240 | |
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241 | Informally, our main intensional result links the time difference in a source |
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242 | code execution to the time difference in the object code, expressing the time |
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243 | for the source by summing the values for the cost labels in the trace, and the |
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244 | time for the target by a clock built in to the 8051 executable semantics. |
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245 | |
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246 | The availability of precise timing information for 8501 |
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247 | implementations and the design of the compiler allow it to give exact |
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248 | time costs in terms of processor cycles. However, these exact results |
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249 | are only available if the subtrace we measure starts and ends at |
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250 | suitable points. In particular, pure computation with no observable |
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251 | effects may be reordered and moved past cost labels, so we cannot |
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252 | measure time between arbitrary statements in the program. |
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253 | |
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254 | There is also a constraint on the subtraces that we |
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255 | measure due to the requirements of the correctness proof for the |
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256 | object code timing analysis. To be sure that the timings are assigned |
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257 | to the correct cost label, we need to know that each return from a |
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258 | function call must go to the correct return address. It is difficult |
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259 | to observe this property locally in the object code because it relies |
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260 | on much earlier stages in the compiler. To convey this information to |
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261 | the timing analysis extra structure is imposed on the subtraces, which |
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262 | we will give more details on in Section~\ref{sec:fegoals}. |
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263 | |
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264 | % Regarding the footnote, would there even be much point? |
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265 | These restrictions are reflected in the subtraces that we give timing |
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266 | guarantees on; they must start at a cost label and end at the return |
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267 | of the enclosing function of the cost label\footnote{We expect that |
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268 | this would generalise to subtraces between cost labels in the same |
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269 | function, but could not justify the extra complexity that would be |
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270 | required to show this.}. A typical example of such a subtrace is |
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271 | the execution of an entire function from the cost label at the start |
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272 | of the function until it returns. We call such any such subtrace |
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273 | \emph{measurable} if it (and the prefix of the trace before it) can |
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274 | also be executed within the available stack space. |
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275 | |
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276 | Now we can give the main intensional statement for the compiler. |
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277 | Given a \emph{measurable} subtrace for a labelled \textsf{Clight} |
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278 | program, there is a subtrace of the 8051 object code program where the |
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279 | time differences match. Moreover, \emph{observable} parts of the |
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280 | trace also match --- these are the appearance of cost labels and |
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281 | function calls and returns. |
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282 | |
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283 | More formally, the definition of this statement in \matita{} is |
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284 | \begin{lstlisting}[language=matita] |
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285 | definition simulates := |
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286 | $\lambda$p: compiler_output. |
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287 | let initial_status := initialise_status $\dots$ (cm (c_labelled_object_code $\dots$ p)) in |
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288 | $\forall$m1,m2. |
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289 | measurable Clight_pcs (c_labelled_clight $\dots$ p) m1 m2 |
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290 | (lookup_stack_cost (c_stack_cost $\dots$ p)) (c_max_stack $\dots$ p) $\rightarrow$ |
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291 | $\forall$c1,c2. |
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292 | clock_after Clight_pcs (c_labelled_clight $\dots$ p) m1 (c_clight_cost_map $\dots$ p) = OK $\dots$ c1 $\rightarrow$ |
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293 | clock_after Clight_pcs (c_labelled_clight $\dots$ p) (m1+m2) (c_clight_cost_map $\dots$ p) = OK $\dots$ c2 $\rightarrow$ |
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294 | $\exists$n1,n2. |
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295 | observables Clight_pcs (c_labelled_clight $\dots$ p) m1 m2 = |
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296 | observables (OC_preclassified_system (c_labelled_object_code $\dots$ p)) |
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297 | (c_labelled_object_code $\dots$ p) n1 n2 |
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298 | $\wedge$ |
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299 | c2 - c1 = clock $\dots$ (execute n2 ? initial_status) - clock $\dots$ (execute n1 ? initial_status). |
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300 | \end{lstlisting} |
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301 | where the \lstinline'measurable', \lstinline'clock_after' and |
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302 | \lstinline'observables' definitions can be applied to multiple |
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303 | languages; in this case the \lstinline'Clight_pcs' record applies them |
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304 | to \textsf{Clight} programs. |
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305 | |
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306 | There is a second part to the statement, which says that the initial |
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307 | processing of the input program to produce the cost labelled version |
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308 | does not affect the semantics of the program: |
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309 | % Yes, I'm paraphrasing the result a tiny bit to remove the observe non-function |
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310 | \begin{lstlisting}[language=matita] |
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311 | $\forall$input_program,output. |
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312 | compile input_program = return output $\rightarrow$ |
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313 | not_wrong … (exec_inf … clight_fullexec input_program) $\rightarrow$ |
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314 | sim_with_labels |
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315 | (exec_inf … clight_fullexec input_program) |
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316 | (exec_inf … clight_fullexec (c_labelled_clight … output)) |
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317 | \end{lstlisting} |
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318 | That is, any successful compilation produces a labelled program that |
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319 | has identical behaviour to the original, so long as there is no |
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320 | `undefined behaviour'. |
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321 | |
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322 | Note that this provides full functional correctness, including |
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323 | preservation of (non-)termination. The intensional result above does |
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324 | not do this directly --- it does not guarantee the same result or same |
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325 | termination. There are two mitigating factors, however: first, to |
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326 | prove the intensional property you need local simulation results that |
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327 | can be pieced together to form full behavioural equivalence, only time |
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328 | constraints have prevented us from doing so. Second, if we wish to |
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329 | confirm a result, termination, or non-termination we could add an |
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330 | observable witness, such as a function that is only called if the |
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331 | correct result is given. The intensional result guarantees that the |
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332 | observable witness is preserved, so the program must behave correctly. |
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333 | |
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334 | \section{Intermediate goals for the front-end} |
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335 | \label{sec:fegoals} |
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336 | |
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337 | The essential parts of the intensional proof were outlined during work |
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338 | on a toy |
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339 | compiler~\cite{d2.1,springerlink:10.1007/978-3-642-32469-7_3}. These |
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340 | are |
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341 | \begin{enumerate} |
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342 | \item functional correctness, in particular preserving the trace of |
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343 | cost labels, |
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344 | \item the \emph{soundness} and \emph{precision} of the cost labelling |
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345 | on the object code, and |
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346 | \item the timing analysis on the object code produces a correct |
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347 | mapping from cost labels to time. |
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348 | \end{enumerate} |
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349 | |
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350 | However, that toy development did not include function calls. For the |
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351 | full \cerco{} compiler we also need to maintain the invariant that |
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352 | functions return to the correct program location in the caller, as we |
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353 | mentioned in the previous section. During work on the back-end timing |
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354 | analysis (describe in more detail in the companion deliverable, D4.4) |
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355 | the notion of a \emph{structured trace} was developed to enforce this |
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356 | return property, and also most of the cost labelling properties too. |
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357 | |
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358 | \begin{figure} |
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359 | \begin{center} |
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360 | \includegraphics[width=0.5\linewidth]{compiler.pdf} |
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361 | \end{center} |
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362 | \caption{The compiler and proof outline} |
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363 | \label{fig:compiler} |
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364 | \end{figure} |
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365 | |
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366 | Jointly, we generalised the structured traces to apply to any of the |
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367 | intermediate languages with some idea of program counter. This means |
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368 | that they are introduced part way through the compiler, see |
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369 | Figure~\ref{fig:compiler}. Proving that a structured trace can be |
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370 | constructed at \textsf{RTLabs} has several virtues: |
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371 | \begin{itemize} |
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372 | \item This is the first language where every operation has its own |
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373 | unique, easily addressable, statement. |
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374 | \item Function calls and returns are still handled in the language and |
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375 | so the structural properties are ensured by the semantics. |
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376 | \item Many of the back-end languages from \textsf{RTL} share a common |
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377 | core set of definitions, and using structured traces throughout |
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378 | increases this uniformity. |
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379 | \end{itemize} |
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380 | |
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381 | \begin{figure} |
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382 | \begin{center} |
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383 | \includegraphics[width=0.6\linewidth]{strtraces.pdf} |
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384 | \end{center} |
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385 | \caption{Nesting of functions in structured traces} |
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386 | \label{fig:strtrace} |
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387 | \end{figure} |
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388 | A structured trace is a mutually inductive data type which principally |
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389 | contains the steps from a normal program trace, but arranged into a |
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390 | nested structure which groups entire function calls together and |
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391 | aggregates individual steps between cost labels (or between the final |
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392 | cost label and the return from the function), see |
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393 | Figure~\ref{fig:strtrace}. This capture the idea that the cost labels |
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394 | only represent costs \emph{within} a function --- calls to other |
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395 | functions are accounted for in the trace for their execution, and we |
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396 | can locally regard function calls as a single step. |
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397 | |
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398 | These structured traces form the core part of the intermediate results |
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399 | that we must prove so that the back-end can complete the main |
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400 | intensional result stated above. In full, we provide the back-end |
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401 | with |
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402 | \begin{enumerate} |
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403 | \item A normal trace of the \textbf{prefix} of the program's execution |
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404 | before reaching the measurable subtrace. (This needs to be |
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405 | preserved so that we know that the stack space consumed is correct.) |
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406 | \item The \textbf{structured trace} corresponding to the measurable |
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407 | subtrace. |
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408 | \item An additional property about the structured trace that no |
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409 | `program counter' is \textbf{repeated} between cost labels. Together with |
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410 | the structure in the trace, this takes over from showing that |
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411 | cost labelling is sound and precise. |
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412 | \item A proof that the \textbf{observables} have been preserved. |
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413 | \item A proof that the \textbf{stack limit} is still observed by the prefix and |
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414 | the structure trace. (This is largely a consequence of the |
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415 | preservation of observables.) |
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416 | \end{enumerate} |
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417 | |
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418 | Following the outline in Figure~\ref{fig:compiler}, we will first deal |
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419 | with the transformations in \textsf{Clight} that produce the source |
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420 | program with cost labels, then show that measurable traces can be |
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421 | lifted to \textsf{RTLabs}, and finally that we can construct the |
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422 | properties listed above ready for the back-end proofs. |
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423 | |
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424 | \section{Input code to cost labelled program} |
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425 | |
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426 | The simple form of labelling used in the formalised compiler is not |
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427 | quite capable of capturing costs arising from complex C |
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428 | \lstinline[language=C]'switch' statements, largely due to the |
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429 | fall-through behaviour. Our first pass replaces these statements with |
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430 | simpler C code, allowing our second pass to perform the cost |
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431 | labelling. We show that the behaviour of programs is unchanged by |
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432 | these passes. |
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433 | |
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434 | TODO: both give one-step-sim-by-many forward sim results; switch |
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435 | removal tricky, uses aux var to keep result of expr, not central to |
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436 | intensional correctness so curtailed proof effort once reasonable |
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437 | level of confidence in code gained; labelling much simpler; don't care |
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438 | what the labels are at this stage, just need to know when to go |
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439 | through extra steps. Rolled up into a single result with a cofixpoint |
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440 | to obtain coinductive statement of equivalence (show). |
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441 | |
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442 | \section{Finding corresponding measurable subtraces} |
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443 | |
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444 | \section{Existence of a structured trace} |
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445 | \label{sec:structuretrace} |
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446 | |
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447 | \section{Conclusion} |
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448 | |
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449 | TODO |
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450 | |
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451 | \bibliographystyle{plain} |
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452 | \bibliography{report} |
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453 | |
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454 | \end{document} |
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