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{stmaryrd} |
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5 | \usepackage{listings} |
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6 | \usepackage{../../style/cerco} |
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7 | \newcommand{\ocaml}{OCaml} |
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8 | \newcommand{\clight}{Clight} |
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9 | \newcommand{\matita}{Matita} |
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10 | \newcommand{\sdcc}{\texttt{sdcc}} |
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11 | |
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12 | \newcommand{\textSigma}{\ensuremath{\Sigma}} |
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13 | |
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14 | % LaTeX Companion, p 74 |
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15 | \newcommand{\todo}[1]{\marginpar{\raggedright - #1}} |
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17 | \lstdefinelanguage{coq} |
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18 | {keywords={Definition,Lemma,Theorem,Remark,Qed,Save,Inductive,Record}, |
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19 | morekeywords={[2]if,then,else}, |
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20 | } |
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21 | |
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22 | \lstdefinelanguage{matita} |
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23 | {keywords={definition,lemma,theorem,remark,inductive,record,qed,let,rec,match,with,Type,and,on}, |
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24 | morekeywords={[2]whd,normalize,elim,cases,destruct}, |
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25 | mathescape=true, |
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26 | morecomment=[n]{(*}{*)}, |
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27 | } |
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28 | |
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29 | \lstset{language=matita,basicstyle=\small\tt,columns=flexible,breaklines=false, |
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30 | keywordstyle=\color{red}\bfseries, |
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31 | keywordstyle=[2]\color{blue}, |
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32 | commentstyle=\color{green}, |
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33 | stringstyle=\color{blue}, |
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34 | showspaces=false,showstringspaces=false} |
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35 | |
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36 | \lstset{extendedchars=false} |
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37 | \lstset{inputencoding=utf8x} |
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38 | \DeclareUnicodeCharacter{8797}{:=} |
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39 | \DeclareUnicodeCharacter{10746}{++} |
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40 | \DeclareUnicodeCharacter{9001}{\ensuremath{\langle}} |
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41 | \DeclareUnicodeCharacter{9002}{\ensuremath{\rangle}} |
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42 | |
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43 | |
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44 | \title{ |
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45 | INFORMATION AND COMMUNICATION TECHNOLOGIES\\ |
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46 | (ICT)\\ |
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47 | PROGRAMME\\ |
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48 | \vspace*{1cm}Project FP7-ICT-2009-C-243881 \cerco{}} |
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49 | |
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50 | \date{ } |
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51 | \author{} |
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52 | |
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53 | \begin{document} |
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54 | \thispagestyle{empty} |
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55 | |
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56 | \vspace*{-1cm} |
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57 | \begin{center} |
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58 | \includegraphics[width=0.6\textwidth]{../../style/cerco_logo.png} |
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59 | \end{center} |
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60 | |
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61 | \begin{minipage}{\textwidth} |
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62 | \maketitle |
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63 | \end{minipage} |
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64 | |
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65 | |
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66 | \vspace*{0.5cm} |
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67 | \begin{center} |
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68 | \begin{LARGE} |
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69 | \bf |
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70 | Report n. D3.4\\ |
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71 | Front-end Correctness Proofs\\ |
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72 | \end{LARGE} |
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73 | \end{center} |
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74 | |
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75 | \vspace*{2cm} |
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76 | \begin{center} |
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77 | \begin{large} |
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78 | Version 1.0 |
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79 | \end{large} |
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80 | \end{center} |
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81 | |
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82 | \vspace*{0.5cm} |
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83 | \begin{center} |
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84 | \begin{large} |
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85 | Authors:\\ |
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86 | Brian~Campbell, Ilias~Garnier, James~McKinna, Ian~Stark |
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87 | \end{large} |
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88 | \end{center} |
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89 | |
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90 | \vspace*{\fill} |
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91 | \noindent |
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92 | Project Acronym: \cerco{}\\ |
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93 | Project full title: Certified Complexity\\ |
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94 | Proposal/Contract no.: FP7-ICT-2009-C-243881 \cerco{}\\ |
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95 | |
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96 | \clearpage \pagestyle{myheadings} \markright{\cerco{}, FP7-ICT-2009-C-243881} |
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97 | |
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98 | \newpage |
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99 | |
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100 | \section*{Executive Summary} |
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101 | \addcontentsline{toc}{section}{Executive Summary} |
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102 | |
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103 | |
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104 | \cerco{} Work Package 3, \emph{Verified Compiler - front end} aims |
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105 | for formalise and verify the front-end of the \cerco{} cost lifting |
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106 | compiler. This document accompanies the final deliverable, |
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107 | \textbf{D3.4: Front-end Correctness Proofs}. The deliverable consists |
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108 | of the formal correctness proofs for the front-end, written in the |
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109 | \matita{} proof assistant and this document provides report on the |
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110 | work carried out for it. |
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111 | |
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112 | \vspace*{1cm} |
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113 | |
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114 | \paragraph{Abstract} |
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115 | |
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116 | We report on the correctness proofs for the front-end of the \cerco{} |
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117 | cost lifting compiler. First, we identify the core result we wish to |
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118 | prove, which says that the we correctly predict the precise execution |
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119 | time for particular parts of the execution called \emph{measurable} |
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120 | subtraces. Then we consider the three distinct parts of the task: |
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121 | showing that the \emph{annotated source code} output by the compiler |
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122 | has equivalent behaviour to the original input (up to the |
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123 | annotations); showing that a measurable subtrace of the |
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124 | annotated source code corresponds to an equivalent measurable subtrace |
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125 | in the code produced by the front-end, including costs; and finally |
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126 | showing that the enriched \emph{structured} execution traces required |
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127 | for cost correctness in the back-end can be constructed from the |
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128 | properties of the code produced by the front-end. |
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129 | |
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130 | A key part of our work is that the intensional correctness results which show |
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131 | that we get consistent cost measurements throughout the intermediate languages |
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132 | of the compiler can be layered on top of normal forward simulation results, |
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133 | if we split those results into local call-structure preserving simulations. |
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134 | This split allowed us to concentrate on the \textbf{intensional} proofs by |
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135 | axiomatising some of the extensional simulation results that are very similar to |
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136 | existing compiler correctness results, such as CompCert. |
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137 | |
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138 | This report is about the correctness results that are deliverable |
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139 | D3.4, which are about the formalised compiler described in D3.2, using |
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140 | the source language semantics from D3.1 and intermediate language |
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141 | semantics from D3.3. It builds on earlier work on the correctness of |
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142 | a toy compiler built to test the labelling approach in D2.1. Together |
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143 | with the companion deliverable about the correctness of the back-end, |
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144 | D4.4, we obtain results about the whole formalised compiler. |
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145 | |
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146 | \newpage |
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147 | |
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148 | \tableofcontents |
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149 | |
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150 | % CHECK: clear up any -ize vs -ise |
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151 | % CHECK: clear up any "front end" vs "front-end" |
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152 | % CHECK: clear up any mentions of languages that aren't textsf'd. |
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153 | % CHECK: fix unicode in listings |
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154 | |
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155 | \section{Introduction} |
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156 | |
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157 | The \cerco{} compiler produces a version of the source code containing |
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158 | annotations describing the timing behaviour of the object code, as |
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159 | well as the object code itself. It compiles C code, targeting |
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160 | microcontrollers implementing the Intel 8051 architecture. There are |
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161 | two versions: first, an initial prototype was implemented in |
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162 | \ocaml{}~\cite{d2.2}, then a version was formalised in the \matita{} |
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163 | proof assistant~\cite{d3.2,d4.2} and extracted to \ocaml{} code to |
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164 | produce an executable compiler. In this document we present results |
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165 | from Deliverable 3.4, the formalised proofs in \matita{} about the |
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166 | front-end of the latter version of the compiler (culminating in the |
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167 | \lstinline'front_end_correct' lemma), and describe how that fits |
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168 | into the verification of the whole compiler. |
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169 | |
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170 | A key part of this work was to layer the \emph{intensional} correctness |
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171 | results that show that the costs produced are correct on top of the |
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172 | proofs about the compiled code's \emph{extensional} behaviour (that is, the |
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173 | functional correctness of the compiler). Unfortunately, the ambitious |
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174 | goal of completely verifying the entire compiler was not feasible |
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175 | within the time available, but thanks to this separation of |
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176 | extensional and intensional proofs we are able to axiomatise some extensional |
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177 | simulation results which are similar to those in other compiler verification |
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178 | projects and concentrate on the novel intensional proofs. We were |
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179 | also able to add stack space costs to obtain a stronger result. The |
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180 | proofs were made more tractable by introducing compile-time checks for |
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181 | the `sound and precise' cost labelling properties rather than proving |
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182 | that they are preserved throughout. |
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183 | |
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184 | The overall statement of correctness says that the annotated program has the |
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185 | same behaviour as the input, and that for any suitably well-structured part of |
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186 | the execution (which we call \emph{measurable}), the object code will execute |
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187 | the same behaviour taking precisely the time given by the cost annotations in |
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188 | the annotated source program. |
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189 | |
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190 | In the next section we recall the structure of the compiler and make the overall |
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191 | statement more precise. Following that, in Section~\ref{sec:fegoals} we |
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192 | describe the statements we need to prove about the intermediate \textsf{RTLabs} |
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193 | programs for the back-end proofs. |
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194 | Section~\ref{sec:inputtolabelling} covers the compiler passes which produce the |
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195 | annotated source program and Section~\ref{sec:measurablelifting} the rest |
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196 | of the transformations in the front-end. Then the compile-time checks |
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197 | for good cost labelling are detailed in Section~\ref{sec:costchecks} |
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198 | and the proofs that the structured traces required by the back-end |
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199 | exist are discussed in Section~\ref{sec:structuredtrace}. |
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200 | |
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201 | \section{The compiler and its correctness statement} |
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202 | |
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203 | The uncertified prototype \ocaml{} \cerco{} compiler was originally described |
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204 | in Deliverables 2.1 and 2.2. Its design was replicated in the formal |
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205 | \matita{} code, which was presented in Deliverables 3.2 and 4.2, for |
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206 | the front-end and back-end respectively. |
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207 | |
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208 | \begin{figure} |
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209 | \begin{center} |
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210 | \includegraphics[width=0.5\linewidth]{compiler-plain.pdf} |
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211 | \end{center} |
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212 | \caption{Languages in the \cerco{} compiler} |
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213 | \label{fig:compilerlangs} |
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214 | \end{figure} |
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215 | |
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216 | The compiler uses a number of intermediate languages, as outlined the |
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217 | middle two lines of Figure~\ref{fig:compilerlangs}. The upper line |
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218 | represents the front-end of the compiler, and the lower the back-end, |
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219 | finishing with Intel 8051 binary code. Not all of the front-end compiler passes |
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220 | introduce a new language, and Figure~\ref{fig:summary} presents a |
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221 | list of every pass involved. |
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222 | |
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223 | \begin{figure} |
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224 | \begin{center} |
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225 | \begin{minipage}{.8\linewidth} |
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226 | \begin{tabbing} |
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227 | \quad \= $\downarrow$ \quad \= \kill |
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228 | \textsf{C} (unformalised)\\ |
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229 | \> $\downarrow$ \> CIL parser (unformalised \ocaml)\\ |
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230 | \textsf{Clight}\\ |
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231 | %\> $\downarrow$ \> add runtime functions\\ |
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232 | \> $\downarrow$ \> \lstinline[language=C]'switch' removal\\ |
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233 | \> $\downarrow$ \> labelling\\ |
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234 | \> $\downarrow$ \> cast removal\\ |
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235 | \> $\downarrow$ \> stack variable allocation and control structure |
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236 | simplification\\ |
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237 | \textsf{Cminor}\\ |
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238 | %\> $\downarrow$ \> generate global variable initialization code\\ |
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239 | \> $\downarrow$ \> transform to RTL graph\\ |
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240 | \textsf{RTLabs}\\ |
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241 | \> $\downarrow$ \> check cost labelled properties of RTL graph\\ |
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242 | \> $\downarrow$ \> start of target specific back-end\\ |
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243 | \>\quad \vdots |
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244 | \end{tabbing} |
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245 | \end{minipage} |
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246 | \end{center} |
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247 | \caption{Front-end languages and compiler passes} |
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248 | \label{fig:summary} |
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249 | \end{figure} |
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250 | |
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251 | \label{page:switchintro} |
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252 | The annotated source code is produced by the cost labelling phase. |
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253 | Note that there is a pass to replace C \lstinline[language=C]'switch' |
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254 | statements before labelling --- we need to remove them because the |
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255 | simple form of labelling used in the formalised compiler is not quite |
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256 | capable of capturing their execution time costs, largely due to C's |
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257 | `fall-through' behaviour where execution from one branch continues in |
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258 | the next unless there is an explicit \lstinline[language=C]'break'. |
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259 | |
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260 | The cast removal phase which follows cost labelling simplifies |
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261 | expressions to prevent unnecessary arithmetic promotion, which is |
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262 | specified by the C standard but costly for an 8-bit target. The |
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263 | transformation to \textsf{Cminor} and subsequently \textsf{RTLabs} |
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264 | bear considerable resemblance to some passes of the CompCert |
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265 | compiler~\cite{Blazy-Leroy-Clight-09,Leroy-backend}, although we use a simpler \textsf{Cminor} where |
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266 | all loops use \lstinline[language=C]'goto' statements, and the |
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267 | \textsf{RTLabs} language retains a target-independent flavour. The |
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268 | back-end takes \textsf{RTLabs} code as input. |
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269 | |
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270 | The whole compilation function returns the following information on success: |
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271 | \begin{lstlisting}[language=matita] |
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272 | record compiler_output : Type[0] := |
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273 | { c_labelled_object_code: labelled_object_code |
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274 | ; c_stack_cost: stack_cost_model |
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275 | ; c_max_stack: nat |
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276 | ; c_init_costlabel: costlabel |
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277 | ; c_labelled_clight: clight_program |
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278 | ; c_clight_cost_map: clight_cost_map |
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279 | }. |
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280 | \end{lstlisting} |
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281 | It consists of annotated 8051 object code, a mapping from function |
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282 | identifiers to the function's stack space usage, the space available for the |
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283 | stack after global variable allocation, a cost label covering the |
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284 | execution time for the initialisation of global variables and the call |
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285 | to the \lstinline[language=C]'main' function, the annotated source |
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286 | code, and finally a mapping from cost labels to actual execution time |
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287 | costs. |
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288 | |
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289 | An \ocaml{} pretty printer is used to provide a concrete version of |
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290 | the output code and annotated source code. In the case of the |
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291 | annotated source code, it also inserts the actual costs alongside the |
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292 | cost labels, and optionally adds a global cost variable and |
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293 | instrumentation to support further reasoning in external tools such as |
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294 | Frama-C. |
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295 | |
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296 | \subsection{Revisions to the prototype compiler} |
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297 | |
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298 | Our focus on intensional properties prompted us to consider whether we |
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299 | could incorporate stack space into the costs presented to the user. |
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300 | We only allocate one fixed-size frame per function, so modelling this |
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301 | was relatively simple. It is the only form of dynamic memory |
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302 | allocation provided by the compiler, so we were able to strengthen the |
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303 | statement of the goal to guarantee successful execution whenever the |
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304 | stack space obeys the \lstinline'c_max_stack' bound calculated by |
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305 | subtracting the global variable requirements from the total memory |
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306 | available. |
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307 | |
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308 | The cost labelling checks at the end of Figure~\ref{fig:summary} have been |
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309 | introduced to reduce the proof burden, and are described in |
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310 | Section~\ref{sec:costchecks}. |
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311 | |
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312 | The use of dependent types to capture simple intermediate language |
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313 | invariants makes every front-end pass a total function, except |
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314 | \textsf{Clight} to \textsf{Cminor} and the cost checks. Hence various |
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315 | well-formedness and type safety checks are performed only once between |
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316 | \textsf{Clight} and \textsf{Cminor}, and the invariants rule out any |
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317 | difficulties in the later stages. With the benefit of hindsight we |
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318 | would have included an initial checking phase to produce a |
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319 | `well-formed' variant of \textsf{Clight}, conjecturing that this would |
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320 | simplify various parts of the proofs for the \textsf{Clight} stages |
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321 | which deal with potentially ill-formed code. |
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322 | |
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323 | Following D2.2, we previously generated code for global variable |
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324 | initialisation in \textsf{Cminor}, for which we reserved a cost label |
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325 | to represent the execution time for initialisation. However, the |
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326 | back-end must also add an initial call to the main function, whose |
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327 | cost must also be accounted for, so we decided to move the |
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328 | initialisation code to the back-end and merge the costs. |
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329 | |
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330 | \subsection{Main correctness statement} |
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331 | |
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332 | Informally, our main intensional result links the time difference in a source |
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333 | code execution to the time difference in the object code, expressing the time |
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334 | for the source by summing the values for the cost labels in the trace, and the |
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335 | time for the target by a clock built in to the 8051 executable semantics. |
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336 | |
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337 | The availability of precise timing information for 8501 |
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338 | implementations and the design of the compiler allow it to give exact |
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339 | time costs in terms of processor cycles, not just upper bounds. |
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340 | However, these exact results are only available if the subtrace we |
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341 | measure starts and ends at suitable points. In particular, pure |
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342 | computation with no observable effects may be reordered and moved past |
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343 | cost labels, so we cannot measure time between arbitrary statements in |
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344 | the program. |
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345 | |
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346 | There is also a constraint on the subtraces that we |
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347 | measure due to the requirements of the correctness proof for the |
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348 | object code timing analysis. To be sure that the timings are assigned |
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349 | to the correct cost label, we need to know that each return from a |
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350 | function call must go to the correct return address. It is difficult |
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351 | to observe this property locally in the object code because it relies |
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352 | on much earlier stages in the compiler. To convey this information to |
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353 | the timing analysis extra structure is imposed on the subtraces, which |
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354 | is described in Section~\ref{sec:fegoals}. |
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355 | |
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356 | % Regarding the footnote, would there even be much point? |
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357 | % TODO: this might be quite easy to add ('just' subtract the |
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358 | % measurable subtrace from the second label to the end). Could also |
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359 | % measure other traces in this manner. |
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360 | These restrictions are reflected in the subtraces that we give timing |
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361 | guarantees on; they must start at a cost label and end at the return |
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362 | of the enclosing function of the cost label\footnote{We expect that |
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363 | this would generalise to more general subtraces by subtracting costs |
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364 | for unwanted measurable suffixes of a measurable subtrace.}. A |
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365 | typical example of such a subtrace is the execution of an entire |
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366 | function from the cost label at the start of the function until it |
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367 | returns. We call such any such subtrace \emph{measurable} if it (and |
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368 | the prefix of the trace from the start to the subtrace) can also be |
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369 | executed within the available stack space. |
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370 | |
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371 | Now we can give the main intensional statement for the compiler. |
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372 | Given a \emph{measurable} subtrace for a labelled \textsf{Clight} |
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373 | program, there is a subtrace of the 8051 object code program where the |
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374 | time differences match. Moreover, \emph{observable} parts of the |
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375 | trace also match --- these are the appearance of cost labels and |
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376 | function calls and returns. |
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377 | |
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378 | |
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379 | |
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380 | More formally, the definition of this statement in \matita{} is |
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381 | \begin{lstlisting}[language=matita] |
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382 | definition simulates := |
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383 | $\lambda$p: compiler_output. |
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384 | let initial_status := initialise_status $...$ (cm (c_labelled_object_code $...$ p)) in |
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385 | $\forall$m1,m2. |
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386 | measurable Clight_pcs (c_labelled_clight $...$ p) m1 m2 |
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387 | (stack_sizes (c_stack_cost $...$ p)) (c_max_stack $...$ p) $\rightarrow$ |
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388 | $\forall$c1,c2. |
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389 | clock_after Clight_pcs (c_labelled_clight $...$ p) m1 (c_clight_cost_map $...$ p) = OK $...$ c1 $\rightarrow$ |
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390 | clock_after Clight_pcs (c_labelled_clight $...$ p) (m1+m2) (c_clight_cost_map $...$ p) = OK $...$ c2 $\rightarrow$ |
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391 | $\exists$n1,n2. |
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392 | observables Clight_pcs (c_labelled_clight $...$ p) m1 m2 = |
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393 | observables (OC_preclassified_system (c_labelled_object_code $...$ p)) |
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394 | (c_labelled_object_code $...$ p) n1 n2 |
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395 | $\wedge$ |
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396 | clock ?? (execute (n1+n2) ? initial_status) = |
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397 | clock ?? (execute n1 ? initial_status) + (c2-c1). |
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398 | \end{lstlisting} |
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399 | where the \lstinline'measurable', \lstinline'clock_after' and |
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400 | \lstinline'observables' definitions are generic definitions for multiple |
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401 | languages; in this case the \lstinline'Clight_pcs' record applies them |
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402 | to \textsf{Clight} programs. |
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403 | |
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404 | There is a second part to the statement, which says that the initial |
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405 | processing of the input program to produce the cost labelled version |
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406 | does not affect the semantics of the program: |
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407 | % Yes, I'm paraphrasing the result a tiny bit to remove the observe non-function |
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408 | \begin{lstlisting}[language=matita] |
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409 | $\forall$input_program,output. |
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410 | compile input_program = return output $\rightarrow$ |
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411 | not_wrong $...$ (exec_inf $...$ clight_fullexec input_program) $\rightarrow$ |
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412 | sim_with_labels |
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413 | (exec_inf $...$ clight_fullexec input_program) |
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414 | (exec_inf $...$ clight_fullexec (c_labelled_clight $...$ output)) |
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415 | \end{lstlisting} |
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416 | That is, any successful compilation produces a labelled program that |
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417 | has identical behaviour to the original, so long as there is no |
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418 | `undefined behaviour'. |
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419 | |
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420 | Note that this statement provides full functional correctness, including |
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421 | preservation of (non-)termination. The intensional result above does |
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422 | not do this directly --- it does not guarantee the same result or same |
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423 | termination. There are two mitigating factors, however: first, to |
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424 | prove the intensional property you need local simulation results --- these |
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425 | can be pieced together to form full behavioural equivalence, only time |
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426 | constraints have prevented us from doing so. Second, if we wish to |
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427 | confirm a result, termination, or non-termination we could add an |
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428 | observable witness, such as a function that is only called if the |
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429 | correct result is given. The intensional result guarantees that the |
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430 | observable witness is preserved, so the program must behave correctly. |
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431 | |
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432 | These two results are combined in the the \lstinline'correct' |
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433 | theorem in the file \lstinline'correctness.ma'. |
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434 | |
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435 | \section{Correctness statement for the front-end} |
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436 | \label{sec:fegoals} |
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437 | |
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438 | The essential parts of the intensional proof were outlined during work |
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439 | on a toy compiler in Task |
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440 | 2.1~\cite{d2.1,springerlink:10.1007/978-3-642-32469-7_3}. These are |
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441 | \begin{enumerate} |
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442 | \item functional correctness, in particular preserving the trace of |
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443 | cost labels, |
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444 | \item the \emph{soundness} and \emph{precision} of the cost labelling |
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445 | on the object code, and |
---|
446 | \item the timing analysis on the object code produces a correct |
---|
447 | mapping from cost labels to time. |
---|
448 | \end{enumerate} |
---|
449 | |
---|
450 | However, that toy development did not include function calls. For the |
---|
451 | full \cerco{} compiler we also need to maintain the invariant that |
---|
452 | functions return to the correct program location in the caller, as we |
---|
453 | mentioned in the previous section. During work on the back-end timing |
---|
454 | analysis (describe in more detail in the companion deliverable, D4.4) |
---|
455 | the notion of a \emph{structured trace} was developed to enforce this |
---|
456 | return property, and also most of the cost labelling properties too. |
---|
457 | |
---|
458 | \begin{figure} |
---|
459 | \begin{center} |
---|
460 | \includegraphics[width=0.5\linewidth]{compiler.pdf} |
---|
461 | \end{center} |
---|
462 | \caption{The compiler and proof outline} |
---|
463 | \label{fig:compiler} |
---|
464 | \end{figure} |
---|
465 | |
---|
466 | Jointly, we generalised the structured traces to apply to any of the |
---|
467 | intermediate languages which have some idea of program counter. This means |
---|
468 | that they are introduced part way through the compiler, see |
---|
469 | Figure~\ref{fig:compiler}. Proving that a structured trace can be |
---|
470 | constructed at \textsf{RTLabs} has several virtues: |
---|
471 | \begin{itemize} |
---|
472 | \item This is the first language where every operation has its own |
---|
473 | unique, easily addressable, statement. |
---|
474 | \item Function calls and returns are still handled implicitly in the |
---|
475 | language and so the structural properties are ensured by the |
---|
476 | semantics. |
---|
477 | \item Many of the back-end languages from \textsf{RTL} onwards share a common |
---|
478 | core set of definitions, and using structured traces throughout |
---|
479 | increases this uniformity. |
---|
480 | \end{itemize} |
---|
481 | |
---|
482 | \begin{figure} |
---|
483 | \begin{center} |
---|
484 | \includegraphics[width=0.6\linewidth]{strtraces.pdf} |
---|
485 | \end{center} |
---|
486 | \caption{Nesting of functions in structured traces} |
---|
487 | \label{fig:strtrace} |
---|
488 | \end{figure} |
---|
489 | A structured trace is a mutually inductive data type which |
---|
490 | contains the steps from a normal program trace, but arranged into a |
---|
491 | nested structure which groups entire function calls together and |
---|
492 | aggregates individual steps between cost labels (or between the final |
---|
493 | cost label and the return from the function), see |
---|
494 | Figure~\ref{fig:strtrace}. This captures the idea that the cost labels |
---|
495 | only represent costs \emph{within} a function --- calls to other |
---|
496 | functions are accounted for in the nested trace for their execution, and we |
---|
497 | can locally regard function calls as a single step. |
---|
498 | |
---|
499 | These structured traces form the core part of the intermediate results |
---|
500 | that we must prove so that the back-end can complete the main |
---|
501 | intensional result stated above. In full, we provide the back-end |
---|
502 | with |
---|
503 | \begin{enumerate} |
---|
504 | \item A normal trace of the \textbf{prefix} of the program's execution |
---|
505 | before reaching the measurable subtrace. (This needs to be |
---|
506 | preserved so that we know that the stack space consumed is correct, |
---|
507 | and to set up the simulation results.) |
---|
508 | \item The \textbf{structured trace} corresponding to the measurable |
---|
509 | subtrace. |
---|
510 | \item An additional property about the structured trace that no |
---|
511 | `program counter' is \textbf{repeated} between cost labels. Together with |
---|
512 | the structure in the trace, this takes over from showing that |
---|
513 | cost labelling is sound and precise. |
---|
514 | \item A proof that the \textbf{observables} have been preserved. |
---|
515 | \item A proof that the \textbf{stack limit} is still observed by the prefix and |
---|
516 | the structure trace. (This is largely a consequence of the |
---|
517 | preservation of observables.) |
---|
518 | \end{enumerate} |
---|
519 | The \lstinline'front_end_correct' lemma in the |
---|
520 | \lstinline'correctness.ma' file provides a record containing these. |
---|
521 | |
---|
522 | Following the outline in Figure~\ref{fig:compiler}, we will first deal |
---|
523 | with the transformations in \textsf{Clight} that produce the source |
---|
524 | program with cost labels, then show that measurable traces can be |
---|
525 | lifted to \textsf{RTLabs}, and finally show that we can construct the |
---|
526 | properties listed above ready for the back-end proofs. |
---|
527 | |
---|
528 | \section{Input code to cost labelled program} |
---|
529 | \label{sec:inputtolabelling} |
---|
530 | |
---|
531 | As explained on page~\pageref{page:switchintro}, the costs of complex |
---|
532 | C \lstinline[language=C]'switch' statements cannot be represented with |
---|
533 | the simple labelling used in the formalised compiler. Our first pass |
---|
534 | replaces these statements with simpler C code, allowing our second |
---|
535 | pass to perform the cost labelling. We show that the behaviour of |
---|
536 | programs is unchanged by these passes using forward |
---|
537 | simulations\footnote{All of our languages are deterministic, which can |
---|
538 | be seen directly from their executable definitions. Thus we know that |
---|
539 | forward simulations are sufficient because the target cannot have any |
---|
540 | other behaviour.}. |
---|
541 | |
---|
542 | \subsection{Switch removal} |
---|
543 | |
---|
544 | We compile away \lstinline[language=C]'switch' statements into more |
---|
545 | basic \textsf{Clight} code. |
---|
546 | Note that this transformation does not necessarily deteriorate the |
---|
547 | efficiency of the generated code. For instance, compilers such as GCC |
---|
548 | introduce balanced trees of ``if-then-else'' constructs for small |
---|
549 | switches. However, our implementation strategy is much simpler. Let |
---|
550 | us consider the following input statement. |
---|
551 | |
---|
552 | \begin{lstlisting}[language=C] |
---|
553 | switch(e) { |
---|
554 | case v1: |
---|
555 | stmt1; |
---|
556 | case v2: |
---|
557 | stmt2; |
---|
558 | default: |
---|
559 | stmt_default; |
---|
560 | } |
---|
561 | \end{lstlisting} |
---|
562 | |
---|
563 | Note that \textsf{stmt1}, \textsf{stmt2}, \ldots \textsf{stmt\_default} |
---|
564 | may contain \lstinline[language=C]'break' statements, which have the |
---|
565 | effect of exiting the switch statement. In the absence of break, the |
---|
566 | execution falls through each case sequentially. In our implementation, |
---|
567 | we produce an equivalent sequence of ``if-then'' chained by gotos: |
---|
568 | \begin{lstlisting}[language=C] |
---|
569 | fresh = e; |
---|
570 | if(fresh == v1) { |
---|
571 | $\llbracket$stmt1$\rrbracket$; |
---|
572 | goto lbl_case2; |
---|
573 | }; |
---|
574 | if(fresh == v2) { |
---|
575 | lbl_case2: |
---|
576 | $\llbracket$stmt2$\rrbracket$; |
---|
577 | goto lbl_case2; |
---|
578 | }; |
---|
579 | $\llbracket$stmt_default$\rrbracket$; |
---|
580 | exit_label: |
---|
581 | \end{lstlisting} |
---|
582 | |
---|
583 | The proof had to tackle the following points: |
---|
584 | \begin{itemize} |
---|
585 | \item the source and target memories are not the same (due to the fresh variable), |
---|
586 | \item the flow of control is changed in a non-local way (e.g. \textbf{goto} |
---|
587 | instead of \textbf{break}). |
---|
588 | \end{itemize} |
---|
589 | In order to tackle the first point, we implemented a version of memory |
---|
590 | extensions similar to those of CompCert. |
---|
591 | |
---|
592 | For the simulation we decided to prove a sufficient amount to give us |
---|
593 | confidence in the definitions and approach, but to curtail the proof |
---|
594 | because this pass does not contribute to the intensional correctness |
---|
595 | result. We tackled several simple cases, that do not interact with |
---|
596 | the switch removal per se, to show that the definitions were usable, |
---|
597 | and part of the switch case to check that the approach is |
---|
598 | reasonable. This comprises propagating the memory extension through |
---|
599 | each statement (except switch), as well as various invariants that are |
---|
600 | needed for the switch case (in particular, freshness hypotheses). The |
---|
601 | details of the evaluation process for the source switch statement and |
---|
602 | its target counterpart can be found in the file |
---|
603 | \lstinline'switchRemoval.ma', along more details on the transformation |
---|
604 | itself. |
---|
605 | |
---|
606 | Proving the correctness of the second point would require reasoning on the |
---|
607 | semantics of \lstinline[language=C]'goto' statements. In the \textsf{Clight} |
---|
608 | semantics, this is implemented as a function-wide lookup of the target label. |
---|
609 | The invariant we would need is the fact that a global label lookup on a freshly |
---|
610 | created goto is equivalent to a local lookup. This would in turn require the |
---|
611 | propagation of some freshness hypotheses on labels. As discussed, |
---|
612 | we decided to omit this part of the correctness proof. |
---|
613 | |
---|
614 | \subsection{Cost labelling} |
---|
615 | |
---|
616 | The simulation for the cost labelling pass is the simplest in the |
---|
617 | front-end. The main argument is that any step of the source program |
---|
618 | is simulated by the same step of the labelled one, plus any extra |
---|
619 | steps for the added cost labels. The extra instructions do not change |
---|
620 | the memory or local environments, although we have to keep track of |
---|
621 | the extra instructions that appear in continuations, for example |
---|
622 | during the execution of a \lstinline[language=C]'while' loop. |
---|
623 | |
---|
624 | We do not attempt to capture any cost properties of the labelling\footnote{We describe how the cost properties are |
---|
625 | established in Section~\ref{sec:costchecks}.} in |
---|
626 | the simulation proof, which allows the proof to be oblivious to the choice |
---|
627 | of cost labels. Hence we do not have to reason about the threading of |
---|
628 | name generation through the labelling function, greatly reducing the |
---|
629 | amount of effort required. |
---|
630 | |
---|
631 | %TODO: both give one-step-sim-by-many forward sim results; switch |
---|
632 | %removal tricky, uses aux var to keep result of expr, not central to |
---|
633 | %intensional correctness so curtailed proof effort once reasonable |
---|
634 | %level of confidence in code gained; labelling much simpler; don't care |
---|
635 | %what the labels are at this stage, just need to know when to go |
---|
636 | %through extra steps. Rolled up into a single result with a cofixpoint |
---|
637 | %to obtain coinductive statement of equivalence (show). |
---|
638 | |
---|
639 | \section{Finding corresponding measurable subtraces} |
---|
640 | \label{sec:measurablelifting} |
---|
641 | |
---|
642 | There follow the three main passes of the front-end: |
---|
643 | \begin{enumerate} |
---|
644 | \item simplification of casts in \textsf{Clight} code |
---|
645 | \item \textsf{Clight} to \textsf{Cminor} translation, performing stack |
---|
646 | variable allocation and simplifying control structures |
---|
647 | \item transformation to \textsf{RTLabs} control flow graph |
---|
648 | \end{enumerate} |
---|
649 | We have taken a common approach to |
---|
650 | each pass: first we build (or axiomatise) forward simulation results |
---|
651 | that are similar to normal compiler proofs, but which are slightly more |
---|
652 | fine-grained so that we can see that the call structure and relative |
---|
653 | placement of cost labels is preserved. |
---|
654 | |
---|
655 | Then we instantiate a general result which shows that we can find a |
---|
656 | \emph{measurable} subtrace in the target of the pass that corresponds |
---|
657 | to the measurable subtrace in the source. By repeated application of |
---|
658 | this result we can find a measurable subtrace of the execution of the |
---|
659 | \textsf{RTLabs} code, suitable for the construction of a structured |
---|
660 | trace (see Section~\ref{sec:structuredtrace}). This is essentially an |
---|
661 | extra layer on top of the simulation proofs that provides us with the |
---|
662 | additional information required for our intensional correctness proof. |
---|
663 | |
---|
664 | \subsection{Generic measurable subtrace lifting proof} |
---|
665 | |
---|
666 | Our generic proof is parametrised on a record containing small-step |
---|
667 | semantics for the source and target language, a classification of |
---|
668 | states (the same form of classification is used when defining |
---|
669 | structured traces), a simulation relation which respects the |
---|
670 | classification and cost labelling and |
---|
671 | four simulation results. The simulations are split by the starting state's |
---|
672 | classification and whether it is a cost label, which will allow us to |
---|
673 | observe that the call structure is preserved. They are: |
---|
674 | \begin{enumerate} |
---|
675 | \item a step from a `normal' state (which is not classified as a call |
---|
676 | or return) which is not a cost label is simulated by zero or more |
---|
677 | `normal' steps; |
---|
678 | \item a step from a `call' state followed by a cost label step is |
---|
679 | simulated by a step from a `call' state, a corresponding label step, |
---|
680 | then zero or more `normal' steps; |
---|
681 | \item a step from a `call' state not followed by a cost label |
---|
682 | similarly (note that this case cannot occur in a well-labelled |
---|
683 | program, but we do not have enough information locally to exploit |
---|
684 | this); and |
---|
685 | \item a cost label step is simulated by a cost label step. |
---|
686 | \end{enumerate} |
---|
687 | Finally, we need to know that a successfully translated program will |
---|
688 | have an initial state in the simulation relation with the original |
---|
689 | program's initial state. |
---|
690 | |
---|
691 | The back-end has similar requirements for lifting simulations to |
---|
692 | structured traces. Fortunately, our treatment of calls and returns |
---|
693 | can be slightly simpler because we have special call and return states |
---|
694 | that correspond to function entry and return that are separate from |
---|
695 | the actual instructions. This was originally inherited from our port |
---|
696 | of CompCert's \textsf{Clight} semantics, but proves useful here |
---|
697 | because we only need to consider adding extra steps \emph{after} a |
---|
698 | call or return state, because the instruction step deals with extra |
---|
699 | steps that occur before. The back-end makes all of the call and |
---|
700 | return machinery explicit, and thus needs more complex statements |
---|
701 | about extra steps before and after each call and return. |
---|
702 | |
---|
703 | \begin{figure} |
---|
704 | \begin{center} |
---|
705 | \includegraphics[width=0.5\linewidth]{meassim.pdf} |
---|
706 | \end{center} |
---|
707 | \caption{Tiling of simulation for a measurable subtrace} |
---|
708 | \label{fig:tiling} |
---|
709 | \end{figure} |
---|
710 | |
---|
711 | To find the measurable subtrace in the target program's execution we |
---|
712 | walk along the original program's execution trace applying the |
---|
713 | appropriate simulation result by induction on the number of steps. |
---|
714 | While the number of steps taken varies, the overall structure is |
---|
715 | preserved, as illustrated in Figure~\ref{fig:tiling}. By preserving |
---|
716 | the structure we also maintain the same intensional observables. One |
---|
717 | delicate point is that the cost label following a call must remain |
---|
718 | directly afterwards\footnote{The prototype compiler allowed some |
---|
719 | straight-line code to appear before the cost label until a later |
---|
720 | stage of the compiler, but we must move the requirement forward to |
---|
721 | fit with the structured traces.} |
---|
722 | % Damn it, I should have just moved the cost label forwards in RTLabs, |
---|
723 | % like the prototype does in RTL to ERTL; the result would have been |
---|
724 | % simpler. Or was there some reason not to do that? |
---|
725 | (both in the program code and in the execution trace), even if we |
---|
726 | introduce extra steps, for example to store parameters in memory in |
---|
727 | \textsf{Cminor}. Thus we have a version of the call simulation |
---|
728 | that deals with both the call and the cost label in one result. |
---|
729 | |
---|
730 | In addition to the subtrace we are interested in measuring, we must |
---|
731 | prove that the earlier part of the trace is also preserved in |
---|
732 | order to use the simulation from the initial state. This proof also |
---|
733 | guarantees that we do not run out of stack space before the subtrace |
---|
734 | we are interested in. The lemmas for this prefix and the measurable |
---|
735 | subtrace are similar, following the pattern above. However, the |
---|
736 | measurable subtrace also requires us to rebuild the termination |
---|
737 | proof. This is defined recursively: |
---|
738 | \label{prog:terminationproof} |
---|
739 | \begin{lstlisting}[language=matita] |
---|
740 | let rec will_return_aux C (depth:nat) |
---|
741 | (trace:list (cs_state $...$ C $\times$ trace)) on trace : bool := |
---|
742 | match trace with |
---|
743 | [ nil $\Rightarrow$ false |
---|
744 | | cons h tl $\Rightarrow$ |
---|
745 | let $\langle$s,tr$\rangle$ := h in |
---|
746 | match cs_classify C s with |
---|
747 | [ cl_call $\Rightarrow$ will_return_aux C (S depth) tl |
---|
748 | | cl_return $\Rightarrow$ |
---|
749 | match depth with |
---|
750 | [ O $\Rightarrow$ match tl with [ nil $\Rightarrow$ true | _ $\Rightarrow$ false ] |
---|
751 | | S d $\Rightarrow$ will_return_aux C d tl |
---|
752 | ] |
---|
753 | | _ $\Rightarrow$ will_return_aux C depth tl |
---|
754 | ] |
---|
755 | ]. |
---|
756 | \end{lstlisting} |
---|
757 | The \lstinline'depth' is the number of return states we need to see |
---|
758 | before we have returned to the original function (initially zero) and |
---|
759 | \lstinline'trace' the measurable subtrace obtained from the running |
---|
760 | the semantics for the correct number of steps. This definition |
---|
761 | unfolds tail recursively for each step, and once the corresponding |
---|
762 | simulation result has been applied a new one for the target can be |
---|
763 | asserted by unfolding and applying the induction hypothesis on the |
---|
764 | shorter trace. |
---|
765 | |
---|
766 | Combining the lemmas about the prefix and the measurable subtrace |
---|
767 | requires a little care because the states joining the two might not be |
---|
768 | related in the simulation. In particular, if the measurable subtrace |
---|
769 | starts from the cost label at the beginning of the function there may |
---|
770 | be some extra instructions in the target code to execute to complete |
---|
771 | function entry before the states are back in the relation. Hence we |
---|
772 | carefully phrased the lemmas to allow for such extra steps. |
---|
773 | |
---|
774 | Together, these then gives us an overall result for any simulation fitting the |
---|
775 | requirements above (contained in the \lstinline'meas_sim' record): |
---|
776 | \begin{lstlisting}[language=matita] |
---|
777 | theorem measured_subtrace_preserved : |
---|
778 | $\forall$MS:meas_sim. |
---|
779 | $\forall$p1,p2,m,n,stack_cost,max. |
---|
780 | ms_compiled MS p1 p2 $\rightarrow$ |
---|
781 | measurable (ms_C1 MS) p1 m n stack_cost max $\rightarrow$ |
---|
782 | $\exists$m',n'. |
---|
783 | measurable (ms_C2 MS) p2 m' n' stack_cost max $\wedge$ |
---|
784 | observables (ms_C1 MS) p1 m n = observables (ms_C2 MS) p2 m' n'. |
---|
785 | \end{lstlisting} |
---|
786 | The stack space requirement that is embedded in \lstinline'measurable' |
---|
787 | is a consequence of the preservation of observables, because it is |
---|
788 | determined by the functions called and returned from, which are observable. |
---|
789 | |
---|
790 | \subsection{Simulation results for each pass} |
---|
791 | |
---|
792 | We now consider the simulation results for the passes, each of which |
---|
793 | is used to instantiate the |
---|
794 | \lstinline[language=matita]'measured_subtrace_preserved' theorem to |
---|
795 | construct the measurable subtrace for the next language. |
---|
796 | |
---|
797 | \subsubsection{Cast simplification} |
---|
798 | |
---|
799 | The parser used in \cerco{} introduces a lot of explicit type casts. |
---|
800 | If left as they are, these constructs can greatly hamper the |
---|
801 | quality of the generated code -- especially as the architecture |
---|
802 | we consider is an $8$-bit one. In \textsf{Clight}, casts are |
---|
803 | expressions. Hence, most of the work of this transformation |
---|
804 | proceeds on expressions. The tranformation proceeds by recursively |
---|
805 | trying to coerce an expression to a particular integer type, which |
---|
806 | is in practice smaller than the original one. This functionality |
---|
807 | is implemented by two mutually recursive functions whose signature |
---|
808 | is the following. |
---|
809 | |
---|
810 | \begin{lstlisting}[language=matita] |
---|
811 | let rec simplify_expr (e:expr) (target_sz:intsize) (target_sg:signedness) |
---|
812 | : $\Sigma$result:bool$\times$expr. |
---|
813 | $\forall$ge,en,m. simplify_inv ge en m e (\snd result) target_sz target_sg (\fst result) := $\ldots$ |
---|
814 | |
---|
815 | and simplify_inside (e:expr) : $\Sigma$result:expr. conservation e result := $\ldots$ |
---|
816 | \end{lstlisting} |
---|
817 | |
---|
818 | The \textsf{simplify\_inside} acts as a wrapper for |
---|
819 | \textsf{simplify\_expr}. Whenever \textsf{simplify\_inside} encounters |
---|
820 | a \textsf{Ecast} expression, it tries to coerce the sub-expression |
---|
821 | to the desired type using \textsf{simplify\_expr}, which tries to |
---|
822 | perform the actual coercion. In return, \textsf{simplify\_expr} calls |
---|
823 | back \textsf{simplify\_inside} in some particular positions, where we |
---|
824 | decided to be conservative in order to simplify the proofs. However, |
---|
825 | the current design allows to incrementally revert to a more aggressive |
---|
826 | version, by replacing recursive calls to \textsf{simplify\_inside} by |
---|
827 | calls to \textsf{simplify\_expr} \emph{and} proving the corresponding |
---|
828 | invariants -- where possible. |
---|
829 | |
---|
830 | The \textsf{simplify\_inv} invariant encodes either the conservation |
---|
831 | of the semantics during the transformation corresponding to the failure |
---|
832 | of the coercion (\textsf{Inv\_eq} constructor), or the successful |
---|
833 | downcast of the considered expression to the target type |
---|
834 | (\textsf{Inv\_coerce\_ok}). |
---|
835 | |
---|
836 | \begin{lstlisting}[language=matita] |
---|
837 | inductive simplify_inv |
---|
838 | (ge : genv) (en : env) (m : mem) |
---|
839 | (e1 : expr) (e2 : expr) (target_sz : intsize) (target_sg : signedness) : bool $\rightarrow$ Prop := |
---|
840 | | Inv_eq : $\forall$result_flag. $\ldots$ |
---|
841 | simplify_inv ge en m e1 e2 target_sz target_sg result_flag |
---|
842 | | Inv_coerce_ok : $\forall$src_sz,src_sg. |
---|
843 | typeof e1 = Tint src_sz src_sg $\rightarrow$ |
---|
844 | typeof e2 = Tint target_sz target_sg $\rightarrow$ |
---|
845 | smaller_integer_val src_sz target_sz src_sg (exec_expr ge en m e1) (exec_expr ge en m e2) $\rightarrow$ |
---|
846 | simplify_inv ge en m e1 e2 target_sz target_sg true. |
---|
847 | \end{lstlisting} |
---|
848 | |
---|
849 | The \textsf{conservation} invariant for \textsf{simplify\_inside} simply states the conservation |
---|
850 | of the semantics, as in the \textsf{Inv\_eq} constructor of the previous |
---|
851 | invariant. |
---|
852 | |
---|
853 | \begin{lstlisting}[language=matita] |
---|
854 | definition conservation := $\lambda$e,result. $\forall$ge,en,m. |
---|
855 | res_sim ? (exec_expr ge en m e) (exec_expr ge en m result) |
---|
856 | $\wedge$ res_sim ? (exec_lvalue ge en m e) (exec_lvalue ge en m result) |
---|
857 | $\wedge$ typeof e = typeof result. |
---|
858 | \end{lstlisting} |
---|
859 | |
---|
860 | This invariant is then easily lifted to statement evaluations. |
---|
861 | The main problem encountered with this particular pass was dealing with |
---|
862 | inconsistently typed programs, a canonical case being a particular |
---|
863 | integer constant of a certain size typed with another size. This |
---|
864 | prompted the need to introduce numerous type checks, making |
---|
865 | both the implementation and the proof more complex, even though more |
---|
866 | comprehensive checks are made in the next stage. |
---|
867 | %\todo{Make this a particular case of the more general statement on baking more invariants in the Clight language} |
---|
868 | |
---|
869 | \subsubsection{Clight to Cminor} |
---|
870 | |
---|
871 | This pass is the last one operating on the \textsf{Clight} language. |
---|
872 | Its input is a full \textsf{Clight} program, and its output is a |
---|
873 | \textsf{Cminor} program. Note that we do not use an equivalent of |
---|
874 | CompCert's \textsf{C\#minor} language: we translate directly to a |
---|
875 | variant of \textsf{Cminor}. This presents the advantage of not |
---|
876 | requiring the special loop constructs, nor the explicit block |
---|
877 | structure. Another salient point of our approach is that a significant |
---|
878 | number of the properties needed for the simulation proof were directly |
---|
879 | encoded in dependently typed translation functions. In particular, |
---|
880 | freshness conditions and well-typedness conditions are included. The |
---|
881 | main effects of the transformation from \textsf{Clight} to |
---|
882 | \textsf{Cminor} are listed below. |
---|
883 | |
---|
884 | \begin{itemize} |
---|
885 | \item Variables are classified as being either globals, stack-allocated |
---|
886 | locals or potentially register-allocated locals. The value of register-allocated |
---|
887 | local variables is moved out of the modelled memory and stored in a |
---|
888 | dedicated environment. |
---|
889 | \item In \textsf{Clight}, each local variable has a dedicated memory block, whereas |
---|
890 | stack-allocated locals are bundled together on a function-by-function basis. |
---|
891 | \item Loops are converted to jumps. |
---|
892 | \end{itemize} |
---|
893 | |
---|
894 | The first two points require memory injections which are more flexible that those |
---|
895 | needed in the switch removal case. In the remainder of this section, we briefly |
---|
896 | discuss our implementation of memory injections, and then the simulation proof. |
---|
897 | |
---|
898 | \paragraph{Memory injections.} |
---|
899 | |
---|
900 | Our memory injections are modelled after the work of Blazy \& Leroy. |
---|
901 | However, the corresponding paper is based on the first version of the |
---|
902 | CompCert memory model~\cite{2008-Leroy-Blazy-memory-model}, whereas we use a much more concrete model, allowing byte-level |
---|
903 | manipulations (as in the later version of CompCert's memory model). We proved |
---|
904 | roughly 80 \% of the required lemmas. Notably, some of the difficulties encountered were |
---|
905 | due to overly relaxed conditions on pointer validity (fixed during development). |
---|
906 | Some more side conditions had to be added to take care of possible overflows when converting |
---|
907 | from \textbf{Z} block bounds to $16$ bit pointer offsets (in practice, such overflows |
---|
908 | only occur in edge cases that are easily ruled out -- but this fact is not visible |
---|
909 | in memory injections). Concretely, some of the lemmas on the preservation of simulation of |
---|
910 | loads after writes were axiomatised, due to a lack of time. |
---|
911 | |
---|
912 | \paragraph{Simulation proof.} |
---|
913 | |
---|
914 | We proved the simulation result for expressions and a representative |
---|
915 | selection of statements. In particular we tackled |
---|
916 | \lstinline[language=C]'while' statements to ensure that we correctly |
---|
917 | translate loops because our approach differs from CompCert by |
---|
918 | converting directly to \textsf{Cminor} \lstinline[language=C]'goto's |
---|
919 | rather than maintaining a notion of loop in \textsf{Cminor}. We also have a partial |
---|
920 | proof for function entry, covering the setup of the memory injection, |
---|
921 | but not function exit. Exits, and the remaining statements, have been |
---|
922 | axiomatised. |
---|
923 | |
---|
924 | Careful management of the proof state was required because proof terms |
---|
925 | are embedded in \textsf{Cminor} code to show that invariants are |
---|
926 | respected. These proof terms appear in the proof state when inverting |
---|
927 | the translation functions, and they can be large and awkward. While |
---|
928 | generalising them away is usually sufficient, it can be difficult when |
---|
929 | they appear under a binder. |
---|
930 | |
---|
931 | %The correctness proof for this transformation was not completed. We proved the |
---|
932 | %simulation result for expressions and for some subset of the critical statement cases. |
---|
933 | %Notably lacking are the function entry and exit, where the memory injection is |
---|
934 | %properly set up. As would be expected, a significant amount of work has to be performed |
---|
935 | %to show the conservation of all invariants at each simulation step. |
---|
936 | |
---|
937 | %\todo{list cases, explain while loop, explain labeling problem} |
---|
938 | |
---|
939 | \subsubsection{Cminor to RTLabs} |
---|
940 | |
---|
941 | The translation from \textsf{Cminor} to \textsf{RTLabs} is a fairly |
---|
942 | routine control flow graph (CFG) construction. As such, we chose to |
---|
943 | axiomatise the associated extensional simulation results. However, we did prove several |
---|
944 | properties of the generated programs: |
---|
945 | \begin{itemize} |
---|
946 | \item All statements are type correct with respect to the declared |
---|
947 | pseudo-register type environment. |
---|
948 | \item The CFG is closed, and has a distinguished entry node and a |
---|
949 | unique exit node. |
---|
950 | \end{itemize} |
---|
951 | |
---|
952 | These properties rely on similar properties about type safety and the |
---|
953 | presence of \lstinline[language=C]'goto'-labels for \textsf{Cminor} programs |
---|
954 | which are checked at the preceding stage. As a result, this |
---|
955 | transformation is total and any compilation failures must occur when |
---|
956 | the corresponding \textsf{Clight} source is available and a better |
---|
957 | error message can be generated. |
---|
958 | |
---|
959 | The proof obligations for these properties include many instances of |
---|
960 | graph inclusion. We automated these proofs using a small amount of |
---|
961 | reflection, making the obligations much easier to handle. One |
---|
962 | drawback to enforcing invariants throughout is that temporarily |
---|
963 | breaking them can be awkward. For example, \lstinline'return' |
---|
964 | statements were originally used as placeholders for |
---|
965 | \lstinline[language=C]'goto' destinations that had not yet been |
---|
966 | translated. However, this made establishing the single exit node |
---|
967 | property rather difficult, and a different placeholder was chosen |
---|
968 | instead. In other circumstances it is possible to prove a more |
---|
969 | complex invariant then simplify it at the end of the transformation. |
---|
970 | |
---|
971 | \section{Checking cost labelling properties} |
---|
972 | \label{sec:costchecks} |
---|
973 | |
---|
974 | Ideally, we would provide proofs that the cost labelling pass always |
---|
975 | produces programs that are soundly and precisely labelled and that |
---|
976 | each subsequent pass preserves these properties. This would match our |
---|
977 | use of dependent types to eliminate impossible sources of errors |
---|
978 | during compilation, in particular retaining intermediate language type |
---|
979 | information. |
---|
980 | |
---|
981 | However, given the limited amount of time available we realised that |
---|
982 | implementing a compile-time check for a sound and precise labelling of |
---|
983 | the \textsf{RTLabs} intermediate code would reduce the proof burden |
---|
984 | considerably. This is similar in spirit to the use of translation |
---|
985 | validation in certified compilation, which makes a similar trade-off |
---|
986 | between the potential for compile-time failure and the volume of proof |
---|
987 | required. |
---|
988 | |
---|
989 | The check cannot be pushed into a later stage of the compiler because |
---|
990 | much of the information is embedded into the structured traces. |
---|
991 | However, if an alternative method was used to show that function |
---|
992 | returns in the compiled code are sufficiently well-behaved, then we |
---|
993 | could consider pushing the cost property checks into the timing |
---|
994 | analysis itself. We leave this as a possible area for future work. |
---|
995 | |
---|
996 | \subsection{Implementation and correctness} |
---|
997 | \label{sec:costchecksimpl} |
---|
998 | |
---|
999 | For a cost labelling to be sound and precise we need a cost label at |
---|
1000 | the start of each function, after each branch and at least one in |
---|
1001 | every loop. The first two parts are trivial to check by examining the |
---|
1002 | code. In \textsf{RTLabs} the last part is specified by saying |
---|
1003 | that there is a bound on the number of successive instruction nodes in |
---|
1004 | the CFG that you can follow before you encounter a cost label, and |
---|
1005 | checking this is more difficult. |
---|
1006 | |
---|
1007 | The implementation progresses through the set of nodes in the graph, |
---|
1008 | following successors until a cost label is found or a label-free cycle |
---|
1009 | is discovered (in which case the property does not hold and we return |
---|
1010 | an error). This is made easier by the prior knowledge that every |
---|
1011 | successor of a branch instruction is a cost label, so we do not need |
---|
1012 | to search each branch. When a label is found, we remove the chain of |
---|
1013 | program counters from the set and continue from another node in the |
---|
1014 | set until it is empty, at which point we know that there is a bound |
---|
1015 | for every node in the graph. |
---|
1016 | |
---|
1017 | Directly reasoning about the function that implements this procedure would be |
---|
1018 | rather awkward, so an inductive specification of a single step of its |
---|
1019 | behaviour was written and proved to match the implementation. This |
---|
1020 | was then used to prove the implementation sound and complete. |
---|
1021 | |
---|
1022 | While we have not attempted to prove that the cost labelled properties |
---|
1023 | are established and preserved earlier in the compiler, we expect that |
---|
1024 | the effort for the \textsf{Cminor} to \textsf{RTLabs} stage alone |
---|
1025 | would be similar to the work outlined above, because it involves the |
---|
1026 | change from requiring a cost label at particular positions to |
---|
1027 | requiring cost labels to break loops in the CFG. As there are another |
---|
1028 | three passes to consider (including the labelling itself), we believe |
---|
1029 | that using the check above is much simpler overall. |
---|
1030 | |
---|
1031 | % TODO? Found some Clight to Cminor bugs quite quickly |
---|
1032 | |
---|
1033 | \section{Existence of a structured trace} |
---|
1034 | \label{sec:structuredtrace} |
---|
1035 | |
---|
1036 | The \emph{structured trace} idea introduced in |
---|
1037 | Section~\ref{sec:fegoals} enriches the execution trace of a program by |
---|
1038 | nesting function calls in a mixed-step style and embedding the cost |
---|
1039 | labelling properties of the program. See Figure~\ref{fig:strtrace} on |
---|
1040 | page~\pageref{fig:strtrace} for an illustration of a structured trace. |
---|
1041 | It was originally designed to support the proof of correctness for the |
---|
1042 | timing analysis of the object code in the back-end, then generalised |
---|
1043 | to provide a common structure to use from the end of the front-end to |
---|
1044 | the object code. |
---|
1045 | |
---|
1046 | To make the definition generic we abstract over the semantics of the |
---|
1047 | language, |
---|
1048 | \begin{lstlisting}[language=matita] |
---|
1049 | record abstract_status : Type[1] := |
---|
1050 | { as_status :> Type[0] |
---|
1051 | ; as_execute : relation as_status |
---|
1052 | ; as_pc : DeqSet |
---|
1053 | ; as_pc_of : as_status $\rightarrow$ as_pc |
---|
1054 | ; as_classify : as_status $\rightarrow$ status_class |
---|
1055 | ; as_label_of_pc : as_pc $\rightarrow$ option costlabel |
---|
1056 | ; as_after_return : ($\Sigma$s:as_status. as_classify s = cl_call) $\rightarrow$ as_status $\rightarrow$ Prop |
---|
1057 | ; as_result: as_status $\rightarrow$ option int |
---|
1058 | ; as_call_ident : ($\Sigma$s:as_status.as_classify s = cl_call) $\rightarrow$ ident |
---|
1059 | ; as_tailcall_ident : ($\Sigma$s:as_status.as_classify s = cl_tailcall) $\rightarrow$ ident |
---|
1060 | }. |
---|
1061 | \end{lstlisting} |
---|
1062 | which requires a type of states, an execution relation\footnote{All of |
---|
1063 | our semantics are executable, but using a relation was simpler in |
---|
1064 | the abstraction.}, some notion of abstract |
---|
1065 | program counter with decidable equality, the classification of states, |
---|
1066 | and functions to extract the observable intensional information (cost |
---|
1067 | labels and the identity of functions that are called). The |
---|
1068 | \lstinline'as_after_return' property links the state before a function |
---|
1069 | call with the state after return, providing the evidence that |
---|
1070 | execution returns to the correct place. The precise form varies |
---|
1071 | between stages; in \textsf{RTLabs} it insists the CFG, the pointer to |
---|
1072 | the CFG node to execute next, and some call stack information is |
---|
1073 | preserved. |
---|
1074 | |
---|
1075 | The structured traces are defined using three mutually inductive |
---|
1076 | types. The core data structure is \lstinline'trace_any_label', which |
---|
1077 | captures some straight-line execution until the next cost label or the |
---|
1078 | return from the enclosing function. Any function calls are embedded as |
---|
1079 | a single step, with its own trace nested inside and the before and |
---|
1080 | after states linked by \lstinline'as_after_return'; and states |
---|
1081 | classified as a `jump' (in particular branches) must be followed by a |
---|
1082 | cost label. |
---|
1083 | |
---|
1084 | The second type, \lstinline'trace_label_label', is a |
---|
1085 | \lstinline'trace_any_label' where the initial state is cost labelled. |
---|
1086 | Thus a trace in this type identifies a series of steps whose cost is |
---|
1087 | entirely accounted for by the label at the start. |
---|
1088 | |
---|
1089 | Finally, \lstinline'trace_label_return' is a sequence of |
---|
1090 | \lstinline'trace_label_label' values which end in the return from the |
---|
1091 | function. These correspond to a measurable subtrace, and in |
---|
1092 | particular include executions of an entire function call (and so are |
---|
1093 | used for the nested calls in \lstinline'trace_any_label'). |
---|
1094 | |
---|
1095 | \subsection{Construction} |
---|
1096 | |
---|
1097 | The construction of the structured trace replaces syntactic cost |
---|
1098 | labelling properties, which place requirements on where labels appear |
---|
1099 | in the program, with semantic properties that constrain the execution |
---|
1100 | traces of the program. The construction begins by defining versions |
---|
1101 | of the sound and precise labelling properties on states and global |
---|
1102 | environments (for the code that appears in each of them) rather than |
---|
1103 | whole programs, and showing that these are preserved by steps of the |
---|
1104 | \textsf{RTLabs} semantics. |
---|
1105 | |
---|
1106 | Then we show that each cost labelling property required by the |
---|
1107 | definition of structured traces is locally satisfied. These proofs are |
---|
1108 | broken up by the classification of states. Similarly, we prove a |
---|
1109 | step-by-step stack preservation result, which states that the |
---|
1110 | \textsf{RTLabs} semantics never changes the lower parts of the stack. |
---|
1111 | |
---|
1112 | The core part of the construction of a structured trace is to use the |
---|
1113 | proof of termination from the measurable trace (defined on |
---|
1114 | page~\pageref{prog:terminationproof}) to `fold up' the execution into |
---|
1115 | the nested form. The results outlined above fill in the proof |
---|
1116 | obligations for the cost labelling properties and the stack |
---|
1117 | preservation result shows that calls return to the correct location. |
---|
1118 | |
---|
1119 | The structured trace alone is not sufficient to capture the property |
---|
1120 | that the program is soundly labelled. While the structured trace |
---|
1121 | guarantees termination, it still permits a loop to be executed a |
---|
1122 | finite number of times without encountering a cost label. We |
---|
1123 | eliminate this by proving that no `program counter' repeats within any |
---|
1124 | \lstinline'trace_any_label' section by showing that it is incompatible |
---|
1125 | with the property that there is a bound on the number of successor |
---|
1126 | instructions you can follow in the CFG before you encounter a cost |
---|
1127 | label (from Section~\ref{sec:costchecksimpl}). |
---|
1128 | |
---|
1129 | \subsubsection{Complete execution structured traces} |
---|
1130 | |
---|
1131 | The development of the construction above started relatively early, |
---|
1132 | before the measurable subtrace preservation proofs. To be confident |
---|
1133 | that the traces were well-formed at that time, we also developed a |
---|
1134 | complete execution form that embeds the traces above. This includes |
---|
1135 | non-terminating program executions, where an infinite number of the terminating |
---|
1136 | structured traces are embedded. This construction confirmed that our |
---|
1137 | definition of structured traces was consistent, although we later |
---|
1138 | found that we did not require the whole execution version for the |
---|
1139 | compiler correctness results. |
---|
1140 | |
---|
1141 | To construct these we need to know whether each function call will |
---|
1142 | eventually terminate, requiring the use of the excluded middle. This |
---|
1143 | classical reasoning is local to the construction of whole program |
---|
1144 | traces and is not necessary for our main results. |
---|
1145 | |
---|
1146 | \section{Conclusion} |
---|
1147 | |
---|
1148 | In combination with the work on the CerCo back-end and by |
---|
1149 | concentrating on the novel intensional parts of the proof, we have |
---|
1150 | shown that it is possible to construct certifying compilers that |
---|
1151 | correctly report execution time and stack space costs. The layering |
---|
1152 | of intensional correctness proofs on top of normal simulation results |
---|
1153 | provides a useful separation of concerns, and could permit the reuse |
---|
1154 | of existing results. |
---|
1155 | |
---|
1156 | \appendix |
---|
1157 | |
---|
1158 | \section{Files} |
---|
1159 | |
---|
1160 | The following table gives a high-level overview of the \matita{} |
---|
1161 | source files in Deliverable 3.4: |
---|
1162 | |
---|
1163 | \bigskip |
---|
1164 | |
---|
1165 | \begin{tabular}{rp{.7\linewidth}} |
---|
1166 | \lstinline'compiler.ma' & Top-level compiler definitions, in particular |
---|
1167 | \lstinline'front_end', and the whole compiler definition |
---|
1168 | \lstinline'compile'. \\ |
---|
1169 | \lstinline'correctness.ma' & Correctness results: \lstinline'front_end_correct' |
---|
1170 | and \lstinline'correct', respectively. \\ |
---|
1171 | \lstinline'Clight/*' & \textsf{Clight}: proofs for switch |
---|
1172 | removal, cost labelling, cast simplification and conversion to |
---|
1173 | \textsf{Cminor}. \\ |
---|
1174 | \lstinline'Cminor/*' & \textsf{Cminor}: axioms of conversion to |
---|
1175 | \textsf{RTLabs}. \\ |
---|
1176 | \lstinline'RTLabs/*' & \textsf{RTLabs}: definitions and proofs for |
---|
1177 | compile-time cost labelling checks, construction of structured traces. |
---|
1178 | \\ |
---|
1179 | \lstinline'common/Measurable.ma' & Definitions for measurable |
---|
1180 | subtraces. \\ |
---|
1181 | \lstinline'common/FEMeasurable.ma' & Generic measurable subtrace |
---|
1182 | lifting proof. \\ |
---|
1183 | \lstinline'common/*' & Other common definitions relevant to many parts |
---|
1184 | of the compiler and proof. \\ |
---|
1185 | \lstinline'utilities/*' & General purpose definitions used throughout, |
---|
1186 | including extensions to the standard \matita{} library. |
---|
1187 | \end{tabular} |
---|
1188 | |
---|
1189 | \bibliographystyle{plain} |
---|
1190 | \bibliography{report} |
---|
1191 | |
---|
1192 | \end{document} |
---|