1 | \documentclass{llncs} |
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2 | \usepackage{hyperref} |
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3 | \usepackage{graphicx} |
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4 | \usepackage{color} |
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5 | \usepackage{listings} |
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6 | \usepackage{bcprules}%\bcprulessavespace |
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7 | \usepackage{verbatim} |
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8 | \usepackage{alltt} |
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9 | \usepackage{subcaption} |
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10 | \usepackage{listings} |
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11 | \usepackage{amssymb} |
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12 | % \usepackage{amsmath} |
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15 | \providecommand{\eqref}[1]{(\ref{#1})} |
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19 | %\newtheorem{theorem}{Theorem} |
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20 | \newtheorem{condition}{Condition} |
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22 | \lstdefinelanguage{coq} |
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23 | {keywords={Definition,Lemma,Theorem,Remark,Qed,Save,Inductive,Record}, |
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24 | morekeywords={[2]if,then,else,forall,Prop,match,with,end,let}, |
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25 | } |
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27 | \lstdefinelanguage[mine]{C}[ANSI]{C}{ |
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28 | morekeywords={int8_t} |
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29 | } |
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31 | \lstset{language=[mine]C,basicstyle=\footnotesize\tt,columns=flexible,breaklines=false, |
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32 | keywordstyle=\color{red}\bfseries, |
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33 | keywordstyle=[2]\color{blue}, |
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36 | showspaces=false,showstringspaces=false, |
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37 | xleftmargin=1em} |
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38 | |
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39 | \usepackage{tikz} |
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40 | \usetikzlibrary{positioning,calc,patterns,chains,shapes.geometric,scopes} |
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41 | \makeatletter |
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42 | \pgfutil@ifundefined{pgf@arrow@code@implies}{% supply for lack of double arrow special arrow tip if it is not there% |
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53 | \pgfpathmoveto{\pgfpoint{-1.5pt}{2.5pt}}% |
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56 | \pgfusepathqstroke% |
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59 | \makeatother |
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61 | \tikzset{state/.style={inner sep = 0, outer sep = 2pt, draw, fill}, |
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62 | every node/.style={inner sep=2pt}, |
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63 | every on chain/.style = {inner sep = 0, outer sep = 2pt}, |
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64 | join all/.style = {every on chain/.append style={join}}, |
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83 | double equal sign distance/.prefix style={double distance=1.5pt}, %% if already defined (newest version of pgf) it should be ignored%} |
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95 | } |
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96 | |
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97 | \def\L{\mathrel{\mathcal L}} |
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98 | \def\S{\mathrel{\mathcal S}} |
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99 | \def\R{\mathrel{\mathcal R}} |
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100 | \def\C{\mathrel{\mathcal C}} |
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101 | |
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102 | \newsavebox{\execbox} |
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103 | \savebox{\execbox}{\tikz[baseline=-.5ex]\draw [-stealth] (0,0) -- ++(1em, 0);} |
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104 | \newcommand{\exec}{\ensuremath{\mathrel{\usebox{\execbox}}}} |
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105 | \let\ar\rightsquigarrow |
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106 | \renewcommand{\verb}{\lstinline[mathescape]} |
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107 | |
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108 | \let\class\triangleright |
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109 | \let\andalso\quad |
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110 | |
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111 | \newcommand{\append}{\mathbin{@}} |
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112 | \newcommand{\iffdef}{\mathrel{:\Leftrightarrow}} |
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113 | |
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114 | \begin{document} |
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115 | \pagestyle{plain} |
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116 | |
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117 | \title{Certification of the Preservation of Structure by a Compiler's Back-end Pass\thanks{The project CerCo acknowledges the financial support of the Future and |
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118 | Emerging Technologies (FET) programme within the Seventh Framework |
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119 | Programme for Research of the European Commission, under FET-Open grant |
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120 | number: 243881}} |
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121 | \author{Paolo Tranquilli \and Claudio Sacerdoti Coen} |
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122 | \institute{Department of Computer Science and Engineering, University of Bologna,\\\email{Paolo.Tranquilli@unibo.it}, \email{Claudio.SacerdotiCoen@unibo.it}} |
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123 | \maketitle |
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124 | \begin{abstract} |
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125 | The labelling approach is a technique to lift cost models for non-functional |
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126 | properties of programs from the object code to the source code. It is based |
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127 | on the preservation of the structure of the high level program in every |
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128 | intermediate language used by the compiler. Such structure is captured by |
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129 | observables that are added to the semantics and that needs to be preserved |
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130 | by the forward simulation proof of correctness of the compiler. Additional |
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131 | special observables are required for function calls. In this paper we |
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132 | present a generic forward simulation proof that preserves all these observables. |
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133 | The proof statement is based on a new mechanised semantics that traces the |
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134 | structure of execution when the language is unstructured. The generic semantics |
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135 | and simulation proof have been mechanised in the interactive theorem prover |
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136 | Matita. |
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137 | \end{abstract} |
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138 | |
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139 | \section{Introduction} |
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140 | The \emph{labelling approach} has been introduced in~\cite{easylabelling} as |
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141 | a technique to \emph{lift} cost models for non-functional properties of programs |
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142 | from the object code to the source code. Examples of non-functional properties |
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143 | are execution time, amount of stack/heap space consumed and energy required for |
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144 | communication. The basic idea of the approach is that it is impossible to |
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145 | provide a \emph{uniform} cost model for an high level language that is preserved |
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146 | \emph{precisely} by a compiler. For instance, two instances of an assignment |
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147 | $x = y$ in the source code can be compiled very differently according to the |
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148 | place (registers vs stack) where $x$ and $y$ are stored at the moment of |
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149 | execution. Therefore a precise cost model must assign a different cost |
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150 | to every occurrence, and the exact cost can only be known after compilation. |
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151 | |
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152 | According to the labelling approach, the compiler is free to compile and optimise |
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153 | the source code without any major restriction, but it must keep trace |
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154 | of what happens to basic blocks during the compilation. The cost model is |
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155 | then computed on the object code. It assigns a cost to every basic block. |
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156 | Finally, the compiler propagates back the cost model to the source level, |
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157 | assigning a cost to each basic block of the source code. |
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158 | |
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159 | Implementing the labelling approach in a certified compiler |
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160 | allows to reason formally on the high level source code of a program to prove |
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161 | non-functional properties that are granted to be preserved by the compiler |
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162 | itself. The trusted code base is then reduced to 1) the interactive theorem |
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163 | prover (or its kernel) used in the certification of the compiler and |
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164 | 2) the software used to certify the property on the source language, that |
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165 | can be itself certified further reducing the trusted code base. |
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166 | In~\cite{easylabelling} the authors provide an example of a simple |
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167 | certified compiler that implements the labelling approach for the |
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168 | imperative \texttt{While} language~\cite{while}, that does not have |
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169 | pointers and function calls. |
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170 | |
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171 | The labelling approach has been shown to scale to more interesting scenarios. |
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172 | In particular in~\cite{functionallabelling} it has been applied to a functional |
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173 | language and in~\cite{loopoptimizations} it has been shown that the approach |
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174 | can be slightly complicated to handle loop optimisations and, more generally, |
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175 | program optimisations that do not preserve the structure of basic blocks. |
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176 | On-going work also shows that the labelling approach is also compatible with |
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177 | the complex analyses required to obtain a cost model for object code |
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178 | on processors that implement advanced features like pipelining, superscalar |
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179 | architectures and caches. |
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180 | |
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181 | In the European Project CerCo (Certified Complexity~\footnote{\url{http://cerco.cs.unibo.it}})~\cite{cerco} we are certifying a labelling approach based compiler for a large subset of C to |
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182 | 8051 object code. The compiler is |
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183 | moderately optimising and implements a compilation chain that is largely |
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184 | inspired to that of CompCert~\cite{compcert1,compcert2}. Compared to work done in~\cite{easylabelling}, the main novelty and source of difficulties is due to the presence |
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185 | of function calls. Surprisingly, the addition of function calls require a |
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186 | revisitation of the proof technique given in~\cite{easylabelling}. In |
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187 | particular, at the core of the labelling approach there is a forward |
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188 | simulation proof that, in the case of \texttt{While}, is only minimally |
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189 | more complex than the proof required for the preservation of the |
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190 | functional properties only. In the case of a programming language with |
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191 | function calls, instead, it turns out that the forward simulation proof for |
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192 | the back-end languages must grant a whole new set of invariants. |
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193 | |
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194 | In this paper we present a formalisation in the Matita interactive theorem |
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195 | prover~\cite{matita1,matita2} of a generic version of the simulation proof required for unstructured |
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196 | languages. All back-end languages of the CerCo compiler are unstructured |
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197 | languages, so the proof covers half of the correctness of the compiler. |
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198 | The statement of the generic proof is based on a new semantics |
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199 | for imperative unstructured languages that is based on \emph{structured |
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200 | traces} and that restores the preservation of structure in the observables of |
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201 | the semantics. The generic proof allows to almost completely split the |
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202 | part of the simulation that deals with functional properties only from the |
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203 | part that deals with the preservation of structure. |
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204 | |
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205 | The plan of this paper is the following. In Section~\ref{labelling} we |
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206 | sketch the labelling method and the problems derived from the application |
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207 | to languages with function calls. In Section~\ref{semantics} we introduce |
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208 | a generic description of an unstructured imperative language and the |
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209 | corresponding structured traces (the novel semantics). In |
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210 | Section~\ref{simulation} we describe the forward simulation proof. |
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211 | Conclusions and future works are in Section~\ref{conclusions} |
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212 | |
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213 | \section{The labelling approach} |
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214 | \label{labelling} |
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215 | We briefly sketch here a simplified version of the labelling approach as |
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216 | introduced in~\cite{easylabelling}. The simplification strengthens the |
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217 | sufficient conditions given in~\cite{easylabelling} to allow a simpler |
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218 | explanation. The simplified conditions given here are also used in the |
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219 | CerCo compiler to simplify the proof. |
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220 | |
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221 | Let $\mathcal{P}$ be a programming language whose semantics is given in |
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222 | terms of observables: a run of a program yields a finite or infinite |
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223 | stream of observables. We also assume for the time being that function |
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224 | calls are not available in $\mathcal{P}$. We want to associate a cost |
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225 | model to a program $P$ written in $\mathcal{P}$. The first step is to |
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226 | extend the syntax of $\mathcal{P}$ with a new construct $\texttt{emit L}$ |
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227 | where $L$ is a label distinct from all observables of $\mathcal{P}$. |
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228 | The semantics of $\texttt{emit L}$ is the emission of the observable |
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229 | \texttt{L} that is meant to signal the beginning of a basic block. |
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230 | |
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231 | There exists an automatic procedure that injects into the program $P$ an |
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232 | $\texttt{emit L}$ at the beginning of each basic block, using a fresh |
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233 | \texttt{L} for each block. In particular, the bodies of loops, both branches |
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234 | of \texttt{if-then-else}s and the targets of \texttt{goto}s must all start |
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235 | with an emission statement. |
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236 | |
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237 | Let now $C$ be a compiler from $\mathcal{P}$ to the object code $\mathcal{M}$, |
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238 | that is organised in passes. Let $\mathcal{Q}_i$ be the $i$-th intermediate |
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239 | language used by the compiler. We can easily extend every |
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240 | intermediate language (and its semantics) with an $\texttt{emit L}$ statement |
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241 | as we did for $\mathcal{P}$. The same is possible for $\mathcal{M}$ too, with |
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242 | the additional difficulty that the syntax of object code is given as a |
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243 | sequence of bytes. The injection of an emission statement in the object code |
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244 | can be done using a map that maps two consecutive code addresses with the |
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245 | statement. The intended semantics is that, if $(pc_1,pc_2) \mapsto \texttt{emit L}$ then the observable \texttt{L} is emitted after the execution of the |
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246 | instruction stored at $pc_1$ and before the execution of the instruction |
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247 | stored at $pc_2$. The two program counters are necessary because the |
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248 | instruction stored at $pc_1$ can have multiple possible successors (e.g. |
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249 | in case of a conditional branch or an indirect call). Dually, the instruction |
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250 | stored at $pc_2$ can have multiple possible predecessors (e.g. if it is the |
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251 | target of a jump). |
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252 | |
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253 | The compiler, to be functionally correct, must preserve the observational |
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254 | equivalence, i.e. executing the program after each compiler pass should |
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255 | yield the same stream of observables. After the injection of emission |
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256 | statements, observables now capture both functional and non-functional |
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257 | behaviours. |
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258 | This correctness property is called in the literature a forward simulation |
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259 | and is sufficient for correctness when the target language is |
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260 | deterministic~\cite{compcert3}. |
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261 | We also require a stronger, non-functional preservation property: after each |
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262 | pass all basic blocks must start with an emission statement, and all labels |
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263 | \texttt{L} must be unique. |
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264 | |
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265 | Now let $M$ be the object code obtained for the program $P$. Let us suppose |
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266 | that we can statically inspect the code $M$ and associate to each basic block |
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267 | a cost (e.g. the number of clock cycles required to execute all instructions |
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268 | in the basic block, or an upper bound to that time). Every basic block is |
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269 | labelled with an unique label \texttt{L}, thus we can actually associate the |
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270 | cost to \texttt{L}. Let call it $k(\texttt{L})$. |
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271 | |
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272 | The function $k$ is defined as the cost model for the object code control |
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273 | blocks. It can be equally used as well as the cost model for the source |
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274 | control blocks. Indeed, if the semantics of $P$ is the stream |
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275 | $L_1 L_2 \ldots$, then, because of forward simulation, the semantics of $M$ is |
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276 | also $L_1 L_2 \ldots$ and its actual execution cost is $\Sigma_i k(L_i)$ because |
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277 | every instruction belongs to a control block and every control block is |
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278 | labelled. Thus it is correct to say that the execution cost of $P$ is also |
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279 | $\Sigma_i k(L_i)$. In other words, we have obtained a cost model $k$ for |
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280 | the blocks of the high level program $P$ that is preserved by compilation. |
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281 | |
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282 | How can the user profit from the high level cost model? Suppose, for instance, |
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283 | that he wants to prove that the WCET of his program is bounded by $c$. It |
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284 | is sufficient for him to prove that $\Sigma_i k(L_i) \leq c$, which is now |
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285 | a purely functional property of the code. He can therefore use any technique |
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286 | available to certify functional properties of the source code. |
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287 | What is suggested in~\cite{easylabelling} is to actually instrument the |
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288 | source code $P$ by replacing every label emission statement |
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289 | $\texttt{emit L}$ with the instruction $\texttt{cost += k(L)}$ that increments |
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290 | a global fresh variable \texttt{cost}. The bound is now proved by establishing |
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291 | the program invariant $\texttt{cost} \leq c$, which can be done for example |
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292 | using the Frama-C~\cite{framaC} suite if the source code is some variant of |
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293 | C. |
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294 | |
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295 | \subsection{Labelling function calls} |
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296 | We now want to extend the labelling approach to support function calls. |
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297 | On the high level, \emph{structured} programming language $\mathcal{P}$ there |
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298 | is not much to change. |
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299 | When a function is invoked, the current basic block is temporarily exited |
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300 | and the basic block the function starts with take control. When the function |
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301 | returns, the execution of the original basic block is resumed. Thus the only |
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302 | significant change is that basic blocks can now be nested. Let \texttt{E} |
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303 | be the label of the external block and \texttt{I} the label of a nested one. |
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304 | Since the external starts before the internal, the semantics observed will be |
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305 | \texttt{E I} and the cost associated to it on the source language will be |
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306 | $k(\texttt{E}) + k(\texttt{I})$, i.e. the cost of executing all instructions |
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307 | in the block \texttt{E} first plus the cost of executing all the instructions in |
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308 | the block \texttt{I}. However, we know that some instructions in \texttt{E} are |
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309 | executed after the last instruction in \texttt{I}. This is actually irrelevant |
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310 | because we are here assuming that costs are additive, so that we can freely |
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311 | permute them\footnote{The additivity assumption fails on modern processors that have stateful subsystems, like caches and pipelines. The extension of the labelling approach to those systems is therefore non trivial and under development in the CerCo project.}. Note that, in the present discussion, we are assuming that |
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312 | the function call terminates and yields back control to the basic block |
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313 | \texttt{E}. If the call diverges, the instrumentation |
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314 | $\texttt{cost += k(E)}$ executed at the beginning of \texttt{E} is still valid, |
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315 | but just as an upper bound to the real execution cost: only precision is lost. |
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316 | |
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317 | Let now consider what happens when we move down the compilation chain to an |
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318 | unstructured intermediate or final language. Here unstructured means that |
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319 | the only control operators are conditional and unconditional jumps, function |
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320 | calls and returns. Unlike a structured language, though, there is no guarantee |
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321 | that a function will return control just after the function call point. |
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322 | The semantics of the return statement, indeed, consists in fetching the |
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323 | return address from some internal structure (typically the control stack) and |
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324 | jumping directly to it. The code can freely manipulate the control stack to |
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325 | make the procedure returns to whatever position. Indeed, it is also possible |
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326 | to break the well nesting of function calls/returns. |
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327 | |
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328 | Is it the case that the code produced by a correct compiler must respect the |
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329 | additional property that every function returns just after its function call |
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330 | point? The answer is negative and the property is not implied by forward |
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331 | simulation proofs. For instance, imagine to modify a correct compiler pass |
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332 | by systematically adding one to the return address on the stack and by |
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333 | putting a \texttt{NOP} (or any other instruction that takes one byte) after |
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334 | every function call. The obtained code will be functionally indistinguishable, |
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335 | and the added instructions will all be dead code. |
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336 | |
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337 | This lack of structure in the semantics badly interferes with the labelling |
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338 | approach. The reason is the following: when a basic block labelled with |
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339 | \texttt{E} contains a function call, it no longer makes any sense to associate |
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340 | to a label \texttt{E} the sum of the costs of all the instructions in the block. |
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341 | Indeed, there is no guarantee that the function will return into the block and |
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342 | that the instructions that will be executed after the return will be the ones |
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343 | we are paying for in the cost model. |
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344 | |
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345 | How can we make the labelling approach work in this scenario? We only see two |
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346 | possible ways. The first one consists in injecting an emission statement after |
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347 | every function call: basic blocks no longer contain function calls, but are now |
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348 | terminated by them. This completely solves the problem and allows the compiler |
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349 | to break the structure of function calls/returns at will. However, the |
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350 | technique has several drawbacks. First of all, it greatly augments the number |
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351 | of cost labels that are injected in the source code and that become |
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352 | instrumentation statements. Thus, when reasoning on the source code to prove |
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353 | non-functional properties, the user (or the automation tool) will have to handle |
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354 | larger expressions. Second, the more labels are emitted, the more difficult it |
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355 | becomes to implement powerful optimisations respecting the code structure. |
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356 | Indeed, function calls are usually implemented in such a way that most registers |
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357 | are preserved by the call, so that the static analysis of the block is not |
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358 | interrupted by the call and an optimisation can involve both the code before |
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359 | and after the function call. Third, instrumenting the source code may require |
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360 | unpleasant modification of it. Take, for example, the code |
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361 | \texttt{f(g(x));}. We need to inject an emission statement/instrumentation |
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362 | instruction just after the execution of \texttt{g}. The only way to do that |
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363 | is to rewrite the code as \texttt{y = g(x); emit L; f(y);} for some fresh |
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364 | variable \texttt{y}. It is pretty clear how in certain situations the obtained |
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365 | code would be more obfuscated and then more difficult to manually reason on. |
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366 | |
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367 | For the previous reasons, in this paper and in the CerCo project we adopt a |
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368 | different approach. We do not inject emission statements after every |
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369 | function call. However, we want to propagate a strong additional invariant in |
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370 | the forward simulation proof. The invariant is the propagation of the structure |
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371 | of the original high level code, even if the target language is unstructured. |
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372 | The structure we want to propagate, that will become more clear in the next |
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373 | section, comprises 1) the property that every function should return just after |
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374 | the function call point, which in turns imply well nesting of function calls; |
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375 | 2) the property that every basic block starts with a code emission statement. |
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376 | |
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377 | In the original labelling approach of~\cite{easylabelling}, the second property |
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378 | was granted syntactically as a property of the generated code. |
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379 | In our revised approach, instead, we will impose the property on the runs: |
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380 | it will be possible to generate code that does not respect the syntactic |
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381 | property, as soon as all possible runs respect it. For instance, dead code will no longer |
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382 | be required to have all basic blocks correctly labelled. The switch is suggested |
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383 | from the fact that the first of the two properties --- that related to |
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384 | function calls/returns --- can only be defined as property of runs, |
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385 | not of the static code. The switch is |
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386 | beneficial to the proof because the original proof was made of two parts: |
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387 | the forward simulation proof and the proof that the static property was granted. |
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388 | In our revised approach the latter disappears and only the forward simulation |
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389 | is kept. |
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390 | |
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391 | In order to capture the structure semantics so that it is preserved |
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392 | by a forward simulation argument, we need to make the structure observable |
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393 | in the semantics. This is the topic of the next section. |
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394 | |
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395 | \section{Structured traces} |
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396 | \label{semantics} |
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397 | The program semantics adopted in the traditional labelling approach is based |
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398 | on labelled deductive systems. Given a set of observables $\mathcal{O}$ and |
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399 | a set of states $\S$, the semantics of one deterministic execution |
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400 | step is |
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401 | defined as a function $S \to S \times O^*$ where $O^*$ is a (finite) stream of |
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402 | observables. The semantics is then lifted compositionally to multiple (finite |
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403 | or infinite) execution steps. |
---|
404 | Finally, the semantics of a a whole program execution is obtained by forgetting |
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405 | about the final state (if any), yielding a function $S \to O^*$ that given an |
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406 | initial status returns the finite or infinite stream of observables in output. |
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407 | |
---|
408 | We present here a new definition of semantics where the structure of execution, |
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409 | as defined in the previous section, is now observable. The idea is to replace |
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410 | the stream of observables with a structured data type that makes explicit |
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411 | function call and returns and that grants some additional invariants by |
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412 | construction. The data structure, called \emph{structured traces}, is |
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413 | defined inductively for terminating programs and coinductively for diverging |
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414 | ones. In the paper we focus only on the inductive structure, i.e. we assume |
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415 | that all programs that are given a semantics are total. The Matita formalisation |
---|
416 | also shows the coinductive definitions. The semantics of a program is then |
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417 | defined as a function that maps an initial state into a structured trace. |
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418 | |
---|
419 | In order to have a definition that works on multiple intermediate languages, |
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420 | we abstract the type of structure traces over an abstract data type of |
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421 | abstract statuses, which we aptly call $\texttt{abstract\_status}$. The fields |
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422 | of this record are the following. |
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423 | \begin{itemize} |
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424 | \item \verb+S : Type[0]+, the type of states. |
---|
425 | \item \verb+as_execute : S $\to$ S $\to$ Prop+, a binary predicate stating |
---|
426 | an execution step. We write $s_1\exec s_2$ for $\verb+as_execute+~s_1~s_2$. |
---|
427 | \item \verb+as_classifier : S $\to$ classification+, a function tagging all |
---|
428 | states with a class in |
---|
429 | $\{\texttt{cl\_return,cl\_jump,cl\_call,cl\_other}\}$, depending on the instruction |
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430 | that is about to be executed (we omit tail-calls for simplicity). We will |
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431 | use $s \class c$ as a shorthand for both $\texttt{as\_classifier}~s=c$ |
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432 | (if $c$ is a classification) and $\texttt{as\_classifier}~s\in c$ |
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433 | (if $c$ is a set of classifications). |
---|
434 | \item \verb+as_label : S $\to$ option label+, telling whether the |
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435 | next instruction to be executed in $s$ is a cost emission statement, |
---|
436 | and if yes returning the associated cost label. Our shorthand for this function |
---|
437 | will be $\ell$, and we will also abuse the notation by using $\ell~s$ as a |
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438 | predicate stating that $s$ is labelled. |
---|
439 | \item \verb+as_call_ident : ($\Sigma$s:S. s $\class$ cl_call) $\to$ label+, |
---|
440 | telling the identifier of the function which is being called in a |
---|
441 | \verb+cl_call+ state. We will use the shorthand $s\uparrow f$ for |
---|
442 | $\verb+as_call_ident+~s = f$. |
---|
443 | \item \verb+as_after_return : ($\Sigma$s:S. s $\class$ cl_call) $\to$ S $\to$ Prop+, |
---|
444 | which holds on the \verb+cl_call+ state $s_1$ and a state $s_2$ when the |
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445 | instruction to be executed in $s_2$ follows the function call to be |
---|
446 | executed in (the witness of the $\Sigma$-type) $s_1$. We will use the notation |
---|
447 | $s_1\ar s_2$ for this relation. |
---|
448 | \end{itemize} |
---|
449 | |
---|
450 | % \begin{alltt} |
---|
451 | % record abstract_status := \{ S: Type[0]; |
---|
452 | % as_execute: S \(\to\) S \(\to\) Prop; as_classifier: S \(\to\) classification; |
---|
453 | % as_label: S \(\to\) option label; as_called: (\(\Sigma\)s:S. c s = cl_call) \(\to\) label; |
---|
454 | % as_after_return: (\(\Sigma\)s:S. c s = cl_call) \(\to\) S \(\to\) Prop \} |
---|
455 | % \end{alltt} |
---|
456 | |
---|
457 | The inductive type for structured traces is actually made by three multiple |
---|
458 | inductive types with the following semantics: |
---|
459 | \begin{enumerate} |
---|
460 | \item $(\texttt{trace\_label\_return}~s_1~s_2)$ (shorthand $\verb+TLR+~s_1~s_2$) |
---|
461 | is a trace that begins in |
---|
462 | the state $s_1$ (included) and ends just before the state $s_2$ (excluded) |
---|
463 | such that the instruction to be executed in $s_1$ is a label emission |
---|
464 | statement and the one to be executed in the state before $s_2$ is a return |
---|
465 | statement. Thus $s_2$ is the state after the return. The trace |
---|
466 | may contain other label emission statements. It captures the structure of |
---|
467 | the execution of function bodies: they must start with a cost emission |
---|
468 | statement and must end with a return; they are obtained by concatenating |
---|
469 | one or more basic blocks, all starting with a label emission |
---|
470 | (e.g. in case of loops). |
---|
471 | \item $(\texttt{trace\_any\_label}~b~s_1~s_2)$ (shorthand $\verb+TAL+~b~s_1~s_2$) |
---|
472 | is a trace that begins in |
---|
473 | the state $s_1$ (included) and ends just before the state $s_2$ (excluded) |
---|
474 | such that the instruction to be executed in $s_2$/in the state before |
---|
475 | $s_2$ is either a label emission statement or |
---|
476 | or a return, according to the boolean $b$. It must not contain |
---|
477 | any label emission statement. It captures the notion of a suffix of a |
---|
478 | basic block. |
---|
479 | \item $(\texttt{trace\_label\_label}~b~s_1~s_2)$ (shorthand $\verb+TLL+~b~s_1~s_2$ is the special case of |
---|
480 | $\verb+TAL+~b~s_1~s_2)$ such that the instruction to be |
---|
481 | executed in $s_1$ is a label emission statement. It captures the notion of |
---|
482 | a basic block. |
---|
483 | \end{enumerate} |
---|
484 | |
---|
485 | \begin{multicols}{3} |
---|
486 | \infrule[\verb+tlr_base+] |
---|
487 | {\texttt{TLL}~true~s_1~s_2} |
---|
488 | {\texttt{TLR}~s_1~s_2} |
---|
489 | |
---|
490 | \infrule[\verb+tlr_step+] |
---|
491 | {\texttt{TLL}~false~s_1~s_2 \andalso |
---|
492 | \texttt{TLR}~s_2~s_3 |
---|
493 | } |
---|
494 | {\texttt{TLR}~s_1~s_3} |
---|
495 | |
---|
496 | \infrule[\verb+tll_base+] |
---|
497 | {\texttt{TAL}~b~s_1~s_2 \andalso |
---|
498 | \ell~s_1 |
---|
499 | } |
---|
500 | {\texttt{TLL}~b~s_1~s_2} |
---|
501 | \end{multicols} |
---|
502 | |
---|
503 | \infrule[\verb+tal_base_not_return+] |
---|
504 | {s_1\exec s_2 \andalso |
---|
505 | s_1\class\{\verb+cl_jump+, \verb+cl_other+\}\andalso |
---|
506 | \ell~s_2 |
---|
507 | } |
---|
508 | {\texttt{TAL}~false~s_1~s_2} |
---|
509 | |
---|
510 | \infrule[\verb+tal_base_return+] |
---|
511 | {s_1\exec s_2 \andalso |
---|
512 | s_1 \class \texttt{cl\_return} |
---|
513 | } |
---|
514 | {\texttt{TAL}~true~s_1~s_2} |
---|
515 | |
---|
516 | \infrule[\verb+tal_base_call+] |
---|
517 | {s_1\exec s_2 \andalso |
---|
518 | s_1 \class \texttt{cl\_call} \andalso |
---|
519 | s_1\ar s_3 \andalso |
---|
520 | \texttt{TLR}~s_2~s_3 \andalso |
---|
521 | \ell~s_3 |
---|
522 | } |
---|
523 | {\texttt{TAL}~false~s_1~s_3} |
---|
524 | |
---|
525 | \infrule[\verb+tal_step_call+] |
---|
526 | {s_1\exec s_2 \andalso |
---|
527 | s_1 \class \texttt{cl\_call} \andalso |
---|
528 | s_1\ar s_3 \andalso |
---|
529 | \texttt{TLR}~s_2~s_3 \andalso |
---|
530 | \texttt{TAL}~b~s_3~s_4 |
---|
531 | } |
---|
532 | {\texttt{TAL}~b~s_1~s_4} |
---|
533 | |
---|
534 | \infrule[\verb+tal_step_default+] |
---|
535 | {s_1\exec s_2 \andalso |
---|
536 | \lnot \ell~s_2 \andalso |
---|
537 | \texttt{TAL}~b~s_2~s_3\andalso |
---|
538 | s_1 \class \texttt{cl\_other} |
---|
539 | } |
---|
540 | {\texttt{TAL}~b~s_1~s_3} |
---|
541 | \begin{comment} |
---|
542 | \begin{verbatim} |
---|
543 | inductive trace_label_return (S:abstract_status) : S → S → Type[0] ≝ |
---|
544 | | tlr_base: |
---|
545 | ∀status_before: S. |
---|
546 | ∀status_after: S. |
---|
547 | trace_label_label S ends_with_ret status_before status_after → |
---|
548 | trace_label_return S status_before status_after |
---|
549 | | tlr_step: |
---|
550 | ∀status_initial: S. |
---|
551 | ∀status_labelled: S. |
---|
552 | ∀status_final: S. |
---|
553 | trace_label_label S doesnt_end_with_ret status_initial status_labelled → |
---|
554 | trace_label_return S status_labelled status_final → |
---|
555 | trace_label_return S status_initial status_final |
---|
556 | with trace_label_label: trace_ends_with_ret → S → S → Type[0] ≝ |
---|
557 | | tll_base: |
---|
558 | ∀ends_flag: trace_ends_with_ret. |
---|
559 | ∀start_status: S. |
---|
560 | ∀end_status: S. |
---|
561 | trace_any_label S ends_flag start_status end_status → |
---|
562 | as_costed S start_status → |
---|
563 | trace_label_label S ends_flag start_status end_status |
---|
564 | with trace_any_label: trace_ends_with_ret → S → S → Type[0] ≝ |
---|
565 | (* Single steps within a function which reach a label. |
---|
566 | Note that this is the only case applicable for a jump. *) |
---|
567 | | tal_base_not_return: |
---|
568 | ∀start_status: S. |
---|
569 | ∀final_status: S. |
---|
570 | as_execute S start_status final_status → |
---|
571 | (as_classifier S start_status cl_jump ∨ |
---|
572 | as_classifier S start_status cl_other) → |
---|
573 | as_costed S final_status → |
---|
574 | trace_any_label S doesnt_end_with_ret start_status final_status |
---|
575 | | tal_base_return: |
---|
576 | ∀start_status: S. |
---|
577 | ∀final_status: S. |
---|
578 | as_execute S start_status final_status → |
---|
579 | as_classifier S start_status cl_return → |
---|
580 | trace_any_label S ends_with_ret start_status final_status |
---|
581 | (* A call followed by a label on return. *) |
---|
582 | | tal_base_call: |
---|
583 | ∀status_pre_fun_call: S. |
---|
584 | ∀status_start_fun_call: S. |
---|
585 | ∀status_final: S. |
---|
586 | as_execute S status_pre_fun_call status_start_fun_call → |
---|
587 | ∀H:as_classifier S status_pre_fun_call cl_call. |
---|
588 | as_after_return S «status_pre_fun_call, H» status_final → |
---|
589 | trace_label_return S status_start_fun_call status_final → |
---|
590 | as_costed S status_final → |
---|
591 | trace_any_label S doesnt_end_with_ret status_pre_fun_call status_final |
---|
592 | (* A call followed by a non-empty trace. *) |
---|
593 | | tal_step_call: |
---|
594 | ∀end_flag: trace_ends_with_ret. |
---|
595 | ∀status_pre_fun_call: S. |
---|
596 | ∀status_start_fun_call: S. |
---|
597 | ∀status_after_fun_call: S. |
---|
598 | ∀status_final: S. |
---|
599 | as_execute S status_pre_fun_call status_start_fun_call → |
---|
600 | ∀H:as_classifier S status_pre_fun_call cl_call. |
---|
601 | as_after_return S «status_pre_fun_call, H» status_after_fun_call → |
---|
602 | trace_label_return S status_start_fun_call status_after_fun_call → |
---|
603 | ¬ as_costed S status_after_fun_call → |
---|
604 | trace_any_label S end_flag status_after_fun_call status_final → |
---|
605 | trace_any_label S end_flag status_pre_fun_call status_final |
---|
606 | | tal_step_default: |
---|
607 | ∀end_flag: trace_ends_with_ret. |
---|
608 | ∀status_pre: S. |
---|
609 | ∀status_init: S. |
---|
610 | ∀status_end: S. |
---|
611 | as_execute S status_pre status_init → |
---|
612 | trace_any_label S end_flag status_init status_end → |
---|
613 | as_classifier S status_pre cl_other → |
---|
614 | ¬ (as_costed S status_init) → |
---|
615 | trace_any_label S end_flag status_pre status_end. |
---|
616 | \end{verbatim} |
---|
617 | \end{comment} |
---|
618 | A \texttt{trace\_label\_return} is isomorphic to a list of |
---|
619 | \texttt{trace\_label\_label}s that ends with a cost emission followed by a |
---|
620 | return terminated \texttt{trace\_label\_label}. |
---|
621 | The interesting cases are those of $\texttt{trace\_any\_label}~b~s_1~s_2$. |
---|
622 | A \texttt{trace\_any\_label} is a sequence of steps built by a syntax directed |
---|
623 | definition on the classification of $s_1$. The constructors of the datatype |
---|
624 | impose several invariants that are meant to impose a structure to the |
---|
625 | otherwise unstructured execution. In particular, the following invariants are |
---|
626 | imposed: |
---|
627 | \begin{enumerate} |
---|
628 | \item the trace is never empty; it ends with a return iff $b$ is |
---|
629 | true |
---|
630 | \item a jump must always be the last instruction of the trace, and it must |
---|
631 | be followed by a cost emission statement; i.e. the target of a jump |
---|
632 | is always the beginning of a new basic block; as such it must start |
---|
633 | with a cost emission statement |
---|
634 | \item a cost emission statement can never occur inside the trace, only in |
---|
635 | the status immediately after |
---|
636 | \item the trace for a function call step is made of a subtrace for the |
---|
637 | function body of type |
---|
638 | $\texttt{trace\_label\_return}~s_1~s_2$, possibly followed by the |
---|
639 | rest of the trace for this basic block. The subtrace represents the |
---|
640 | function execution. Being an inductive datum, it grants totality of |
---|
641 | the function call. The status $s_2$ is the one that follows the return |
---|
642 | statement. The next instruction of $s_2$ must follow the function call |
---|
643 | instruction. As a consequence, function calls are also well nested. |
---|
644 | \end{enumerate} |
---|
645 | |
---|
646 | There are three mutual structural recursive functions, one for each of |
---|
647 | \verb+TLR+, \verb+TLL+ and \verb+TAL+, for which we use the same notation |
---|
648 | $|\,.\,|$: the \emph{flattening} of the traces. These functions |
---|
649 | allow to extract from a structured trace the list of emitted cost labels. |
---|
650 | % We only show here the type of one |
---|
651 | % of them: |
---|
652 | % \begin{alltt} |
---|
653 | % flatten_trace_label_return: |
---|
654 | % \(\forall\)S: abstract_status. \(\forall\)\(s_1,s_2\). |
---|
655 | % trace_label_return \(s_1\) \(s_2\) \(\to\) list (as_cost_label S) |
---|
656 | % \end{alltt} |
---|
657 | |
---|
658 | \paragraph{Cost prediction on structured traces.} |
---|
659 | |
---|
660 | The first main theorem of CerCo about traces |
---|
661 | (theorem \texttt{compute\_max\_trace\_label\_return\_cost\_ok\_with\_trace}) |
---|
662 | holds for the |
---|
663 | instantiation |
---|
664 | of the structured traces to the concrete status of object code programs. |
---|
665 | Simplifying a bit, it states that |
---|
666 | \begin{equation}\label{th1} |
---|
667 | \begin{array}{l}\forall s_1,s_2. \forall \tau: \texttt{TLR}~s_1~s_2.~ |
---|
668 | \texttt{clock}~s_2 = \texttt{clock}~s_1 + |
---|
669 | \Sigma_{\alpha \in |\tau|}\;k(\alpha) |
---|
670 | \end{array} |
---|
671 | \end{equation} |
---|
672 | where the cost model $k$ is statically computed from the object code |
---|
673 | by associating to each label $\alpha$ the sum of the cost of the instructions |
---|
674 | in the basic block that starts at $\alpha$ and ends before the next labelled |
---|
675 | instruction. The theorem is proved by structural induction over the structured |
---|
676 | trace, and is based on the invariant that |
---|
677 | iff the function that computes the cost model has analysed the instruction |
---|
678 | to be executed at $s_2$ after the one to be executed at $s_1$, and if |
---|
679 | the structured trace starts with $s_1$, then eventually it will contain also |
---|
680 | $s_2$. When $s_1$ is not a function call, the result holds trivially because |
---|
681 | of the $s_1\exec s_2$ condition obtained by inversion on |
---|
682 | the trace. The only non |
---|
683 | trivial case is the one of function calls: the cost model computation function |
---|
684 | does recursion on the first instruction that follows that function call; the |
---|
685 | \texttt{as\_after\_return} condition of the \texttt{tal\_base\_call} and |
---|
686 | \texttt{tal\_step\_call} grants exactly that the execution will eventually reach |
---|
687 | this state. |
---|
688 | |
---|
689 | \paragraph{Structured traces similarity and cost prediction invariance.} |
---|
690 | |
---|
691 | A compiler pass maps source to object code and initial states to initial |
---|
692 | states. The source code and initial state uniquely determine the structured |
---|
693 | trace of a program, if it exists. The structured trace fails to exists iff |
---|
694 | the structural conditions are violated by the program execution (e.g. a function |
---|
695 | body does not start with a cost emission statement). Let us assume that the |
---|
696 | target structured trace exists. |
---|
697 | |
---|
698 | What is the relation between the source and target structured traces? |
---|
699 | In general, the two traces can be arbitrarily different. However, we are |
---|
700 | interested only in those compiler passes that maps a trace $\tau_1$ to a trace |
---|
701 | $\tau_2$ such that |
---|
702 | \begin{equation}|\tau_1| = |\tau_2|.\label{th2}\end{equation} |
---|
703 | The reason is that the combination of~\eqref{th1} with~\eqref{th2} yields the |
---|
704 | corollary |
---|
705 | \begin{equation}\label{th3} |
---|
706 | \forall s_1,s_2. \forall \tau: \texttt{TLR}~s_1~s_2.~ |
---|
707 | \texttt{clock}~s_2 - \texttt{clock}~s_1 = |
---|
708 | \Sigma_{\alpha \in |\tau_1|}\;k(\alpha) = |
---|
709 | \Sigma_{\alpha \in |\tau_2|}\;k(\alpha). |
---|
710 | \end{equation} |
---|
711 | This corollary states that the actual execution time of the program can be computed equally well on the source or target language. Thus it becomes possible to |
---|
712 | transfer the cost model from the target to the source code and reason on the |
---|
713 | source code only. |
---|
714 | |
---|
715 | We are therefore interested in conditions stronger than~\eqref{th2}. |
---|
716 | Therefore we introduce here a similarity relation between traces with |
---|
717 | the same structure. Theorem~\texttt{tlr\_rel\_to\_traces\_same\_flatten} |
---|
718 | in the Matita formalisation shows that~\eqref{th2} holds for every pair |
---|
719 | $(\tau_1,\tau_2)$ of similar traces. |
---|
720 | |
---|
721 | Intuitively, two traces are similar when one can be obtained from |
---|
722 | the other by erasing or inserting silent steps, i.e. states that are |
---|
723 | not \texttt{as\_costed} and that are classified as \texttt{cl\_other}. |
---|
724 | Silent steps do not alter the structure of the traces. |
---|
725 | In particular, |
---|
726 | the relation maps function calls to function calls to the same function, |
---|
727 | label emission statements to emissions of the same label, concatenation of |
---|
728 | subtraces to concatenation of subtraces of the same length and starting with |
---|
729 | the same emission statement, etc. |
---|
730 | |
---|
731 | In the formalisation the three similarity relations --- one for each trace |
---|
732 | kind --- are defined by structural recursion on the first trace and pattern |
---|
733 | matching over the second. Here we turn |
---|
734 | the definition into the inference rules shown in \autoref{fig:txx_rel} |
---|
735 | for the sake of readability. We also omit from trace constructors all arguments, |
---|
736 | but those that are traces or that |
---|
737 | are used in the premises of the rules. By abuse of notation we denote all three |
---|
738 | relations by infixing $\approx$. |
---|
739 | |
---|
740 | \begin{figure} |
---|
741 | \begin{multicols}{2} |
---|
742 | \infrule |
---|
743 | {tll_1\approx tll_2 |
---|
744 | } |
---|
745 | {\texttt{tlr\_base}~tll_1 \approx \texttt{tlr\_base}~tll_2} |
---|
746 | |
---|
747 | \infrule |
---|
748 | {tll_1 \approx tll_2 \andalso |
---|
749 | tlr_1 \approx tlr_2 |
---|
750 | } |
---|
751 | {\texttt{tlr\_step}~tll_1~tlr_1 \approx \texttt{tlr\_step}~tll_2~tlr_2} |
---|
752 | \end{multicols} |
---|
753 | \vspace{3ex} |
---|
754 | \begin{multicols}{2} |
---|
755 | \infrule |
---|
756 | {\ell~s_1 = \ell~s_2 \andalso |
---|
757 | tal_1\approx tal_2 |
---|
758 | } |
---|
759 | {\texttt{tll\_base}~s_1~tal_1 \approx \texttt{tll\_base}~s_2~tal_2} |
---|
760 | |
---|
761 | \infrule |
---|
762 | {tal_1\approx tal_2 |
---|
763 | } |
---|
764 | {\texttt{tal\_step\_default}~tal_1 \approx tal_2} |
---|
765 | \end{multicols} |
---|
766 | \vspace{3ex} |
---|
767 | \infrule |
---|
768 | {} |
---|
769 | {\texttt{tal\_base\_not\_return}\approx taa \append \texttt{tal\_base\_not\_return}} |
---|
770 | \vspace{1ex} |
---|
771 | \infrule |
---|
772 | {} |
---|
773 | {\texttt{tal\_base\_return}\approx taa \append \texttt{tal\_base\_return}} |
---|
774 | \vspace{1ex} |
---|
775 | \infrule |
---|
776 | {tlr_1\approx tlr_2 \andalso |
---|
777 | s_1 \uparrow f \andalso s_2\uparrow f |
---|
778 | } |
---|
779 | {\texttt{tal\_base\_call}~s_1~tlr_1\approx taa \append \texttt{tal\_base\_call}~s_2~tlr_2} |
---|
780 | \vspace{1ex} |
---|
781 | \infrule |
---|
782 | {tlr_1\approx tlr_2 \andalso |
---|
783 | s_1 \uparrow f \andalso s_2\uparrow f \andalso |
---|
784 | \texttt{tal\_collapsable}~tal_2 |
---|
785 | } |
---|
786 | {\texttt{tal\_base\_call}~s_1~tlr_1 \approx taa \append \texttt{tal\_step\_call}~s_2~tlr_2~tal_2)} |
---|
787 | \vspace{1ex} |
---|
788 | \infrule |
---|
789 | {tlr_1\approx tlr_2 \andalso |
---|
790 | s_1 \uparrow f \andalso s_2\uparrow f \andalso |
---|
791 | \texttt{tal\_collapsable}~tal_1 |
---|
792 | } |
---|
793 | {\texttt{tal\_step\_call}~s_1~tlr_1~tal_1 \approx taa \append \texttt{tal\_base\_call}~s_2~tlr_2)} |
---|
794 | \vspace{1ex} |
---|
795 | \infrule |
---|
796 | {tlr_1 \approx tlr_2 \andalso |
---|
797 | s_1 \uparrow f \andalso s_2\uparrow f\andalso |
---|
798 | tal_1 \approx tal_2 \andalso |
---|
799 | } |
---|
800 | {\texttt{tal\_step\_call}~s_1~tlr_1~tal_1 \approx taa \append \texttt{tal\_step\_call}~s_2~tlr_2~tal_2} |
---|
801 | \caption{The inference rule for the relation $\approx$.} |
---|
802 | \label{fig:txx_rel} |
---|
803 | \end{figure} |
---|
804 | % |
---|
805 | \begin{comment} |
---|
806 | \begin{verbatim} |
---|
807 | let rec tlr_rel S1 st1 st1' S2 st2 st2' |
---|
808 | (tlr1 : trace_label_return S1 st1 st1') |
---|
809 | (tlr2 : trace_label_return S2 st2 st2') on tlr1 : Prop ≝ |
---|
810 | match tlr1 with |
---|
811 | [ tlr_base st1 st1' tll1 ⇒ |
---|
812 | match tlr2 with |
---|
813 | [ tlr_base st2 st2' tll2 ⇒ tll_rel … tll1 tll2 |
---|
814 | | _ ⇒ False |
---|
815 | ] |
---|
816 | | tlr_step st1 st1' st1'' tll1 tl1 ⇒ |
---|
817 | match tlr2 with |
---|
818 | [ tlr_step st2 st2' st2'' tll2 tl2 ⇒ |
---|
819 | tll_rel … tll1 tll2 ∧ tlr_rel … tl1 tl2 |
---|
820 | | _ ⇒ False |
---|
821 | ] |
---|
822 | ] |
---|
823 | and tll_rel S1 fl1 st1 st1' S2 fl2 st2 st2' |
---|
824 | (tll1 : trace_label_label S1 fl1 st1 st1') |
---|
825 | (tll2 : trace_label_label S2 fl2 st2 st2') on tll1 : Prop ≝ |
---|
826 | match tll1 with |
---|
827 | [ tll_base fl1 st1 st1' tal1 H ⇒ |
---|
828 | match tll2 with |
---|
829 | [ tll_base fl2 st2 st2 tal2 G ⇒ |
---|
830 | as_label_safe … («?, H») = as_label_safe … («?, G») ∧ |
---|
831 | tal_rel … tal1 tal2 |
---|
832 | ] |
---|
833 | ] |
---|
834 | and tal_rel S1 fl1 st1 st1' S2 fl2 st2 st2' |
---|
835 | (tal1 : trace_any_label S1 fl1 st1 st1') |
---|
836 | (tal2 : trace_any_label S2 fl2 st2 st2') |
---|
837 | on tal1 : Prop ≝ |
---|
838 | match tal1 with |
---|
839 | [ tal_base_not_return st1 st1' _ _ _ ⇒ |
---|
840 | fl2 = doesnt_end_with_ret ∧ |
---|
841 | ∃st2mid,taa,H,G,K. |
---|
842 | tal2 ≃ taa_append_tal ? st2 ??? taa |
---|
843 | (tal_base_not_return ? st2mid st2' H G K) |
---|
844 | | tal_base_return st1 st1' _ _ ⇒ |
---|
845 | fl2 = ends_with_ret ∧ |
---|
846 | ∃st2mid,taa,H,G. |
---|
847 | tal2 ≃ taa_append_tal ? st2 ? st2mid st2' taa |
---|
848 | (tal_base_return ? st2mid st2' H G) |
---|
849 | | tal_base_call st1 st1' st1'' _ prf _ tlr1 _ ⇒ |
---|
850 | fl2 = doesnt_end_with_ret ∧ |
---|
851 | ∃st2mid,G.as_call_ident S2 («st2mid, G») = as_call_ident ? «st1, prf» ∧ |
---|
852 | ∃taa : trace_any_any ? st2 st2mid.∃st2mid',H. |
---|
853 | (* we must allow a tal_base_call to be similar to a call followed |
---|
854 | by a collapsable trace (trace_any_any followed by a base_not_return; |
---|
855 | we cannot use trace_any_any as it disallows labels in the end as soon |
---|
856 | as it is non-empty) *) |
---|
857 | (∃K.∃tlr2 : trace_label_return ? st2mid' st2'.∃L. |
---|
858 | tal2 ≃ taa @ (tal_base_call … H G K tlr2 L) ∧ tlr_rel … tlr1 tlr2) ∨ |
---|
859 | ∃st2mid'',K.∃tlr2 : trace_label_return ? st2mid' st2mid''.∃L. |
---|
860 | ∃tl2 : trace_any_label … doesnt_end_with_ret st2mid'' st2'. |
---|
861 | tal2 ≃ taa @ (tal_step_call … H G K tlr2 L tl2) ∧ |
---|
862 | tlr_rel … tlr1 tlr2 ∧ tal_collapsable … tl2 |
---|
863 | | tal_step_call fl1 st1 st1' st1'' st1''' _ prf _ tlr1 _ tl1 ⇒ |
---|
864 | ∃st2mid,G.as_call_ident S2 («st2mid, G») = as_call_ident ? «st1, prf» ∧ |
---|
865 | ∃taa : trace_any_any ? st2 st2mid.∃st2mid',H. |
---|
866 | (fl2 = doesnt_end_with_ret ∧ ∃K.∃tlr2 : trace_label_return ? st2mid' st2'.∃L. |
---|
867 | tal2 ≃ taa @ tal_base_call … H G K tlr2 L ∧ |
---|
868 | tal_collapsable … tl1 ∧ tlr_rel … tlr1 tlr2) ∨ |
---|
869 | ∃st2mid'',K.∃tlr2 : trace_label_return ? st2mid' st2mid''.∃L. |
---|
870 | ∃tl2 : trace_any_label ? fl2 st2mid'' st2'. |
---|
871 | tal2 ≃ taa @ (tal_step_call … H G K tlr2 L tl2) ∧ |
---|
872 | tal_rel … tl1 tl2 ∧ tlr_rel … tlr1 tlr2 |
---|
873 | | tal_step_default fl1 st1 st1' st1'' _ tl1 _ _ ⇒ |
---|
874 | tal_rel … tl1 tal2 (* <- this makes it many to many *) |
---|
875 | ]. |
---|
876 | \end{verbatim} |
---|
877 | \end{comment} |
---|
878 | % |
---|
879 | In the preceding rules, a $taa$ is an inhabitant of the |
---|
880 | $\texttt{trace\_any\_any}~s_1~s_2$ (shorthand $\texttt{TAA}~s_1~s_2$), |
---|
881 | an inductive data type whose definition |
---|
882 | is not in the paper for lack of space. It is the type of valid |
---|
883 | prefixes (even empty ones) of \texttt{TAL}'s that do not contain |
---|
884 | any function call. Therefore it |
---|
885 | is possible to concatenate (using ``$\append$'') a \texttt{TAA} to the |
---|
886 | left of a \texttt{TAL}. A \texttt{TAA} captures |
---|
887 | a sequence of silent moves. |
---|
888 | The \texttt{tal\_collapsable} unary predicate over \texttt{TAL}'s |
---|
889 | holds when the argument does not contain any function call and it ends |
---|
890 | with a label (not a return). The intuition is that after a function call we |
---|
891 | can still perform a sequence of silent actions while remaining similar. |
---|
892 | |
---|
893 | As should be expected, even though the rules are asymmetric $\approx$ is in fact |
---|
894 | an equivalence relation. |
---|
895 | \section{Forward simulation} |
---|
896 | \label{simulation} |
---|
897 | |
---|
898 | We summarise here the results of the previous sections. Each intermediate |
---|
899 | unstructured language can be given a semantics based on structured traces, |
---|
900 | that single out those runs that respect a certain number of invariants. |
---|
901 | A cost model can be computed on the object code and it can be used to predict |
---|
902 | the execution costs of runs that produce structured traces. The cost model |
---|
903 | can be lifted from the target to the source code of a pass if the pass maps |
---|
904 | structured traces to similar structured traces. The latter property is called |
---|
905 | a \emph{forward simulation}. |
---|
906 | |
---|
907 | As for labelled transition systems, in order to establish the forward |
---|
908 | simulation we are interested in (preservation of observables), we are |
---|
909 | forced to prove a stronger notion of forward simulation that introduces |
---|
910 | an explicit relation between states. The classical notion of a 1-to-many |
---|
911 | forward simulation is the existence of a relation $\S$ over states such that |
---|
912 | if $s_1 \S s_2$ and $s_1 \to^1 s_1'$ then there exists an $s_2'$ such that |
---|
913 | $s_2 \to^* s_2'$ and $s_1' \S s_2'$. In our context, we need to replace the |
---|
914 | one and multi step transition relations $\to^n$ with the existence of |
---|
915 | a structured trace between the two states, and we need to add the request that |
---|
916 | the two structured traces are similar. Thus what we would like to state is |
---|
917 | something like:\\ |
---|
918 | for all $s_1,s_2,s_1'$ such that there is a $\tau_1$ from |
---|
919 | $s_1$ to $s_1'$ and $s_1 \S s_2$ there exists an $s_2'$ such that |
---|
920 | $s_1' \S s_2'$ and a $\tau_2$ from $s_2$ to $s_2'$ such that |
---|
921 | $\tau_1$ is similar to $\tau_2$. We call this particular form of forward |
---|
922 | simulation \emph{trace reconstruction}. |
---|
923 | |
---|
924 | The statement just introduced, however, is too simplistic and not provable |
---|
925 | in the general case. To understand why, consider the case of a function call |
---|
926 | and the pass that fixes the parameter passing conventions. A function |
---|
927 | call in the source code takes in input an arbitrary number of pseudo-registers (the actual parameters to pass) and returns an arbitrary number of pseudo-registers (where the result is stored). A function call in the target language has no |
---|
928 | input nor output parameters. The pass must add explicit code before and after |
---|
929 | the function call to move the pseudo-registers content from/to the hardware |
---|
930 | registers or the stack in order to implement the parameter passing strategy. |
---|
931 | Similarly, each function body must be augmented with a preamble and a postamble |
---|
932 | to complete/initiate the parameter passing strategy for the call/return phase. |
---|
933 | Therefore what used to be a call followed by the next instruction to execute |
---|
934 | after the function return, now becomes a sequence of instructions, followed by |
---|
935 | a call, followed by another sequence. The two states at the beginning of the |
---|
936 | first sequence and at the end of the second sequence are in relation with |
---|
937 | the status before/after the call in the source code, like in an usual forward |
---|
938 | simulation. How can we prove however the additional condition for function calls |
---|
939 | that asks that when the function returns the instruction immediately after the |
---|
940 | function call is called? To grant this invariant, there must be another relation |
---|
941 | between the address of the function call in the source and in the target code. |
---|
942 | This additional relation is to be used in particular to relate the two stacks. |
---|
943 | |
---|
944 | Another example is given by preservation of code emission statements. A single |
---|
945 | code emission instruction can be simulated by a sequence of steps, followed |
---|
946 | by a code emission, followed by another sequence. Clearly the initial and final |
---|
947 | statuses of the sequence are to be in relation with the status before/after the |
---|
948 | code emission in the source code. In order to preserve the structured traces |
---|
949 | invariants, however, we must consider a second relation between states that |
---|
950 | traces the preservation of the code emission statement. |
---|
951 | |
---|
952 | Therefore we now introduce an abstract notion of relation set between abstract |
---|
953 | statuses and an abstract notion of 1-to-many forward simulation conditions. |
---|
954 | These two definitions enjoy the following remarkable properties: |
---|
955 | \begin{enumerate} |
---|
956 | \item they are generic enough to accommodate all passes of the CerCo compiler |
---|
957 | \item the conjunction of the 1-to-many forward simulation conditions are |
---|
958 | just slightly stricter than the statement of a 1-to-many forward |
---|
959 | simulation in the classical case. In particular, they only require |
---|
960 | the construction of very simple forms of structured traces made of |
---|
961 | silent states only. |
---|
962 | \item they allow to prove our main result of the paper: the 1-to-many |
---|
963 | forward simulation conditions are sufficient to prove the trace |
---|
964 | reconstruction theorem |
---|
965 | \end{enumerate} |
---|
966 | |
---|
967 | Point 3. is the important one. First of all it means that we have reduced |
---|
968 | the complex problem of trace reconstruction to a much simpler one that, |
---|
969 | moreover, can be solved with slight adaptations of the forward simulation proof |
---|
970 | that is performed for a compiler that only cares about functional properties. |
---|
971 | Therefore we have successfully splitted as much as possible the proof of |
---|
972 | preservation of functional properties from that of non-functional ones. |
---|
973 | Secondly, combined with the results in the previous section, it implies |
---|
974 | that the cost model can be computed on the object code and lifted to the |
---|
975 | source code to reason on non-functional properties, assuming that |
---|
976 | the 1-to-many forward simulation conditions are fulfilled for every |
---|
977 | compiler pass. |
---|
978 | |
---|
979 | \paragraph{Relation sets.} |
---|
980 | |
---|
981 | We introduce now the four relations $\mathcal{S,C,L,R}$ between abstract |
---|
982 | statuses that are used to correlate the corresponding statues before and |
---|
983 | after a compiler pass. The first two are abstract and must be instantiated |
---|
984 | by every pass. The remaining two are derived relations. |
---|
985 | |
---|
986 | The $\S$ relation between states is the classical relation used |
---|
987 | in forward simulation proofs. It correlates the data of the status |
---|
988 | (e.g. registers, memory, etc.). |
---|
989 | |
---|
990 | The $\C$ relation correlates call states. It allows to track the |
---|
991 | position in the target code of every call in the source code. |
---|
992 | |
---|
993 | The $\L$ relation simply says that the two states are both label |
---|
994 | emitting states that emit the same label, \emph{i.e.}\ $s_1\L s_2\iffdef \ell~s_1=\ell~s_2$. |
---|
995 | It allows to track the position in |
---|
996 | the target code of every cost emitting statement in the source code. |
---|
997 | |
---|
998 | Finally the $\R$ relation is the more complex one. Two states |
---|
999 | $s_1$ and $s_2$ are $\R$ correlated if every time $s_1$ is the |
---|
1000 | successors of a call state that is $\C$-related to a call state |
---|
1001 | $s_2'$ in the target code, then $s_2$ is the successor of $s_2'$. Formally: |
---|
1002 | $$s_1\R s_2 \iffdef \forall s_1',s_2'.s_1'\C s_2' \to s_1'\ar s_1 \to s_2' \ar s_2.$$ |
---|
1003 | We will require all pairs of states that follow a related call to be |
---|
1004 | $\R$-related. This is the fundamental requirement granting |
---|
1005 | that the target trace is well structured, \emph{i.e.}\ that calls are well |
---|
1006 | nested and returning where they are supposed to. |
---|
1007 | |
---|
1008 | % \begin{alltt} |
---|
1009 | % record status_rel (S1,S2 : abstract_status) : Type[1] := \{ |
---|
1010 | % \(\S\): S1 \(\to\) S2 \(\to\) Prop; |
---|
1011 | % \(\C\): (\(\Sigma\)s.as_classifier S1 s cl_call) \(\to\) |
---|
1012 | % (\(\Sigma\)s.as_classifier S2 s cl_call) \(\to\) Prop \}. |
---|
1013 | % |
---|
1014 | % definition \(\L\) S1 S2 st1 st2 := as_label S1 st1 = as_label S2 st2. |
---|
1015 | % |
---|
1016 | % definition \(\R\) S1 S2 (R: status_rel S1 S2) s1_ret s2_ret ≝ |
---|
1017 | % \(\forall\)s1_pre,s2_pre. |
---|
1018 | % as_after_return s1_pre s1_ret \(\to\) s1_pre \(\R\) s2_pre \(\to\) |
---|
1019 | % as_after_return s2_pre s2_ret. |
---|
1020 | % \end{alltt} |
---|
1021 | |
---|
1022 | \begin{figure} |
---|
1023 | \centering |
---|
1024 | \begin{tabular}{@{}c@{}c@{}c@{}} |
---|
1025 | % \begin{subfigure}{.475\linewidth} |
---|
1026 | % \centering |
---|
1027 | % \begin{tikzpicture}[every join/.style={ar}, join all, thick, |
---|
1028 | % every label/.style=overlay, node distance=10mm] |
---|
1029 | % \matrix [diag] (m) {% |
---|
1030 | % \node (s1) [is jump] {}; & \node [fill=white] (t1) {};\\ |
---|
1031 | % \node (s2) {}; & \node (t2) {}; \\ |
---|
1032 | % }; |
---|
1033 | % \node [above=0 of t1, overlay] {$\alpha$}; |
---|
1034 | % {[-stealth] |
---|
1035 | % \draw (s1) -- (t1); |
---|
1036 | % \draw [new] (s2) -- node [above] {$*$} (t2); |
---|
1037 | % } |
---|
1038 | % \draw (s1) to node [rel] {$\S$} (s2); |
---|
1039 | % \draw [new] (t1) to node [rel] {$\S,\L$} (t2); |
---|
1040 | % \end{tikzpicture} |
---|
1041 | % \caption{The \texttt{cl\_jump} case.} |
---|
1042 | % \label{subfig:cl_jump} |
---|
1043 | % \end{subfigure} |
---|
1044 | % & |
---|
1045 | \begin{subfigure}{.25\linewidth} |
---|
1046 | \centering |
---|
1047 | \begin{tikzpicture}[every join/.style={ar}, join all, thick, |
---|
1048 | every label/.style=overlay, node distance=10mm] |
---|
1049 | \matrix [diag] (m) {% |
---|
1050 | \node (s1) {}; & \node (t1) {};\\ |
---|
1051 | \node (s2) {}; & \node (t2) {}; \\ |
---|
1052 | }; |
---|
1053 | {[-stealth] |
---|
1054 | \draw (s1) -- (t1); |
---|
1055 | \draw [new] (s2) -- node [above] {$*$} (t2); |
---|
1056 | } |
---|
1057 | \draw (s1) to node [rel] {$\S$} (s2); |
---|
1058 | \draw [new] (t1) to node [rel] {$\S,\L$} (t2); |
---|
1059 | \end{tikzpicture} |
---|
1060 | \caption{The \texttt{cl\_oher} and \texttt{cl\_jump} cases.} |
---|
1061 | \label{subfig:cl_other_jump} |
---|
1062 | \end{subfigure} |
---|
1063 | & |
---|
1064 | \begin{subfigure}{.375\linewidth} |
---|
1065 | \centering |
---|
1066 | \begin{tikzpicture}[every join/.style={ar}, join all, thick, |
---|
1067 | every label/.style=overlay, node distance=10mm] |
---|
1068 | \matrix [diag, small vgap] (m) {% |
---|
1069 | \node (t1) {}; \\ |
---|
1070 | \node (s1) [is call] {}; \\ |
---|
1071 | & \node (l) {}; & \node (t2) {};\\ |
---|
1072 | \node (s2) {}; & \node (c) [is call] {};\\ |
---|
1073 | }; |
---|
1074 | {[-stealth] |
---|
1075 | \draw (s1) -- node [left] {$f$} (t1); |
---|
1076 | \draw [new] (s2) -- node [above] {$*$} (c); |
---|
1077 | \draw [new] (c) -- node [right] {$f$} (l); |
---|
1078 | \draw [new] (l) -- node [above] {$*$} (t2); |
---|
1079 | } |
---|
1080 | \draw (s1) to node [rel] {$\S$} (s2); |
---|
1081 | \draw [new] (t1) to [bend left] node [rel] {$\S$} (t2); |
---|
1082 | \draw [new] (t1) to [bend left] node [rel] {$\L$} (l); |
---|
1083 | \draw [new] (t1) to node [rel] {$\C$} (c); |
---|
1084 | \end{tikzpicture} |
---|
1085 | \caption{The \texttt{cl\_call} case.} |
---|
1086 | \label{subfig:cl_call} |
---|
1087 | \end{subfigure} |
---|
1088 | & |
---|
1089 | \begin{subfigure}{.375\linewidth} |
---|
1090 | \centering |
---|
1091 | \begin{tikzpicture}[every join/.style={ar}, join all, thick, |
---|
1092 | every label/.style=overlay, node distance=10mm] |
---|
1093 | \matrix [diag, small vgap] (m) {% |
---|
1094 | \node (s1) [is ret] {}; \\ |
---|
1095 | \node (t1) {}; \\ |
---|
1096 | \node (s2) {}; & \node (c) [is ret] {};\\ |
---|
1097 | & \node (r) {}; & \node (t2) {}; \\ |
---|
1098 | }; |
---|
1099 | {[-stealth] |
---|
1100 | \draw (s1) -- (t1); |
---|
1101 | \draw [new] (s2) -- node [above] {$*$} (c); |
---|
1102 | \draw [new] (c) -- (r); |
---|
1103 | \draw [new] (r) -- node [above] {$*$} (t2); |
---|
1104 | } |
---|
1105 | \draw (s1) to [bend right=45] node [rel] {$\S$} (s2); |
---|
1106 | \draw [new, overlay] (t1) to [bend left=90, looseness=1] node [rel] {$\S,\L$} (t2); |
---|
1107 | \draw [new, overlay] (t1) to [bend left=90, looseness=1.2] node [rel] {$\R$} (r); |
---|
1108 | \end{tikzpicture} |
---|
1109 | \caption{The \texttt{cl\_return} case.} |
---|
1110 | \label{subfig:cl_return} |
---|
1111 | \end{subfigure} |
---|
1112 | \end{tabular} |
---|
1113 | \caption{Mnemonic diagrams depicting the hypotheses for the preservation of structured traces. |
---|
1114 | Dashed lines |
---|
1115 | and arrows indicates how the diagrams must be closed when solid relations |
---|
1116 | are present.} |
---|
1117 | \label{fig:forwardsim} |
---|
1118 | \end{figure} |
---|
1119 | |
---|
1120 | \paragraph{1-to-many forward simulation conditions.} |
---|
1121 | \begin{condition}[Cases \texttt{cl\_other} and \texttt{cl\_jump}] |
---|
1122 | For all $s_1,s_1',s_2$ such that $s_1 \S s_1'$, and |
---|
1123 | $s_1\exec s_1'$, and either $s_1 \class \texttt{cl\_other}$ or |
---|
1124 | both $s_1\class\texttt{cl\_other}\}$ and $\ell~s_1'$, |
---|
1125 | there exists an $s_2'$ and a $\texttt{trace\_any\_any\_free}~s_2~s_2'$ called $taaf$ |
---|
1126 | such that $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and either |
---|
1127 | $taaf$ is non empty, or one among $s_1$ and $s_1'$ is \texttt{as\_costed}. |
---|
1128 | \end{condition} |
---|
1129 | |
---|
1130 | In the above condition depicted in \autoref{subfig:cl_other_jump}, |
---|
1131 | a $\texttt{trace\_any\_any\_free}~s_1~s_2$ (which from now on |
---|
1132 | will be shorthanded as \verb+TAAF+) is an |
---|
1133 | inductive type of structured traces that do not contain function calls or |
---|
1134 | cost emission statements. Differently from a \verb+TAA+, the |
---|
1135 | instruction to be executed in the lookahead state $s_2$ may be a cost emission |
---|
1136 | statement. |
---|
1137 | |
---|
1138 | The intuition of the condition is that one step can be replaced with zero or more steps if it |
---|
1139 | preserves the relation between the data and if the two final statuses are |
---|
1140 | labelled in the same way. Moreover, we must take special care of the empty case |
---|
1141 | to avoid collapsing two consecutive states that emit a label, missing one of the two emissions. |
---|
1142 | |
---|
1143 | \begin{condition}[Case \texttt{cl\_call}] |
---|
1144 | For all $s_1,s_1',s_2$ s.t. $s_1 \S s_1'$ and |
---|
1145 | $s_1\exec s_1'$ and $s_1 \class \texttt{cl\_call}$, there exists $s_a, s_b, s_2'$, a |
---|
1146 | $\verb+TAA+~s_2~s_a$, and a |
---|
1147 | $\verb+TAAF+~s_b~s_2'$ such that: |
---|
1148 | $s_a\class\texttt{cl\_call}$, the \texttt{as\_call\_ident}'s of |
---|
1149 | the two call states are the same, $s_1 \C s_a$, |
---|
1150 | $s_a\exec s_b$, $s_1' \L s_b$ and |
---|
1151 | $s_1' \S s_2'$. |
---|
1152 | \end{condition} |
---|
1153 | |
---|
1154 | The condition, depicted in \autoref{subfig:cl_call} says that, to simulate a function call, we can perform a |
---|
1155 | sequence of silent actions before and after the function call itself. |
---|
1156 | The old and new call states must be $\C$-related, the old and new |
---|
1157 | states at the beginning of the function execution must be $\L$-related |
---|
1158 | and, finally, the two initial and final states must be $\S$-related |
---|
1159 | as usual. |
---|
1160 | |
---|
1161 | \begin{condition}[Case \texttt{cl\_return}] |
---|
1162 | For all $s_1,s_1',s_2$ s.t. $s_1 \S s_1'$, |
---|
1163 | $s_1\exec s_1'$ and $s_1 \class \texttt{cl\_return}$, there exists $s_a, s_b, s_2'$, a |
---|
1164 | $\verb+TAA+~s_2~s_a$, a |
---|
1165 | $\verb+TAAF+~s_b~s_2'$ called $taaf$ such that: |
---|
1166 | $s_a\class\texttt{cl\_return}$, |
---|
1167 | $s_a\exec s_b$, |
---|
1168 | $s_1' \R s_b$ and |
---|
1169 | $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and either |
---|
1170 | $taaf$ is non empty, or $\lnot \ell~s_a$. |
---|
1171 | \end{condition} |
---|
1172 | |
---|
1173 | Similarly to the call condition, to simulate a return we can perform a |
---|
1174 | sequence of silent actions before and after the return statement itself, |
---|
1175 | as depicted in \autoref{subfig:cl_return}. |
---|
1176 | The old and the new statements after the return must be $\R$-related, |
---|
1177 | to grant that they returned to corresponding calls. |
---|
1178 | The two initial and final states must be $\S$-related |
---|
1179 | as usual and, moreover, they must exhibit the same labels. Finally, when |
---|
1180 | the suffix is non empty we must take care of not inserting a new |
---|
1181 | unmatched cost emission statement just after the return statement. |
---|
1182 | |
---|
1183 | \begin{comment} |
---|
1184 | \begin{verbatim} |
---|
1185 | definition status_simulation ≝ |
---|
1186 | λS1 : abstract_status. |
---|
1187 | λS2 : abstract_status. |
---|
1188 | λsim_status_rel : status_rel S1 S2. |
---|
1189 | ∀st1,st1',st2.as_execute S1 st1 st1' → |
---|
1190 | sim_status_rel st1 st2 → |
---|
1191 | match as_classify … st1 with |
---|
1192 | [ None ⇒ True |
---|
1193 | | Some cl ⇒ |
---|
1194 | match cl with |
---|
1195 | [ cl_call ⇒ ∀prf. |
---|
1196 | (* |
---|
1197 | st1' ------------S----------\ |
---|
1198 | ↑ \ \ |
---|
1199 | st1 \--L--\ \ |
---|
1200 | | \ \ \ |
---|
1201 | S \-C-\ st2_after_call →taa→ st2' |
---|
1202 | | \ ↑ |
---|
1203 | st2 →taa→ st2_pre_call |
---|
1204 | *) |
---|
1205 | ∃st2_pre_call. |
---|
1206 | as_call_ident ? st2_pre_call = as_call_ident ? («st1, prf») ∧ |
---|
1207 | call_rel ?? sim_status_rel «st1, prf» st2_pre_call ∧ |
---|
1208 | ∃st2_after_call,st2'. |
---|
1209 | ∃taa2 : trace_any_any … st2 st2_pre_call. |
---|
1210 | ∃taa2' : trace_any_any … st2_after_call st2'. |
---|
1211 | as_execute … st2_pre_call st2_after_call ∧ |
---|
1212 | sim_status_rel st1' st2' ∧ |
---|
1213 | label_rel … st1' st2_after_call |
---|
1214 | | cl_return ⇒ |
---|
1215 | (* |
---|
1216 | st1 |
---|
1217 | / ↓ |
---|
1218 | | st1'----------S,L------------\ |
---|
1219 | S \ \ |
---|
1220 | \ \-----R-------\ | |
---|
1221 | \ | | |
---|
1222 | st2 →taa→ st2_ret | | |
---|
1223 | ↓ / | |
---|
1224 | st2_after_ret →taaf→ st2' |
---|
1225 | |
---|
1226 | we also ask that st2_after_ret be not labelled if the taaf tail is |
---|
1227 | not empty |
---|
1228 | *) |
---|
1229 | ∃st2_ret,st2_after_ret,st2'. |
---|
1230 | ∃taa2 : trace_any_any … st2 st2_ret. |
---|
1231 | ∃taa2' : trace_any_any_free … st2_after_ret st2'. |
---|
1232 | (if taaf_non_empty … taa2' then ¬as_costed … st2_after_ret else True) ∧ |
---|
1233 | as_classifier … st2_ret cl_return ∧ |
---|
1234 | as_execute … st2_ret st2_after_ret ∧ sim_status_rel st1' st2' ∧ |
---|
1235 | ret_rel … sim_status_rel st1' st2_after_ret ∧ |
---|
1236 | label_rel … st1' st2' |
---|
1237 | | cl_other ⇒ |
---|
1238 | (* |
---|
1239 | st1 → st1' |
---|
1240 | | \ |
---|
1241 | S S,L |
---|
1242 | | \ |
---|
1243 | st2 →taaf→ st2' |
---|
1244 | |
---|
1245 | the taaf can be empty (e.g. tunneling) but we ask it must not be the |
---|
1246 | case when both st1 and st1' are labelled (we would be able to collapse |
---|
1247 | labels otherwise) |
---|
1248 | *) |
---|
1249 | ∃st2'. |
---|
1250 | ∃taa2 : trace_any_any_free … st2 st2'. |
---|
1251 | (if taaf_non_empty … taa2 then True else (¬as_costed … st1 ∨ ¬as_costed … st1')) ∧ |
---|
1252 | sim_status_rel st1' st2' ∧ |
---|
1253 | label_rel … st1' st2' |
---|
1254 | | cl_jump ⇒ |
---|
1255 | (* just like cl_other, but with a hypothesis more *) |
---|
1256 | as_costed … st1' → |
---|
1257 | ∃st2'. |
---|
1258 | ∃taa2 : trace_any_any_free … st2 st2'. |
---|
1259 | (if taaf_non_empty … taa2 then True else (¬as_costed … st1 ∨ ¬as_costed … st1')) ∧ |
---|
1260 | sim_status_rel st1' st2' ∧ |
---|
1261 | label_rel … st1' st2' |
---|
1262 | ] |
---|
1263 | ]. |
---|
1264 | \end{verbatim} |
---|
1265 | \end{comment} |
---|
1266 | |
---|
1267 | \paragraph{Main result: the 1-to-many forward simulation conditions |
---|
1268 | are sufficient to trace reconstruction} |
---|
1269 | |
---|
1270 | Let us assume that a relation set is given such that the 1-to-many |
---|
1271 | forward simulation conditions are satisfied. Under this assumption we |
---|
1272 | can prove the following three trace reconstruction theorems by mutual |
---|
1273 | structural induction over the traces given in input between the |
---|
1274 | $s_1$ and $s_1'$ states. |
---|
1275 | |
---|
1276 | In particular, the \texttt{status\_simulation\_produce\_tlr} theorem |
---|
1277 | applied to the \texttt{main} function of the program and equal |
---|
1278 | $s_{2_b}$ and $s_2$ states shows that, for every initial state in the |
---|
1279 | source code that induces a structured trace in the source code, |
---|
1280 | the compiled code produces a similar structured trace. |
---|
1281 | |
---|
1282 | \begin{theorem}[\texttt{status\_simulation\_produce\_tlr}] |
---|
1283 | For every $s_1,s_1',s_{2_b},s_2$ s.t. |
---|
1284 | there is a $\texttt{TLR}~s_1~s_1'$ called $tlr_1$ and a |
---|
1285 | $\verb+TAA+~s_{2_b}~s_2$ and $s_1 \L s_{2_b}$ and |
---|
1286 | $s_1 \S s_2$, there exists $s_{2_m},s_2'$ s.t. |
---|
1287 | there is a $\texttt{TLR}~s_{2_b}~s_{2_m}$ called $tlr_2$ and |
---|
1288 | there is a $\verb+TAAF+~s_{2_m}~s_2'$ called $taaf$ |
---|
1289 | s.t. if $taaf$ is non empty then $\lnot (\ell~s_{2_m})$, |
---|
1290 | and $tlr_1\approx tlr_2$ |
---|
1291 | and $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and |
---|
1292 | $s_1' \R s_{2_m}$. |
---|
1293 | \end{theorem} |
---|
1294 | |
---|
1295 | The theorem states that a \texttt{trace\_label\_return} in the source code |
---|
1296 | together with a precomputed preamble of silent states |
---|
1297 | (the \verb+TAA+) in the target code induces a |
---|
1298 | similar \texttt{trace\_label\_return} in the target code which can be |
---|
1299 | followed by a sequence of silent states. Note that the statement does not |
---|
1300 | require the produced \texttt{trace\_label\_return} to start with the |
---|
1301 | precomputed preamble, even if this is likely to be the case in concrete |
---|
1302 | implementations. The preamble in input is necessary for compositionality, e.g. |
---|
1303 | because the 1-to-many forward simulation conditions allow in the |
---|
1304 | case of function calls to execute a preamble of silent instructions just after |
---|
1305 | the function call. |
---|
1306 | |
---|
1307 | Clearly similar results are also available for the other two types of structured |
---|
1308 | traces (in fact, they are all proved simultaneously by mutual induction). |
---|
1309 | % \begin{theorem}[\texttt{status\_simulation\_produce\_tll}] |
---|
1310 | % For every $s_1,s_1',s_{2_b},s_2$ s.t. |
---|
1311 | % there is a $\texttt{TLL}~b~s_1~s_1'$ called $tll_1$ and a |
---|
1312 | % $\verb+TAA+~s_{2_b}~s_2$ and $s_1 \L s_{2_b}$ and |
---|
1313 | % $s_1 \S s_2$, there exists $s_{2_m},s_2'$ s.t. |
---|
1314 | % \begin{itemize} |
---|
1315 | % \item if $b$ (the trace ends with a return) then there exists $s_{2_m},s_2'$ |
---|
1316 | % and a trace $\texttt{TLL}~b~s_{2_b}~s_{2_m}$ called $tll_2$ |
---|
1317 | % and a $\texttt{TAAF}~s_{2_m}~s_2'$ called $taa_2$ s.t. |
---|
1318 | % $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and |
---|
1319 | % $s_1' \R s_{2_m}$ and |
---|
1320 | % $tll_1\approx tll_2$ and |
---|
1321 | % if $taa_2$ is non empty then $\lnot \ell~s_{2_m}$; |
---|
1322 | % \item else there exists $s_2'$ and a |
---|
1323 | % $\texttt{TLL}~b~s_{2_b}~s_2'$ called $tll_2$ such that |
---|
1324 | % $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and |
---|
1325 | % $tll_1\approx tll_2$. |
---|
1326 | % \end{itemize} |
---|
1327 | % \end{theorem} |
---|
1328 | % |
---|
1329 | % The statement is similar to the previous one: a source |
---|
1330 | % \texttt{trace\_label\_label} and a given target preamble of silent states |
---|
1331 | % in the target code induce a similar \texttt{trace\_label\_label} in the |
---|
1332 | % target code, possibly followed by a sequence of silent moves that become the |
---|
1333 | % preamble for the next \texttt{trace\_label\_label} translation. |
---|
1334 | % |
---|
1335 | % \begin{theorem}[\texttt{status\_simulation\_produce\_tal}] |
---|
1336 | % For every $s_1,s_1',s_2$ s.t. |
---|
1337 | % there is a $\texttt{TAL}~b~s_1~s_1'$ called $tal_1$ and |
---|
1338 | % $s_1 \S s_2$ |
---|
1339 | % \begin{itemize} |
---|
1340 | % \item if $b$ (the trace ends with a return) then there exists $s_{2_m},s_2'$ |
---|
1341 | % and a trace $\texttt{TAL}~b~s_2~s_{2_m}$ called $tal_2$ and a |
---|
1342 | % $\texttt{TAAF}~s_{2_m}~s_2'$ called $taa_2$ s.t. |
---|
1343 | % $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and |
---|
1344 | % $s_1' \R s_{2_m}$ and |
---|
1345 | % $tal_1 \approx tal_2$ and |
---|
1346 | % if $taa_2$ is non empty then $\lnot \ell~s_{2_m}$; |
---|
1347 | % \item else there exists $s_2'$ and a |
---|
1348 | % $\texttt{TAL}~b~s_2~s_2'$ called $tal_2$ such that |
---|
1349 | % either $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and |
---|
1350 | % $tal_1\approx tal_2$ |
---|
1351 | % or $s_1' \mathrel{{\S} \cap {\L}} s_2$ and |
---|
1352 | % $\texttt{tal\_collapsable}~tal_1$ and $\lnot \ell~s_1$. |
---|
1353 | % \end{itemize} |
---|
1354 | % \end{theorem} |
---|
1355 | % |
---|
1356 | % The statement is also similar to the previous ones, but for the lack of |
---|
1357 | % the target code preamble. |
---|
1358 | |
---|
1359 | \begin{comment} |
---|
1360 | \begin{corollary} |
---|
1361 | For every $s_1,s_1',s_2$ s.t. |
---|
1362 | there is a $\texttt{trace\_label\_return}~s_1~s_1'$ called $tlr_1$ and |
---|
1363 | $s_1 (\L \cap \S) s_2$ |
---|
1364 | there exists $s_{2_m},s_2'$ s.t. |
---|
1365 | there is a $\texttt{trace\_label\_return}~s_2~s_{2_m}$ called $tlr_2$ and |
---|
1366 | there is a $\texttt{trace\_any\_any\_free}~s_{2_m}~s_2'$ called $taaf$ |
---|
1367 | s.t. if $taaf$ is non empty then $\lnot (\texttt{as\_costed}~s_{2_m})$, |
---|
1368 | and $\texttt{tlr\_rel}~tlr_1~tlr_2$ |
---|
1369 | and $s_1' (\S \cap \L) s_2'$ and |
---|
1370 | $s_1' \R s_{2_m}$. |
---|
1371 | \end{corollary} |
---|
1372 | \end{comment} |
---|
1373 | |
---|
1374 | \begin{comment} |
---|
1375 | \begin{verbatim} |
---|
1376 | status_simulation_produce_tlr S1 S2 R |
---|
1377 | (* we start from this situation |
---|
1378 | st1 →→→→tlr→→→→ st1' |
---|
1379 | | \ |
---|
1380 | L \---S--\ |
---|
1381 | | \ |
---|
1382 | st2_lab →taa→ st2 (the taa preamble is in general either empty or given |
---|
1383 | by the preceding call) |
---|
1384 | |
---|
1385 | and we produce |
---|
1386 | st1 →→→→tlr→→→→ st1' |
---|
1387 | \\ / \ |
---|
1388 | // R \-L,S-\ |
---|
1389 | \\ | \ |
---|
1390 | st2_lab →tlr→ st2_mid →taaf→ st2' |
---|
1391 | *) |
---|
1392 | st1 st1' st2_lab st2 |
---|
1393 | (tlr1 : trace_label_return S1 st1 st1') |
---|
1394 | (taa2_pre : trace_any_any S2 st2_lab st2) |
---|
1395 | (sim_execute : status_simulation S1 S2 R) |
---|
1396 | on tlr1 : R st1 st2 → label_rel … st1 st2_lab → |
---|
1397 | ∃st2_mid.∃st2'. |
---|
1398 | ∃tlr2 : trace_label_return S2 st2_lab st2_mid. |
---|
1399 | ∃taa2 : trace_any_any_free … st2_mid st2'. |
---|
1400 | (if taaf_non_empty … taa2 then ¬as_costed … st2_mid else True) ∧ |
---|
1401 | R st1' st2' ∧ ret_rel … R st1' st2_mid ∧ label_rel … st1' st2' ∧ |
---|
1402 | tlr_rel … tlr1 tlr2 |
---|
1403 | \end{verbatim} |
---|
1404 | \end{comment} |
---|
1405 | |
---|
1406 | \section{Conclusions and future works} |
---|
1407 | \label{conclusions} |
---|
1408 | The labelling approach is a technique to implement compilers that induce on |
---|
1409 | the source code a non uniform cost model determined from the object code |
---|
1410 | produced. The cost model assigns a cost to each basic block of the program. |
---|
1411 | The main theorem of the approach says that there is an exact |
---|
1412 | correspondence between the sequence of basic blocks started in the source |
---|
1413 | and object code, and that no instruction in the source or object code is |
---|
1414 | executed outside a basic block. Thus the cost of object code execution |
---|
1415 | can be computed precisely on the source. |
---|
1416 | |
---|
1417 | In this paper we scale the labelling approach to cover a programming language |
---|
1418 | with function calls. This introduces new difficulties only when the language |
---|
1419 | is unstructured, i.e. it allows function calls to return anywhere in the code, |
---|
1420 | destroying the hope of a static prediction of the cost of basic blocks. |
---|
1421 | We restore static predictability by introducing a new semantics for unstructured |
---|
1422 | programs that single outs well structured executions. The latter are represented |
---|
1423 | by structured traces, a generalisation of streams of observables that capture |
---|
1424 | several structural invariants of the execution, like well nesting of functions |
---|
1425 | or the fact that every basic block must start with a code emission statement. |
---|
1426 | We show that structured traces are sufficiently structured to statically compute |
---|
1427 | a precise cost model on the object code. |
---|
1428 | |
---|
1429 | We introduce a similarity relation on structured traces that must hold between |
---|
1430 | source and target traces. When the relation holds for every program, we prove |
---|
1431 | that the cost model can be lifted from the object to the source code. |
---|
1432 | |
---|
1433 | In order to prove that similarity holds, we present a generic proof of forward |
---|
1434 | simulation that is aimed at pulling apart as much as possible the part of the |
---|
1435 | simulation related to non-functional properties (preservation of structure) |
---|
1436 | from that related to functional properties. In particular, we reduce the |
---|
1437 | problem of preservation of structure to that of showing a 1-to-many |
---|
1438 | forward simulation that only adds a few additional proof obligations to those |
---|
1439 | of a traditional, function properties only, proof. |
---|
1440 | |
---|
1441 | All results presented in the paper are part of a larger certification of a |
---|
1442 | C compiler which is based on the labelling approach. The certification, done |
---|
1443 | in Matita, is the main deliverable of the FET-Open Certified Complexity (CerCo). |
---|
1444 | |
---|
1445 | The short term future work consists in the completion of the certification of |
---|
1446 | the CerCo compiler exploiting the main theorem of this paper. |
---|
1447 | |
---|
1448 | \paragraph{Related works.} |
---|
1449 | CerCo is the first project that explicitly tries to induce a |
---|
1450 | precise cost model on the source code in order to establish non-functional |
---|
1451 | properties of programs on an high level language. Traditional certifications |
---|
1452 | of compilers, like~\cite{compcert2,piton}, only explicitly prove preservation |
---|
1453 | of the functional properties. |
---|
1454 | |
---|
1455 | Usually forward simulations take the following form: for each transition |
---|
1456 | from $s_1$ to $s_2$ in the source code, there exists an equivalent sequence of |
---|
1457 | transitions in the target code of length $n$. The number $n$ of transition steps |
---|
1458 | in the target code can just be the witness of the existential statement. |
---|
1459 | An equivalent alternative when the proof of simulation is constructive consists |
---|
1460 | in providing an explicit function, called \emph{clock function} in the |
---|
1461 | literature~\cite{clockfunctions}, that computes $n$ from $s_1$. Every clock |
---|
1462 | function constitutes then a cost model for the source code, in the spirit of |
---|
1463 | what we are doing in CerCo. However, we believe our solution to be superior |
---|
1464 | in the following respects: 1) the machinery of the labelling approach is |
---|
1465 | insensible to the resource being measured. Indeed, any cost model computed on |
---|
1466 | the object code can be lifted to the source code (e.g. stack space used, |
---|
1467 | energy consumed, etc.). On the contrary, clock functions only talk about |
---|
1468 | number of transition steps. In order to extend the approach with clock functions |
---|
1469 | to other resources, additional functions must be introduced. Moreover, the |
---|
1470 | additional functions would be handled differently in the proof. |
---|
1471 | 2) the cost models induced by the labelling approach have a simple presentation. |
---|
1472 | In particular, they associate a number to each basic block. More complex |
---|
1473 | models can be induced when the approach is scaled to cover, for instance, |
---|
1474 | loop optimisations~\cite{loopoptimizations}, but the costs are still meant to |
---|
1475 | be easy to understand and manipulate in an interactive theorem prover or |
---|
1476 | in Frama-C. |
---|
1477 | On the contrary, a clock function is a complex function of the state $s_1$ |
---|
1478 | which, as a function, is an opaque object that is difficult to reify as |
---|
1479 | source code in order to reason on it. |
---|
1480 | |
---|
1481 | \bibliographystyle{splncs03} |
---|
1482 | \bibliography{ccexec} |
---|
1483 | |
---|
1484 | % \appendix |
---|
1485 | % \section{Notes for the reviewers} |
---|
1486 | % |
---|
1487 | % The results described in the paper are part of a larger formalization |
---|
1488 | % (the certification of the CerCo compiler). At the moment of the submission |
---|
1489 | % we need to single out from the CerCo formalization the results presented here. |
---|
1490 | % Before the 16-th of February we will submit an attachment that contains the |
---|
1491 | % minimal subset of the CerCo formalization that allows to prove those results. |
---|
1492 | % At that time it will also be possible to measure exactly the size of the |
---|
1493 | % formalization described here. At the moment a rough approximation suggests |
---|
1494 | % about 2700 lines of Matita code. |
---|
1495 | % |
---|
1496 | % We will also attach the development version of the interactive theorem |
---|
1497 | % prover Matita that compiles the submitted formalization. Another possibility |
---|
1498 | % is to backport the development to the last released version of the system |
---|
1499 | % to avoid having to re-compile Matita from scratch. |
---|
1500 | % |
---|
1501 | % The programming and certification style used in the formalization heavily |
---|
1502 | % exploit dependent types. Dependent types are used: 1) to impose invariants |
---|
1503 | % by construction on the data types and operations (e.g. a traces from a state |
---|
1504 | % $s_1$ to a state $s_2$ can be concatenad to a trace from a state |
---|
1505 | % $s_2'$ to a state $s_3$ only if $s_2$ is convertible with $s_2'$); 2) |
---|
1506 | % to state and prove the theorems by using the Russell methodology of |
---|
1507 | % Matthieu Sozeau\footnote{Subset Coercions in Coq in TYPES'06. Matthieu Sozeau. Thorsten Altenkirch and Conor McBride (Eds). Volume 4502 of Lecture Notes in Computer Science. Springer, 2007, pp.237-252. |
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1508 | % }, better known in the Coq world as ``\texttt{Program}'' and reimplemented in a simpler way in Matita using coercion propagations\footnote{Andrea Asperti, Wilmer Ricciotti, Claudio Sacerdoti Coen, Enrico Tassi: A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions. Logical Methods in Computer Science 8(1) (2012)}. However, no result presented depends |
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1509 | % mandatorily on dependent types: it should be easy to adapt the technique |
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1510 | % and results presented in the paper to HOL. |
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1511 | % |
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1512 | % Finally, Matita and Coq are based on minor variations of the Calculus of |
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1513 | % (Co)Inductive Constructions. These variations do not affect the CerCo |
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1514 | % formalization. Therefore a porting of the proofs and ideas to Coq would be |
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1515 | % rather straightforward. |
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1516 | |
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1517 | \end{document} |
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