Changeset 1360
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 Oct 12, 2011, 11:47:21 AM (8 years ago)
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Deliverables/D4.24.3/reports/D42.tex
r1356 r1360 150 150 \label{sect.backend.intermediate.languages.matita} 151 151 152 We now discuss the encoding of the compiler backend languages in the Calculus of Constructions proper. 153 We pay particular heed to changes that we made from the O'Caml prototype. 154 In particular, many aspects of the backend languages have been unified into a single `joint' language. 155 We have also made heavy use of dependent types to reduce `spurious partiality' and to encode invariants. 156 157 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 158 % SECTION. % 159 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 160 \subsection{Abstracting related languages} 161 \label{subsect.abstracting.related.languages} 162 163 The O'Caml compiler is written in the following manner. 164 Each intermediate language has its own dedicated syntax, notions of internal function, and so on. 165 Translations map syntaxes to syntaxes, and internal functions to internal functions explicitly. 166 167 This is a perfectly valid way to write a compiler, where everything is made explicit, but writing a \emph{verified} compiler poses new challenges. 168 In particular, we must look ahead to see how our choice of encodings will affect the size and complexity of the forthcoming proofs of correctness. 169 We now discuss some abstractions, introduced in the Matita code, which we hope will make our proofs shorter, amongst other benefits. 170 171 \paragraph{Shared code} 172 Many features of individual backend intermediate languages are shared with other intermediate languages. 173 For instance, RTLabs, RTL, ERTL and LTL are all graph based languages, where functions are represented as a graph of statements that form their bodies. 174 Functions for adding statements to a graph, searching the graph, and so on, are remarkably similar across all languages, but are duplicated in the O'Caml code. 175 152 176 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 153 177 % SECTION. % … … 156 180 \label{subsect.use.of.dependent.types} 157 181 158 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 159 % SECTION. % 160 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 161 \subsection{Abstracting related languages} 162 \label{subsect.abstracting.related.languages} 182 We use dependent types in the backend for three reasons. 183 184 First, we encode informal invariants, or uses of \texttt{assert false} in the O'Caml code, with dependent types, converting partial functions into total functions. 185 There are numerous examples of this throughout the backend. 186 For example, in the \texttt{RTLabs} to \texttt{RTL} transformation pass, many functions only `make sense' when lists of registers passed to them as arguments conform to some specific length. 187 For instance, the \texttt{translate\_negint} function, which translates a negative integer constant: 188 \begin{lstlisting} 189 definition translate_negint ≝ 190 $\lambda$globals: list ident. 191 $\lambda$destrs: list register. 192 $\lambda$srcrs: list register. 193 $\lambda$start_lbl: label. 194 $\lambda$dest_lbl: label. 195 $\lambda$def: rtl_internal_function globals. 196 $\lambda$prf: destrs = srcrs. (* assert in caml code *) 197 ... 198 \end{lstlisting} 199 The last argument to the function, \texttt{prf}, is a proof that the lengths of the lists of source and destination registers are the same. 200 This was an assertion in the O'Caml code. 201 202 Secondly, we make use of dependent types to make the Matita code easier to read, and eventually the proofs of correctness for the compiler easier to write. 203 For instance, many intermediate languages in the backend of the compiler, from RTLabs to LTL, are graph based languages. 204 Here, function definitions consist of a graph (i.e. a map from labels to statements) and a pair of labels denoting the entry and exit points of this graph. 205 Practically, we would always like to ensure that the entry and exit labels are present in the statement graph. 206 We ensure that this is so with a dependent sum type in the \texttt{joint\_internal\_function} record, which all graph based languages specialise to obtain their own internal function representation: 207 \begin{lstlisting} 208 record joint_internal_function (globals: list ident) (p: params globals): Type[0] := 209 { 210 ... 211 joint_if_code : codeT $\ldots$ p; 212 joint_if_entry : $\Sigma$l: label. lookup $\ldots$ joint_if_code l $\neq$ None $\ldots$; 213 ... 214 }. 215 \end{lstlisting} 216 Here, \texttt{codeT} is a parameterised type representing the `structure' of the function's body (a graph in graph based languages, and a list in the linearised LIN language). 217 Specifically, the \texttt{joint\_if\_entry} is a dependent pair consisting of a label and a proof that the label in question is a vertex in the function's graph. 218 A similar device exists for the exit label. 219 220 Finally, we make use of dependent types for another reason: experimentation. 221 Namely, CompCert makes little use of dependent types to encode invariants. 222 In contrast, we wish to make as much use of dependent types as possible, both to experiment with different ways of encoding compilers in a proof assistant, but also as a way of `stress testing' Matita's support for dependent types. 163 223 164 224 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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