Changeset 3657
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 Mar 16, 2017, 12:39:35 PM (3 months ago)
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 Papers/jarcerco2017
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Papers/jarcerco2017/cerco.tex
r3656 r3657 70 70 \DeclareUnicodeCharacter{9001}{\ensuremath{\langle}} 71 71 \DeclareUnicodeCharacter{9002}{\ensuremath{\rangle}} 72 73 \newcommand{\cerco}{CerCo} 74 \newcommand{\ocaml}{OCaml} 75 \newcommand{\clight}{Clight} 76 \newcommand{\matita}{Matita} 77 \newcommand{\sdcc}{\texttt{sdcc}} 72 78 73 79 \title{CerCo: Certified Complexity\thanks{The project CerCo acknowledges the 
Papers/jarcerco2017/lstgrafite.tex
r3622 r3657 8 8 notation, interpretation, lapply, repeat, try, as, clear, in, change, whd, exists}, 9 9 morekeywords={[2]include}, 10 keywordstyle=\color{blue}\bfseries, 10 11 %emph={[1]Type, Prop, nat, real}, emphstyle={[1]\textit}, 11 12 % literate= … … 150 151 captionpos=b, 151 152 mathescape=true, 152 keywordstyle=\bfseries,153 keywordstyle=[2]\bfseries,154 keywordstyle=[3]\bfseries,153 %keywordstyle=\bfseries, 154 %keywordstyle=[2]\bfseries, 155 %keywordstyle=[3]\bfseries, 155 156 %backgroundcolor=\color{gray}, 156 157 frame=tblr, 
Papers/jarcerco2017/proof.tex
r3656 r3657 8 8 \section{Compiler proof} 9 9 \label{sect.compiler.proof} 10 11 \section{Introduction} 12 13 The \cerco{} compiler produces a version of the source code containing 14 annotations describing the timing behaviour of the object code, as 15 well as the object code itself. It compiles C code, targeting 16 microcontrollers implementing the Intel 8051 architecture. There are 17 two versions: first, an initial prototype was implemented in 18 \ocaml{}~\cite{d2.2}, then a version was formalised in the \matita{} 19 proof assistant~\cite{d3.2,d4.2} and extracted to \ocaml{} code to 20 produce an executable compiler. In this document we present results 21 from Deliverable 3.4, the formalised proofs in \matita{} about the 22 frontend of the latter version of the compiler (culminating in the 23 \lstinline'front_end_correct' lemma), and describe how that fits 24 into the verification of the whole compiler. 25 26 A key part of this work was to layer the \emph{intensional} correctness 27 results that show that the costs produced are correct on top of the 28 proofs about the compiled code's \emph{extensional} behaviour (that is, the 29 functional correctness of the compiler). Unfortunately, the ambitious 30 goal of completely verifying the entire compiler was not feasible 31 within the time available, but thanks to this separation of 32 extensional and intensional proofs we are able to axiomatise some extensional 33 simulation results which are similar to those in other compiler verification 34 projects and concentrate on the novel intensional proofs. We were 35 also able to add stack space costs to obtain a stronger result. The 36 proofs were made more tractable by introducing compiletime checks for 37 the `sound and precise' cost labelling properties rather than proving 38 that they are preserved throughout. 39 40 The overall statement of correctness says that the annotated program has the 41 same behaviour as the input, and that for any suitably wellstructured part of 42 the execution (which we call \emph{measurable}), the object code will execute 43 the same behaviour taking precisely the time given by the cost annotations in 44 the annotated source program. 45 46 In the next section we recall the structure of the compiler and make the overall 47 statement more precise. Following that, in Section~\ref{sec:fegoals} we 48 describe the statements we need to prove about the intermediate \textsf{RTLabs} 49 programs for the backend proofs. 50 Section~\ref{sec:inputtolabelling} covers the compiler passes which produce the 51 annotated source program and Section~\ref{sec:measurablelifting} the rest 52 of the transformations in the frontend. Then the compiletime checks 53 for good cost labelling are detailed in Section~\ref{sec:costchecks} 54 and the proofs that the structured traces required by the backend 55 exist are discussed in Section~\ref{sec:structuredtrace}. 56 57 \section{The compiler and its correctness statement} 58 59 The uncertified prototype \ocaml{} \cerco{} compiler was originally described 60 in Deliverables 2.1 and 2.2. Its design was replicated in the formal 61 \matita{} code, which was presented in Deliverables 3.2 and 4.2, for 62 the frontend and backend respectively. 63 64 \begin{figure} 65 \begin{center} 66 \includegraphics[width=0.5\linewidth]{compilerplain.pdf} 67 \end{center} 68 \caption{Languages in the \cerco{} compiler} 69 \label{fig:compilerlangs} 70 \end{figure} 71 72 The compiler uses a number of intermediate languages, as outlined the 73 middle two lines of Figure~\ref{fig:compilerlangs}. The upper line 74 represents the frontend of the compiler, and the lower the backend, 75 finishing with Intel 8051 binary code. Not all of the frontend compiler passes 76 introduce a new language, and Figure~\ref{fig:summary} presents a 77 list of every pass involved. 78 79 \begin{figure} 80 \begin{center} 81 \begin{minipage}{.8\linewidth} 82 \begin{tabbing} 83 \quad \= $\downarrow$ \quad \= \kill 84 \textsf{C} (unformalised)\\ 85 \> $\downarrow$ \> CIL parser (unformalised \ocaml)\\ 86 \textsf{Clight}\\ 87 %\> $\downarrow$ \> add runtime functions\\ 88 \> $\downarrow$ \> \lstinline[language=C]'switch' removal\\ 89 \> $\downarrow$ \> labelling\\ 90 \> $\downarrow$ \> cast removal\\ 91 \> $\downarrow$ \> stack variable allocation and control structure 92 simplification\\ 93 \textsf{Cminor}\\ 94 %\> $\downarrow$ \> generate global variable initialization code\\ 95 \> $\downarrow$ \> transform to RTL graph\\ 96 \textsf{RTLabs}\\ 97 \> $\downarrow$ \> check cost labelled properties of RTL graph\\ 98 \> $\downarrow$ \> start of target specific backend\\ 99 \>\quad \vdots 100 \end{tabbing} 101 \end{minipage} 102 \end{center} 103 \caption{Frontend languages and compiler passes} 104 \label{fig:summary} 105 \end{figure} 106 107 \label{page:switchintro} 108 The annotated source code is produced by the cost labelling phase. 109 Note that there is a pass to replace C \lstinline[language=C]'switch' 110 statements before labelling  we need to remove them because the 111 simple form of labelling used in the formalised compiler is not quite 112 capable of capturing their execution time costs, largely due to C's 113 `fallthrough' behaviour where execution from one branch continues in 114 the next unless there is an explicit \lstinline[language=C]'break'. 115 116 The cast removal phase which follows cost labelling simplifies 117 expressions to prevent unnecessary arithmetic promotion, which is 118 specified by the C standard but costly for an 8bit target. The 119 transformation to \textsf{Cminor} and subsequently \textsf{RTLabs} 120 bear considerable resemblance to some passes of the CompCert 121 compiler~\cite{BlazyLeroyClight09,Leroybackend}, although we use a simpler \textsf{Cminor} where 122 all loops use \lstinline[language=C]'goto' statements, and the 123 \textsf{RTLabs} language retains a targetindependent flavour. The 124 backend takes \textsf{RTLabs} code as input. 125 126 The whole compilation function returns the following information on success: 127 \begin{lstlisting}[language=Grafite] 128 record compiler_output : Type[0] := 129 { c_labelled_object_code: labelled_object_code 130 ; c_stack_cost: stack_cost_model 131 ; c_max_stack: nat 132 ; c_init_costlabel: costlabel 133 ; c_labelled_clight: clight_program 134 ; c_clight_cost_map: clight_cost_map 135 }. 136 \end{lstlisting} 137 It consists of annotated 8051 object code, a mapping from function 138 identifiers to the function's stack space usage, the space available for the 139 stack after global variable allocation, a cost label covering the 140 execution time for the initialisation of global variables and the call 141 to the \lstinline[language=C]'main' function, the annotated source 142 code, and finally a mapping from cost labels to actual execution time 143 costs. 144 145 An \ocaml{} pretty printer is used to provide a concrete version of 146 the output code and annotated source code. In the case of the 147 annotated source code, it also inserts the actual costs alongside the 148 cost labels, and optionally adds a global cost variable and 149 instrumentation to support further reasoning in external tools such as 150 FramaC. 151 152 \subsection{Revisions to the prototype compiler} 153 154 Our focus on intensional properties prompted us to consider whether we 155 could incorporate stack space into the costs presented to the user. 156 We only allocate one fixedsize frame per function, so modelling this 157 was relatively simple. It is the only form of dynamic memory 158 allocation provided by the compiler, so we were able to strengthen the 159 statement of the goal to guarantee successful execution whenever the 160 stack space obeys the \lstinline'c_max_stack' bound calculated by 161 subtracting the global variable requirements from the total memory 162 available. 163 164 The cost labelling checks at the end of Figure~\ref{fig:summary} have been 165 introduced to reduce the proof burden, and are described in 166 Section~\ref{sec:costchecks}. 167 168 The use of dependent types to capture simple intermediate language 169 invariants makes every frontend pass a total function, except 170 \textsf{Clight} to \textsf{Cminor} and the cost checks. Hence various 171 wellformedness and type safety checks are performed only once between 172 \textsf{Clight} and \textsf{Cminor}, and the invariants rule out any 173 difficulties in the later stages. With the benefit of hindsight we 174 would have included an initial checking phase to produce a 175 `wellformed' variant of \textsf{Clight}, conjecturing that this would 176 simplify various parts of the proofs for the \textsf{Clight} stages 177 which deal with potentially illformed code. 178 179 Following D2.2, we previously generated code for global variable 180 initialisation in \textsf{Cminor}, for which we reserved a cost label 181 to represent the execution time for initialisation. However, the 182 backend must also add an initial call to the main function, whose 183 cost must also be accounted for, so we decided to move the 184 initialisation code to the backend and merge the costs. 185 186 \subsection{Main correctness statement} 187 188 Informally, our main intensional result links the time difference in a source 189 code execution to the time difference in the object code, expressing the time 190 for the source by summing the values for the cost labels in the trace, and the 191 time for the target by a clock built in to the 8051 executable semantics. 192 193 The availability of precise timing information for 8501 194 implementations and the design of the compiler allow it to give exact 195 time costs in terms of processor cycles, not just upper bounds. 196 However, these exact results are only available if the subtrace we 197 measure starts and ends at suitable points. In particular, pure 198 computation with no observable effects may be reordered and moved past 199 cost labels, so we cannot measure time between arbitrary statements in 200 the program. 201 202 There is also a constraint on the subtraces that we 203 measure due to the requirements of the correctness proof for the 204 object code timing analysis. To be sure that the timings are assigned 205 to the correct cost label, we need to know that each return from a 206 function call must go to the correct return address. It is difficult 207 to observe this property locally in the object code because it relies 208 on much earlier stages in the compiler. To convey this information to 209 the timing analysis extra structure is imposed on the subtraces, which 210 is described in Section~\ref{sec:fegoals}. 211 212 % Regarding the footnote, would there even be much point? 213 % TODO: this might be quite easy to add ('just' subtract the 214 % measurable subtrace from the second label to the end). Could also 215 % measure other traces in this manner. 216 These restrictions are reflected in the subtraces that we give timing 217 guarantees on; they must start at a cost label and end at the return 218 of the enclosing function of the cost label\footnote{We expect that 219 this would generalise to more general subtraces by subtracting costs 220 for unwanted measurable suffixes of a measurable subtrace.}. A 221 typical example of such a subtrace is the execution of an entire 222 function from the cost label at the start of the function until it 223 returns. We call such any such subtrace \emph{measurable} if it (and 224 the prefix of the trace from the start to the subtrace) can also be 225 executed within the available stack space. 226 227 Now we can give the main intensional statement for the compiler. 228 Given a \emph{measurable} subtrace for a labelled \textsf{Clight} 229 program, there is a subtrace of the 8051 object code program where the 230 time differences match. Moreover, \emph{observable} parts of the 231 trace also match  these are the appearance of cost labels and 232 function calls and returns. 233 234 235 236 More formally, the definition of this statement in \matita{} is 237 \begin{lstlisting}[language=Grafite] 238 definition simulates := 239 $\lambda$p: compiler_output. 240 let initial_status := initialise_status $...$ (cm (c_labelled_object_code $...$ p)) in 241 $\forall$m1,m2. 242 measurable Clight_pcs (c_labelled_clight $...$ p) m1 m2 243 (stack_sizes (c_stack_cost $...$ p)) (c_max_stack $...$ p) $\rightarrow$ 244 $\forall$c1,c2. 245 clock_after Clight_pcs (c_labelled_clight $...$ p) m1 (c_clight_cost_map $...$ p) = OK $...$ c1 $\rightarrow$ 246 clock_after Clight_pcs (c_labelled_clight $...$ p) (m1+m2) (c_clight_cost_map $...$ p) = OK $...$ c2 $\rightarrow$ 247 $\exists$n1,n2. 248 observables Clight_pcs (c_labelled_clight $...$ p) m1 m2 = 249 observables (OC_preclassified_system (c_labelled_object_code $...$ p)) 250 (c_labelled_object_code $...$ p) n1 n2 251 $\wedge$ 252 clock ?? (execute (n1+n2) ? initial_status) = 253 clock ?? (execute n1 ? initial_status) + (c2c1). 254 \end{lstlisting} 255 where the \lstinline'measurable', \lstinline'clock_after' and 256 \lstinline'observables' definitions are generic definitions for multiple 257 languages; in this case the \lstinline'Clight_pcs' record applies them 258 to \textsf{Clight} programs. 259 260 There is a second part to the statement, which says that the initial 261 processing of the input program to produce the cost labelled version 262 does not affect the semantics of the program: 263 % Yes, I'm paraphrasing the result a tiny bit to remove the observe nonfunction 264 \begin{lstlisting}[language=Grafite] 265 $\forall$input_program,output. 266 compile input_program = return output $\rightarrow$ 267 not_wrong $...$ (exec_inf $...$ clight_fullexec input_program) $\rightarrow$ 268 sim_with_labels 269 (exec_inf $...$ clight_fullexec input_program) 270 (exec_inf $...$ clight_fullexec (c_labelled_clight $...$ output)) 271 \end{lstlisting} 272 That is, any successful compilation produces a labelled program that 273 has identical behaviour to the original, so long as there is no 274 `undefined behaviour'. 275 276 Note that this statement provides full functional correctness, including 277 preservation of (non)termination. The intensional result above does 278 not do this directly  it does not guarantee the same result or same 279 termination. There are two mitigating factors, however: first, to 280 prove the intensional property you need local simulation results  these 281 can be pieced together to form full behavioural equivalence, only time 282 constraints have prevented us from doing so. Second, if we wish to 283 confirm a result, termination, or nontermination we could add an 284 observable witness, such as a function that is only called if the 285 correct result is given. The intensional result guarantees that the 286 observable witness is preserved, so the program must behave correctly. 287 288 These two results are combined in the the \lstinline'correct' 289 theorem in the file \lstinline'correctness.ma'. 290 291 \section{Correctness statement for the frontend} 292 \label{sec:fegoals} 293 294 The essential parts of the intensional proof were outlined during work 295 on a toy compiler in Task 296 2.1~\cite{d2.1,springerlink:10.1007/9783642324697_3}. These are 297 \begin{enumerate} 298 \item functional correctness, in particular preserving the trace of 299 cost labels, 300 \item the \emph{soundness} and \emph{precision} of the cost labelling 301 on the object code, and 302 \item the timing analysis on the object code produces a correct 303 mapping from cost labels to time. 304 \end{enumerate} 305 306 However, that toy development did not include function calls. For the 307 full \cerco{} compiler we also need to maintain the invariant that 308 functions return to the correct program location in the caller, as we 309 mentioned in the previous section. During work on the backend timing 310 analysis (describe in more detail in the companion deliverable, D4.4) 311 the notion of a \emph{structured trace} was developed to enforce this 312 return property, and also most of the cost labelling properties too. 313 314 \begin{figure} 315 \begin{center} 316 \includegraphics[width=0.5\linewidth]{compiler.pdf} 317 \end{center} 318 \caption{The compiler and proof outline} 319 \label{fig:compiler} 320 \end{figure} 321 322 Jointly, we generalised the structured traces to apply to any of the 323 intermediate languages which have some idea of program counter. This means 324 that they are introduced part way through the compiler, see 325 Figure~\ref{fig:compiler}. Proving that a structured trace can be 326 constructed at \textsf{RTLabs} has several virtues: 327 \begin{itemize} 328 \item This is the first language where every operation has its own 329 unique, easily addressable, statement. 330 \item Function calls and returns are still handled implicitly in the 331 language and so the structural properties are ensured by the 332 semantics. 333 \item Many of the backend languages from \textsf{RTL} onwards share a common 334 core set of definitions, and using structured traces throughout 335 increases this uniformity. 336 \end{itemize} 337 338 \begin{figure} 339 \begin{center} 340 \includegraphics[width=0.6\linewidth]{strtraces.pdf} 341 \end{center} 342 \caption{Nesting of functions in structured traces} 343 \label{fig:strtrace} 344 \end{figure} 345 A structured trace is a mutually inductive data type which 346 contains the steps from a normal program trace, but arranged into a 347 nested structure which groups entire function calls together and 348 aggregates individual steps between cost labels (or between the final 349 cost label and the return from the function), see 350 Figure~\ref{fig:strtrace}. This captures the idea that the cost labels 351 only represent costs \emph{within} a function  calls to other 352 functions are accounted for in the nested trace for their execution, and we 353 can locally regard function calls as a single step. 354 355 These structured traces form the core part of the intermediate results 356 that we must prove so that the backend can complete the main 357 intensional result stated above. In full, we provide the backend 358 with 359 \begin{enumerate} 360 \item A normal trace of the \textbf{prefix} of the program's execution 361 before reaching the measurable subtrace. (This needs to be 362 preserved so that we know that the stack space consumed is correct, 363 and to set up the simulation results.) 364 \item The \textbf{structured trace} corresponding to the measurable 365 subtrace. 366 \item An additional property about the structured trace that no 367 `program counter' is \textbf{repeated} between cost labels. Together with 368 the structure in the trace, this takes over from showing that 369 cost labelling is sound and precise. 370 \item A proof that the \textbf{observables} have been preserved. 371 \item A proof that the \textbf{stack limit} is still observed by the prefix and 372 the structure trace. (This is largely a consequence of the 373 preservation of observables.) 374 \end{enumerate} 375 The \lstinline'front_end_correct' lemma in the 376 \lstinline'correctness.ma' file provides a record containing these. 377 378 Following the outline in Figure~\ref{fig:compiler}, we will first deal 379 with the transformations in \textsf{Clight} that produce the source 380 program with cost labels, then show that measurable traces can be 381 lifted to \textsf{RTLabs}, and finally show that we can construct the 382 properties listed above ready for the backend proofs. 383 384 \section{Input code to cost labelled program} 385 \label{sec:inputtolabelling} 386 387 As explained on page~\pageref{page:switchintro}, the costs of complex 388 C \lstinline[language=C]'switch' statements cannot be represented with 389 the simple labelling used in the formalised compiler. Our first pass 390 replaces these statements with simpler C code, allowing our second 391 pass to perform the cost labelling. We show that the behaviour of 392 programs is unchanged by these passes using forward 393 simulations\footnote{All of our languages are deterministic, which can 394 be seen directly from their executable definitions. Thus we know that 395 forward simulations are sufficient because the target cannot have any 396 other behaviour.}. 397 398 \subsection{Switch removal} 399 400 We compile away \lstinline[language=C]'switch' statements into more 401 basic \textsf{Clight} code. 402 Note that this transformation does not necessarily deteriorate the 403 efficiency of the generated code. For instance, compilers such as GCC 404 introduce balanced trees of ``ifthenelse'' constructs for small 405 switches. However, our implementation strategy is much simpler. Let 406 us consider the following input statement. 407 408 \begin{lstlisting}[language=C] 409 switch(e) { 410 case v1: 411 stmt1; 412 case v2: 413 stmt2; 414 default: 415 stmt_default; 416 } 417 \end{lstlisting} 418 419 Note that \textsf{stmt1}, \textsf{stmt2}, \ldots \textsf{stmt\_default} 420 may contain \lstinline[language=C]'break' statements, which have the 421 effect of exiting the switch statement. In the absence of break, the 422 execution falls through each case sequentially. In our implementation, 423 we produce an equivalent sequence of ``ifthen'' chained by gotos: 424 \begin{lstlisting}[language=C] 425 fresh = e; 426 if(fresh == v1) { 427 $\llbracket$stmt1$\rrbracket$; 428 goto lbl_case2; 429 }; 430 if(fresh == v2) { 431 lbl_case2: 432 $\llbracket$stmt2$\rrbracket$; 433 goto lbl_case2; 434 }; 435 $\llbracket$stmt_default$\rrbracket$; 436 exit_label: 437 \end{lstlisting} 438 439 The proof had to tackle the following points: 440 \begin{itemize} 441 \item the source and target memories are not the same (due to the fresh variable), 442 \item the flow of control is changed in a nonlocal way (e.g. \textbf{goto} 443 instead of \textbf{break}). 444 \end{itemize} 445 In order to tackle the first point, we implemented a version of memory 446 extensions similar to those of CompCert. 447 448 For the simulation we decided to prove a sufficient amount to give us 449 confidence in the definitions and approach, but to curtail the proof 450 because this pass does not contribute to the intensional correctness 451 result. We tackled several simple cases, that do not interact with 452 the switch removal per se, to show that the definitions were usable, 453 and part of the switch case to check that the approach is 454 reasonable. This comprises propagating the memory extension through 455 each statement (except switch), as well as various invariants that are 456 needed for the switch case (in particular, freshness hypotheses). The 457 details of the evaluation process for the source switch statement and 458 its target counterpart can be found in the file 459 \lstinline'switchRemoval.ma', along more details on the transformation 460 itself. 461 462 Proving the correctness of the second point would require reasoning on the 463 semantics of \lstinline[language=C]'goto' statements. In the \textsf{Clight} 464 semantics, this is implemented as a functionwide lookup of the target label. 465 The invariant we would need is the fact that a global label lookup on a freshly 466 created goto is equivalent to a local lookup. This would in turn require the 467 propagation of some freshness hypotheses on labels. As discussed, 468 we decided to omit this part of the correctness proof. 469 470 \subsection{Cost labelling} 471 472 The simulation for the cost labelling pass is the simplest in the 473 frontend. The main argument is that any step of the source program 474 is simulated by the same step of the labelled one, plus any extra 475 steps for the added cost labels. The extra instructions do not change 476 the memory or local environments, although we have to keep track of 477 the extra instructions that appear in continuations, for example 478 during the execution of a \lstinline[language=C]'while' loop. 479 480 We do not attempt to capture any cost properties of the labelling\footnote{We describe how the cost properties are 481 established in Section~\ref{sec:costchecks}.} in 482 the simulation proof, which allows the proof to be oblivious to the choice 483 of cost labels. Hence we do not have to reason about the threading of 484 name generation through the labelling function, greatly reducing the 485 amount of effort required. 486 487 %TODO: both give onestepsimbymany forward sim results; switch 488 %removal tricky, uses aux var to keep result of expr, not central to 489 %intensional correctness so curtailed proof effort once reasonable 490 %level of confidence in code gained; labelling much simpler; don't care 491 %what the labels are at this stage, just need to know when to go 492 %through extra steps. Rolled up into a single result with a cofixpoint 493 %to obtain coinductive statement of equivalence (show). 494 495 \section{Finding corresponding measurable subtraces} 496 \label{sec:measurablelifting} 497 498 There follow the three main passes of the frontend: 499 \begin{enumerate} 500 \item simplification of casts in \textsf{Clight} code 501 \item \textsf{Clight} to \textsf{Cminor} translation, performing stack 502 variable allocation and simplifying control structures 503 \item transformation to \textsf{RTLabs} control flow graph 504 \end{enumerate} 505 We have taken a common approach to 506 each pass: first we build (or axiomatise) forward simulation results 507 that are similar to normal compiler proofs, but which are slightly more 508 finegrained so that we can see that the call structure and relative 509 placement of cost labels is preserved. 510 511 Then we instantiate a general result which shows that we can find a 512 \emph{measurable} subtrace in the target of the pass that corresponds 513 to the measurable subtrace in the source. By repeated application of 514 this result we can find a measurable subtrace of the execution of the 515 \textsf{RTLabs} code, suitable for the construction of a structured 516 trace (see Section~\ref{sec:structuredtrace}). This is essentially an 517 extra layer on top of the simulation proofs that provides us with the 518 additional information required for our intensional correctness proof. 519 520 \subsection{Generic measurable subtrace lifting proof} 521 522 Our generic proof is parametrised on a record containing smallstep 523 semantics for the source and target language, a classification of 524 states (the same form of classification is used when defining 525 structured traces), a simulation relation which respects the 526 classification and cost labelling and 527 four simulation results. The simulations are split by the starting state's 528 classification and whether it is a cost label, which will allow us to 529 observe that the call structure is preserved. They are: 530 \begin{enumerate} 531 \item a step from a `normal' state (which is not classified as a call 532 or return) which is not a cost label is simulated by zero or more 533 `normal' steps; 534 \item a step from a `call' state followed by a cost label step is 535 simulated by a step from a `call' state, a corresponding label step, 536 then zero or more `normal' steps; 537 \item a step from a `call' state not followed by a cost label 538 similarly (note that this case cannot occur in a welllabelled 539 program, but we do not have enough information locally to exploit 540 this); and 541 \item a cost label step is simulated by a cost label step. 542 \end{enumerate} 543 Finally, we need to know that a successfully translated program will 544 have an initial state in the simulation relation with the original 545 program's initial state. 546 547 The backend has similar requirements for lifting simulations to 548 structured traces. Fortunately, our treatment of calls and returns 549 can be slightly simpler because we have special call and return states 550 that correspond to function entry and return that are separate from 551 the actual instructions. This was originally inherited from our port 552 of CompCert's \textsf{Clight} semantics, but proves useful here 553 because we only need to consider adding extra steps \emph{after} a 554 call or return state, because the instruction step deals with extra 555 steps that occur before. The backend makes all of the call and 556 return machinery explicit, and thus needs more complex statements 557 about extra steps before and after each call and return. 558 559 \begin{figure} 560 \begin{center} 561 \includegraphics[width=0.5\linewidth]{meassim.pdf} 562 \end{center} 563 \caption{Tiling of simulation for a measurable subtrace} 564 \label{fig:tiling} 565 \end{figure} 566 567 To find the measurable subtrace in the target program's execution we 568 walk along the original program's execution trace applying the 569 appropriate simulation result by induction on the number of steps. 570 While the number of steps taken varies, the overall structure is 571 preserved, as illustrated in Figure~\ref{fig:tiling}. By preserving 572 the structure we also maintain the same intensional observables. One 573 delicate point is that the cost label following a call must remain 574 directly afterwards\footnote{The prototype compiler allowed some 575 straightline code to appear before the cost label until a later 576 stage of the compiler, but we must move the requirement forward to 577 fit with the structured traces.} 578 % Damn it, I should have just moved the cost label forwards in RTLabs, 579 % like the prototype does in RTL to ERTL; the result would have been 580 % simpler. Or was there some reason not to do that? 581 (both in the program code and in the execution trace), even if we 582 introduce extra steps, for example to store parameters in memory in 583 \textsf{Cminor}. Thus we have a version of the call simulation 584 that deals with both the call and the cost label in one result. 585 586 In addition to the subtrace we are interested in measuring, we must 587 prove that the earlier part of the trace is also preserved in 588 order to use the simulation from the initial state. This proof also 589 guarantees that we do not run out of stack space before the subtrace 590 we are interested in. The lemmas for this prefix and the measurable 591 subtrace are similar, following the pattern above. However, the 592 measurable subtrace also requires us to rebuild the termination 593 proof. This is defined recursively: 594 \label{prog:terminationproof} 595 \begin{lstlisting}[language=Grafite] 596 let rec will_return_aux C (depth:nat) 597 (trace:list (cs_state $...$ C $\times$ trace)) on trace : bool := 598 match trace with 599 [ nil $\Rightarrow$ false 600  cons h tl $\Rightarrow$ 601 let $\langle$s,tr$\rangle$ := h in 602 match cs_classify C s with 603 [ cl_call $\Rightarrow$ will_return_aux C (S depth) tl 604  cl_return $\Rightarrow$ 605 match depth with 606 [ O $\Rightarrow$ match tl with [ nil $\Rightarrow$ true  _ $\Rightarrow$ false ] 607  S d $\Rightarrow$ will_return_aux C d tl 608 ] 609  _ $\Rightarrow$ will_return_aux C depth tl 610 ] 611 ]. 612 \end{lstlisting} 613 The \lstinline'depth' is the number of return states we need to see 614 before we have returned to the original function (initially zero) and 615 \lstinline'trace' the measurable subtrace obtained from the running 616 the semantics for the correct number of steps. This definition 617 unfolds tail recursively for each step, and once the corresponding 618 simulation result has been applied a new one for the target can be 619 asserted by unfolding and applying the induction hypothesis on the 620 shorter trace. 621 622 Combining the lemmas about the prefix and the measurable subtrace 623 requires a little care because the states joining the two might not be 624 related in the simulation. In particular, if the measurable subtrace 625 starts from the cost label at the beginning of the function there may 626 be some extra instructions in the target code to execute to complete 627 function entry before the states are back in the relation. Hence we 628 carefully phrased the lemmas to allow for such extra steps. 629 630 Together, these then gives us an overall result for any simulation fitting the 631 requirements above (contained in the \lstinline'meas_sim' record): 632 \begin{lstlisting}[language=Grafite] 633 theorem measured_subtrace_preserved : 634 $\forall$MS:meas_sim. 635 $\forall$p1,p2,m,n,stack_cost,max. 636 ms_compiled MS p1 p2 $\rightarrow$ 637 measurable (ms_C1 MS) p1 m n stack_cost max $\rightarrow$ 638 $\exists$m',n'. 639 measurable (ms_C2 MS) p2 m' n' stack_cost max $\wedge$ 640 observables (ms_C1 MS) p1 m n = observables (ms_C2 MS) p2 m' n'. 641 \end{lstlisting} 642 The stack space requirement that is embedded in \lstinline'measurable' 643 is a consequence of the preservation of observables, because it is 644 determined by the functions called and returned from, which are observable. 645 646 \subsection{Simulation results for each pass} 647 648 We now consider the simulation results for the passes, each of which 649 is used to instantiate the 650 \lstinline[language=Grafite]'measured_subtrace_preserved' theorem to 651 construct the measurable subtrace for the next language. 652 653 \subsubsection{Cast simplification} 654 655 The parser used in \cerco{} introduces a lot of explicit type casts. 656 If left as they are, these constructs can greatly hamper the 657 quality of the generated code  especially as the architecture 658 we consider is an $8$bit one. In \textsf{Clight}, casts are 659 expressions. Hence, most of the work of this transformation 660 proceeds on expressions. The transformation proceeds by recursively 661 trying to coerce an expression to a particular integer type, which 662 is in practice smaller than the original one. This functionality 663 is implemented by two mutually recursive functions whose signature 664 is the following. 665 666 \begin{lstlisting}[language=Grafite] 667 let rec simplify_expr (e:expr) (target_sz:intsize) (target_sg:signedness) 668 : $\Sigma$result:bool$\times$expr. 669 $\forall$ge,en,m. simplify_inv ge en m e (\snd result) target_sz target_sg (\fst result) := $\ldots$ 670 671 and simplify_inside (e:expr) : $\Sigma$result:expr. conservation e result := $\ldots$ 672 \end{lstlisting} 673 674 The \textsf{simplify\_inside} acts as a wrapper for 675 \textsf{simplify\_expr}. Whenever \textsf{simplify\_inside} encounters 676 a \textsf{Ecast} expression, it tries to coerce the subexpression 677 to the desired type using \textsf{simplify\_expr}, which tries to 678 perform the actual coercion. In return, \textsf{simplify\_expr} calls 679 back \textsf{simplify\_inside} in some particular positions, where we 680 decided to be conservative in order to simplify the proofs. However, 681 the current design allows to incrementally revert to a more aggressive 682 version, by replacing recursive calls to \textsf{simplify\_inside} by 683 calls to \textsf{simplify\_expr} \emph{and} proving the corresponding 684 invariants  where possible. 685 686 The \textsf{simplify\_inv} invariant encodes either the conservation 687 of the semantics during the transformation corresponding to the failure 688 of the coercion (\textsf{Inv\_eq} constructor), or the successful 689 downcast of the considered expression to the target type 690 (\textsf{Inv\_coerce\_ok}). 691 692 \begin{lstlisting}[language=Grafite] 693 inductive simplify_inv 694 (ge : genv) (en : env) (m : mem) 695 (e1 : expr) (e2 : expr) (target_sz : intsize) (target_sg : signedness) : bool $\rightarrow$ Prop := 696  Inv_eq : $\forall$result_flag. $\ldots$ 697 simplify_inv ge en m e1 e2 target_sz target_sg result_flag 698  Inv_coerce_ok : $\forall$src_sz,src_sg. 699 typeof e1 = Tint src_sz src_sg $\rightarrow$ 700 typeof e2 = Tint target_sz target_sg $\rightarrow$ 701 smaller_integer_val src_sz target_sz src_sg (exec_expr ge en m e1) (exec_expr ge en m e2) $\rightarrow$ 702 simplify_inv ge en m e1 e2 target_sz target_sg true. 703 \end{lstlisting} 704 705 The \textsf{conservation} invariant for \textsf{simplify\_inside} simply states the conservation 706 of the semantics, as in the \textsf{Inv\_eq} constructor of the previous 707 invariant. 708 709 \begin{lstlisting}[language=Grafite] 710 definition conservation := $\lambda$e,result. $\forall$ge,en,m. 711 res_sim ? (exec_expr ge en m e) (exec_expr ge en m result) 712 $\wedge$ res_sim ? (exec_lvalue ge en m e) (exec_lvalue ge en m result) 713 $\wedge$ typeof e = typeof result. 714 \end{lstlisting} 715 716 This invariant is then easily lifted to statement evaluations. 717 The main problem encountered with this particular pass was dealing with 718 inconsistently typed programs, a canonical case being a particular 719 integer constant of a certain size typed with another size. This 720 prompted the need to introduce numerous type checks, making 721 both the implementation and the proof more complex, even though more 722 comprehensive checks are made in the next stage. 723 %\todo{Make this a particular case of the more general statement on baking more invariants in the Clight language} 724 725 \subsubsection{Clight to Cminor} 726 727 This pass is the last one operating on the \textsf{Clight} language. 728 Its input is a full \textsf{Clight} program, and its output is a 729 \textsf{Cminor} program. Note that we do not use an equivalent of 730 CompCert's \textsf{C\#minor} language: we translate directly to a 731 variant of \textsf{Cminor}. This presents the advantage of not 732 requiring the special loop constructs, nor the explicit block 733 structure. Another salient point of our approach is that a significant 734 number of the properties needed for the simulation proof were directly 735 encoded in dependently typed translation functions. In particular, 736 freshness conditions and welltypedness conditions are included. The 737 main effects of the transformation from \textsf{Clight} to 738 \textsf{Cminor} are listed below. 739 740 \begin{itemize} 741 \item Variables are classified as being either globals, stackallocated 742 locals or potentially registerallocated locals. The value of registerallocated 743 local variables is moved out of the modelled memory and stored in a 744 dedicated environment. 745 \item In \textsf{Clight}, each local variable has a dedicated memory block, whereas 746 stackallocated locals are bundled together on a functionbyfunction basis. 747 \item Loops are converted to jumps. 748 \end{itemize} 749 750 The first two points require memory injections which are more flexible that those 751 needed in the switch removal case. In the remainder of this section, we briefly 752 discuss our implementation of memory injections, and then the simulation proof. 753 754 \paragraph{Memory injections.} 755 756 Our memory injections are modelled after the work of Blazy \& Leroy. 757 However, the corresponding paper is based on the first version of the 758 CompCert memory model~\cite{2008LeroyBlazymemorymodel}, whereas we use a much more concrete model, allowing bytelevel 759 manipulations (as in the later version of CompCert's memory model). We proved 760 roughly 80 \% of the required lemmas. Notably, some of the difficulties encountered were 761 due to overly relaxed conditions on pointer validity (fixed during development). 762 Some more side conditions had to be added to take care of possible overflows when converting 763 from \textbf{Z} block bounds to $16$ bit pointer offsets (in practice, such overflows 764 only occur in edge cases that are easily ruled out  but this fact is not visible 765 in memory injections). Concretely, some of the lemmas on the preservation of simulation of 766 loads after writes were axiomatised, due to a lack of time. 767 768 \paragraph{Simulation proof.} 769 770 We proved the simulation result for expressions and a representative 771 selection of statements. In particular we tackled 772 \lstinline[language=C]'while' statements to ensure that we correctly 773 translate loops because our approach differs from CompCert by 774 converting directly to \textsf{Cminor} \lstinline[language=C]'goto's 775 rather than maintaining a notion of loop in \textsf{Cminor}. We also have a partial 776 proof for function entry, covering the setup of the memory injection, 777 but not function exit. Exits, and the remaining statements, have been 778 axiomatised. 779 780 Careful management of the proof state was required because proof terms 781 are embedded in \textsf{Cminor} code to show that invariants are 782 respected. These proof terms appear in the proof state when inverting 783 the translation functions, and they can be large and awkward. While 784 generalising them away is usually sufficient, it can be difficult when 785 they appear under a binder. 786 787 %The correctness proof for this transformation was not completed. We proved the 788 %simulation result for expressions and for some subset of the critical statement cases. 789 %Notably lacking are the function entry and exit, where the memory injection is 790 %properly set up. As would be expected, a significant amount of work has to be performed 791 %to show the conservation of all invariants at each simulation step. 792 793 %\todo{list cases, explain while loop, explain labeling problem} 794 795 \subsubsection{Cminor to RTLabs} 796 797 The translation from \textsf{Cminor} to \textsf{RTLabs} is a fairly 798 routine control flow graph (CFG) construction. As such, we chose to 799 axiomatise the associated extensional simulation results. However, we did prove several 800 properties of the generated programs: 801 \begin{itemize} 802 \item All statements are type correct with respect to the declared 803 pseudoregister type environment. 804 \item The CFG is closed, and has a distinguished entry node and a 805 unique exit node. 806 \end{itemize} 807 808 These properties rely on similar properties about type safety and the 809 presence of \lstinline[language=C]'goto'labels for \textsf{Cminor} programs 810 which are checked at the preceding stage. As a result, this 811 transformation is total and any compilation failures must occur when 812 the corresponding \textsf{Clight} source is available and a better 813 error message can be generated. 814 815 The proof obligations for these properties include many instances of 816 graph inclusion. We automated these proofs using a small amount of 817 reflection, making the obligations much easier to handle. One 818 drawback to enforcing invariants throughout is that temporarily 819 breaking them can be awkward. For example, \lstinline'return' 820 statements were originally used as placeholders for 821 \lstinline[language=C]'goto' destinations that had not yet been 822 translated. However, this made establishing the single exit node 823 property rather difficult, and a different placeholder was chosen 824 instead. In other circumstances it is possible to prove a more 825 complex invariant then simplify it at the end of the transformation. 826 827 \section{Checking cost labelling properties} 828 \label{sec:costchecks} 829 830 Ideally, we would provide proofs that the cost labelling pass always 831 produces programs that are soundly and precisely labelled and that 832 each subsequent pass preserves these properties. This would match our 833 use of dependent types to eliminate impossible sources of errors 834 during compilation, in particular retaining intermediate language type 835 information. 836 837 However, given the limited amount of time available we realised that 838 implementing a compiletime check for a sound and precise labelling of 839 the \textsf{RTLabs} intermediate code would reduce the proof burden 840 considerably. This is similar in spirit to the use of translation 841 validation in certified compilation, which makes a similar tradeoff 842 between the potential for compiletime failure and the volume of proof 843 required. 844 845 The check cannot be pushed into a later stage of the compiler because 846 much of the information is embedded into the structured traces. 847 However, if an alternative method was used to show that function 848 returns in the compiled code are sufficiently wellbehaved, then we 849 could consider pushing the cost property checks into the timing 850 analysis itself. We leave this as a possible area for future work. 851 852 \subsection{Implementation and correctness} 853 \label{sec:costchecksimpl} 854 855 For a cost labelling to be sound and precise we need a cost label at 856 the start of each function, after each branch and at least one in 857 every loop. The first two parts are trivial to check by examining the 858 code. In \textsf{RTLabs} the last part is specified by saying 859 that there is a bound on the number of successive instruction nodes in 860 the CFG that you can follow before you encounter a cost label, and 861 checking this is more difficult. 862 863 The implementation progresses through the set of nodes in the graph, 864 following successors until a cost label is found or a labelfree cycle 865 is discovered (in which case the property does not hold and we return 866 an error). This is made easier by the prior knowledge that every 867 successor of a branch instruction is a cost label, so we do not need 868 to search each branch. When a label is found, we remove the chain of 869 program counters from the set and continue from another node in the 870 set until it is empty, at which point we know that there is a bound 871 for every node in the graph. 872 873 Directly reasoning about the function that implements this procedure would be 874 rather awkward, so an inductive specification of a single step of its 875 behaviour was written and proved to match the implementation. This 876 was then used to prove the implementation sound and complete. 877 878 While we have not attempted to prove that the cost labelled properties 879 are established and preserved earlier in the compiler, we expect that 880 the effort for the \textsf{Cminor} to \textsf{RTLabs} stage alone 881 would be similar to the work outlined above, because it involves the 882 change from requiring a cost label at particular positions to 883 requiring cost labels to break loops in the CFG. As there are another 884 three passes to consider (including the labelling itself), we believe 885 that using the check above is much simpler overall. 886 887 % TODO? Found some Clight to Cminor bugs quite quickly 888 889 \section{Existence of a structured trace} 890 \label{sec:structuredtrace} 891 892 The \emph{structured trace} idea introduced in 893 Section~\ref{sec:fegoals} enriches the execution trace of a program by 894 nesting function calls in a mixedstep style and embedding the cost 895 labelling properties of the program. See Figure~\ref{fig:strtrace} on 896 page~\pageref{fig:strtrace} for an illustration of a structured trace. 897 It was originally designed to support the proof of correctness for the 898 timing analysis of the object code in the backend, then generalised 899 to provide a common structure to use from the end of the frontend to 900 the object code. 901 902 To make the definition generic we abstract over the semantics of the 903 language, 904 \begin{lstlisting}[language=Grafite] 905 record abstract_status : Type[1] := 906 { as_status :> Type[0] 907 ; as_execute : relation as_status 908 ; as_pc : DeqSet 909 ; as_pc_of : as_status $\rightarrow$ as_pc 910 ; as_classify : as_status $\rightarrow$ status_class 911 ; as_label_of_pc : as_pc $\rightarrow$ option costlabel 912 ; as_after_return : ($\Sigma$s:as_status. as_classify s = cl_call) $\rightarrow$ as_status $\rightarrow$ Prop 913 ; as_result: as_status $\rightarrow$ option int 914 ; as_call_ident : ($\Sigma$s:as_status.as_classify s = cl_call) $\rightarrow$ ident 915 ; as_tailcall_ident : ($\Sigma$s:as_status.as_classify s = cl_tailcall) $\rightarrow$ ident 916 }. 917 \end{lstlisting} 918 which requires a type of states, an execution relation\footnote{All of 919 our semantics are executable, but using a relation was simpler in 920 the abstraction.}, some notion of abstract 921 program counter with decidable equality, the classification of states, 922 and functions to extract the observable intensional information (cost 923 labels and the identity of functions that are called). The 924 \lstinline'as_after_return' property links the state before a function 925 call with the state after return, providing the evidence that 926 execution returns to the correct place. The precise form varies 927 between stages; in \textsf{RTLabs} it insists the CFG, the pointer to 928 the CFG node to execute next, and some call stack information is 929 preserved. 930 931 The structured traces are defined using three mutually inductive 932 types. The core data structure is \lstinline'trace_any_label', which 933 captures some straightline execution until the next cost label or the 934 return from the enclosing function. Any function calls are embedded as 935 a single step, with its own trace nested inside and the before and 936 after states linked by \lstinline'as_after_return'; and states 937 classified as a `jump' (in particular branches) must be followed by a 938 cost label. 939 940 The second type, \lstinline'trace_label_label', is a 941 \lstinline'trace_any_label' where the initial state is cost labelled. 942 Thus a trace in this type identifies a series of steps whose cost is 943 entirely accounted for by the label at the start. 944 945 Finally, \lstinline'trace_label_return' is a sequence of 946 \lstinline'trace_label_label' values which end in the return from the 947 function. These correspond to a measurable subtrace, and in 948 particular include executions of an entire function call (and so are 949 used for the nested calls in \lstinline'trace_any_label'). 950 951 \subsection{Construction} 952 953 The construction of the structured trace replaces syntactic cost 954 labelling properties, which place requirements on where labels appear 955 in the program, with semantic properties that constrain the execution 956 traces of the program. The construction begins by defining versions 957 of the sound and precise labelling properties on states and global 958 environments (for the code that appears in each of them) rather than 959 whole programs, and showing that these are preserved by steps of the 960 \textsf{RTLabs} semantics. 961 962 Then we show that each cost labelling property required by the 963 definition of structured traces is locally satisfied. These proofs are 964 broken up by the classification of states. Similarly, we prove a 965 stepbystep stack preservation result, which states that the 966 \textsf{RTLabs} semantics never changes the lower parts of the stack. 967 968 The core part of the construction of a structured trace is to use the 969 proof of termination from the measurable trace (defined on 970 page~\pageref{prog:terminationproof}) to `fold up' the execution into 971 the nested form. The results outlined above fill in the proof 972 obligations for the cost labelling properties and the stack 973 preservation result shows that calls return to the correct location. 974 975 The structured trace alone is not sufficient to capture the property 976 that the program is soundly labelled. While the structured trace 977 guarantees termination, it still permits a loop to be executed a 978 finite number of times without encountering a cost label. We 979 eliminate this by proving that no `program counter' repeats within any 980 \lstinline'trace_any_label' section by showing that it is incompatible 981 with the property that there is a bound on the number of successor 982 instructions you can follow in the CFG before you encounter a cost 983 label (from Section~\ref{sec:costchecksimpl}). 984 985 \subsubsection{Complete execution structured traces} 986 987 The development of the construction above started relatively early, 988 before the measurable subtrace preservation proofs. To be confident 989 that the traces were wellformed at that time, we also developed a 990 complete execution form that embeds the traces above. This includes 991 nonterminating program executions, where an infinite number of the terminating 992 structured traces are embedded. This construction confirmed that our 993 definition of structured traces was consistent, although we later 994 found that we did not require the whole execution version for the 995 compiler correctness results. 996 997 To construct these we need to know whether each function call will 998 eventually terminate, requiring the use of the excluded middle. This 999 classical reasoning is local to the construction of whole program 1000 traces and is not necessary for our main results. 1001 1002 \section{Conclusion} 1003 1004 In combination with the work on the CerCo backend and by 1005 concentrating on the novel intensional parts of the proof, we have 1006 shown that it is possible to construct certifying compilers that 1007 correctly report execution time and stack space costs. The layering 1008 of intensional correctness proofs on top of normal simulation results 1009 provides a useful separation of concerns, and could permit the reuse 1010 of existing results. 1011 1012 \section{Files} 1013 1014 The following table gives a highlevel overview of the \matita{} 1015 source files in Deliverable 3.4: 1016 1017 \bigskip 1018 1019 \begin{tabular}{rp{.7\linewidth}} 1020 \lstinline'compiler.ma' & Toplevel compiler definitions, in particular 1021 \lstinline'front_end', and the whole compiler definition 1022 \lstinline'compile'. \\ 1023 \lstinline'correctness.ma' & Correctness results: \lstinline'front_end_correct' 1024 and \lstinline'correct', respectively. \\ 1025 \lstinline'Clight/*' & \textsf{Clight}: proofs for switch 1026 removal, cost labelling, cast simplification and conversion to 1027 \textsf{Cminor}. \\ 1028 \lstinline'Cminor/*' & \textsf{Cminor}: axioms of conversion to 1029 \textsf{RTLabs}. \\ 1030 \lstinline'RTLabs/*' & \textsf{RTLabs}: definitions and proofs for 1031 compiletime cost labelling checks, construction of structured traces. 1032 \\ 1033 \lstinline'common/Measurable.ma' & Definitions for measurable 1034 subtraces. \\ 1035 \lstinline'common/FEMeasurable.ma' & Generic measurable subtrace 1036 lifting proof. \\ 1037 \lstinline'common/*' & Other common definitions relevant to many parts 1038 of the compiler and proof. \\ 1039 \lstinline'utilities/*' & General purpose definitions used throughout, 1040 including extensions to the standard \matita{} library. 1041 \end{tabular} 10 1042 11 1043 \subsection{A brief overview of the backend compilation chain} … … 80 1112 81 1113 Our abstraction takes the following form: 82 \begin{lstlisting} 1114 \begin{lstlisting}[language=Grafite] 83 1115 inductive joint_instruction (p: params__) (globals: list ident): Type[0] := 84 1116  COMMENT: String $\rightarrow$ joint_instruction p globals … … 86 1118  INT: generic_reg p $\rightarrow$ Byte $\rightarrow$ joint_instruction p globals 87 1119 ... 88  OP1: Op1 $ 89 ightarrow$ acc_a_reg p $ 90 ightarrow$ acc_a_reg p $ 91 ightarrow$ joint_instruction p globals 1120  OP1: Op1 $\rightarrow$ acc_a_reg p $\rightarrow$ acc_a_reg p $\rightarrow$ joint_instruction p globals 92 1121 ... 93 1122  extension: extend_statements p $\rightarrow$ joint_instruction p globals. … … 106 1135 As \texttt{p} varies between intermediate languages, we can provide language specific extensions to the syntax of the joint language. 107 1136 For example, ERTL's extended syntax consists of the following extra statements: 108 \begin{lstlisting} 1137 \begin{lstlisting}[language=Grafite] 109 1138 inductive ertl_statement_extension: Type[0] := 110 1139  ertl_st_ext_new_frame: ertl_statement_extension … … 113 1142 \end{lstlisting} 114 1143 These are further packaged into an ERTL specific instance of \texttt{params\_\_} as follows: 115 \begin{lstlisting} 1144 \begin{lstlisting}[language=Grafite] 116 1145 definition ertl_params__: params__ := 117 1146 mk_params__ register register ... ertl_statement_extension. … … 129 1158 130 1159 Our joint internal function record looks like so: 131 \begin{lstlisting} 1160 \begin{lstlisting}[language=Grafite] 132 1161 record joint_internal_function (globals: list ident) (p:params globals) : Type[0] := 133 1162 { … … 150 1179 The `joint' syntax makes this especially clear. 151 1180 For instance, in the definition: 152 \begin{lstlisting} 1181 \begin{lstlisting}[language=Grafite] 153 1182 inductive joint_instruction (p:params__) (globals: list ident): Type[0] := 154 1183 ... … … 232 1261 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. 233 1262 For instance, the \texttt{translate\_negint} function, which translates a negative integer constant: 234 \begin{lstlisting} 1263 \begin{lstlisting}[language=Grafite] 235 1264 definition translate_negint := 236 1265 $\lambda$globals: list ident. … … 251 1280 Practically, we would always like to ensure that the entry and exit labels are present in the statement graph. 252 1281 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: 253 \begin{lstlisting} 1282 \begin{lstlisting}[language=Grafite] 254 1283 record joint_internal_function (globals: list ident) (p: params globals): Type[0] := 255 1284 { … … 389 1418 We begin the abstraction process with the \texttt{params\_\_} record. 390 1419 This holds the types of the representations of the different register varieties in the intermediate languages: 391 \begin{lstlisting} 1420 \begin{lstlisting}[language=Grafite] 392 1421 record params__: Type[1] := 393 1422 { … … 421 1450 422 1451 As mentioned in the report for Deliverable D4.2, the record \texttt{params\_\_} is enough to be able to specify the instructions of the joint languages: 423 \begin{lstlisting} 1452 \begin{lstlisting}[language=Grafite] 424 1453 inductive joint_instruction (p: params__) (globals: list ident): Type[0] := 425 1454  COMMENT: String $\rightarrow$ joint_instruction p globals … … 435 1464 Naturally, as some intermediate languages are graph based, and others linearised, the passing act of passing control on to the `successor' instruction can either be the act of following a graph edge in a control flow graph, or incrementing an index into a list. 436 1465 We make a distinction between instructions that pass control onto their immediate successors, and those that jump elsewhere in the program, through the use of \texttt{succ}, denoting the immediate successor of the current instruction, in the \texttt{params\_} record described below. 437 \begin{lstlisting} 1466 \begin{lstlisting}[language=Grafite] 438 1467 record params_: Type[1] := 439 1468 { … … 444 1473 The type \texttt{succ} corresponds to labels, in the case of control flow graph based languages, or is instantiated to the unit type for the linearised language, LIN. 445 1474 Using \texttt{param\_} we can define statements of the joint language: 446 \begin{lstlisting} 1475 \begin{lstlisting}[language=Grafite] 447 1476 inductive joint_statement (p:params_) (globals: list ident): Type[0] := 448 1477  sequential: joint_instruction p globals $\rightarrow$ succ p $\rightarrow$ joint_statement p globals … … 457 1486 For the semantics, we need further parametererised types. 458 1487 In particular, we parameterise the result and parameter type of an internal function call in \texttt{params0}: 459 \begin{lstlisting} 1488 \begin{lstlisting}[language=Grafite] 460 1489 record params0: Type[1] := 461 1490 { … … 468 1497 469 1498 We further extend \texttt{params0} with a type for local variables in internal function calls: 470 \begin{lstlisting} 1499 \begin{lstlisting}[language=Grafite] 471 1500 record params1 : Type[1] := 472 1501 { … … 478 1507 Further, we hypothesise a generic method for looking up the next instruction in the graph, called \texttt{lookup}. 479 1508 Note that \texttt{lookup} may fail, and returns an \texttt{option} type: 480 \begin{lstlisting} 1509 \begin{lstlisting}[language=Grafite] 481 1510 record params (globals: list ident): Type[1] := 482 1511 { … … 492 1521 In particular, we have a description of the result, parameters and the local variables of a function. 493 1522 Note also that we have lifted the hypothesised \texttt{lookup} function from \texttt{params} into a dependent sigma type, which combines a label (the entry and exit points of the control flow graph or list) combined with a proof that the label is in the graph structure: 494 \begin{lstlisting} 1523 \begin{lstlisting}[language=Grafite] 495 1524 record joint_internal_function (globals: list ident) (p:params globals) : Type[0] := 496 1525 { … … 509 1538 The reason is because some intermediate languages share a host of parameters, and only differ on some others. 510 1539 For instance, in instantiating the ERTL language, certain parameters are shared with RTL, whilst others are ERTL specific: 511 \begin{lstlisting} 1540 \begin{lstlisting}[language=Grafite] 512 1541 ... 513 1542 definition ertl_params__: params__ := … … 516 1545 ... 517 1546 definition ertl_params1: params1 := rtl_ertl_params1 ertl_params0. 518 definition ertl_params: ∀globals. params globals := rtl_ertl_params ertl_params0.1547 definition ertl_params: $\forall$globals. params globals := rtl_ertl_params ertl_params0. 519 1548 ... 520 1549 definition ertl_statement := joint_statement ertl_params_. … … 524 1553 \end{lstlisting} 525 1554 Here, \texttt{rtl\_ertl\_params1} are the common parameters of the ERTL and RTL languages: 526 \begin{lstlisting} 1555 \begin{lstlisting}[language=Grafite] 527 1556 definition rtl_ertl_params1 := $\lambda$pars0. mk_params1 pars0 (list register). 528 1557 \end{lstlisting} 529 1558 530 1559 The record \texttt{more\_sem\_params} bundles together functions that store and retrieve values in various forms of register: 531 \begin{lstlisting} 1560 \begin{lstlisting}[language=Grafite] 532 1561 record more_sem_params (p:params_): Type[1] := 533 1562 { … … 565 1594 566 1595 We extend \texttt{more\_sem\_params} with yet more parameters via \texttt{more\_sem\_params2}: 567 \begin{lstlisting} 1596 \begin{lstlisting}[language=Grafite] 568 1597 record more_sem_params1 (globals: list ident) (p: params globals) : Type[1] := 569 1598 { … … 605 1634 We bundle \texttt{params} and \texttt{sem\_params} together into a single record. 606 1635 This will be used in the function \texttt{eval\_statement} which executes a single statement of the joint language: 607 \begin{lstlisting} 1636 \begin{lstlisting}[language=Grafite] 608 1637 record sem_params2 (globals: list ident): Type[1] := 609 1638 { … … 614 1643 \noindent 615 1644 The \texttt{state} record holds the current state of the interpreter: 616 \begin{lstlisting} 1645 \begin{lstlisting}[language=Grafite] 617 1646 record state (p: sem_params): Type[0] := 618 1647 { … … 632 1661 633 1662 We use the function \texttt{eval\_statement} to evaluate a single joint statement: 634 \begin{lstlisting} 1663 \begin{lstlisting}[language=Grafite] 635 1664 definition eval_statement: 636 1665 $\forall$globals: list ident.$\forall$p:sem_params2 globals. … … 663 1692 We call this function \texttt{return} and say that it must have type $\alpha \rightarrow M \alpha$, where $M$ is the name of the monad. 664 1693 In our example, the `lifting' function for the \texttt{option} monad can be implemented as: 665 \begin{lstlisting} 1694 \begin{lstlisting}[language=Grafite] 666 1695 let return x = Some x 667 1696 \end{lstlisting} … … 671 1700 We can see that bind `unpacks' a monadic value, applies a function after unpacking, and `repacks' the new value in the monad. 672 1701 In our example, the sequencing function for the \texttt{option} monad can be implemented as: 673 \begin{lstlisting} 1702 \begin{lstlisting}[language=Grafite] 674 1703 let bind o f = 675 1704 match o with … … 687 1716 Here, the monad's sequencing operation ensures that cost label emissions and function calls are maintained in the correct order. 688 1717 We have already seen how the \texttt{eval\_statement} function of the joint language is monadic, with type: 689 \begin{lstlisting} 1718 \begin{lstlisting}[language=Grafite] 690 1719 definition eval_statement: 691 1720 $\forall$globals: list ident.$\forall$p:sem_params2 globals. … … 695 1724 If we examine the body of \texttt{eval\_statement}, we may also see how the monad sequences effects. 696 1725 For instance, in the case for the \texttt{LOAD} statement, we have the following: 697 \begin{lstlisting} 1726 \begin{lstlisting}[language=Grafite] 698 1727 definition eval_statement: 699 1728 $\forall$globals: list ident. $\forall$p:sem_params2 globals. … … 702 1731 ... 703 1732 match s with 704  LOAD dst addrl addrh ⇒1733  LOAD dst addrl addrh $\Rightarrow$ 705 1734 ! vaddrh $\leftarrow$ dph_retrieve ... st addrh; 706 1735 ! vaddrl $\leftarrow$ dpl_retrieve ... st addrl; … … 713 1742 Here, we employ a certain degree of syntactic sugaring. 714 1743 The syntax 715 \begin{lstlisting} 1744 \begin{lstlisting}[language=Grafite] 716 1745 ... 717 1746 ! vaddrh $\leftarrow$ dph_retrieve ... st addrh; … … 721 1750 is sugaring for the \texttt{IO} monad's binding operation. 722 1751 We can expand this sugaring to the following much more verbose code: 723 \begin{lstlisting} 1752 \begin{lstlisting}[language=Grafite] 724 1753 ... 725 1754 bind (dph_retrieve ... st addrh) ($\lambda$vaddrh. bind (dpl_retrieve ... st addrl)
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