source: Papers/itp-2013/ccexec2.tex @ 3356

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[3343]12% \usepackage{amsmath}
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106\def\L{\mathrel{\mathcal L}}
107\def\S{\mathrel{\mathcal S}}
108\def\R{\mathrel{\mathcal R}}
109\def\C{\mathrel{\mathcal C}}
[3347]111\def\Labels{\mathbb L}
112\def\Functions{\mathbb F}
115\savebox{\execbox}{\tikz[baseline=-.5ex]\draw [-stealth] (0,0) -- ++(1em, 0);}
141\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
142Emerging Technologies (FET) programme within the Seventh Framework
143Programme for Research of the European Commission, under FET-Open grant
144number: 243881}}
145\author{Paolo Tranquilli \and Claudio Sacerdoti Coen}
146\institute{Department of Computer Science and Engineering, University of Bologna,\\\email{}, \email{}}
149The labelling approach is a technique to lift cost models for non-functional
150properties of programs from the object code to the source code. It is based
151on the preservation of the structure of the high level program in every
152intermediate language used by the compiler. Such structure is captured by
153observables that are added to the semantics and that needs to be preserved
154by the forward simulation proof of correctness of the compiler. Additional
155special observables are required for function calls. In this paper we
156present a generic forward simulation proof that preserves all these observables.
157The proof statement is based on a new mechanised semantics that traces the
158structure of execution when the language is unstructured. The generic semantics
159and simulation proof have been mechanised in the interactive theorem prover
164The \emph{labelling approach} has been introduced in~\cite{easylabelling} as
165a technique to \emph{lift} cost models for non-functional properties of programs
166from the object code to the source code. Examples of non-functional properties
167are execution time, amount of stack/heap space consumed and energy required for
[3355]168communication. The basic premise of the approach is that it is impossible to
[3343]169provide a \emph{uniform} cost model for an high level language that is preserved
170\emph{precisely} by a compiler. For instance, two instances of an assignment
171$x = y$ in the source code can be compiled very differently according to the
172place (registers vs stack) where $x$ and $y$ are stored at the moment of
173execution. Therefore a precise cost model must assign a different cost
174to every occurrence, and the exact cost can only be known after compilation.
176According to the labelling approach, the compiler is free to compile and optimise
177the source code without any major restriction, but it must keep trace
178of what happens to basic blocks during the compilation. The cost model is
179then computed on the object code. It assigns a cost to every basic block.
180Finally, the compiler propagates back the cost model to the source level,
181assigning a cost to each basic block of the source code.
183Implementing the labelling approach in a certified compiler
184allows to reason formally on the high level source code of a program to prove
185non-functional properties that are granted to be preserved by the compiler
186itself. The trusted code base is then reduced to 1) the interactive theorem
187prover (or its kernel) used in the certification of the compiler and
1882) the software used to certify the property on the source language, that
189can be itself certified further reducing the trusted code base.
190In~\cite{easylabelling} the authors provide an example of a simple
191certified compiler that implements the labelling approach for the
[3347]192imperative \verb+While+ language~\cite{while}, that does not have
[3343]193pointers and function calls.
195The labelling approach has been shown to scale to more interesting scenarios.
196In particular in~\cite{functionallabelling} it has been applied to a functional
197language and in~\cite{loopoptimizations} it has been shown that the approach
198can be slightly complicated to handle loop optimisations and, more generally,
199program optimisations that do not preserve the structure of basic blocks.
200On-going work also shows that the labelling approach is also compatible with
201the complex analyses required to obtain a cost model for object code
202on processors that implement advanced features like pipelining, superscalar
203architectures and caches.
[3355]205In the European Project CerCo (Certified Complexity\footnote{\url{}})~\cite{cerco} we are certifying a labelling approach based compiler for a large subset of C to
[3343]2068051 object code. The compiler is
207moderately optimising and implements a compilation chain that is largely
208inspired 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
209of function calls. Surprisingly, the addition of function calls require a
210revisitation of the proof technique given in~\cite{easylabelling}. In
211particular, at the core of the labelling approach there is a forward
[3347]212simulation proof that, in the case of \verb+While+, is only minimally
[3343]213more complex than the proof required for the preservation of the
214functional properties only. In the case of a programming language with
215function calls, instead, it turns out that the forward simulation proof for
[3355]216the back-end languages, which are unstructured, must grant a whole new set of invariants.
218In this paper we present a formalisation in the Matita interactive theorem
219prover~\cite{matita1,matita2} of a generic version of the simulation proof required for unstructured
220languages. All back-end languages of the CerCo compiler are unstructured
221languages, so the proof covers half of the correctness of the compiler.
222The statement of the generic proof is based on a new semantics
223for imperative unstructured languages that is based on \emph{structured
224traces} and that restores the preservation of structure in the observables of
225the semantics. The generic proof allows to almost completely split the
226part of the simulation that deals with functional properties only from the
227part that deals with the preservation of structure.
229The plan of this paper is the following. In Section~\ref{labelling} we
230sketch the labelling method and the problems derived from the application
231to languages with function calls. In Section~\ref{semantics} we introduce
232a generic description of an unstructured imperative language and the
233corresponding structured traces (the novel semantics). In
234Section~\ref{simulation} we describe the forward simulation proof.
235Conclusions and future works are in Section~\ref{conclusions}
237\section{The labelling approach}
[3347]239% \subsection{A brief introduction to the labelling approach}
[3356]240We briefly explain the labelling approach as introduced in~\cite{easylabelling}
241on the example in \autoref{examplewhile}.
[3347]242The user wants to analyse the execution time of the program (the black lines in
243\autoref{subfig:example_input}). He compiles the program using
[3344]244a special compiler that first inserts in the code three label emission
[3356]245statements (\verb+EMIT L_i+) to mark the beginning of basic blocks
[3347]247then the compiler compiles the code to machine code (\autoref{subfig:example_oc}),
248granting that the execution of the source and object
[3344]249code emits the same sequence of labels ($L_1; L_2; L_2; L_3$ in the example).
250This is achieved by keeping track of basic blocks during compilation, avoiding
251all optimizations that alter the control flow. The latter can be recovered with
[3356]252a more refined version of the labelling approach~\cite{loopoptimizations}, but in the
[3344]253present paper we stick to this simple variant for simplicity. Once the object
254code is produced, the compiler runs a static code analyzer to associate to
255each label $L_1, \ldots, L_3$ the cost (in clock cycles) of the instructions
256that belong to the corresponding basic block. For example, the cost $k_1$
257associated to $L_1$ is the number of cycles required to execute the block
[3347]258$I_3$ and \verb+COND l_2+, while the cost $k_2$ associated to $L_2$ counts the
[3350]259cycles required by the block $I_4$, \verb+GOTO l_1+ and \verb+COND l_2+. The compiler also guarantees
[3344]260that every executed instruction is in the scope of some code emission label,
261that each scope does not contain loops (to associate a finite cost), and that
262both branches of a conditional statement are followed by a code emission
263statement. Under these assumptions it is true that the total execution cost
264of the program $\Delta_t$ is equal to the sum over the sequence of emitted
265labels of the cost associated to every label:
[3347]266$\Delta_t = k(L_1; L_2; L_2; L_3) = k_1 + k_2 + k_2 + k_3$.
[3344]267Finally, the compiler emits an instrumented version of the source code
[3347]268(\autoref{subfig:example_instrument}) where label emission statements are replaced
269by increments of a global variable \verb+cost+ that, before every increment, holds the
[3344]270exact number of clock cycles spent by the microprocessor so far:
[3347]271the difference $\Deltacost$ between the final and initial value of the
272internal clock is $\Deltacost = k_1 + k_2 + k_2 + k_3 = \Delta_t$. Finally, the
[3344]273user can employ any available method (e.g. Hoare logic, invariant generators,
[3347]274abstract interpretation and automated provers) to certify that $\Deltacost$
[3344]275never exceeds a certain bound~\cite{cerco}, which is now a functional property
276of the code.
282|EMIT L_1;|
284for (i=0; i<2; i++) {
285  |EMIT L_2;|
286  $I_2$;
288|EMIT L_3;|
290\caption{The input program (black lines) with its labelling (red lines).}
296     EMIT L_1
297     $I_3$
298l_1: COND l_2
299     EMIT L_2
300     $I_4$
301     GOTO l_1
302l_2: EMIT L_3
304\caption{The output labelled object code.}
310cost += $k_1$;
312for (i=0; i<2; i++) {
313cost += $k_2$;
314  $I_2$;
316cost += $k_3$;           
318\caption{The output instrumented code.}
321% \begin{verbatim}
322% EMIT L_1;                         EMIT L_1         cost += k_1;
323% I_1;                              I_3              I_1;
324% for (i=0; i<2; i++) {        l_1: COND l_2         for (i=0; i<2; i++) {
325%   EMIT L_2;                       EMIT L_2           cost += k_2;           
326%   I_2;                            I_4                I_2;               
327%  }                                GOTO l_1          }                   
328% EMIT L_3;                    l_2: EMIT L_3         cost += k_3;           
329% \end{verbatim}
330\caption{The labelling approach applied to a simple program.\label{examplewhile}. The $I_i$ are sequences of instructions not containing jumps or loops. }
[3356]333\section{Extending the labelling approach to function calls}
[3344]335Let's now consider a simple program written in C that contains a function
[3347]336pointer call inside the scope of the cost label $L_1$, in \autoref{subfig:call_input}.
[3344]337The labelling method works exactly as before, inserting
[3347]338code emission statements/\verb+cost+ variable increments at the beginning
[3344]339of every basic block and at the beginning of every function. The compiler
340still grants that the sequence of labels observed on the two programs are
[3347]341the same. A new difficulty appears when the compiler needs to statically
[3344]342analyze the object code to assign a cost to every label. What should the scope
[3347]343of the $L_1$ label be? After executing the $I_4$ block, the \verb+CALL+
[3344]344statement passes control to a function that cannot be determined statically.
345Therefore the cost of executing the body must be paid by some other label
346(hence the requirement that every function starts with a code emission
347statement). What label should pay for the cost for the block $I_5$? The only
348reasonable answer is $L_1$, i.e. \emph{the scope of labels should extend to the
[3347]349next label emission statement or the end of the function, stepping over function calls}.
356void main() {
[3356]357  EMIT $L_1$;
[3347]358  $I_1$;
359  (*f)();
360  $I_2$;
[3347]363void g() {
[3356]364  EMIT $L_2$;
[3347]365  $I_3$;
368\caption{The input labelled C program.}
[3356]376  EMIT $L_1$
[3347]377  $I_4$
378  CALL
379  $I_5$
[3356]383  EMIT $L_2$
[3347]384  $I_6$
387\caption{The output labelled object code.}
391\caption{An example compilation of a simple program with a function pointer
392         call.}
396The latter definition of scope is adeguate on the source level because
397C is a structured language that guarantees that every function call, if it
398returns, passes control to the first instruction that follows the call. However,
399this is not guaranteed for object code, the backend languages of a compiler
400and, more generally, for unstructured
401languages that use a writable control stack to store the return address of
402calls. For example, $I_6$ could increment by $1$ the return address on the
[3347]403stack so that the next \verb+RETURN+ would start at the second instruction
[3344]404of $I_5$. The compiler would still be perfectly correct if a random, dead
[3356]405code instruction was added after the \verb+CALL+ as the first instruction of $I_5$. More generally,
[3344]406\emph{there is no guarantee that a correct compiler that respects the functional
407behaviour of a program also respects the calling structure of the source code}.
408Without such an assumption, however, it may not be true that the execution cost
409of the program is the sum of the costs associated to the labels emitted. In our
410example, the cost of $I_5$ is paid by $L_1$, but in place of $I_5$ the processor could execute any other code after $g$ returns.
412Obviously, any reasonably written compiler produces object code that behaves
413as if the language was structured (i.e. by properly nesting function
414calls/returns and without tampering with the return addresses on the control
415stack). This property, however, is a property of the runs of object code
416programs, and not a property of the object code that can be easily statically
[3356]417verified (as the ones required in \autoref{labelling} in absence of function calls).
[3344]418Therefore, we now need to single out those runs whose cost behaviour can be
419statically predicted, and we need to prove that every run of programs generated
420by our compiler are of that type. We call them \emph{structured} since their
421main property is to respect properties that hold for free on the source code
[3347]422because of structure. Moreover, in order to avoid proving
[3344]423too many preservation properties of our compiler, we drop the original
[3347]424requirements on the object code (all instructons must be in scope of some labels,
425no loops inside a scope, etc.) in favour of the corresponding requirement
426for structured runs (a structured run must start with a label emission, no
427instruction can be executed twice between two emissions, etc.).
429We will therefore proceed as follows. In the following section
4301) we formally introduce the notion of
431structured trace, which captures structured runs in the style of labelled
432transition systems; 2) we show that on the object code we can correctly
433compute the execution time of a structured run from the sequence of labels
[3356]434observed; 3) we show that on the source code we can correctly compute the
[3344]435execution time of a program if the compiler produces object code whose
[3356]436runs are weakly similar to the source code runs and structured.
[3356]438The proof of correctness of such a compiler is harder than a traditional
439proof of preservation of functional properties, and a standard forward
440simulation argument does not work. In \autoref{simulation} we present
441a refinement of forward simulation that grants all required correctness
444All the definitions and theorems presented in the paper have been formalized
445in the interactive theorem prover Matita and are being used to certify
446the complexity preserving compiler developed in the CerCo project~\cite{cerco}.
447The formalization can be
448found at~\ref{YYY} and it heavily relies on algebraic and dependent types for
449both structured traces and the definition of weak similarity. In the paper
450we did not try to stay close to the formalization. On the contrary,
451the definitions given in the paper are the result of a significant
452simplification effort for
453the sake of presentation and to make easier the re-implementation of the
454concepts in a proof assistant which is not based on the Calculus of Inductive
[3347]455Constructions. In any case the formalization is heavily commented to allow the
[3344]456reader to understand the technical details of the formalization.
[3347]459% @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
461% We briefly sketch here a simplified version of the labelling approach as
462% introduced in~\cite{easylabelling}. The simplification strengthens the
463% sufficient conditions given in~\cite{easylabelling} to allow a simpler
464% explanation. The simplified conditions given here are also used in the
465% CerCo compiler to simplify the proof.
467% Let $\mathcal{P}$ be a programming language whose semantics is given in
468% terms of observables: a run of a program yields a finite or infinite
469% stream of observables. We also assume for the time being that function
470% calls are not available in $\mathcal{P}$. We want to associate a cost
471% model to a program $P$ written in $\mathcal{P}$. The first step is to
472% extend the syntax of $\mathcal{P}$ with a new construct $\verb+emit L+$
473% where $L$ is a label distinct from all observables of $\mathcal{P}$.
474% The semantics of $\verb+emit L+$ is the emission of the observable
475% \verb+L+ that is meant to signal the beginning of a basic block.
477% There exists an automatic procedure that injects into the program $P$ an
478% $\verb+emit L+$ at the beginning of each basic block, using a fresh
479% \verb+L+ for each block. In particular, the bodies of loops, both branches
480% of \verb+if-then-else+s and the targets of \verb+goto+s must all start
481% with an emission statement.
483% Let now $C$ be a compiler from $\mathcal{P}$ to the object code $\mathcal{M}$,
484% that is organised in passes. Let $\mathcal{Q}_i$ be the $i$-th intermediate
485% language used by the compiler. We can easily extend every
486% intermediate language (and its semantics) with an $\verb+emit L+$ statement
487% as we did for $\mathcal{P}$. The same is possible for $\mathcal{M}$ too, with
488% the additional difficulty that the syntax of object code is given as a
489% sequence of bytes. The injection of an emission statement in the object code
490% can be done using a map that maps two consecutive code addresses with the
491% statement. The intended semantics is that, if $(pc_1,pc_2) \mapsto \verb+emit L+$ then the observable \verb+L+ is emitted after the execution of the
492% instruction stored at $pc_1$ and before the execution of the instruction
493% stored at $pc_2$. The two program counters are necessary because the
494% instruction stored at $pc_1$ can have multiple possible successors (e.g.
495% in case of a conditional branch or an indirect call). Dually, the instruction
496% stored at $pc_2$ can have multiple possible predecessors (e.g. if it is the
497% target of a jump).
499% The compiler, to be functionally correct, must preserve the observational
500% equivalence, i.e. executing the program after each compiler pass should
501% yield the same stream of observables. After the injection of emission
502% statements, observables now capture both functional and non-functional
503% behaviours.
504% This correctness property is called in the literature a forward simulation
505% and is sufficient for correctness when the target language is
506% deterministic~\cite{compcert3}.
507% We also require a stronger, non-functional preservation property: after each
508% pass all basic blocks must start with an emission statement, and all labels
509% \verb+L+ must be unique.
511% Now let $M$ be the object code obtained for the program $P$. Let us suppose
512% that we can statically inspect the code $M$ and associate to each basic block
513% a cost (e.g. the number of clock cycles required to execute all instructions
514% in the basic block, or an upper bound to that time). Every basic block is
515% labelled with an unique label \verb+L+, thus we can actually associate the
516% cost to \verb+L+. Let call it $k(\verb+L+)$.
518% The function $k$ is defined as the cost model for the object code control
519% blocks. It can be equally used as well as the cost model for the source
520% control blocks. Indeed, if the semantics of $P$ is the stream
521% $L_1 L_2 \ldots$, then, because of forward simulation, the semantics of $M$ is
522% also $L_1 L_2 \ldots$ and its actual execution cost is $\Sigma_i k(L_i)$ because
523% every instruction belongs to a control block and every control block is
524% labelled. Thus it is correct to say that the execution cost of $P$ is also
525% $\Sigma_i k(L_i)$. In other words, we have obtained a cost model $k$ for
526% the blocks of the high level program $P$ that is preserved by compilation.
528% How can the user profit from the high level cost model? Suppose, for instance,
529% that he wants to prove that the WCET of his program is bounded by $c$. It
530% is sufficient for him to prove that $\Sigma_i k(L_i) \leq c$, which is now
531% a purely functional property of the code. He can therefore use any technique
532% available to certify functional properties of the source code.
533% What is suggested in~\cite{easylabelling} is to actually instrument the
534% source code $P$ by replacing every label emission statement
535% $\verb+emit L+$ with the instruction $\verb+cost += k(L)+$ that increments
536% a global fresh variable \verb+cost+. The bound is now proved by establishing
537% the program invariant $\verb+cost+ \leq c$, which can be done for example
538% using the Frama-C~\cite{framaC} suite if the source code is some variant of
539% C.
541% In order to extend the labelling approach to function calls we make
542% \verb+CALL f+ emit the observable \verb+f+ and \verb+RET+ emit a distinguished observable
543% \verb+ret+.
545% For example the following execution history of the program in \autoref{fig:esempio}
546% $$I_1; \verb+CALL f+; \verb+COND l+; \verb+EMIT $\ell_2$+; I_3; \verb+RET+; I_2; \verb+RET+$$
547% emits the trace
548% $$\verb+main+, \verb+f+$$
549% \begin{figure}
550% \hfil
551% \begin{minipage}{.2\linewidth}
552% \begin{lstlisting}
553% main: $\!I_1$
554%       CALL f
555%       $I_2$
556%       RET
557% \end{lstlisting}
558% \end{minipage}
559% \begin{minipage}{.1\linewidth}
560% \begin{lstlisting}
561% main
562% main
563% main
564% main
565% \end{lstlisting}
566% \end{minipage}
567% \hfil
568% \begin{minipage}{.2\linewidth}
569% \begin{lstlisting}
570% f: $\!$COND l
571%    EMIT $\ell_2$
572%    RET
573% l: $\!$EMIT $\ell_3$
574%    $I_3$
575%    RET
576% \end{lstlisting}
577% \end{minipage}
578% \begin{minipage}{.1\linewidth}
579% \begin{lstlisting}
580% f
582% $\ell_2$
584% $\ell_3$
585% $\ell_3$
586% \end{lstlisting}
587% \end{minipage}
588% \hfil{}
589% \caption{}
590% \label{fig:esempio}
591% \end{figure}
594% \subsection{Labelling function calls}
595% We now want to extend the labelling approach to support function calls.
596% On the high level, \emph{structured} programming language $\mathcal{P}$ there
597% is not much to change.
598% When a function is invoked, the current basic block is temporarily exited
599% and the basic block the function starts with take control. When the function
600% returns, the execution of the original basic block is resumed. Thus the only
601% significant change is that basic blocks can now be nested. Let \verb+E+
602% be the label of the external block and \verb+I+ the label of a nested one.
603% Since the external starts before the internal, the semantics observed will be
604% \verb+E I+ and the cost associated to it on the source language will be
605% $k(\verb+E+) + k(\verb+I+)$, i.e. the cost of executing all instructions
606% in the block \verb+E+ first plus the cost of executing all the instructions in
607% the block \verb+I+. However, we know that some instructions in \verb+E+ are
608% executed after the last instruction in \verb+I+. This is actually irrelevant
609% because we are here assuming that costs are additive, so that we can freely
610% 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
611% the function call terminates and yields back control to the basic block
612% \verb+E+. If the call diverges, the instrumentation
613% $\verb+cost += k(E)+$ executed at the beginning of \verb+E+ is still valid,
614% but just as an upper bound to the real execution cost: only precision is lost.
616% Let now consider what happens when we move down the compilation chain to an
617% unstructured intermediate or final language. Here unstructured means that
618% the only control operators are conditional and unconditional jumps, function
619% calls and returns. Unlike a structured language, though, there is no guarantee
620% that a function will return control just after the function call point.
621% The semantics of the return statement, indeed, consists in fetching the
622% return address from some internal structure (typically the control stack) and
623% jumping directly to it. The code can freely manipulate the control stack to
624% make the procedure returns to whatever position. Indeed, it is also possible
625% to break the well nesting of function calls/returns.
627% Is it the case that the code produced by a correct compiler must respect the
628% additional property that every function returns just after its function call
629% point? The answer is negative and the property is not implied by forward
630% simulation proofs. For instance, imagine to modify a correct compiler pass
631% by systematically adding one to the return address on the stack and by
632% putting a \verb+NOP+ (or any other instruction that takes one byte) after
633% every function call. The obtained code will be functionally indistinguishable,
634% and the added instructions will all be dead code.
636% This lack of structure in the semantics badly interferes with the labelling
637% approach. The reason is the following: when a basic block labelled with
638% \verb+E+ contains a function call, it no longer makes any sense to associate
639% to a label \verb+E+ the sum of the costs of all the instructions in the block.
640% Indeed, there is no guarantee that the function will return into the block and
641% that the instructions that will be executed after the return will be the ones
642% we are paying for in the cost model.
644% How can we make the labelling approach work in this scenario? We only see two
645% possible ways. The first one consists in injecting an emission statement after
646% every function call: basic blocks no longer contain function calls, but are now
647% terminated by them. This completely solves the problem and allows the compiler
648% to break the structure of function calls/returns at will. However, the
649% technique has several drawbacks. First of all, it greatly augments the number
650% of cost labels that are injected in the source code and that become
651% instrumentation statements. Thus, when reasoning on the source code to prove
652% non-functional properties, the user (or the automation tool) will have to handle
653% larger expressions. Second, the more labels are emitted, the more difficult it
654% becomes to implement powerful optimisations respecting the code structure.
655% Indeed, function calls are usually implemented in such a way that most registers
656% are preserved by the call, so that the static analysis of the block is not
657% interrupted by the call and an optimisation can involve both the code before
658% and after the function call. Third, instrumenting the source code may require
659% unpleasant modification of it. Take, for example, the code
660% \verb+f(g(x));+. We need to inject an emission statement/instrumentation
661% instruction just after the execution of \verb+g+. The only way to do that
662% is to rewrite the code as \verb+y = g(x); emit L; f(y);+ for some fresh
663% variable \verb+y+. It is pretty clear how in certain situations the obtained
664% code would be more obfuscated and then more difficult to manually reason on.
666% For the previous reasons, in this paper and in the CerCo project we adopt a
667% different approach. We do not inject emission statements after every
668% function call. However, we want to propagate a strong additional invariant in
669% the forward simulation proof. The invariant is the propagation of the structure
670%  of the original high level code, even if the target language is unstructured.
671% The structure we want to propagate, that will become more clear in the next
672% section, comprises 1) the property that every function should return just after
673% the function call point, which in turns imply well nesting of function calls;
674% 2) the property that every basic block starts with a code emission statement.
676% In the original labelling approach of~\cite{easylabelling}, the second property
677% was granted syntactically as a property of the generated code.
678% In our revised approach, instead, we will impose the property on the runs:
679% it will be possible to generate code that does not respect the syntactic
680% property, as soon as all possible runs respect it. For instance, dead code will no longer
681% be required to have all basic blocks correctly la. The switch is suggested
682% from the fact that the first of the two properties --- that related to
683% function calls/returns --- can only be defined as property of runs,
684% not of the static code. The switch is
685% beneficial to the proof because the original proof was made of two parts:
686% the forward simulation proof and the proof that the static property was granted.
687% In our revised approach the latter disappears and only the forward simulation
688% is kept.
690% In order to capture the structure semantics so that it is preserved
691% by a forward simulation argument, we need to make the structure observable
692% in the semantics. This is the topic of the next section.
[3356]694\subsection{Structured traces}
697Let's consider a generic unstructured language already equipped with a
698small step structured operational semantics (SOS). We introduce a
[3350]699deterministic labelled transition system~\cite{LTS} $(S,\Lambda,\to)$
[3345]700that refines the
[3356]701SOS by observing function calls/returns and the beginning of basic blocks.
[3350]702$S$ is the set of states of the program and
[3347]703$\Lambda = \{ \tau, RET \} \cup \Labels \cup \Functions$
704where $\Functions$ is the set of names of functions that can occur in the
705program, $\Labels$ is a set of labels disjoint from $\Functions$
[3348]706and $\tau$ and $RET$ do not belong to $\Functions \cup \Labels$. Moreover there
707is an injective function $\ell : \Functions \to \Labels$ that tells the
708starting label of the body of each function, and $\ell(\Functions)\subseteq \Labels$
709denotes the image of this function.
[3347]710The transition function is defined as $s_1 \to[o] s_2$ if
[3350]711$s_1$ moves to $s_2$ according to the SOS and $o = f \in \Functions$ if
[3347]712the function $f$ is called, $o = RET$ if a \verb+RETURN+ is executed,
713$o = L \in \Labels$ if an \verb+EMIT $L$+ is executed to signal the
[3345]714beginning of a basic block, and $o = \tau$ in all other cases.
715Because we assume the language to be deterministic, the label emitted can
[3347]716actually be computed simply observing $s_1$. Finally, $S$ is also endowed with
[3350]717a relation $s\ar s'$ ($s'$ \emph{follows} $s$) that holds when the instruction
718to be executed in $s'$ follows syntactically the one in $s$ in the source program.
[3350]720In the rest of the paper we write $s_0 \to^{*} s_n$ for the finite execution
721fragment $T = s_0 \to[o_0] s_1 \to[o_1] \ldots \to[o_{n-1}] s_n$
722and, we call \emph{weak trace} of $T$ (denoted as $|T|$) the
723subsequence $o_{i_0} \ldots o_{i_m}$ of $o_0 \ldots o_{n-1}$ obtained dropping
724every internal action $\tau$.
726%Let $k$ be a cost model for observables actions that maps elements of
727%$\Lambda \setminus \{\tau\}$ to any commutative cost monoid
728%(e.g. natural numbers). We extend the domain of $k$ to executable fragments
729%by posing $k(T) = \Sigma_{o \in |T|} k(o)$.
731\paragraph{Structured execution fragments}
[3347]732Among all possible finite execution fragments we want to
[3345]733identify the ones that satisfy the requirements we sketched in the previous
[3347]734section. We say that an execution fragment
[3350]735$s_0 \to^{*} s_n$
736is \emph{structured} (and we denote it as $s_0 \To s_n$) iff the following conditions
[3347]737are met.
[3347]739 \item For every $i$, if $s_i \to[f] s_{i+1}$ then there is a
740   label $L$ and a $k\ge i+2$ such that
[3348]741    $s_{i+1} \to[\ell(f)] s_{i+2} \To s_k \to[RET] s_{k+1}$, with
[3347]742    $s_i \ar s_{k+1}$.
[3348]743   In other words, $s_{i+1}$ must start execution with \verb+EMIT $\ell(f)$+
[3350]744   --- so that no instruction falls outside the scope of every label ---
[3347]745   and then continue with a structured fragment returning control
746   to the instruction immediately following the call.
748   The condition also enforces convergence of every function call, which is
749   necessary to bound the cost of the fragment. Note that
750   non convergent programs may still have structured execution fragments
751   that are worth measuring. For example, we can measure the reaction time
752   of a server implemented as an unbounded loop whose body waits for an
753   input, process it and performs an output before looping: the processing
754   steps form a structured execution fragment.
[3347]755 \item The number of $RET$'s in the fragment is equal to the number of
[3350]756   calls performed. In combination with the previous condition, this ensures
757   well-backeting of function calls.
758 \item
759   \label{req3}
760   For every $i$ and $f$, if $s_{i+1}\to[\ell(f)]s_{i+2}$ then
761   $s_i\to[f]s_{i+1}$. This is a technical condition needed to ensure that a
762   label associated with a function is only used at the beginning of its
763   body. Its use will become clear in~\autoref{simulation}.
[3347]764 \item For every $i$, if the instruction to be executed in $s_i$ is a
765   conditional branch, then there is an $L$ such that $s_{i+1} \to[L] s_{i+2}$ or, equivalently, that $s_{i+1}$ must start execution with an
766   \verb+EMIT $L$+. This captures the requirement that every branch which is
[3350]767   live code must start with a label emission. Otherwise, it would be possible
768   to have conditional instructions whose branches are assigned different
769   costs, making impossible to assign a single cost to the label whose scope
770   contains the jump.
[3348]772One might wonder why $f$ and $\ell(f)$, that aways appear in this order, are not
[3350]773collapsed into a single observable. This would simplify some aspects of the
774formalisation at the price of others. For example, we should add special
775cases when the fragment starts at the beginning of a function body
776(e.g. the one of \texttt{main}) because in that case nobody would have emitted
[3356]777the observable $\ell(f)$. We plan to compare the two approaches in the future.
[3350]779\paragraph{Measurable execution fragments and their cost prediction.}
[3356]780The first main theorem of CerCo deals with the object code.
[3350]781It states that the execution cost of
782certain execution fragments, that we call \emph{measurable fragments}, can be
783computed from their weak trace by choosing the cost model $k$ that assigns to
784any label the cost (in clock cycles) of the instructions in its scope, and
785$0$ to function calls and $RET$ observables.
788 \label{static}
789 for all measurable fragment $T = s_0 \to^{*} s_n$,\\
790 $$\Delta_t := \verb+clock+_{s_n} - \verb+clock+_{s_0} = \Sigma_{o \in |T|} k(o)$$
[3350]793An execution fragment $s_0 \to^{*} s_n$ is
794measurable if it is structured (up to a possible final \texttt{RETURN}) and
795if it does not start or end in the middle of a basic block.
796Ending in the middle of a block would mean that the last label encountered
797would have pre-paid more instructions than the ones executed; starting in the
798middle would mean not paying any instruction up to the first label emission.
800Formally, $s_0 \to^{*} s_n$ is measurable iff $o_0 \in \Labels$ (or equivalently
[3347]801in $s_0$ the program must emit a label) and either
[3350]802$s_0 \To s_{n-1}$ and $s_{n-1}\to[RET]s_n$ or
803$s_0 \To s_n$ and $s_n$ must be a label emission statement.
[3356]805%\textbf{CSC: PROVA----------------------}
[3350]806% The theorem is proved by structural induction over the structured
807% trace, and is based on the invariant that
808% iff the function that computes the cost model has analysed the instruction
809% to be executed at $s_2$ after the one to be executed at $s_1$, and if
810% the structured trace starts with $s_1$, then eventually it will contain also
811% $s_2$. When $s_1$ is not a function call, the result holds trivially because
812% of the $s_1\exec s_2$ condition obtained by inversion on
813% the trace. The only non
814% trivial case is the one of function calls: the cost model computation function
815% does recursion on the first instruction that follows that function call; the
816% \verb+as_after_return+ condition of the \verb+tal_base_call+ and
817% \verb+tal_step_call+ grants exactly that the execution will eventually reach
818% this state.
820\paragraph{Weak similarity and cost invariance.}
[3346]821Given two deterministic unstructured programming languages with their own
[3350]822operational semantics, we say that two execution fragments are
823\emph{weakly trace equivalent} if their weak traces are equal.
[3350]825A compiler (pass) that preserves the program semantics also preserves weak
826traces and propagates measurability iff for every measurable
827fragment $T_1 = s_1 \to^{*} s_1'$ of the source code, the corresponding
828execution fragment $T_2 = s_2 \to^{*} s_2'$ of the object code is measurable
829and $T_1$ and $T_2$ are weakly trace equivalent. The very intuitive notion of
830``corresponding fragment'' is made clear in the forward simulation proof of
831preservation of the semantics of the program by saying that $s_2$ and $s_1$
832are in a certain relation. Clearly the property holds for a compiler if it
833holds for each compiler pass.
835Having proved in~\autoref{static} that the statically computed cost model is
836accurate for the object code, we get as a corollary that it is also accurate
837for the source code if the compiler preserves weak traces and
[3356]838propagates measurability. Thus, as prescribed by the CerCo's methodology~\cite{fopara}, it becomes possible to compute cost models
[3350]839on the object code, transfer it to the source code and then reason comfortably
840on the source code only.
843Given a compiler that preserves weak traces and propagates measurability,
844for all measurable execution fragment $T_1 = s_1 \to^{*} s_1'$ of the source
845code such that $T_2 = s_2 \to^{*} s_2'$ is the corresponding fragment of the
846object code,
848$$\Delta_t := \verb+clock+_{s_2'} - \verb+clock+_{s_2} = \Sigma_{o \in |T_2|} k(o) = \Sigma_{o \in |T_1|} k(o)$$
[3356]851\section{Proving the compiler correctness}
853Because of \autoref{preservation}, to certify a compiler for the labelling
854approach we need to both prove that it respects the functional semantics of the
855program, and that it preserves weak traces and propagates measurability.
[3356]856We achieve this by independently proving the three properties for each compiler
[3350]858The first property is standard and can be proved by means of a forward simulation argument (see for example~\cite{compcert}) that runs like this.
859First a relation between the corresponding
[3356]860source and target states is defined. Then a lemma establishes
[3350]861a local simulation condition: given two states in relation, if the source
862one performs one step then the target one performs zero or more steps and
863the two resulting states are synchronized again according to the relation.
[3356]864No requirements are imposed on the intermediate target states.
[3350]865Finally, the lemma is iterated over the execution trace to establish the
866final result.
868In principle, preservation of weak traces could be easily shown with the same
869argument (and also at the same time). Surprisingly, propagation of
870measurability cannot. What makes the standard forward
871simulation proof work is the fact that usually a compiler pass performs some
872kind of local or global analysis of the code followed by a compositional, order
873preserving translation of every instruction. In order to produce structured
[3356]874traces, however, code emission cannot be fully compositional any longer,
875and requirements need to be enforced on intermediate target states.
877For example, consider~requirement \ref{req3} that asks every function body
878to start with a label emission statement. Some compiler passes must
879add preambles to functions, for example to take care of the parameter passing
880convention. In order to not violate the requirement, the preamble must be
881inserted after the label emission. In the forward simulation proof, however,
882function call steps in the source language are simulated by the new function
883call followed by the execution of the preamble, and only at the end of the
884preamble the reached states are again in the expected relation. In the meantime,
885however, the object code has already performed the label emission statement,
886that still needs to be executed in the source code, breaking forward simulation.
888Another reason why the standard argument breaks is due to the requirement that
[3351]889function calls should yield back control after the calling point. Some passes
890need to translate a function call to a function call followed by some
891instructions (for example to restore caller-saved registers in the pass that
892sets the parameter convenction). In the forward simulation proof, these
893instructions are taken care of when simulating the \texttt{RETURN} step:
894the state just after the return in the source code is matched by the state $s_2$
895after these steps in the object code. However, the aforementioned requirement
[3356]896does not involve $s_2$, but the intermediate state reached after the return in the object
897code. Therefore this requirement too
[3351]898cannot be enforced with the standard forward simulation argument.
[3351]900In this section we present now a modified forward simulation argument that
901can be used to prove at once that a compiler preserves the semantics of the
902program, its weak traces and that the compiler propagates measurability.
[3347]904% @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
[3347]906% The program semantics adopted in the traditional labelling approach is based
907% on labelled deductive systems. Given a set of observables $\mathcal{O}$ and
908% a set of states $\S$, the semantics of one deterministic execution
909% step is
910% defined as a function $S \to S \times O^*$ where $O^*$ is a (finite) stream of
911% observables. The semantics is then lifted compositionally to multiple (finite
912% or infinite) execution steps.
913% Finally, the semantics of a a whole program execution is obtained by forgetting
914% about the final state (if any), yielding a function $S \to O^*$ that given an
915% initial status returns the finite or infinite stream of observables in output.
[3347]917% We present here a new definition of semantics where the structure of execution,
918% as defined in the previous section, is now observable. The idea is to replace
919% the stream of observables with a structured data type that makes explicit
920% function call and returns and that grants some additional invariants by
921% construction. The data structure, called \emph{structured traces}, is
922% defined inductively for terminating programs and coinductively for diverging
923% ones. In the paper we focus only on the inductive structure, i.e. we assume
924% that all programs that are given a semantics are total. The Matita formalisation
925% also shows the coinductive definitions. The semantics of a program is then
926% defined as a function that maps an initial state into a structured trace.
928% In order to have a definition that works on multiple intermediate languages,
929% we abstract the type of structure traces over an abstract data type of
930% abstract statuses, which we aptly call $\verb+abstract_status+$. The fields
931% of this record are the following.
932% \begin{itemize}
933%  \item \verb+S : Type[0]+, the type of states.
934%  \item \verb+as_execute : S $\to$ S $\to$ Prop+, a binary predicate stating
935%  an execution step. We write $s_1\exec s_2$ for $\verb+as_execute+~s_1~s_2$.
936%  \item \verb+as_classifier : S $\to$ classification+, a function tagging all
937%  states with a class in
938%  $\{\verb+cl_return,cl_jump,cl_call,cl_other+\}$, depending on the instruction
939%  that is about to be executed (we omit tail-calls for simplicity). We will
940%  use $s \class c$ as a shorthand for both $\verb+as_classifier+~s=c$
941%  (if $c$ is a classification) and $\verb+as_classifier+~s\in c$
942%  (if $c$ is a set of classifications).
943%  \item \verb+as_label : S $\to$ option label+, telling whether the
944%  next instruction to be executed in $s$ is a cost emission statement,
945%  and if yes returning the associated cost label. Our shorthand for this function
946%  will be $\ell$, and we will also abuse the notation by using $\ell~s$ as a
947%  predicate stating that $s$ is labelled.
948%  \item \verb+as_call_ident : ($\Sigma$s:S. s $\class$ cl_call) $\to$ label+,
949%  telling the identifier of the function which is being called in a
950%  \verb+cl_call+ state. We will use the shorthand $s\uparrow f$ for
951%  $\verb+as_call_ident+~s = f$.
952%  \item \verb+as_after_return : ($\Sigma$s:S. s $\class$ cl_call) $\to$ S $\to$ Prop+,
953%  which holds on the \verb+cl_call+ state $s_1$ and a state $s_2$ when the
954%  instruction to be executed in $s_2$ follows the function call to be
955%  executed in (the witness of the $\Sigma$-type) $s_1$. We will use the notation
956%  $s_1\ar s_2$ for this relation.
957% \end{itemize}
959% % \begin{alltt}
960% % record abstract_status := \{ S: Type[0];
961% %  as_execute: S \(\to\) S \(\to\) Prop;   as_classifier: S \(\to\) classification;
962% %  as_label: S \(\to\) option label;    as_called: (\(\Sigma\)s:S. c s = cl_call) \(\to\) label;
963% %  as_after_return: (\(\Sigma\)s:S. c s = cl_call) \(\to\) S \(\to\) Prop \}
964% % \end{alltt}
966% The inductive type for structured traces is actually made by three multiple
967% inductive types with the following semantics:
968% \begin{enumerate}
969%  \item $(\verb+trace_label_return+~s_1~s_2)$ (shorthand $\verb+TLR+~s_1~s_2$)
970%    is a trace that begins in
971%    the state $s_1$ (included) and ends just before the state $s_2$ (excluded)
972%    such that the instruction to be executed in $s_1$ is a label emission
973%    statement and the one to be executed in the state before $s_2$ is a return
974%    statement. Thus $s_2$ is the state after the return. The trace
975%    may contain other label emission statements. It captures the structure of
976%    the execution of function bodies: they must start with a cost emission
977%    statement and must end with a return; they are obtained by concatenating
978%    one or more basic blocks, all starting with a label emission
979%    (e.g. in case of loops).
980%  \item $(\verb+trace_any_label+~b~s_1~s_2)$ (shorthand $\verb+TAL+~b~s_1~s_2$)
981%    is a trace that begins in
982%    the state $s_1$ (included) and ends just before the state $s_2$ (excluded)
983%    such that the instruction to be executed in $s_2$/in the state before
984%    $s_2$ is either a label emission statement or
985%    or a return, according to the boolean $b$. It must not contain
986%    any label emission statement. It captures the notion of a suffix of a
987%    basic block.
988%  \item $(\verb+trace_label_label+~b~s_1~s_2)$ (shorthand $\verb+TLL+~b~s_1~s_2$ is the special case of
989%    $\verb+TAL+~b~s_1~s_2)$ such that the instruction to be
990%    executed in $s_1$ is a label emission statement. It captures the notion of
991%    a basic block.
992% \end{enumerate}
994% \begin{multicols}{3}
995% \infrule[\verb+tlr_base+]
996%  {\verb+TLL+~true~s_1~s_2}
997%  {\verb+TLR+~s_1~s_2}
999% \infrule[\verb+tlr_step+]
1000%  {\verb+TLL+~false~s_1~s_2 \andalso
1001%   \verb+TLR+~s_2~s_3
1002%  }
1003%  {\verb+TLR+~s_1~s_3}
1005% \infrule[\verb+tll_base+]
1006%  {\verb+TAL+~b~s_1~s_2 \andalso
1007%   \ell~s_1
1008%  }
1009%  {\verb+TLL+~b~s_1~s_2}
1010% \end{multicols}
1012% \infrule[\verb+tal_base_not_return+]
1013%  {s_1\exec s_2 \andalso
1014%   s_1\class\{\verb+cl_jump+, \verb+cl_other+\}\andalso
1015%   \ell~s_2
1016%  }
1017%  {\verb+TAL+~false~s_1~s_2}
1019% \infrule[\verb+tal_base_return+]
1020%  {s_1\exec s_2 \andalso
1021%   s_1 \class \verb+cl_return+
1022%  }
1023%  {\verb+TAL+~true~s_1~s_2}
1025% \infrule[\verb+tal_base_call+]
1026%  {s_1\exec s_2 \andalso
1027%   s_1 \class \verb+cl_call+ \andalso
1028%   s_1\ar s_3 \andalso
1029%   \verb+TLR+~s_2~s_3 \andalso
1030%   \ell~s_3
1031%  }
1032%  {\verb+TAL+~false~s_1~s_3}
1034% \infrule[\verb+tal_step_call+]
1035%  {s_1\exec s_2 \andalso
1036%   s_1 \class \verb+cl_call+ \andalso
1037%   s_1\ar s_3 \andalso
1038%   \verb+TLR+~s_2~s_3 \andalso
1039%   \verb+TAL+~b~s_3~s_4
1040%  }
1041%  {\verb+TAL+~b~s_1~s_4}
1043% \infrule[\verb+tal_step_default+]
1044%  {s_1\exec s_2 \andalso
1045%   \lnot \ell~s_2 \andalso
1046%   \verb+TAL+~b~s_2~s_3\andalso
1047%   s_1 \class \verb+cl_other+
1048%  }
1049%  {\verb+TAL+~b~s_1~s_3}
1050% \begin{comment}
1051% \begin{verbatim}
1052% inductive trace_label_return (S:abstract_status) : S → S → Type[0] ≝
1053%   | tlr_base:
1054%       ∀status_before: S.
1055%       ∀status_after: S.
1056%         trace_label_label S ends_with_ret status_before status_after →
1057%         trace_label_return S status_before status_after
1058%   | tlr_step:
1059%       ∀status_initial: S.
1060%       ∀status_labelled: S.
1061%       ∀status_final: S.
1062%         trace_label_label S doesnt_end_with_ret status_initial status_labelled →
1063%         trace_label_return S status_labelled status_final →
1064%           trace_label_return S status_initial status_final
1065% with trace_label_label: trace_ends_with_ret → S → S → Type[0] ≝
1066%   | tll_base:
1067%       ∀ends_flag: trace_ends_with_ret.
1068%       ∀start_status: S.
1069%       ∀end_status: S.
1070%         trace_any_label S ends_flag start_status end_status →
1071%         as_costed S start_status →
1072%           trace_label_label S ends_flag start_status end_status
1073% with trace_any_label: trace_ends_with_ret → S → S → Type[0] ≝
1074%   (* Single steps within a function which reach a label.
1075%      Note that this is the only case applicable for a jump. *)
1076%   | tal_base_not_return:
1077%       ∀start_status: S.
1078%       ∀final_status: S.
1079%         as_execute S start_status final_status →
1080%         (as_classifier S start_status cl_jump ∨
1081%          as_classifier S start_status cl_other) →
1082%         as_costed S final_status →
1083%           trace_any_label S doesnt_end_with_ret start_status final_status
1084%   | tal_base_return:
1085%       ∀start_status: S.
1086%       ∀final_status: S.
1087%         as_execute S start_status final_status →
1088%         as_classifier S start_status cl_return →
1089%           trace_any_label S ends_with_ret start_status final_status
1090%   (* A call followed by a label on return. *)
1091%   | tal_base_call:
1092%       ∀status_pre_fun_call: S.
1093%       ∀status_start_fun_call: S.
1094%       ∀status_final: S.
1095%         as_execute S status_pre_fun_call status_start_fun_call →
1096%         ∀H:as_classifier S status_pre_fun_call cl_call.
1097%           as_after_return S «status_pre_fun_call, H» status_final →
1098%           trace_label_return S status_start_fun_call status_final →
1099%           as_costed S status_final →
1100%             trace_any_label S doesnt_end_with_ret status_pre_fun_call status_final
1101%   (* A call followed by a non-empty trace. *)
1102%   | tal_step_call:
1103%       ∀end_flag: trace_ends_with_ret.
1104%       ∀status_pre_fun_call: S.
1105%       ∀status_start_fun_call: S.
1106%       ∀status_after_fun_call: S.
1107%       ∀status_final: S.
1108%         as_execute S status_pre_fun_call status_start_fun_call →
1109%         ∀H:as_classifier S status_pre_fun_call cl_call.
1110%           as_after_return S «status_pre_fun_call, H» status_after_fun_call →
1111%           trace_label_return S status_start_fun_call status_after_fun_call →
1112%           ¬ as_costed S status_after_fun_call →
1113%           trace_any_label S end_flag status_after_fun_call status_final →
1114%             trace_any_label S end_flag status_pre_fun_call status_final
1115%   | tal_step_default:
1116%       ∀end_flag: trace_ends_with_ret.
1117%       ∀status_pre: S.
1118%       ∀status_init: S.
1119%       ∀status_end: S.
1120%         as_execute S status_pre status_init →
1121%         trace_any_label S end_flag status_init status_end →
1122%         as_classifier S status_pre cl_other →
1123%         ¬ (as_costed S status_init) →
1124%           trace_any_label S end_flag status_pre status_end.
1125% \end{verbatim}
1126% \end{comment}
1127% A \verb+trace_label_return+ is isomorphic to a list of
1128% \verb+trace_label_label+s that ends with a cost emission followed by a
1129% return terminated \verb+trace_label_label+.
1130% The interesting cases are those of $\verb+trace_any_label+~b~s_1~s_2$.
1131% A \verb+trace_any_label+ is a sequence of steps built by a syntax directed
1132% definition on the classification of $s_1$. The constructors of the datatype
1133% impose several invariants that are meant to impose a structure to the
1134% otherwise unstructured execution. In particular, the following invariants are
1135% imposed:
1136% \begin{enumerate}
1137%  \item the trace is never empty; it ends with a return iff $b$ is
1138%        true
1139%  \item a jump must always be the last instruction of the trace, and it must
1140%        be followed by a cost emission statement; i.e. the target of a jump
1141%        is always the beginning of a new basic block; as such it must start
1142%        with a cost emission statement
1143%  \item a cost emission statement can never occur inside the trace, only in
1144%        the status immediately after
1145%  \item the trace for a function call step is made of a subtrace for the
1146%        function body of type
1147%        $\verb+trace_label_return+~s_1~s_2$, possibly followed by the
1148%        rest of the trace for this basic block. The subtrace represents the
1149%        function execution. Being an inductive datum, it grants totality of
1150%        the function call. The status $s_2$ is the one that follows the return
1151%        statement. The next instruction of $s_2$ must follow the function call
1152%        instruction. As a consequence, function calls are also well nested.
1153% \end{enumerate}
1155% There are three mutual structural recursive functions, one for each of
1156% \verb+TLR+, \verb+TLL+ and \verb+TAL+, for which we use the same notation
1157% $|\,.\,|$: the \emph{flattening} of the traces. These functions
1158% allow to extract from a structured trace the list of emitted cost labels.
1159% %  We only show here the type of one
1160% % of them:
1161% % \begin{alltt}
1162% % flatten_trace_label_return:
1163% %  \(\forall\)S: abstract_status. \(\forall\)\(s_1,s_2\).
1164% %   trace_label_return \(s_1\) \(s_2\) \(\to\) list (as_cost_label S)
1165% % \end{alltt}
1167% \paragraph{Structured traces similarity and cost prediction invariance.}
1169% A compiler pass maps source to object code and initial states to initial
1170% states. The source code and initial state uniquely determine the structured
1171% trace of a program, if it exists. The structured trace fails to exists iff
1172% the structural conditions are violated by the program execution (e.g. a function
1173% body does not start with a cost emission statement). Let us assume that the
1174% target structured trace exists.
1176% What is the relation between the source and target structured traces?
1177% In general, the two traces can be arbitrarily different. However, we are
1178% interested only in those compiler passes that maps a trace $\tau_1$ to a trace
1179% $\tau_2$ such that
1180% \begin{equation}|\tau_1| = |\tau_2|.\label{th2}\end{equation}
1181% The reason is that the combination of~\eqref{th1} with~\eqref{th2} yields the
1182% corollary
1183% \begin{equation}\label{th3}
1184% \forall s_1,s_2. \forall \tau: \verb+TLR+~s_1~s_2.~
1185%   \verb+clock+~s_2 - \verb+clock+~s_1 =
1186%   \Sigma_{\alpha \in |\tau_1|}\;k(\alpha) =
1187%   \Sigma_{\alpha \in |\tau_2|}\;k(\alpha).
1188% \end{equation}
1189% 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
1190% transfer the cost model from the target to the source code and reason on the
1191% source code only.
1193% We are therefore interested in conditions stronger than~\eqref{th2}.
1194% Therefore we introduce here a similarity relation between traces with
1195% the same structure. Theorem~\verb+tlr_rel_to_traces_same_flatten+
1196% in the Matita formalisation shows that~\eqref{th2} holds for every pair
1197% $(\tau_1,\tau_2)$ of similar traces.
1199% Intuitively, two traces are similar when one can be obtained from
1200% the other by erasing or inserting silent steps, i.e. states that are
1201% not \verb+as_costed+ and that are classified as \verb+cl_other+.
1202% Silent steps do not alter the structure of the traces.
1203% In particular,
1204% the relation maps function calls to function calls to the same function,
1205% label emission statements to emissions of the same label, concatenation of
1206% subtraces to concatenation of subtraces of the same length and starting with
1207% the same emission statement, etc.
1209% In the formalisation the three similarity relations --- one for each trace
1210% kind --- are defined by structural recursion on the first trace and pattern
1211% matching over the second. Here we turn
1212% the definition into the inference rules shown in \autoref{fig:txx_rel}
1213% for the sake of readability. We also omit from trace constructors all arguments,
1214% but those that are traces or that
1215% are used in the premises of the rules. By abuse of notation we denote all three
1216% relations by infixing $\approx$.
1218% \begin{figure}
1219% \begin{multicols}{2}
1220% \infrule
1221%  {tll_1\approx tll_2
1222%  }
1223%  {\verb+tlr_base+~tll_1 \approx \verb+tlr_base+~tll_2}
1225% \infrule
1226%  {tll_1 \approx tll_2 \andalso
1227%   tlr_1 \approx tlr_2
1228%  }
1229%  {\verb+tlr_step+~tll_1~tlr_1 \approx \verb+tlr_step+~tll_2~tlr_2}
1230% \end{multicols}
1231% \vspace{3ex}
1232% \begin{multicols}{2}
1233% \infrule
1234%  {\ell~s_1 = \ell~s_2 \andalso
1235%   tal_1\approx tal_2
1236%  }
1237%  {\verb+tll_base+~s_1~tal_1 \approx \verb+tll_base+~s_2~tal_2}
1239% \infrule
1240%  {tal_1\approx tal_2
1241%  }
1242%  {\verb+tal_step_default+~tal_1 \approx tal_2}
1243% \end{multicols}
1244% \vspace{3ex}
1245% \infrule
1246%  {}
1247%  {\verb+tal_base_not_return+\approx taa \append \verb+tal_base_not_return+}
1248% \vspace{1ex}
1249% \infrule
1250%  {}
1251%  {\verb+tal_base_return+\approx taa \append \verb+tal_base_return+}
1252% \vspace{1ex}
1253% \infrule
1254%  {tlr_1\approx tlr_2 \andalso
1255%   s_1 \uparrow f \andalso s_2\uparrow f
1256%  }
1257%  {\verb+tal_base_call+~s_1~tlr_1\approx taa \append \verb+tal_base_call+~s_2~tlr_2}
1258% \vspace{1ex}
1259% \infrule
1260%  {tlr_1\approx tlr_2 \andalso
1261%   s_1 \uparrow f \andalso s_2\uparrow f \andalso
1262%   \verb+tal_collapsable+~tal_2
1263%  }
1264%  {\verb+tal_base_call+~s_1~tlr_1 \approx taa \append \verb+tal_step_call+~s_2~tlr_2~tal_2)}
1265% \vspace{1ex}
1266% \infrule
1267%  {tlr_1\approx tlr_2 \andalso
1268%   s_1 \uparrow f \andalso s_2\uparrow f \andalso
1269%   \verb+tal_collapsable+~tal_1
1270%  }
1271%  {\verb+tal_step_call+~s_1~tlr_1~tal_1 \approx taa \append \verb+tal_base_call+~s_2~tlr_2)}
1272% \vspace{1ex}
1273% \infrule
1274%  {tlr_1 \approx tlr_2 \andalso
1275%   s_1 \uparrow f \andalso s_2\uparrow f\andalso
1276%   tal_1 \approx tal_2 \andalso
1277%  }
1278%  {\verb+tal_step_call+~s_1~tlr_1~tal_1 \approx taa \append \verb+tal_step_call+~s_2~tlr_2~tal_2}
1279% \caption{The inference rule for the relation $\approx$.}
1280% \label{fig:txx_rel}
1281% \end{figure}
1282% %
1283% \begin{comment}
1284% \begin{verbatim}
1285% let rec tlr_rel S1 st1 st1' S2 st2 st2'
1286%   (tlr1 : trace_label_return S1 st1 st1')
1287%   (tlr2 : trace_label_return S2 st2 st2') on tlr1 : Prop ≝
1288% match tlr1 with
1289%   [ tlr_base st1 st1' tll1 ⇒
1290%     match tlr2 with
1291%     [ tlr_base st2 st2' tll2 ⇒ tll_rel … tll1 tll2
1292%     | _ ⇒ False
1293%     ]
1294%   | tlr_step st1 st1' st1'' tll1 tl1 ⇒
1295%     match tlr2 with
1296%     [ tlr_step st2 st2' st2'' tll2 tl2 ⇒
1297%       tll_rel … tll1 tll2 ∧ tlr_rel … tl1 tl2
1298%     | _ ⇒ False
1299%     ]
1300%   ]
1301% and tll_rel S1 fl1 st1 st1' S2 fl2 st2 st2'
1302%  (tll1 : trace_label_label S1 fl1 st1 st1')
1303%  (tll2 : trace_label_label S2 fl2 st2 st2') on tll1 : Prop ≝
1304%   match tll1 with
1305%   [ tll_base fl1 st1 st1' tal1 H ⇒
1306%     match tll2 with
1307%     [ tll_base fl2 st2 st2 tal2 G ⇒
1308%       as_label_safe … («?, H») = as_label_safe … («?, G») ∧
1309%       tal_rel … tal1 tal2
1310%     ]
1311%   ]
1312% and tal_rel S1 fl1 st1 st1' S2 fl2 st2 st2'
1313%  (tal1 : trace_any_label S1 fl1 st1 st1')
1314%  (tal2 : trace_any_label S2 fl2 st2 st2')
1315%    on tal1 : Prop ≝
1316%   match tal1 with
1317%   [ tal_base_not_return st1 st1' _ _ _ ⇒
1318%     fl2 = doesnt_end_with_ret ∧
1319%     ∃st2mid,taa,H,G,K.
1320%     tal2 ≃ taa_append_tal ? st2 ??? taa
1321%       (tal_base_not_return ? st2mid st2' H G K)
1322%   | tal_base_return st1 st1' _ _ ⇒
1323%     fl2 = ends_with_ret ∧
1324%     ∃st2mid,taa,H,G.
1325%     tal2 ≃ taa_append_tal ? st2 ? st2mid st2' taa
1326%       (tal_base_return ? st2mid st2' H G)
1327%   | tal_base_call st1 st1' st1'' _ prf _ tlr1 _ ⇒
1328%     fl2 = doesnt_end_with_ret ∧
1329%     ∃st2mid,G.as_call_ident S2 («st2mid, G») = as_call_ident ? «st1, prf» ∧
1330%     ∃taa : trace_any_any ? st2 st2mid.∃st2mid',H.
1331%     (* we must allow a tal_base_call to be similar to a call followed
1332%       by a collapsable trace (trace_any_any followed by a base_not_return;
1333%       we cannot use trace_any_any as it disallows labels in the end as soon
1334%       as it is non-empty) *)
1335%     (∃K.∃tlr2 : trace_label_return ? st2mid' st2'.∃L.
1336%       tal2 ≃ taa @ (tal_base_call … H G K tlr2 L) ∧ tlr_rel … tlr1 tlr2) ∨
1337%     ∃st2mid'',K.∃tlr2 : trace_label_return ? st2mid' st2mid''.∃L.
1338%     ∃tl2 : trace_any_label … doesnt_end_with_ret st2mid'' st2'.
1339%       tal2 ≃ taa @ (tal_step_call … H G K tlr2 L tl2) ∧
1340%       tlr_rel … tlr1 tlr2 ∧ tal_collapsable … tl2
1341%   | tal_step_call fl1 st1 st1' st1'' st1''' _ prf _ tlr1 _ tl1 ⇒
1342%     ∃st2mid,G.as_call_ident S2 («st2mid, G») = as_call_ident ? «st1, prf» ∧
1343%     ∃taa : trace_any_any ? st2 st2mid.∃st2mid',H.
1344%     (fl2 = doesnt_end_with_ret ∧ ∃K.∃tlr2 : trace_label_return ? st2mid' st2'.∃L.
1345%       tal2 ≃ taa @ tal_base_call … H G K tlr2 L ∧
1346%       tal_collapsable … tl1 ∧ tlr_rel … tlr1 tlr2) ∨
1347%     ∃st2mid'',K.∃tlr2 : trace_label_return ? st2mid' st2mid''.∃L.
1348%     ∃tl2 : trace_any_label ? fl2 st2mid'' st2'.
1349%       tal2 ≃ taa @ (tal_step_call … H G K tlr2 L tl2) ∧
1350%       tal_rel … tl1 tl2 ∧ tlr_rel … tlr1 tlr2
1351%   | tal_step_default fl1 st1 st1' st1'' _ tl1 _ _ ⇒
1352%     tal_rel … tl1 tal2 (* <- this makes it many to many *)
1353%   ].
1354% \end{verbatim}
1355% \end{comment}
1356% %
1357% In the preceding rules, a $taa$ is an inhabitant of the
1358% $\verb+trace_any_any+~s_1~s_2$ (shorthand $\verb+TAA+~s_1~s_2$),
1359% an inductive data type whose definition
1360% is not in the paper for lack of space. It is the type of valid
1361% prefixes (even empty ones) of \verb+TAL+'s that do not contain
1362% any function call. Therefore it
1363% is possible to concatenate (using ``$\append$'') a \verb+TAA+ to the
1364% left of a \verb+TAL+. A \verb+TAA+ captures
1365% a sequence of silent moves.
1366% The \verb+tal_collapsable+ unary predicate over \verb+TAL+'s
1367% holds when the argument does not contain any function call and it ends
1368% with a label (not a return). The intuition is that after a function call we
1369% can still perform a sequence of silent actions while remaining similar.
1371% As should be expected, even though the rules are asymmetric $\approx$ is in fact
1372% an equivalence relation.
[3343]1376Therefore we now introduce an abstract notion of relation set between abstract
1377statuses and an abstract notion of 1-to-many forward simulation conditions.
1378These two definitions enjoy the following remarkable properties:
1380 \item they are generic enough to accommodate all passes of the CerCo compiler
1381 \item the conjunction of the 1-to-many forward simulation conditions are
1382       just slightly stricter than the statement of a 1-to-many forward
1383       simulation in the classical case. In particular, they only require
1384       the construction of very simple forms of structured traces made of
1385       silent states only.
1386 \item they allow to prove our main result of the paper: the 1-to-many
1387       forward simulation conditions are sufficient to prove the trace
1388       reconstruction theorem
1391Point 3. is the important one. First of all it means that we have reduced
1392the complex problem of trace reconstruction to a much simpler one that,
1393moreover, can be solved with slight adaptations of the forward simulation proof
1394that is performed for a compiler that only cares about functional properties.
1395Therefore we have successfully splitted as much as possible the proof of
1396preservation of functional properties from that of non-functional ones.
[3343]1400\paragraph{Relation sets.}
[3349]1401Let $S_1$ and $S_2$ be two deterministic labelled transition systems as described
1402in \autoref{semantics}. We introduce now the four relations $\mathcal{S,C,R,L}\subseteq S_1\times S_2$
1403between states of the two systems. The first two are abstract and must be instantiated
1404by every pass. The remaining two are derived.
1406The $\S$ relation between states is the classical relation used
[3349]1407in forward simulation proofs. It correlates the data of the states
[3343]1408(e.g. registers, memory, etc.).
[3349]1410The $\C$ relation correlates states that are about to execute calls.
1411It allows to track the position in the target code of every call in the source code.
[3349]1413% The $\L$ relation simply says that the two states are both label
1414% emitting states that emit the same label, \emph{i.e.}\ $s_1\L s_2\iffdef \ell~s_1=\ell~s_2$.
1415% It allows to track the position in
1416% the target code of every cost emitting statement in the source code.
[3349]1418Two states
1419$s_1$ and $s_2$ are $\R$-related if every time $s_1$ is the
[3356]1420successor of a call state $s_1'$ that is $\C$-related to a call state
[3343]1421$s_2'$ in the target code, then $s_2$ is the successor of $s_2'$. Formally:
1422$$s_1\R s_2 \iffdef \forall s_1',s_2'.s_1'\C s_2' \to s_1'\ar s_1 \to s_2' \ar s_2.$$
[3349]1423We will require all pairs of states that return from related calls to be
[3356]1424$\R$-related. This, in combinantion with a dual requirement on function calls,
1425will grant that calls return exactly where they are supposed to be.
[3349]1427We say states in $s_1\in S_1$ and $s_2\in S_2$ are label-related
1428(marked $s_1\L s_2$) if
1430\item they both emit the same label $L$;
1431\item if $L\in \ell(\Functions)$ then
1432there is a state $s_2'$ such that $s_2\to[L]\to[\tau *]s_2'$ with
1433 $s_1\S s_2'$, otherwise if $L\notin\ell(\Functions)$ then $s_1\S s_2$.
[3356]1436Given the relations $\S$ and $\C$, \autoref{fig:forwardsim} defines a set of
1437local simulation conditions that are sufficient to grant the correctness of
1438the compiler.
1442% \begin{subfigure}{.475\linewidth}
1443% \centering
1444% \begin{tikzpicture}[every join/.style={ar}, join all, thick,
1445%                             every label/.style=overlay, node distance=10mm]
1446%     \matrix [diag] (m) {%
1447%          \node (s1) [is jump] {}; & \node [fill=white] (t1) {};\\
1448%          \node (s2) {}; & \node (t2) {}; \\
1449%     };
1450%     \node [above=0 of t1, overlay] {$\alpha$};
1451%     {[-stealth]
1452%     \draw (s1) -- (t1);
1453%     \draw [new] (s2) -- node [above] {$*$} (t2);
1454%     }
1455%     \draw (s1) to node [rel] {$\S$} (s2);
1456%     \draw [new] (t1) to node [rel] {$\S,\L$} (t2);
1457% \end{tikzpicture}
[3347]1458% \caption{The \verb+cl_jump+ case.}
[3343]1459% \label{subfig:cl_jump}
1460% \end{subfigure}
1461% &
[3349]1463\tikzset{diag/.append style=
1464         {every node/.append style={is other,
1465                                    text height=0,text depth=0,text width=0,
1466                                    text opacity=0}                                                         
1467         },
1468         }
[3343]1470    \matrix [diag] (m) {%
[3348]1471         \node (s1) {s_1}; & \node (t1) {s_1'};\\
1472         \node (s2) {s_2}; & \node (t2) {s_2'}; \\
[3343]1473    };
1474    {[-stealth]
[3348]1475    \draw (s1) -- node [above] {$\tau$} (t1);
1476    \draw [new] (s2) -- node [above] {$\tau *$} (t2);
[3343]1477    }
[3347]1478    \draw (s1) to [bend right, anchor=mid] node [rel] {$\S$} (s2);
[3348]1479    \draw [new] (t1) to [bend left, anchor=mid] node [rel] {$\S$} (t2);
1483    \matrix [diag] (m) {%
1484         \node (s1) {s_1}; & \node (t1) {s_1'};\\
[3349]1485         \node (s2) {s_2}; & \node (mid) {s_a}; & \node (t2) {s_2'}; \\
[3343]1486    };
1487    {[-stealth]
[3348]1488    \draw (s1) -- node [above] {$L$} (t1);
[3349]1489    \draw [new] (s2) -- node [above] {$L$} (mid);
1490    \draw [new] (mid) -- node [above] {$\tau *$} (t2);
[3348]1491    }
1492    \draw (s1) to [bend right, anchor=mid] node [rel] {$\S$} (s2);
1493    \draw [new] (t1) to [bend left, anchor=mid] node [rel] {$\S$} (t2);
1495\text{ if $L\notin\ell(\Functions)$}
1498    \matrix [diag] (m) {%
1499         \node (s1) {s_1}; & \node (t1) {s_1'};\\
1500         \node (s2) {s_2};\\
1501    };
1502    {[-stealth]
1503    \draw (s1) -- node [above] {$\ell(f)$} (t1);
1504    }
1505    \draw (s1) to [bend right, anchor=mid] node [rel] {$\S$} (s2);
1506    \draw [new] (t1) to [bend left, anchor=mid] node [rel] {$\S$} (s2);
1510    \matrix [diag, small vgap] (m) {%
1511         &\node (t1) {s_1'}; \\
1512         \node (s1) {s_1}; \\
1513         && \node (l1) {s_b}; & \node (l2) {s_c}; & \node (t2) {s_2'};\\
1514         \node (s2) {s_2}; & \node (c) {s_a};\\   
1515    };
1516    {[-stealth]
[3347]1517    \draw (s1) -- node [above left] {$f$} (t1);
[3348]1518    \draw [new] (s2) -- node [above] {$\tau *$} (c);
1519    \draw [new] (c) -- node [above left] {$f$} (l1);
1520    \draw [new] (l1) -- node [above] {$\ell(f)$} (l2);
1521    \draw [new] (l2) -- node [above] {$\tau *$} (t2);
[3343]1522    }
[3347]1523    \draw (s1) to [bend right] node [rel] {$\S$} (s2);
[3343]1524    \draw [new] (t1) to [bend left] node [rel] {$\S$} (t2);
[3347]1525    \draw [new] (t1) to [bend right] node [rel] {$\C$} (c);
1529    \matrix [diag, small vgap] (m) {%
1530        \node (s1) {s_1}; \\
1531        &\node (t1) {s_1'}; \\
1532        \node (s2) {s_2}; & \node (c) {s_a};\\
1533        && \node (r) {s_b}; & \node (t2) {s_2'}; \\   
[3343]1534    };
1535    {[-stealth]
[3347]1536    \draw (s1) -- node [above right] {$RET$} (t1);
[3348]1537    \draw [new] (s2) -- node [above] {$\tau *$} (c);
[3347]1538    \draw [new] (c) -- node [below left] {$RET$} (r);
[3348]1539    \draw [new] (r) -- node [above] {$\tau *$} (t2);
[3343]1540    }
[3347]1541    \draw (s1) to [bend right] node [rel] {$\S$} (s2);
[3348]1542    \draw [new, overlay] (t1) to [bend left=60] node [rel] {$\S$} (t2);
[3347]1543    \draw [new, overlay] (t1) to [bend left ] node [rel] {$\R$} (r);
[3356]1545\caption{Local simulation conditions. Each one states that the existence of
1546states in~XXX such that the dashed relations holds is implied by the assumptions
1547drawn as solid lines.}
[3356]1552If $S_1,S_2,\S,\C$ satisfy the diagrams in \autoref{fig:forwardsim},
1553$T_1=s_1\To s_1'$ is a structured fragment not starting with a $\ell(f)$ emission,
1554and $s_1\S s_2$, then there is $T_2=s_2\To s_2'$ with $T\approx T'$ and $s_1'\S s_2'$.
[3356]1557If $S_1,S_2,\S,\C$ satisfy the diagrams in \autoref{fig:forwardsim},
1558$M_1:s_1\to^{*} s_1'$ is a measurable fragment of $S_1$ and $s_2$ is such that
1559$s_1\L s_2$, then there is $M_2:s_2\to^{*} s_2'$ with $M_1\approx M_2$.
1563% \begin{figure}
1564% \centering
1565% \begin{tabular}{@{}c@{}c@{}c@{}}
1566% % \begin{subfigure}{.475\linewidth}
1567% % \centering
1568% % \begin{tikzpicture}[every join/.style={ar}, join all, thick,
1569% %                             every label/.style=overlay, node distance=10mm]
1570% %     \matrix [diag] (m) {%
1571% %          \node (s1) [is jump] {}; & \node [fill=white] (t1) {};\\
1572% %          \node (s2) {}; & \node (t2) {}; \\
1573% %     };
1574% %     \node [above=0 of t1, overlay] {$\alpha$};
1575% %     {[-stealth]
1576% %     \draw (s1) -- (t1);
1577% %     \draw [new] (s2) -- node [above] {$*$} (t2);
1578% %     }
1579% %     \draw (s1) to node [rel] {$\S$} (s2);
1580% %     \draw [new] (t1) to node [rel] {$\S,\L$} (t2);
1581% % \end{tikzpicture}
1582% % \caption{The \verb+cl_jump+ case.}
1583% % \label{subfig:cl_jump}
1584% % \end{subfigure}
1585% % &
1586% \begin{subfigure}{.25\linewidth}
1587% \centering
1588% \begin{tikzpicture}[every join/.style={ar}, join all, thick,
1589%                             every label/.style=overlay, node distance=10mm]
1590%     \matrix [diag] (m) {%
1591%          \node (s1) {}; & \node (t1) {};\\
1592%          \node (s2) {}; & \node (t2) {}; \\
1593%     };
1594%     {[-stealth]
1595%     \draw (s1) -- (t1);
1596%     \draw [new] (s2) -- node [above] {$*$} (t2);
1597%     }
1598%     \draw (s1) to [bend right, anchor=mid] node [rel] {$\S$} (s2);
1599%     \draw [new] (t1) to [bend left, anchor=mid] node [rel] {$\S,\L$} (t2);
1600% \end{tikzpicture}
1601% \caption{The \verb+cl_oher+ and \verb+cl_jump+ cases.}
1602% \label{subfig:cl_other_jump}
1603% \end{subfigure}
1604% &
1605% \begin{subfigure}{.375\linewidth}
1606% \centering
1607% \begin{tikzpicture}[every join/.style={ar}, join all, thick,
1608%                             every label/.style=overlay, node distance=10mm]
1609%     \matrix [diag, small gap] (m) {%
1610%          &\node (t1) {}; \\
1611%          \node (s1) [is call] {}; \\
1612%          && \node (l) {}; & \node (t2) {};\\
1613%          \node (s2) {}; & \node (c) [is call] {};\\   
1614%     };
1615%     {[-stealth]
1616%     \draw (s1) -- node [above left] {$f$} (t1);
1617%     \draw [new] (s2) -- node [above] {$*$} (c);
1618%     \draw [new] (c) -- node [below right] {$f$} (l);
1619%     \draw [new] (l) -- node [above] {$*$} (t2);
1620%     }
1621%     \draw (s1) to [bend right] node [rel] {$\S$} (s2);
1622%     \draw [new] (t1) to [bend left] node [rel] {$\S$} (t2);
1623%     \draw [new] (t1) to [bend left] node [rel] {$\L$} (l);
1624%     \draw [new] (t1) to [bend right] node [rel] {$\C$} (c);
1625%     \end{tikzpicture}
1626% \caption{The \verb+cl_call+ case.}
1627% \label{subfig:cl_call}
1628% \end{subfigure}
1629% &
1630% \begin{subfigure}{.375\linewidth}
1631% \centering
1632% \begin{tikzpicture}[every join/.style={ar}, join all, thick,
1633%                             every label/.style=overlay, node distance=10mm]
1634%     \matrix [diag, small gap] (m) {%
1635%         \node (s1) [is ret] {}; \\
1636%         &\node (t1) {}; \\
1637%         \node (s2) {}; & \node (c) [is ret] {};\\
1638%         && \node (r) {}; & \node (t2) {}; \\   
1639%     };
1640%     {[-stealth]
1641%     \draw (s1) -- node [above right] {$RET$} (t1);
1642%     \draw [new] (s2) -- node [above] {$*$} (c);
1643%     \draw [new] (c) -- node [below left] {$RET$} (r);
1644%     \draw [new] (r) -- node [above] {$*$} (t2);
1645%     }
1646%     \draw (s1) to [bend right] node [rel] {$\S$} (s2);
1647%     \draw [new, overlay] (t1) to [bend left=60] node [rel] {$\S,\L$} (t2);
1648%     \draw [new, overlay] (t1) to [bend left ] node [rel] {$\R$} (r);
1649% \end{tikzpicture}
1650% \caption{The \verb+cl_return+ case.}
1651% \label{subfig:cl_return}
1652% \end{subfigure}
1653% \end{tabular}
1654% \caption{Mnemonic diagrams depicting the hypotheses for the preservation of structured traces.
1655%          Dashed lines
1656%          and arrows indicates how the diagrams must be closed when solid relations
1657%          are present.}
1658% \label{fig:forwardsim}
1659% \end{figure}
[3349]1661% \paragraph{1-to-many forward simulation conditions.}
1662% \begin{condition}[Cases \verb+cl_other+ and \verb+cl_jump+]
1663%  For all $s_1,s_1',s_2$ such that $s_1 \S s_1'$, and
1664%  $s_1\exec s_1'$, and either $s_1 \class \verb+cl_other+$ or
1665%  both $s_1\class\verb+cl_other+\}$ and $\ell~s_1'$,
1666%  there exists an $s_2'$ and a $\verb+trace_any_any_free+~s_2~s_2'$ called $taaf$
1667%  such that $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and either
1668% $taaf$ is non empty, or one among $s_1$ and $s_1'$ is \verb+as_costed+.
1669% \end{condition}
1671% In the above condition depicted in \autoref{subfig:cl_other_jump},
1672% a $\verb+trace_any_any_free+~s_1~s_2$ (which from now on
1673% will be shorthanded as \verb+TAAF+) is an
1674% inductive type of structured traces that do not contain function calls or
1675% cost emission statements. Differently from a \verb+TAA+, the
1676% instruction to be executed in the lookahead state $s_2$ may be a cost emission
1677% statement.
1679% The intuition of the condition is that one step can be replaced with zero or more steps if it
1680% preserves the relation between the data and if the two final statuses are
1681% labelled in the same way. Moreover, we must take special care of the empty case
1682% to avoid collapsing two consecutive states that emit a label, missing one of the two emissions.
1684% \begin{condition}[Case \verb+cl_call+]
1685%  For all $s_1,s_1',s_2$ s.t. $s_1 \S s_1'$ and
1686%  $s_1\exec s_1'$ and $s_1 \class \verb+cl_call+$, there exists $s_a, s_b, s_2'$, a
1687% $\verb+TAA+~s_2~s_a$, and a
1688% $\verb+TAAF+~s_b~s_2'$ such that:
1689% $s_a\class\verb+cl_call+$, the \verb+as_call_ident+'s of
1690% the two call states are the same, $s_1 \C s_a$,
1691% $s_a\exec s_b$, $s_1' \L s_b$ and
1692% $s_1' \S s_2'$.
1693% \end{condition}
1695% The condition, depicted in \autoref{subfig:cl_call} says that, to simulate a function call, we can perform a
1696% sequence of silent actions before and after the function call itself.
1697% The old and new call states must be $\C$-related, the old and new
1698% states at the beginning of the function execution must be $\L$-related
1699% and, finally, the two initial and final states must be $\S$-related
1700% as usual.
1702% \begin{condition}[Case \verb+cl_return+]
1703%  For all $s_1,s_1',s_2$ s.t. $s_1 \S s_1'$,
1704%  $s_1\exec s_1'$ and $s_1 \class \verb+cl_return+$, there exists $s_a, s_b, s_2'$, a
1705% $\verb+TAA+~s_2~s_a$, a
1706% $\verb+TAAF+~s_b~s_2'$ called $taaf$ such that:
1707% $s_a\class\verb+cl_return+$,
1708% $s_a\exec s_b$,
1709% $s_1' \R s_b$ and
1710% $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and either
1711% $taaf$ is non empty, or $\lnot \ell~s_a$.
1712% \end{condition}
1714% Similarly to the call condition, to simulate a return we can perform a
1715% sequence of silent actions before and after the return statement itself,
1716% as depicted in \autoref{subfig:cl_return}.
1717% The old and the new statements after the return must be $\R$-related,
1718% to grant that they returned to corresponding calls.
1719% The two initial and final states must be $\S$-related
1720% as usual and, moreover, they must exhibit the same labels. Finally, when
1721% the suffix is non empty we must take care of not inserting a new
1722% unmatched cost emission statement just after the return statement.
1724% \begin{comment}
1725% \begin{verbatim}
1726% definition status_simulation ≝
1727%   λS1 : abstract_status.
1728%   λS2 : abstract_status.
1729%   λsim_status_rel : status_rel S1 S2.
1730%     ∀st1,st1',st2.as_execute S1 st1 st1' →
1731%     sim_status_rel st1 st2 →
1732%     match as_classify … st1 with
1733%     [ None ⇒ True
1734%     | Some cl ⇒
1735%       match cl with
1736%       [ cl_call ⇒ ∀prf.
1737%         (*
1738%              st1' ------------S----------\
1739%               ↑ \                         \
1740%              st1 \--L--\                   \
1741%               | \       \                   \
1742%               S  \-C-\  st2_after_call →taa→ st2'
1743%               |       \     ↑
1744%              st2 →taa→ st2_pre_call
1745%         *)
1746%         ∃st2_pre_call.
1747%         as_call_ident ? st2_pre_call = as_call_ident ? («st1, prf») ∧
1748%         call_rel ?? sim_status_rel «st1, prf» st2_pre_call ∧
1749%         ∃st2_after_call,st2'.
1750%         ∃taa2 : trace_any_any … st2 st2_pre_call.
1751%         ∃taa2' : trace_any_any … st2_after_call st2'.
1752%         as_execute … st2_pre_call st2_after_call ∧
1753%         sim_status_rel st1' st2' ∧
1754%         label_rel … st1' st2_after_call
1755%       | cl_return ⇒
1756%         (*
1757%              st1
1758%             / ↓
1759%            | st1'----------S,L------------\
1760%            S   \                           \
1761%             \   \-----R-------\            |     
1762%              \                 |           |
1763%              st2 →taa→ st2_ret |           |
1764%                           ↓   /            |
1765%                      st2_after_ret →taaf→ st2'
1767%            we also ask that st2_after_ret be not labelled if the taaf tail is
1768%            not empty
1769%         *)
1770%         ∃st2_ret,st2_after_ret,st2'.
1771%         ∃taa2 : trace_any_any … st2 st2_ret.
1772%         ∃taa2' : trace_any_any_free … st2_after_ret st2'.
1773%         (if taaf_non_empty … taa2' then ¬as_costed … st2_after_ret else True) ∧
1774%         as_classifier … st2_ret cl_return ∧
1775%         as_execute … st2_ret st2_after_ret ∧ sim_status_rel st1' st2' ∧
1776%         ret_rel … sim_status_rel st1' st2_after_ret ∧
1777%         label_rel … st1' st2'
1778%       | cl_other ⇒
1779%           (*         
1780%           st1 → st1'
1781%             |      \
1782%             S      S,L
1783%             |        \
1784%            st2 →taaf→ st2'
1786%            the taaf can be empty (e.g. tunneling) but we ask it must not be the
1787%            case when both st1 and st1' are labelled (we would be able to collapse
1788%            labels otherwise)
1789%          *)
1790%         ∃st2'.
1791%         ∃taa2 : trace_any_any_free … st2 st2'.
1792%         (if taaf_non_empty … taa2 then True else (¬as_costed … st1 ∨ ¬as_costed … st1')) ∧
1793%         sim_status_rel st1' st2' ∧
1794%         label_rel … st1' st2'
1795%       | cl_jump ⇒
1796%         (* just like cl_other, but with a hypothesis more *)
1797%         as_costed … st1' →
1798%         ∃st2'.
1799%         ∃taa2 : trace_any_any_free … st2 st2'.
1800%         (if taaf_non_empty … taa2 then True else (¬as_costed … st1 ∨ ¬as_costed … st1')) ∧
1801%         sim_status_rel st1' st2' ∧
1802%         label_rel … st1' st2'
1803%       ]
1804%     ].
1805% \end{verbatim}
1806% \end{comment}
[3349]1808% \paragraph{Main result: the 1-to-many forward simulation conditions
1809% are sufficient to trace reconstruction}
1811% Let us assume that a relation set is given such that the 1-to-many
1812% forward simulation conditions are satisfied. Under this assumption we
1813% can prove the following three trace reconstruction theorems by mutual
1814% structural induction over the traces given in input between the
1815% $s_1$ and $s_1'$ states.
1817% In particular, the \verb+status_simulation_produce_tlr+ theorem
1818% applied to the \verb+main+ function of the program and equal
1819% $s_{2_b}$ and $s_2$ states shows that, for every initial state in the
1820% source code that induces a structured trace in the source code,
1821% the compiled code produces a similar structured trace.
1823% \begin{theorem}[\verb+status_simulation_produce_tlr+]
1824% For every $s_1,s_1',s_{2_b},s_2$ s.t.
1825% there is a $\verb+TLR+~s_1~s_1'$ called $tlr_1$ and a
1826% $\verb+TAA+~s_{2_b}~s_2$ and $s_1 \L s_{2_b}$ and
1827% $s_1 \S s_2$, there exists $s_{2_m},s_2'$ s.t.
1828% there is a $\verb+TLR+~s_{2_b}~s_{2_m}$ called $tlr_2$ and
1829% there is a $\verb+TAAF+~s_{2_m}~s_2'$ called $taaf$
1830% s.t. if $taaf$ is non empty then $\lnot (\ell~s_{2_m})$,
1831% and $tlr_1\approx tlr_2$
1832% and $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1833% $s_1' \R s_{2_m}$.
1834% \end{theorem}
1836% The theorem states that a \verb+trace_label_return+ in the source code
1837% together with a precomputed preamble of silent states
1838% (the \verb+TAA+) in the target code induces a
1839% similar \verb+trace_label_return+ in the target code which can be
1840% followed by a sequence of silent states. Note that the statement does not
1841% require the produced \verb+trace_label_return+ to start with the
1842% precomputed preamble, even if this is likely to be the case in concrete
1843% implementations. The preamble in input is necessary for compositionality, e.g.
1844% because the 1-to-many forward simulation conditions allow in the
1845% case of function calls to execute a preamble of silent instructions just after
1846% the function call.
1848% Clearly similar results are also available for the other two types of structured
1849% traces (in fact, they are all proved simultaneously by mutual induction).
[3347]1850% \begin{theorem}[\verb+status_simulation_produce_tll+]
[3343]1851% For every $s_1,s_1',s_{2_b},s_2$ s.t.
[3347]1852% there is a $\verb+TLL+~b~s_1~s_1'$ called $tll_1$ and a
[3343]1853% $\verb+TAA+~s_{2_b}~s_2$ and $s_1 \L s_{2_b}$ and
1854% $s_1 \S s_2$, there exists $s_{2_m},s_2'$ s.t.
1855% \begin{itemize}
1856%  \item if $b$ (the trace ends with a return) then there exists $s_{2_m},s_2'$
[3347]1857%        and a trace $\verb+TLL+~b~s_{2_b}~s_{2_m}$ called $tll_2$
1858%        and a $\verb+TAAF+~s_{2_m}~s_2'$ called $taa_2$ s.t.
[3343]1859%        $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1860%        $s_1' \R s_{2_m}$ and
1861%        $tll_1\approx tll_2$ and
1862%        if $taa_2$ is non empty then $\lnot \ell~s_{2_m}$;
1863%  \item else there exists $s_2'$ and a
[3347]1864%        $\verb+TLL+~b~s_{2_b}~s_2'$ called $tll_2$ such that
[3343]1865%        $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1866%        $tll_1\approx tll_2$.
1867% \end{itemize}
1868% \end{theorem}
1870% The statement is similar to the previous one: a source
[3347]1871% \verb+trace_label_label+ and a given target preamble of silent states
1872% in the target code induce a similar \verb+trace_label_label+ in the
[3343]1873% target code, possibly followed by a sequence of silent moves that become the
[3347]1874% preamble for the next \verb+trace_label_label+ translation.
[3347]1876% \begin{theorem}[\verb+status_simulation_produce_tal+]
[3343]1877% For every $s_1,s_1',s_2$ s.t.
[3347]1878% there is a $\verb+TAL+~b~s_1~s_1'$ called $tal_1$ and
[3343]1879% $s_1 \S s_2$
1880% \begin{itemize}
1881%  \item if $b$ (the trace ends with a return) then there exists $s_{2_m},s_2'$
[3347]1882%    and a trace $\verb+TAL+~b~s_2~s_{2_m}$ called $tal_2$ and a
1883%    $\verb+TAAF+~s_{2_m}~s_2'$ called $taa_2$ s.t.
[3343]1884%    $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1885%    $s_1' \R s_{2_m}$ and
1886%    $tal_1 \approx tal_2$ and
1887%    if $taa_2$ is non empty then $\lnot \ell~s_{2_m}$;
1888%  \item else there exists $s_2'$ and a
[3347]1889%    $\verb+TAL+~b~s_2~s_2'$ called $tal_2$ such that
[3343]1890%    either $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1891%        $tal_1\approx tal_2$
1892%    or $s_1' \mathrel{{\S} \cap {\L}} s_2$ and
[3347]1893%    $\verb+tal_collapsable+~tal_1$ and $\lnot \ell~s_1$.
[3343]1894% \end{itemize}
1895% \end{theorem}
1897% The statement is also similar to the previous ones, but for the lack of
1898% the target code preamble.
1902For every $s_1,s_1',s_2$ s.t.
[3347]1903there is a $\verb+trace_label_return+~s_1~s_1'$ called $tlr_1$ and
[3343]1904$s_1 (\L \cap \S) s_2$
1905there exists $s_{2_m},s_2'$ s.t.
[3347]1906there is a $\verb+trace_label_return+~s_2~s_{2_m}$ called $tlr_2$ and
1907there is a $\verb+trace_any_any_free+~s_{2_m}~s_2'$ called $taaf$
1908s.t. if $taaf$ is non empty then $\lnot (\verb+as_costed+~s_{2_m})$,
1909and $\verb+tlr_rel+~tlr_1~tlr_2$
[3343]1910and $s_1' (\S \cap \L) s_2'$ and
1911$s_1' \R s_{2_m}$.
1917status_simulation_produce_tlr S1 S2 R
1918(* we start from this situation
1919     st1 →→→→tlr→→→→ st1'
1920      | \
1921      L  \---S--\
1922      |          \
1923   st2_lab →taa→ st2   (the taa preamble is in general either empty or given
1924                        by the preceding call)
1926   and we produce
1927     st1 →→→→tlr→→→→ st1'
1928             \\      /  \
1929             //     R    \-L,S-\
1930             \\     |           \
1931   st2_lab →tlr→ st2_mid →taaf→ st2'
1933  st1 st1' st2_lab st2
1934  (tlr1 : trace_label_return S1 st1 st1')
1935  (taa2_pre : trace_any_any S2 st2_lab st2)
1936  (sim_execute : status_simulation S1 S2 R)
1937  on tlr1 : R st1 st2 → label_rel … st1 st2_lab →
1938  ∃st2_mid.∃st2'.
1939  ∃tlr2 : trace_label_return S2 st2_lab st2_mid.
1940  ∃taa2 : trace_any_any_free … st2_mid st2'.
1941  (if taaf_non_empty … taa2 then ¬as_costed … st2_mid else True) ∧
1942  R st1' st2' ∧ ret_rel … R st1' st2_mid ∧ label_rel … st1' st2' ∧
1943  tlr_rel … tlr1 tlr2
1947\section{Conclusions and future works}
1951The labelling approach is a technique to implement compilers that induce on
1952the source code a non uniform cost model determined from the object code
1953produced. The cost model assigns a cost to each basic block of the program.
1954The main theorem of the approach says that there is an exact
1955correspondence between the sequence of basic blocks started in the source
1956and object code, and that no instruction in the source or object code is
1957executed outside a basic block. Thus the cost of object code execution
1958can be computed precisely on the source.
[3356]1960In this paper we scaled the labelling approach to cover a programming language
1961with function calls. This introduces new difficulties when the language
[3343]1962is unstructured, i.e. it allows function calls to return anywhere in the code,
1963destroying the hope of a static prediction of the cost of basic blocks.
[3356]1964We restored static predictability by introducing a new semantics for unstructured
[3343]1965programs that single outs well structured executions. The latter are represented
1966by structured traces, a generalisation of streams of observables that capture
1967several structural invariants of the execution, like well nesting of functions
1968or the fact that every basic block must start with a code emission statement.
[3356]1969We showed that structured traces are sufficiently well behaved to statically compute a precise cost model on the object code.
[3356]1971We also proved that the cost model computed on the object code is also valid
1972on the source code if the compiler respects two requirements: the weak execution
1973traces of the source and target code must be the same and the object
1974code execution fragments are structured.
[3356]1976To prove that a compiler respects the requirement we extended the notion
1977of forward simulation proof for a labelled transition system to grant
1978preservation of structured fragments. If the source language of the compiler
1979is structured, all its execution fragments are, allowing to deduce from
1980preservation of structure that object code fragments are structured too.
[3356]1982Finally, we identified measurable execution fragments that are those whose
1983execution time (once compiled) can be exactly computed looking at the object
1984code weak execution traces only. A final instrumentation pass on the source
1985code can then be used to turn the non functional property of having a certain
1986cost into the functional property of granting that a certain global variable
1987incremented at the beginning of every block according to the induced cost model
1988has a certain value.
[3343]1990All results presented in the paper are part of a larger certification of a
1991C compiler which is based on the labelling approach. The certification, done
1992in Matita, is the main deliverable of the FET-Open Certified Complexity (CerCo).
1994The short term future work consists in the completion of the certification of
1995the CerCo compiler exploiting the main theorem of this paper.
1997\paragraph{Related works.}
1998CerCo is the first project that explicitly tries to induce a
1999precise cost model on the source code in order to establish non-functional
2000properties of programs on an high level language. Traditional certifications
2001of compilers, like~\cite{compcert2,piton}, only explicitly prove preservation
2002of the functional properties.
2004Usually forward simulations take the following form: for each transition
2005from $s_1$ to $s_2$ in the source code, there exists an equivalent sequence of
2006transitions in the target code of length $n$. The number $n$ of transition steps
2007in the target code can just be the witness of the existential statement.
2008An equivalent alternative when the proof of simulation is constructive consists
2009in providing an explicit function, called \emph{clock function} in the
2010literature~\cite{clockfunctions}, that computes $n$ from $s_1$. Every clock
2011function constitutes then a cost model for the source code, in the spirit of
2012what we are doing in CerCo. However, we believe our solution to be superior
2013in the following respects: 1) the machinery of the labelling approach is
2014insensible to the resource being measured. Indeed, any cost model computed on
2015the object code can be lifted to the source code (e.g. stack space used,
[3356]2016energy consumed, etc.) simply re-proving an analogue of~\autoref{static}.
2017For example, in CerCo we transported to the source level not only the execution
2018time cost model, but also the amount of stack used by function calls.
2019On the contrary, clock functions only talk about
[3343]2020number of transition steps. In order to extend the approach with clock functions
2021to other resources, additional functions must be introduced. Moreover, the
2022additional functions would be handled differently in the proof.
20232) the cost models induced by the labelling approach have a simple presentation.
2024In particular, they associate a number to each basic block. More complex
2025models can be induced when the approach is scaled to cover, for instance,
2026loop optimisations~\cite{loopoptimizations}, but the costs are still meant to
2027be easy to understand and manipulate in an interactive theorem prover or
2028in Frama-C.
2029On the contrary, a clock function is a complex function of the state $s_1$
2030which, as a function, is an opaque object that is difficult to reify as
2031source code in order to reason on it.
2036% \appendix
2037% \section{Notes for the reviewers}
2039% The results described in the paper are part of a larger formalization
2040% (the certification of the CerCo compiler). At the moment of the submission
2041% we need to single out from the CerCo formalization the results presented here.
2042% Before the 16-th of February we will submit an attachment that contains the
2043% minimal subset of the CerCo formalization that allows to prove those results.
2044% At that time it will also be possible to measure exactly the size of the
2045% formalization described here. At the moment a rough approximation suggests
2046% about 2700 lines of Matita code.
2048% We will also attach the development version of the interactive theorem
2049% prover Matita that compiles the submitted formalization. Another possibility
2050% is to backport the development to the last released version of the system
2051% to avoid having to re-compile Matita from scratch.
2053% The programming and certification style used in the formalization heavily
2054% exploit dependent types. Dependent types are used: 1) to impose invariants
2055% by construction on the data types and operations (e.g. a traces from a state
2056% $s_1$ to a state $s_2$ can be concatenad to a trace from a state
2057% $s_2'$ to a state $s_3$ only if $s_2$ is convertible with $s_2'$); 2)
2058% to state and prove the theorems by using the Russell methodology of
2059% 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.
[3347]2060% }, better known in the Coq world as ``\verb+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
[3343]2061% mandatorily on dependent types: it should be easy to adapt the technique
2062% and results presented in the paper to HOL.
2064% Finally, Matita and Coq are based on minor variations of the Calculus of
2065% (Co)Inductive Constructions. These variations do not affect the CerCo
2066% formalization. Therefore a porting of the proofs and ideas to Coq would be
2067% rather straightforward.
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