source: Papers/itp-2013/ccexec2.tex @ 3344

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98\def\L{\mathrel{\mathcal L}}
99\def\S{\mathrel{\mathcal S}}
100\def\R{\mathrel{\mathcal R}}
101\def\C{\mathrel{\mathcal C}}
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118\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
119Emerging Technologies (FET) programme within the Seventh Framework
120Programme for Research of the European Commission, under FET-Open grant
121number: 243881}}
122\author{Paolo Tranquilli \and Claudio Sacerdoti Coen}
123\institute{Department of Computer Science and Engineering, University of Bologna,\\\email{}, \email{}}
126The labelling approach is a technique to lift cost models for non-functional
127properties of programs from the object code to the source code. It is based
128on the preservation of the structure of the high level program in every
129intermediate language used by the compiler. Such structure is captured by
130observables that are added to the semantics and that needs to be preserved
131by the forward simulation proof of correctness of the compiler. Additional
132special observables are required for function calls. In this paper we
133present a generic forward simulation proof that preserves all these observables.
134The proof statement is based on a new mechanised semantics that traces the
135structure of execution when the language is unstructured. The generic semantics
136and simulation proof have been mechanised in the interactive theorem prover
141The \emph{labelling approach} has been introduced in~\cite{easylabelling} as
142a technique to \emph{lift} cost models for non-functional properties of programs
143from the object code to the source code. Examples of non-functional properties
144are execution time, amount of stack/heap space consumed and energy required for
145communication. The basic idea of the approach is that it is impossible to
146provide a \emph{uniform} cost model for an high level language that is preserved
147\emph{precisely} by a compiler. For instance, two instances of an assignment
148$x = y$ in the source code can be compiled very differently according to the
149place (registers vs stack) where $x$ and $y$ are stored at the moment of
150execution. Therefore a precise cost model must assign a different cost
151to every occurrence, and the exact cost can only be known after compilation.
153According to the labelling approach, the compiler is free to compile and optimise
154the source code without any major restriction, but it must keep trace
155of what happens to basic blocks during the compilation. The cost model is
156then computed on the object code. It assigns a cost to every basic block.
157Finally, the compiler propagates back the cost model to the source level,
158assigning a cost to each basic block of the source code.
160Implementing the labelling approach in a certified compiler
161allows to reason formally on the high level source code of a program to prove
162non-functional properties that are granted to be preserved by the compiler
163itself. The trusted code base is then reduced to 1) the interactive theorem
164prover (or its kernel) used in the certification of the compiler and
1652) the software used to certify the property on the source language, that
166can be itself certified further reducing the trusted code base.
167In~\cite{easylabelling} the authors provide an example of a simple
168certified compiler that implements the labelling approach for the
169imperative \texttt{While} language~\cite{while}, that does not have
170pointers and function calls.
172The labelling approach has been shown to scale to more interesting scenarios.
173In particular in~\cite{functionallabelling} it has been applied to a functional
174language and in~\cite{loopoptimizations} it has been shown that the approach
175can be slightly complicated to handle loop optimisations and, more generally,
176program optimisations that do not preserve the structure of basic blocks.
177On-going work also shows that the labelling approach is also compatible with
178the complex analyses required to obtain a cost model for object code
179on processors that implement advanced features like pipelining, superscalar
180architectures and caches.
182In 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
1838051 object code. The compiler is
184moderately optimising and implements a compilation chain that is largely
185inspired 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
186of function calls. Surprisingly, the addition of function calls require a
187revisitation of the proof technique given in~\cite{easylabelling}. In
188particular, at the core of the labelling approach there is a forward
189simulation proof that, in the case of \texttt{While}, is only minimally
190more complex than the proof required for the preservation of the
191functional properties only. In the case of a programming language with
192function calls, instead, it turns out that the forward simulation proof for
193the back-end languages must grant a whole new set of invariants.
195In this paper we present a formalisation in the Matita interactive theorem
196prover~\cite{matita1,matita2} of a generic version of the simulation proof required for unstructured
197languages. All back-end languages of the CerCo compiler are unstructured
198languages, so the proof covers half of the correctness of the compiler.
199The statement of the generic proof is based on a new semantics
200for imperative unstructured languages that is based on \emph{structured
201traces} and that restores the preservation of structure in the observables of
202the semantics. The generic proof allows to almost completely split the
203part of the simulation that deals with functional properties only from the
204part that deals with the preservation of structure.
206The plan of this paper is the following. In Section~\ref{labelling} we
207sketch the labelling method and the problems derived from the application
208to languages with function calls. In Section~\ref{semantics} we introduce
209a generic description of an unstructured imperative language and the
210corresponding structured traces (the novel semantics). In
211Section~\ref{simulation} we describe the forward simulation proof.
212Conclusions and future works are in Section~\ref{conclusions}
214\section{The labelling approach}
217\section{A brief introduction to the labelling approach}
221EMIT L_1;                         EMIT L_1         cost += k_1;
222I_1;                              I_3              I_1;
223for (i=0; i<2; i++) {        l_1: COND l_2         for (i=0; i<2; i++) {
224  EMIT L_2;                       EMIT L_2           cost += k_2;           
225  I_2;                            I_4                I_2;               
226 }                                GOTO l_1          }                   
227EMIT L_3;                    l_2: EMIT L_3         cost += k_3;           
229\caption{The labelling approach applied to a simple program.\label{examplewhile}. The $I_i$ are sequences of instructions not containing jumps or loops. }
231We briefly explain the labelling approach on the example in Figure~\ref{examplewhile}. The user wants to analyse the execution time of the program made by
232the black lines in the r.h.s. of the figure. He compiles the program using
233a special compiler that first inserts in the code three label emission
234statements (\texttt{EMIT L$_i$}, in red) to mark the beginning of basic blocks;
235then the compiler compiles the code to machine code (in the
236middle of the figure), granting that the execution of the source and object
237code emits the same sequence of labels ($L_1; L_2; L_2; L_3$ in the example).
238This is achieved by keeping track of basic blocks during compilation, avoiding
239all optimizations that alter the control flow. The latter can be recovered with
240a more refined version of the labelling approach~\cite{tranquill}, but in the
241present paper we stick to this simple variant for simplicity. Once the object
242code is produced, the compiler runs a static code analyzer to associate to
243each label $L_1, \ldots, L_3$ the cost (in clock cycles) of the instructions
244that belong to the corresponding basic block. For example, the cost $k_1$
245associated to $L_1$ is the number of cycles required to execute the block
246$I_3$ and $COND l_2$, while the cost $k_2$ associated to $L_2$ counts the
247cycles required by the block $I_4$ and $GOTO l_1$. The compiler also guarantees
248that every executed instruction is in the scope of some code emission label,
249that each scope does not contain loops (to associate a finite cost), and that
250both branches of a conditional statement are followed by a code emission
251statement. Under these assumptions it is true that the total execution cost
252of the program $\Delta_t$ is equal to the sum over the sequence of emitted
253labels of the cost associated to every label:
254$\Delta t = k(L_1; L_2; L_2; L_3) = k_1 + k_2 + k_2 + k_3$.
255Finally, the compiler emits an instrumented version of the source code
256(in the r.h.s. of the figure) where label emission statements are replaced
257by increments of a global variable cost that, before every increment, holds the
258exact number of clock cycles spent by the microprocessor so far:
259the difference $\Delta clock$ between the final and initial value of the variable clock is $\Delta clock = k_1 + k_2 + k_2 + k_3 = \Delta t$. Finally, the
260user can employ any available method (e.g. Hoare logic, invariant generators,
261abstract interpretation and automated provers) to certify that $\Delta clock$
262never exceeds a certain bound~\cite{cerco}, which is now a functional property
263of the code.
265\section{The labelling approach in presence of loops}
267Let's now consider a simple program written in C that contains a function
268pointer call inside the scope of the cost label $L_1$.
270main: EMIT L_1       g: EMIT L_3   EMIT L_1     g: EMIT L_3
271      I_1;              I_3;       I_4;            I_6;
272      (*f)();           return;    CALL            RETURN
273      I_2;                         I_5;
274      EMIT L_2                     EMIT L_2
276The labelling method works exactly as before, inserting
277code emission statements/\texttt{cost} variable increments at the beginning
278of every basic block and at the beginning of every function. The compiler
279still grants that the sequence of labels observed on the two programs are
280the same. A new difficulty happears when the compiler needs to statically
281analyze the object code to assign a cost to every label. What should the scope
282of the $L_1$ label be? After executing the $I_4$ block, the \texttt{CALL}
283statement passes control to a function that cannot be determined statically.
284Therefore the cost of executing the body must be paid by some other label
285(hence the requirement that every function starts with a code emission
286statement). What label should pay for the cost for the block $I_5$? The only
287reasonable answer is $L_1$, i.e. \emph{the scope of labels should extend to the
288next label emission statement, stepping over function calls}.
290The latter definition of scope is adeguate on the source level because
291C is a structured language that guarantees that every function call, if it
292returns, passes control to the first instruction that follows the call. However,
293this is not guaranteed for object code, the backend languages of a compiler
294and, more generally, for unstructured
295languages that use a writable control stack to store the return address of
296calls. For example, $I_6$ could increment by $1$ the return address on the
297stack so that the next \texttt{RETURN} would start at the second instruction
298of $I_5$. The compiler would still be perfectly correct if a random, dead
299code instruction was also added just after each \texttt{CALL}. More generally,
300\emph{there is no guarantee that a correct compiler that respects the functional
301behaviour of a program also respects the calling structure of the source code}.
302Without such an assumption, however, it may not be true that the execution cost
303of the program is the sum of the costs associated to the labels emitted. In our
304example, 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.
306Obviously, any reasonably written compiler produces object code that behaves
307as if the language was structured (i.e. by properly nesting function
308calls/returns and without tampering with the return addresses on the control
309stack). This property, however, is a property of the runs of object code
310programs, and not a property of the object code that can be easily statically
311verified (as the ones we required for the basic labelling method).
312Therefore, we now need to single out those runs whose cost behaviour can be
313statically predicted, and we need to prove that every run of programs generated
314by our compiler are of that type. We call them \emph{structured} since their
315main property is to respect properties that hold for free on the source code
316because the source language is structured. Moreover, in order to avoid proving
317too many preservation properties of our compiler, we drop the original
318requirements on the object code (all instructons must be in scope of some labels, no loops inside a scope, etc.) in favour of the corresponding requirement
319for structured runs (a structured run must start with a label emission, no instruction can be executed twice between two emissions, etc.).
321We will therefore proceed as follows. In the following section
3221) we formally introduce the notion of
323structured trace, which captures structured runs in the style of labelled
324transition systems; 2) we show that on the object code we can correctly
325compute the execution time of a structured run from the sequence of labels
326observed; 3) we give unstructured languages a semantics in terms of structured
327traces; 4) we show that on the source code we can correctly compute the
328execution time of a program if the compiler produces object code whose
329runs are weakly similar to the source code runs.
331The notion of weak bisimulation for structured traces is a global property
332which is hard to prove formally and much more demanding than the simple forward
333simulation required for proofs of preservation of functional properties.
334Therefore in Section~\ref{XXX} we will present a set of local simulation
335conditions that refine the corresponding conditions for forward simulation and
336that are sufficient to grant the production of weakly similar traces.
338All the definitions and theorems presented in the paper have been formalized
339in the interactive theorem prover Matita and are being used to certify
340the complexity preserving compiler developed in the CerCo project~\cite{cerco}.
341The formalization can be
342found at~\ref{YYY} and it heavily relies on algebraic and dependent types for
343both structured traces and the definition of weak similarity. In the paper
344we did not try to stay close to the formalization. On the contrary,
345the definitions given in the paper are the result of a significant
346simplification effort for
347the sake of presentation and to make easier the re-implementation of the
348concepts in a proof assistant which is not based on the Calculus of Inductive
349Constructions. However the formalization is heavily commented to allow the
350reader to understand the technical details of the formalization.
355We briefly sketch here a simplified version of the labelling approach as
356introduced in~\cite{easylabelling}. The simplification strengthens the
357sufficient conditions given in~\cite{easylabelling} to allow a simpler
358explanation. The simplified conditions given here are also used in the
359CerCo compiler to simplify the proof.
361Let $\mathcal{P}$ be a programming language whose semantics is given in
362terms of observables: a run of a program yields a finite or infinite
363stream of observables. We also assume for the time being that function
364calls are not available in $\mathcal{P}$. We want to associate a cost
365model to a program $P$ written in $\mathcal{P}$. The first step is to
366extend the syntax of $\mathcal{P}$ with a new construct $\texttt{emit L}$
367where $L$ is a label distinct from all observables of $\mathcal{P}$.
368The semantics of $\texttt{emit L}$ is the emission of the observable
369\texttt{L} that is meant to signal the beginning of a basic block.
371There exists an automatic procedure that injects into the program $P$ an
372$\texttt{emit L}$ at the beginning of each basic block, using a fresh
373\texttt{L} for each block. In particular, the bodies of loops, both branches
374of \texttt{if-then-else}s and the targets of \texttt{goto}s must all start
375with an emission statement.
377Let now $C$ be a compiler from $\mathcal{P}$ to the object code $\mathcal{M}$,
378that is organised in passes. Let $\mathcal{Q}_i$ be the $i$-th intermediate
379language used by the compiler. We can easily extend every
380intermediate language (and its semantics) with an $\texttt{emit L}$ statement
381as we did for $\mathcal{P}$. The same is possible for $\mathcal{M}$ too, with
382the additional difficulty that the syntax of object code is given as a
383sequence of bytes. The injection of an emission statement in the object code
384can be done using a map that maps two consecutive code addresses with the
385statement. The intended semantics is that, if $(pc_1,pc_2) \mapsto \texttt{emit L}$ then the observable \texttt{L} is emitted after the execution of the
386instruction stored at $pc_1$ and before the execution of the instruction
387stored at $pc_2$. The two program counters are necessary because the
388instruction stored at $pc_1$ can have multiple possible successors (e.g.
389in case of a conditional branch or an indirect call). Dually, the instruction
390stored at $pc_2$ can have multiple possible predecessors (e.g. if it is the
391target of a jump).
393The compiler, to be functionally correct, must preserve the observational
394equivalence, i.e. executing the program after each compiler pass should
395yield the same stream of observables. After the injection of emission
396statements, observables now capture both functional and non-functional
398This correctness property is called in the literature a forward simulation
399and is sufficient for correctness when the target language is
401We also require a stronger, non-functional preservation property: after each
402pass all basic blocks must start with an emission statement, and all labels
403\texttt{L} must be unique.
405Now let $M$ be the object code obtained for the program $P$. Let us suppose
406that we can statically inspect the code $M$ and associate to each basic block
407a cost (e.g. the number of clock cycles required to execute all instructions
408in the basic block, or an upper bound to that time). Every basic block is
409labelled with an unique label \texttt{L}, thus we can actually associate the
410cost to \texttt{L}. Let call it $k(\texttt{L})$.
412The function $k$ is defined as the cost model for the object code control
413blocks. It can be equally used as well as the cost model for the source
414control blocks. Indeed, if the semantics of $P$ is the stream
415$L_1 L_2 \ldots$, then, because of forward simulation, the semantics of $M$ is
416also $L_1 L_2 \ldots$ and its actual execution cost is $\Sigma_i k(L_i)$ because
417every instruction belongs to a control block and every control block is
418labelled. Thus it is correct to say that the execution cost of $P$ is also
419$\Sigma_i k(L_i)$. In other words, we have obtained a cost model $k$ for
420the blocks of the high level program $P$ that is preserved by compilation.
422How can the user profit from the high level cost model? Suppose, for instance,
423that he wants to prove that the WCET of his program is bounded by $c$. It
424is sufficient for him to prove that $\Sigma_i k(L_i) \leq c$, which is now
425a purely functional property of the code. He can therefore use any technique
426available to certify functional properties of the source code.
427What is suggested in~\cite{easylabelling} is to actually instrument the
428source code $P$ by replacing every label emission statement
429$\texttt{emit L}$ with the instruction $\texttt{cost += k(L)}$ that increments
430a global fresh variable \texttt{cost}. The bound is now proved by establishing
431the program invariant $\texttt{cost} \leq c$, which can be done for example
432using the Frama-C~\cite{framaC} suite if the source code is some variant of
435In order to extend the labeling approach to function calls we make
436\verb+CALL f+ emit the observable \verb+f+ and \verb+RET+ emit a distinguished observable
439For example the following execution history of the program in \autoref{fig:esempio}
440$$I_1; \verb+CALL f+; \verb+COND l+; \verb+EMIT $\ell_2$+; I_3; \verb+RET+; I_2; \verb+RET+$$
441emits the trace
442$$\verb+main+, \verb+f+$$
447main: $\!I_1$
448      CALL f
449      $I_2$
450      RET
464f: $\!$COND l
465   EMIT $\ell_2$
466   RET
467l: $\!$EMIT $\ell_3$
468   $I_3$
469   RET
488\subsection{Labelling function calls}
489We now want to extend the labelling approach to support function calls.
490On the high level, \emph{structured} programming language $\mathcal{P}$ there
491is not much to change.
492When a function is invoked, the current basic block is temporarily exited
493and the basic block the function starts with take control. When the function
494returns, the execution of the original basic block is resumed. Thus the only
495significant change is that basic blocks can now be nested. Let \texttt{E}
496be the label of the external block and \texttt{I} the label of a nested one.
497Since the external starts before the internal, the semantics observed will be
498\texttt{E I} and the cost associated to it on the source language will be
499$k(\texttt{E}) + k(\texttt{I})$, i.e. the cost of executing all instructions
500in the block \texttt{E} first plus the cost of executing all the instructions in
501the block \texttt{I}. However, we know that some instructions in \texttt{E} are
502executed after the last instruction in \texttt{I}. This is actually irrelevant
503because we are here assuming that costs are additive, so that we can freely
504permute 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
505the function call terminates and yields back control to the basic block
506\texttt{E}. If the call diverges, the instrumentation
507$\texttt{cost += k(E)}$ executed at the beginning of \texttt{E} is still valid,
508but just as an upper bound to the real execution cost: only precision is lost.
510Let now consider what happens when we move down the compilation chain to an
511unstructured intermediate or final language. Here unstructured means that
512the only control operators are conditional and unconditional jumps, function
513calls and returns. Unlike a structured language, though, there is no guarantee
514that a function will return control just after the function call point.
515The semantics of the return statement, indeed, consists in fetching the
516return address from some internal structure (typically the control stack) and
517jumping directly to it. The code can freely manipulate the control stack to
518make the procedure returns to whatever position. Indeed, it is also possible
519to break the well nesting of function calls/returns.
521Is it the case that the code produced by a correct compiler must respect the
522additional property that every function returns just after its function call
523point? The answer is negative and the property is not implied by forward
524simulation proofs. For instance, imagine to modify a correct compiler pass
525by systematically adding one to the return address on the stack and by
526putting a \texttt{NOP} (or any other instruction that takes one byte) after
527every function call. The obtained code will be functionally indistinguishable,
528and the added instructions will all be dead code.
530This lack of structure in the semantics badly interferes with the labelling
531approach. The reason is the following: when a basic block labelled with
532\texttt{E} contains a function call, it no longer makes any sense to associate
533to a label \texttt{E} the sum of the costs of all the instructions in the block.
534Indeed, there is no guarantee that the function will return into the block and
535that the instructions that will be executed after the return will be the ones
536we are paying for in the cost model.
538How can we make the labelling approach work in this scenario? We only see two
539possible ways. The first one consists in injecting an emission statement after
540every function call: basic blocks no longer contain function calls, but are now
541terminated by them. This completely solves the problem and allows the compiler
542to break the structure of function calls/returns at will. However, the
543technique has several drawbacks. First of all, it greatly augments the number
544of cost labels that are injected in the source code and that become
545instrumentation statements. Thus, when reasoning on the source code to prove
546non-functional properties, the user (or the automation tool) will have to handle
547larger expressions. Second, the more labels are emitted, the more difficult it
548becomes to implement powerful optimisations respecting the code structure.
549Indeed, function calls are usually implemented in such a way that most registers
550are preserved by the call, so that the static analysis of the block is not
551interrupted by the call and an optimisation can involve both the code before
552and after the function call. Third, instrumenting the source code may require
553unpleasant modification of it. Take, for example, the code
554\texttt{f(g(x));}. We need to inject an emission statement/instrumentation
555instruction just after the execution of \texttt{g}. The only way to do that
556is to rewrite the code as \texttt{y = g(x); emit L; f(y);} for some fresh
557variable \texttt{y}. It is pretty clear how in certain situations the obtained
558code would be more obfuscated and then more difficult to manually reason on.
560For the previous reasons, in this paper and in the CerCo project we adopt a
561different approach. We do not inject emission statements after every
562function call. However, we want to propagate a strong additional invariant in
563the forward simulation proof. The invariant is the propagation of the structure
564 of the original high level code, even if the target language is unstructured.
565The structure we want to propagate, that will become more clear in the next
566section, comprises 1) the property that every function should return just after
567the function call point, which in turns imply well nesting of function calls;
5682) the property that every basic block starts with a code emission statement.
570In the original labelling approach of~\cite{easylabelling}, the second property
571was granted syntactically as a property of the generated code.
572In our revised approach, instead, we will impose the property on the runs:
573it will be possible to generate code that does not respect the syntactic
574property, as soon as all possible runs respect it. For instance, dead code will no longer
575be required to have all basic blocks correctly labelled. The switch is suggested
576from the fact that the first of the two properties --- that related to
577function calls/returns --- can only be defined as property of runs,
578not of the static code. The switch is
579beneficial to the proof because the original proof was made of two parts:
580the forward simulation proof and the proof that the static property was granted.
581In our revised approach the latter disappears and only the forward simulation
582is kept.
584In order to capture the structure semantics so that it is preserved
585by a forward simulation argument, we need to make the structure observable
586in the semantics. This is the topic of the next section.
588\section{Structured traces}
590The program semantics adopted in the traditional labelling approach is based
591on labelled deductive systems. Given a set of observables $\mathcal{O}$ and
592a set of states $\S$, the semantics of one deterministic execution
593step is
594defined as a function $S \to S \times O^*$ where $O^*$ is a (finite) stream of
595observables. The semantics is then lifted compositionally to multiple (finite
596or infinite) execution steps.
597Finally, the semantics of a a whole program execution is obtained by forgetting
598about the final state (if any), yielding a function $S \to O^*$ that given an
599initial status returns the finite or infinite stream of observables in output.
601We present here a new definition of semantics where the structure of execution,
602as defined in the previous section, is now observable. The idea is to replace
603the stream of observables with a structured data type that makes explicit
604function call and returns and that grants some additional invariants by
605construction. The data structure, called \emph{structured traces}, is
606defined inductively for terminating programs and coinductively for diverging
607ones. In the paper we focus only on the inductive structure, i.e. we assume
608that all programs that are given a semantics are total. The Matita formalisation
609also shows the coinductive definitions. The semantics of a program is then
610defined as a function that maps an initial state into a structured trace.
612In order to have a definition that works on multiple intermediate languages,
613we abstract the type of structure traces over an abstract data type of
614abstract statuses, which we aptly call $\texttt{abstract\_status}$. The fields
615of this record are the following.
617 \item \verb+S : Type[0]+, the type of states.
618 \item \verb+as_execute : S $\to$ S $\to$ Prop+, a binary predicate stating
619 an execution step. We write $s_1\exec s_2$ for $\verb+as_execute+~s_1~s_2$.
620 \item \verb+as_classifier : S $\to$ classification+, a function tagging all
621 states with a class in
622 $\{\texttt{cl\_return,cl\_jump,cl\_call,cl\_other}\}$, depending on the instruction
623 that is about to be executed (we omit tail-calls for simplicity). We will
624 use $s \class c$ as a shorthand for both $\texttt{as\_classifier}~s=c$
625 (if $c$ is a classification) and $\texttt{as\_classifier}~s\in c$
626 (if $c$ is a set of classifications).
627 \item \verb+as_label : S $\to$ option label+, telling whether the
628 next instruction to be executed in $s$ is a cost emission statement,
629 and if yes returning the associated cost label. Our shorthand for this function
630 will be $\ell$, and we will also abuse the notation by using $\ell~s$ as a
631 predicate stating that $s$ is labelled.
632 \item \verb+as_call_ident : ($\Sigma$s:S. s $\class$ cl_call) $\to$ label+,
633 telling the identifier of the function which is being called in a
634 \verb+cl_call+ state. We will use the shorthand $s\uparrow f$ for
635 $\verb+as_call_ident+~s = f$.
636 \item \verb+as_after_return : ($\Sigma$s:S. s $\class$ cl_call) $\to$ S $\to$ Prop+,
637 which holds on the \verb+cl_call+ state $s_1$ and a state $s_2$ when the
638 instruction to be executed in $s_2$ follows the function call to be
639 executed in (the witness of the $\Sigma$-type) $s_1$. We will use the notation
640 $s_1\ar s_2$ for this relation.
643% \begin{alltt}
644% record abstract_status := \{ S: Type[0];
645%  as_execute: S \(\to\) S \(\to\) Prop;   as_classifier: S \(\to\) classification;
646%  as_label: S \(\to\) option label;    as_called: (\(\Sigma\)s:S. c s = cl_call) \(\to\) label;
647%  as_after_return: (\(\Sigma\)s:S. c s = cl_call) \(\to\) S \(\to\) Prop \}
648% \end{alltt}
650The inductive type for structured traces is actually made by three multiple
651inductive types with the following semantics:
653 \item $(\texttt{trace\_label\_return}~s_1~s_2)$ (shorthand $\verb+TLR+~s_1~s_2$)
654   is a trace that begins in
655   the state $s_1$ (included) and ends just before the state $s_2$ (excluded)
656   such that the instruction to be executed in $s_1$ is a label emission
657   statement and the one to be executed in the state before $s_2$ is a return
658   statement. Thus $s_2$ is the state after the return. The trace
659   may contain other label emission statements. It captures the structure of
660   the execution of function bodies: they must start with a cost emission
661   statement and must end with a return; they are obtained by concatenating
662   one or more basic blocks, all starting with a label emission
663   (e.g. in case of loops).
664 \item $(\texttt{trace\_any\_label}~b~s_1~s_2)$ (shorthand $\verb+TAL+~b~s_1~s_2$)
665   is a trace that begins in
666   the state $s_1$ (included) and ends just before the state $s_2$ (excluded)
667   such that the instruction to be executed in $s_2$/in the state before
668   $s_2$ is either a label emission statement or
669   or a return, according to the boolean $b$. It must not contain
670   any label emission statement. It captures the notion of a suffix of a
671   basic block.
672 \item $(\texttt{trace\_label\_label}~b~s_1~s_2)$ (shorthand $\verb+TLL+~b~s_1~s_2$ is the special case of
673   $\verb+TAL+~b~s_1~s_2)$ such that the instruction to be
674   executed in $s_1$ is a label emission statement. It captures the notion of
675   a basic block.
680 {\texttt{TLL}~true~s_1~s_2}
681 {\texttt{TLR}~s_1~s_2}
684 {\texttt{TLL}~false~s_1~s_2 \andalso
685  \texttt{TLR}~s_2~s_3
686 }
687 {\texttt{TLR}~s_1~s_3}
690 {\texttt{TAL}~b~s_1~s_2 \andalso
691  \ell~s_1
692 }
693 {\texttt{TLL}~b~s_1~s_2}
697 {s_1\exec s_2 \andalso
698  s_1\class\{\verb+cl_jump+, \verb+cl_other+\}\andalso
699  \ell~s_2
700 }
701 {\texttt{TAL}~false~s_1~s_2}
704 {s_1\exec s_2 \andalso
705  s_1 \class \texttt{cl\_return}
706 }
707 {\texttt{TAL}~true~s_1~s_2}
710 {s_1\exec s_2 \andalso
711  s_1 \class \texttt{cl\_call} \andalso
712  s_1\ar s_3 \andalso
713  \texttt{TLR}~s_2~s_3 \andalso
714  \ell~s_3
715 }
716 {\texttt{TAL}~false~s_1~s_3}
719 {s_1\exec s_2 \andalso
720  s_1 \class \texttt{cl\_call} \andalso
721  s_1\ar s_3 \andalso
722  \texttt{TLR}~s_2~s_3 \andalso
723  \texttt{TAL}~b~s_3~s_4
724 }
725 {\texttt{TAL}~b~s_1~s_4}
728 {s_1\exec s_2 \andalso
729  \lnot \ell~s_2 \andalso
730  \texttt{TAL}~b~s_2~s_3\andalso
731  s_1 \class \texttt{cl\_other}
732 }
733 {\texttt{TAL}~b~s_1~s_3}
736inductive trace_label_return (S:abstract_status) : S → S → Type[0] ≝
737  | tlr_base:
738      ∀status_before: S.
739      ∀status_after: S.
740        trace_label_label S ends_with_ret status_before status_after →
741        trace_label_return S status_before status_after
742  | tlr_step:
743      ∀status_initial: S.
744      ∀status_labelled: S.
745      ∀status_final: S.
746        trace_label_label S doesnt_end_with_ret status_initial status_labelled →
747        trace_label_return S status_labelled status_final →
748          trace_label_return S status_initial status_final
749with trace_label_label: trace_ends_with_ret → S → S → Type[0] ≝
750  | tll_base:
751      ∀ends_flag: trace_ends_with_ret.
752      ∀start_status: S.
753      ∀end_status: S.
754        trace_any_label S ends_flag start_status end_status →
755        as_costed S start_status →
756          trace_label_label S ends_flag start_status end_status
757with trace_any_label: trace_ends_with_ret → S → S → Type[0] ≝
758  (* Single steps within a function which reach a label.
759     Note that this is the only case applicable for a jump. *)
760  | tal_base_not_return:
761      ∀start_status: S.
762      ∀final_status: S.
763        as_execute S start_status final_status →
764        (as_classifier S start_status cl_jump ∨
765         as_classifier S start_status cl_other) →
766        as_costed S final_status →
767          trace_any_label S doesnt_end_with_ret start_status final_status
768  | tal_base_return:
769      ∀start_status: S.
770      ∀final_status: S.
771        as_execute S start_status final_status →
772        as_classifier S start_status cl_return →
773          trace_any_label S ends_with_ret start_status final_status
774  (* A call followed by a label on return. *)
775  | tal_base_call:
776      ∀status_pre_fun_call: S.
777      ∀status_start_fun_call: S.
778      ∀status_final: S.
779        as_execute S status_pre_fun_call status_start_fun_call →
780        ∀H:as_classifier S status_pre_fun_call cl_call.
781          as_after_return S «status_pre_fun_call, H» status_final →
782          trace_label_return S status_start_fun_call status_final →
783          as_costed S status_final →
784            trace_any_label S doesnt_end_with_ret status_pre_fun_call status_final
785  (* A call followed by a non-empty trace. *)
786  | tal_step_call:
787      ∀end_flag: trace_ends_with_ret.
788      ∀status_pre_fun_call: S.
789      ∀status_start_fun_call: S.
790      ∀status_after_fun_call: S.
791      ∀status_final: S.
792        as_execute S status_pre_fun_call status_start_fun_call →
793        ∀H:as_classifier S status_pre_fun_call cl_call.
794          as_after_return S «status_pre_fun_call, H» status_after_fun_call →
795          trace_label_return S status_start_fun_call status_after_fun_call →
796          ¬ as_costed S status_after_fun_call →
797          trace_any_label S end_flag status_after_fun_call status_final →
798            trace_any_label S end_flag status_pre_fun_call status_final
799  | tal_step_default:
800      ∀end_flag: trace_ends_with_ret.
801      ∀status_pre: S.
802      ∀status_init: S.
803      ∀status_end: S.
804        as_execute S status_pre status_init →
805        trace_any_label S end_flag status_init status_end →
806        as_classifier S status_pre cl_other →
807        ¬ (as_costed S status_init) →
808          trace_any_label S end_flag status_pre status_end.
811A \texttt{trace\_label\_return} is isomorphic to a list of
812\texttt{trace\_label\_label}s that ends with a cost emission followed by a
813return terminated \texttt{trace\_label\_label}.
814The interesting cases are those of $\texttt{trace\_any\_label}~b~s_1~s_2$.
815A \texttt{trace\_any\_label} is a sequence of steps built by a syntax directed
816definition on the classification of $s_1$. The constructors of the datatype
817impose several invariants that are meant to impose a structure to the
818otherwise unstructured execution. In particular, the following invariants are
821 \item the trace is never empty; it ends with a return iff $b$ is
822       true
823 \item a jump must always be the last instruction of the trace, and it must
824       be followed by a cost emission statement; i.e. the target of a jump
825       is always the beginning of a new basic block; as such it must start
826       with a cost emission statement
827 \item a cost emission statement can never occur inside the trace, only in
828       the status immediately after
829 \item the trace for a function call step is made of a subtrace for the
830       function body of type
831       $\texttt{trace\_label\_return}~s_1~s_2$, possibly followed by the
832       rest of the trace for this basic block. The subtrace represents the
833       function execution. Being an inductive datum, it grants totality of
834       the function call. The status $s_2$ is the one that follows the return
835       statement. The next instruction of $s_2$ must follow the function call
836       instruction. As a consequence, function calls are also well nested.
839There are three mutual structural recursive functions, one for each of
840\verb+TLR+, \verb+TLL+ and \verb+TAL+, for which we use the same notation
841$|\,.\,|$: the \emph{flattening} of the traces. These functions
842allow to extract from a structured trace the list of emitted cost labels.
843%  We only show here the type of one
844% of them:
845% \begin{alltt}
846% flatten_trace_label_return:
847%  \(\forall\)S: abstract_status. \(\forall\)\(s_1,s_2\).
848%   trace_label_return \(s_1\) \(s_2\) \(\to\) list (as_cost_label S)
849% \end{alltt}
851\paragraph{Cost prediction on structured traces.}
853The first main theorem of CerCo about traces
854(theorem \texttt{compute\_max\_trace\_label\_return\_cost\_ok\_with\_trace})
855holds for the
857of the structured traces to the concrete status of object code programs.
858Simplifying a bit, it states that
860\begin{array}{l}\forall s_1,s_2. \forall \tau: \texttt{TLR}~s_1~s_2.~
861  \texttt{clock}~s_2 = \texttt{clock}~s_1 +
862  \Sigma_{\alpha \in |\tau|}\;k(\alpha)
865where the cost model $k$ is statically computed from the object code
866by associating to each label $\alpha$ the sum of the cost of the instructions
867in the basic block that starts at $\alpha$ and ends before the next labelled
868instruction. The theorem is proved by structural induction over the structured
869trace, and is based on the invariant that
870iff the function that computes the cost model has analysed the instruction
871to be executed at $s_2$ after the one to be executed at $s_1$, and if
872the structured trace starts with $s_1$, then eventually it will contain also
873$s_2$. When $s_1$ is not a function call, the result holds trivially because
874of the $s_1\exec s_2$ condition obtained by inversion on
875the trace. The only non
876trivial case is the one of function calls: the cost model computation function
877does recursion on the first instruction that follows that function call; the
878\texttt{as\_after\_return} condition of the \texttt{tal\_base\_call} and
879\texttt{tal\_step\_call} grants exactly that the execution will eventually reach
880this state.
882\paragraph{Structured traces similarity and cost prediction invariance.}
884A compiler pass maps source to object code and initial states to initial
885states. The source code and initial state uniquely determine the structured
886trace of a program, if it exists. The structured trace fails to exists iff
887the structural conditions are violated by the program execution (e.g. a function
888body does not start with a cost emission statement). Let us assume that the
889target structured trace exists.
891What is the relation between the source and target structured traces?
892In general, the two traces can be arbitrarily different. However, we are
893interested only in those compiler passes that maps a trace $\tau_1$ to a trace
894$\tau_2$ such that
895\begin{equation}|\tau_1| = |\tau_2|.\label{th2}\end{equation}
896The reason is that the combination of~\eqref{th1} with~\eqref{th2} yields the
899\forall s_1,s_2. \forall \tau: \texttt{TLR}~s_1~s_2.~
900  \texttt{clock}~s_2 - \texttt{clock}~s_1 =
901  \Sigma_{\alpha \in |\tau_1|}\;k(\alpha) =
902  \Sigma_{\alpha \in |\tau_2|}\;k(\alpha).
904This 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
905transfer the cost model from the target to the source code and reason on the
906source code only.
908We are therefore interested in conditions stronger than~\eqref{th2}.
909Therefore we introduce here a similarity relation between traces with
910the same structure. Theorem~\texttt{tlr\_rel\_to\_traces\_same\_flatten}
911in the Matita formalisation shows that~\eqref{th2} holds for every pair
912$(\tau_1,\tau_2)$ of similar traces.
914Intuitively, two traces are similar when one can be obtained from
915the other by erasing or inserting silent steps, i.e. states that are
916not \texttt{as\_costed} and that are classified as \texttt{cl\_other}.
917Silent steps do not alter the structure of the traces.
918In particular,
919the relation maps function calls to function calls to the same function,
920label emission statements to emissions of the same label, concatenation of
921subtraces to concatenation of subtraces of the same length and starting with
922the same emission statement, etc.
924In the formalisation the three similarity relations --- one for each trace
925kind --- are defined by structural recursion on the first trace and pattern
926matching over the second. Here we turn
927the definition into the inference rules shown in \autoref{fig:txx_rel}
928for the sake of readability. We also omit from trace constructors all arguments,
929but those that are traces or that
930are used in the premises of the rules. By abuse of notation we denote all three
931relations by infixing $\approx$.
936 {tll_1\approx tll_2
937 }
938 {\texttt{tlr\_base}~tll_1 \approx \texttt{tlr\_base}~tll_2}
941 {tll_1 \approx tll_2 \andalso
942  tlr_1 \approx tlr_2
943 }
944 {\texttt{tlr\_step}~tll_1~tlr_1 \approx \texttt{tlr\_step}~tll_2~tlr_2}
949 {\ell~s_1 = \ell~s_2 \andalso
950  tal_1\approx tal_2
951 }
952 {\texttt{tll\_base}~s_1~tal_1 \approx \texttt{tll\_base}~s_2~tal_2}
955 {tal_1\approx tal_2
956 }
957 {\texttt{tal\_step\_default}~tal_1 \approx tal_2}
961 {}
962 {\texttt{tal\_base\_not\_return}\approx taa \append \texttt{tal\_base\_not\_return}}
965 {}
966 {\texttt{tal\_base\_return}\approx taa \append \texttt{tal\_base\_return}}
969 {tlr_1\approx tlr_2 \andalso
970  s_1 \uparrow f \andalso s_2\uparrow f
971 }
972 {\texttt{tal\_base\_call}~s_1~tlr_1\approx taa \append \texttt{tal\_base\_call}~s_2~tlr_2}
975 {tlr_1\approx tlr_2 \andalso
976  s_1 \uparrow f \andalso s_2\uparrow f \andalso
977  \texttt{tal\_collapsable}~tal_2
978 }
979 {\texttt{tal\_base\_call}~s_1~tlr_1 \approx taa \append \texttt{tal\_step\_call}~s_2~tlr_2~tal_2)}
982 {tlr_1\approx tlr_2 \andalso
983  s_1 \uparrow f \andalso s_2\uparrow f \andalso
984  \texttt{tal\_collapsable}~tal_1
985 }
986 {\texttt{tal\_step\_call}~s_1~tlr_1~tal_1 \approx taa \append \texttt{tal\_base\_call}~s_2~tlr_2)}
989 {tlr_1 \approx tlr_2 \andalso
990  s_1 \uparrow f \andalso s_2\uparrow f\andalso
991  tal_1 \approx tal_2 \andalso
992 }
993 {\texttt{tal\_step\_call}~s_1~tlr_1~tal_1 \approx taa \append \texttt{tal\_step\_call}~s_2~tlr_2~tal_2}
994\caption{The inference rule for the relation $\approx$.}
1000let rec tlr_rel S1 st1 st1' S2 st2 st2'
1001  (tlr1 : trace_label_return S1 st1 st1')
1002  (tlr2 : trace_label_return S2 st2 st2') on tlr1 : Prop ≝
1003match tlr1 with
1004  [ tlr_base st1 st1' tll1 ⇒
1005    match tlr2 with
1006    [ tlr_base st2 st2' tll2 ⇒ tll_rel … tll1 tll2
1007    | _ ⇒ False
1008    ]
1009  | tlr_step st1 st1' st1'' tll1 tl1 ⇒
1010    match tlr2 with
1011    [ tlr_step st2 st2' st2'' tll2 tl2 ⇒
1012      tll_rel … tll1 tll2 ∧ tlr_rel … tl1 tl2
1013    | _ ⇒ False
1014    ]
1015  ]
1016and tll_rel S1 fl1 st1 st1' S2 fl2 st2 st2'
1017 (tll1 : trace_label_label S1 fl1 st1 st1')
1018 (tll2 : trace_label_label S2 fl2 st2 st2') on tll1 : Prop ≝
1019  match tll1 with
1020  [ tll_base fl1 st1 st1' tal1 H ⇒
1021    match tll2 with
1022    [ tll_base fl2 st2 st2 tal2 G ⇒
1023      as_label_safe … («?, H») = as_label_safe … («?, G») ∧
1024      tal_rel … tal1 tal2
1025    ]
1026  ]
1027and tal_rel S1 fl1 st1 st1' S2 fl2 st2 st2'
1028 (tal1 : trace_any_label S1 fl1 st1 st1')
1029 (tal2 : trace_any_label S2 fl2 st2 st2')
1030   on tal1 : Prop ≝
1031  match tal1 with
1032  [ tal_base_not_return st1 st1' _ _ _ ⇒
1033    fl2 = doesnt_end_with_ret ∧
1034    ∃st2mid,taa,H,G,K.
1035    tal2 ≃ taa_append_tal ? st2 ??? taa
1036      (tal_base_not_return ? st2mid st2' H G K)
1037  | tal_base_return st1 st1' _ _ ⇒
1038    fl2 = ends_with_ret ∧
1039    ∃st2mid,taa,H,G.
1040    tal2 ≃ taa_append_tal ? st2 ? st2mid st2' taa
1041      (tal_base_return ? st2mid st2' H G)
1042  | tal_base_call st1 st1' st1'' _ prf _ tlr1 _ ⇒
1043    fl2 = doesnt_end_with_ret ∧
1044    ∃st2mid,G.as_call_ident S2 («st2mid, G») = as_call_ident ? «st1, prf» ∧
1045    ∃taa : trace_any_any ? st2 st2mid.∃st2mid',H.
1046    (* we must allow a tal_base_call to be similar to a call followed
1047      by a collapsable trace (trace_any_any followed by a base_not_return;
1048      we cannot use trace_any_any as it disallows labels in the end as soon
1049      as it is non-empty) *)
1050    (∃K.∃tlr2 : trace_label_return ? st2mid' st2'.∃L.
1051      tal2 ≃ taa @ (tal_base_call … H G K tlr2 L) ∧ tlr_rel … tlr1 tlr2) ∨
1052    ∃st2mid'',K.∃tlr2 : trace_label_return ? st2mid' st2mid''.∃L.
1053    ∃tl2 : trace_any_label … doesnt_end_with_ret st2mid'' st2'.
1054      tal2 ≃ taa @ (tal_step_call … H G K tlr2 L tl2) ∧
1055      tlr_rel … tlr1 tlr2 ∧ tal_collapsable … tl2
1056  | tal_step_call fl1 st1 st1' st1'' st1''' _ prf _ tlr1 _ tl1 ⇒
1057    ∃st2mid,G.as_call_ident S2 («st2mid, G») = as_call_ident ? «st1, prf» ∧
1058    ∃taa : trace_any_any ? st2 st2mid.∃st2mid',H.
1059    (fl2 = doesnt_end_with_ret ∧ ∃K.∃tlr2 : trace_label_return ? st2mid' st2'.∃L.
1060      tal2 ≃ taa @ tal_base_call … H G K tlr2 L ∧
1061      tal_collapsable … tl1 ∧ tlr_rel … tlr1 tlr2) ∨
1062    ∃st2mid'',K.∃tlr2 : trace_label_return ? st2mid' st2mid''.∃L.
1063    ∃tl2 : trace_any_label ? fl2 st2mid'' st2'.
1064      tal2 ≃ taa @ (tal_step_call … H G K tlr2 L tl2) ∧
1065      tal_rel … tl1 tl2 ∧ tlr_rel … tlr1 tlr2
1066  | tal_step_default fl1 st1 st1' st1'' _ tl1 _ _ ⇒
1067    tal_rel … tl1 tal2 (* <- this makes it many to many *)
1068  ].
1072In the preceding rules, a $taa$ is an inhabitant of the
1073$\texttt{trace\_any\_any}~s_1~s_2$ (shorthand $\texttt{TAA}~s_1~s_2$),
1074an inductive data type whose definition
1075is not in the paper for lack of space. It is the type of valid
1076prefixes (even empty ones) of \texttt{TAL}'s that do not contain
1077any function call. Therefore it
1078is possible to concatenate (using ``$\append$'') a \texttt{TAA} to the
1079left of a \texttt{TAL}. A \texttt{TAA} captures
1080a sequence of silent moves.
1081The \texttt{tal\_collapsable} unary predicate over \texttt{TAL}'s
1082holds when the argument does not contain any function call and it ends
1083with a label (not a return). The intuition is that after a function call we
1084can still perform a sequence of silent actions while remaining similar.
1086As should be expected, even though the rules are asymmetric $\approx$ is in fact
1087an equivalence relation.
1088\section{Forward simulation}
1091We summarise here the results of the previous sections. Each intermediate
1092unstructured language can be given a semantics based on structured traces,
1093that single out those runs that respect a certain number of invariants.
1094A cost model can be computed on the object code and it can be used to predict
1095the execution costs of runs that produce structured traces. The cost model
1096can be lifted from the target to the source code of a pass if the pass maps
1097structured traces to similar structured traces. The latter property is called
1098a \emph{forward simulation}.
1100As for labelled transition systems, in order to establish the forward
1101simulation we are interested in (preservation of observables), we are
1102forced to prove a stronger notion of forward simulation that introduces
1103an explicit relation between states. The classical notion of a 1-to-many
1104forward simulation is the existence of a relation $\S$ over states such that
1105if $s_1 \S s_2$ and $s_1 \to^1 s_1'$ then there exists an $s_2'$ such that
1106$s_2 \to^* s_2'$ and $s_1' \S s_2'$. In our context, we need to replace the
1107one and multi step transition relations $\to^n$ with the existence of
1108a structured trace between the two states, and we need to add the request that
1109the two structured traces are similar. Thus what we would like to state is
1110something like:\\
1111for all $s_1,s_2,s_1'$ such that there is a $\tau_1$ from
1112$s_1$ to $s_1'$ and $s_1 \S s_2$ there exists an $s_2'$ such that
1113$s_1' \S s_2'$ and a $\tau_2$ from $s_2$ to $s_2'$ such that
1114$\tau_1$ is similar to $\tau_2$. We call this particular form of forward
1115simulation \emph{trace reconstruction}.
1117The statement just introduced, however, is too simplistic and not provable
1118in the general case. To understand why, consider the case of a function call
1119and the pass that fixes the parameter passing conventions. A function
1120call in the source code takes in input an arbitrary number of pseudo-registers (the actual parameters to pass) and returns an arbitrary number of pseudo-registers (where the result is stored). A function call in the target language has no
1121input nor output parameters. The pass must add explicit code before and after
1122the function call to move the pseudo-registers content from/to the hardware
1123registers or the stack in order to implement the parameter passing strategy.
1124Similarly, each function body must be augmented with a preamble and a postamble
1125to complete/initiate the parameter passing strategy for the call/return phase.
1126Therefore what used to be a call followed by the next instruction to execute
1127after the function return, now becomes a sequence of instructions, followed by
1128a call, followed by another sequence. The two states at the beginning of the
1129first sequence and at the end of the second sequence are in relation with
1130the status before/after the call in the source code, like in an usual forward
1131simulation. How can we prove however the additional condition for function calls
1132that asks that when the function returns the instruction immediately after the
1133function call is called? To grant this invariant, there must be another relation
1134between the address of the function call in the source and in the target code.
1135This additional relation is to be used in particular to relate the two stacks.
1137Another example is given by preservation of code emission statements. A single
1138code emission instruction can be simulated by a sequence of steps, followed
1139by a code emission, followed by another sequence. Clearly the initial and final
1140statuses of the sequence are to be in relation with the status before/after the
1141code emission in the source code. In order to preserve the structured traces
1142invariants, however, we must consider a second relation between states that
1143traces the preservation of the code emission statement.
1145Therefore we now introduce an abstract notion of relation set between abstract
1146statuses and an abstract notion of 1-to-many forward simulation conditions.
1147These two definitions enjoy the following remarkable properties:
1149 \item they are generic enough to accommodate all passes of the CerCo compiler
1150 \item the conjunction of the 1-to-many forward simulation conditions are
1151       just slightly stricter than the statement of a 1-to-many forward
1152       simulation in the classical case. In particular, they only require
1153       the construction of very simple forms of structured traces made of
1154       silent states only.
1155 \item they allow to prove our main result of the paper: the 1-to-many
1156       forward simulation conditions are sufficient to prove the trace
1157       reconstruction theorem
1160Point 3. is the important one. First of all it means that we have reduced
1161the complex problem of trace reconstruction to a much simpler one that,
1162moreover, can be solved with slight adaptations of the forward simulation proof
1163that is performed for a compiler that only cares about functional properties.
1164Therefore we have successfully splitted as much as possible the proof of
1165preservation of functional properties from that of non-functional ones.
1166Secondly, combined with the results in the previous section, it implies
1167that the cost model can be computed on the object code and lifted to the
1168source code to reason on non-functional properties, assuming that
1169the 1-to-many forward simulation conditions are fulfilled for every
1170compiler pass.
1172\paragraph{Relation sets.}
1174We introduce now the four relations $\mathcal{S,C,L,R}$ between abstract
1175statuses that are used to correlate the corresponding statues before and
1176after a compiler pass. The first two are abstract and must be instantiated
1177by every pass. The remaining two are derived relations.
1179The $\S$ relation between states is the classical relation used
1180in forward simulation proofs. It correlates the data of the status
1181(e.g. registers, memory, etc.).
1183The $\C$ relation correlates call states. It allows to track the
1184position in the target code of every call in the source code.
1186The $\L$ relation simply says that the two states are both label
1187emitting states that emit the same label, \emph{i.e.}\ $s_1\L s_2\iffdef \ell~s_1=\ell~s_2$.
1188It allows to track the position in
1189the target code of every cost emitting statement in the source code.
1191Finally the $\R$ relation is the more complex one. Two states
1192$s_1$ and $s_2$ are $\R$ correlated if every time $s_1$ is the
1193successors of a call state that is $\C$-related to a call state
1194$s_2'$ in the target code, then $s_2$ is the successor of $s_2'$. Formally:
1195$$s_1\R s_2 \iffdef \forall s_1',s_2'.s_1'\C s_2' \to s_1'\ar s_1 \to s_2' \ar s_2.$$
1196We will require all pairs of states that follow a related call to be
1197$\R$-related. This is the fundamental requirement granting
1198that the target trace is well structured, \emph{i.e.}\ that calls are well
1199nested and returning where they are supposed to.
1201% \begin{alltt}
1202% record status_rel (S1,S2 : abstract_status) : Type[1] := \{
1203%   \(\S\): S1 \(\to\) S2 \(\to\) Prop;
1204%   \(\C\): (\(\Sigma\)s.as_classifier S1 s cl_call) \(\to\)
1205%      (\(\Sigma\)s.as_classifier S2 s cl_call) \(\to\) Prop \}.
1207% definition \(\L\) S1 S2 st1 st2 := as_label S1 st1 = as_label S2 st2.
1209% definition \(\R\) S1 S2 (R: status_rel S1 S2) s1_ret s2_ret ≝
1210%  \(\forall\)s1_pre,s2_pre.
1211%   as_after_return s1_pre s1_ret \(\to\) s1_pre \(\R\) s2_pre \(\to\)
1212%    as_after_return s2_pre s2_ret.
1213% \end{alltt}
1218% \begin{subfigure}{.475\linewidth}
1219% \centering
1220% \begin{tikzpicture}[every join/.style={ar}, join all, thick,
1221%                             every label/.style=overlay, node distance=10mm]
1222%     \matrix [diag] (m) {%
1223%          \node (s1) [is jump] {}; & \node [fill=white] (t1) {};\\
1224%          \node (s2) {}; & \node (t2) {}; \\
1225%     };
1226%     \node [above=0 of t1, overlay] {$\alpha$};
1227%     {[-stealth]
1228%     \draw (s1) -- (t1);
1229%     \draw [new] (s2) -- node [above] {$*$} (t2);
1230%     }
1231%     \draw (s1) to node [rel] {$\S$} (s2);
1232%     \draw [new] (t1) to node [rel] {$\S,\L$} (t2);
1233% \end{tikzpicture}
1234% \caption{The \texttt{cl\_jump} case.}
1235% \label{subfig:cl_jump}
1236% \end{subfigure}
1237% &
1240\begin{tikzpicture}[every join/.style={ar}, join all, thick,
1241                            every label/.style=overlay, node distance=10mm]
1242    \matrix [diag] (m) {%
1243         \node (s1) {}; & \node (t1) {};\\
1244         \node (s2) {}; & \node (t2) {}; \\
1245    };
1246    {[-stealth]
1247    \draw (s1) -- (t1);
1248    \draw [new] (s2) -- node [above] {$*$} (t2);
1249    }
1250    \draw (s1) to node [rel] {$\S$} (s2);
1251    \draw [new] (t1) to node [rel] {$\S,\L$} (t2);
1253\caption{The \texttt{cl\_oher} and \texttt{cl\_jump} cases.}
1259\begin{tikzpicture}[every join/.style={ar}, join all, thick,
1260                            every label/.style=overlay, node distance=10mm]
1261    \matrix [diag, small vgap] (m) {%
1262        \node (t1) {}; \\
1263         \node (s1) [is call] {}; \\
1264         & \node (l) {}; & \node (t2) {};\\
1265         \node (s2) {}; & \node (c) [is call] {};\\   
1266    };
1267    {[-stealth]
1268    \draw (s1) -- node [left] {$f$} (t1);
1269    \draw [new] (s2) -- node [above] {$*$} (c);
1270    \draw [new] (c) -- node [right] {$f$} (l);
1271    \draw [new] (l) -- node [above] {$*$} (t2);
1272    }
1273    \draw (s1) to node [rel] {$\S$} (s2);
1274    \draw [new] (t1) to [bend left] node [rel] {$\S$} (t2);
1275    \draw [new] (t1) to [bend left] node [rel] {$\L$} (l);
1276    \draw [new] (t1) to node [rel] {$\C$} (c);
1277    \end{tikzpicture}
1278\caption{The \texttt{cl\_call} case.}
1284\begin{tikzpicture}[every join/.style={ar}, join all, thick,
1285                            every label/.style=overlay, node distance=10mm]
1286    \matrix [diag, small vgap] (m) {%
1287        \node (s1) [is ret] {}; \\
1288        \node (t1) {}; \\
1289        \node (s2) {}; & \node (c) [is ret] {};\\
1290        & \node (r) {}; & \node (t2) {}; \\   
1291    };
1292    {[-stealth]
1293    \draw (s1) -- (t1);
1294    \draw [new] (s2) -- node [above] {$*$} (c);
1295    \draw [new] (c) -- (r);
1296    \draw [new] (r) -- node [above] {$*$} (t2);
1297    }
1298    \draw (s1) to [bend right=45] node [rel] {$\S$} (s2);
1299    \draw [new, overlay] (t1) to [bend left=90, looseness=1] node [rel] {$\S,\L$} (t2);
1300    \draw [new, overlay] (t1) to [bend left=90, looseness=1.2] node [rel] {$\R$} (r);
1302\caption{The \texttt{cl\_return} case.}
1306\caption{Mnemonic diagrams depicting the hypotheses for the preservation of structured traces.
1307         Dashed lines
1308         and arrows indicates how the diagrams must be closed when solid relations
1309         are present.}
1313\paragraph{1-to-many forward simulation conditions.}
1314\begin{condition}[Cases \texttt{cl\_other} and \texttt{cl\_jump}]
1315 For all $s_1,s_1',s_2$ such that $s_1 \S s_1'$, and
1316 $s_1\exec s_1'$, and either $s_1 \class \texttt{cl\_other}$ or
1317 both $s_1\class\texttt{cl\_other}\}$ and $\ell~s_1'$,
1318 there exists an $s_2'$ and a $\texttt{trace\_any\_any\_free}~s_2~s_2'$ called $taaf$
1319 such that $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and either
1320$taaf$ is non empty, or one among $s_1$ and $s_1'$ is \texttt{as\_costed}.
1323In the above condition depicted in \autoref{subfig:cl_other_jump},
1324a $\texttt{trace\_any\_any\_free}~s_1~s_2$ (which from now on
1325will be shorthanded as \verb+TAAF+) is an
1326inductive type of structured traces that do not contain function calls or
1327cost emission statements. Differently from a \verb+TAA+, the
1328instruction to be executed in the lookahead state $s_2$ may be a cost emission
1331The intuition of the condition is that one step can be replaced with zero or more steps if it
1332preserves the relation between the data and if the two final statuses are
1333labelled in the same way. Moreover, we must take special care of the empty case
1334to avoid collapsing two consecutive states that emit a label, missing one of the two emissions.
1336\begin{condition}[Case \texttt{cl\_call}]
1337 For all $s_1,s_1',s_2$ s.t. $s_1 \S s_1'$ and
1338 $s_1\exec s_1'$ and $s_1 \class \texttt{cl\_call}$, there exists $s_a, s_b, s_2'$, a
1339$\verb+TAA+~s_2~s_a$, and a
1340$\verb+TAAF+~s_b~s_2'$ such that:
1341$s_a\class\texttt{cl\_call}$, the \texttt{as\_call\_ident}'s of
1342the two call states are the same, $s_1 \C s_a$,
1343$s_a\exec s_b$, $s_1' \L s_b$ and
1344$s_1' \S s_2'$.
1347The condition, depicted in \autoref{subfig:cl_call} says that, to simulate a function call, we can perform a
1348sequence of silent actions before and after the function call itself.
1349The old and new call states must be $\C$-related, the old and new
1350states at the beginning of the function execution must be $\L$-related
1351and, finally, the two initial and final states must be $\S$-related
1352as usual.
1354\begin{condition}[Case \texttt{cl\_return}]
1355 For all $s_1,s_1',s_2$ s.t. $s_1 \S s_1'$,
1356 $s_1\exec s_1'$ and $s_1 \class \texttt{cl\_return}$, there exists $s_a, s_b, s_2'$, a
1357$\verb+TAA+~s_2~s_a$, a
1358$\verb+TAAF+~s_b~s_2'$ called $taaf$ such that:
1360$s_a\exec s_b$,
1361$s_1' \R s_b$ and
1362$s_1' \mathrel{{\S} \cap {\L}} s_2'$ and either
1363$taaf$ is non empty, or $\lnot \ell~s_a$.
1366Similarly to the call condition, to simulate a return we can perform a
1367sequence of silent actions before and after the return statement itself,
1368as depicted in \autoref{subfig:cl_return}.
1369The old and the new statements after the return must be $\R$-related,
1370to grant that they returned to corresponding calls.
1371The two initial and final states must be $\S$-related
1372as usual and, moreover, they must exhibit the same labels. Finally, when
1373the suffix is non empty we must take care of not inserting a new
1374unmatched cost emission statement just after the return statement.
1378definition status_simulation ≝
1379  λS1 : abstract_status.
1380  λS2 : abstract_status.
1381  λsim_status_rel : status_rel S1 S2.
1382    ∀st1,st1',st2.as_execute S1 st1 st1' →
1383    sim_status_rel st1 st2 →
1384    match as_classify … st1 with
1385    [ None ⇒ True
1386    | Some cl ⇒
1387      match cl with
1388      [ cl_call ⇒ ∀prf.
1389        (*
1390             st1' ------------S----------\
1391              ↑ \                         \
1392             st1 \--L--\                   \
1393              | \       \                   \
1394              S  \-C-\  st2_after_call →taa→ st2'
1395              |       \     ↑
1396             st2 →taa→ st2_pre_call
1397        *)
1398        ∃st2_pre_call.
1399        as_call_ident ? st2_pre_call = as_call_ident ? («st1, prf») ∧
1400        call_rel ?? sim_status_rel «st1, prf» st2_pre_call ∧
1401        ∃st2_after_call,st2'.
1402        ∃taa2 : trace_any_any … st2 st2_pre_call.
1403        ∃taa2' : trace_any_any … st2_after_call st2'.
1404        as_execute … st2_pre_call st2_after_call ∧
1405        sim_status_rel st1' st2' ∧
1406        label_rel … st1' st2_after_call
1407      | cl_return ⇒
1408        (*
1409             st1
1410            / ↓
1411           | st1'----------S,L------------\
1412           S   \                           \
1413            \   \-----R-------\            |     
1414             \                 |           |
1415             st2 →taa→ st2_ret |           |
1416                          ↓   /            |
1417                     st2_after_ret →taaf→ st2'
1419           we also ask that st2_after_ret be not labelled if the taaf tail is
1420           not empty
1421        *)
1422        ∃st2_ret,st2_after_ret,st2'.
1423        ∃taa2 : trace_any_any … st2 st2_ret.
1424        ∃taa2' : trace_any_any_free … st2_after_ret st2'.
1425        (if taaf_non_empty … taa2' then ¬as_costed … st2_after_ret else True) ∧
1426        as_classifier … st2_ret cl_return ∧
1427        as_execute … st2_ret st2_after_ret ∧ sim_status_rel st1' st2' ∧
1428        ret_rel … sim_status_rel st1' st2_after_ret ∧
1429        label_rel … st1' st2'
1430      | cl_other ⇒
1431          (*         
1432          st1 → st1'
1433            |      \
1434            S      S,L
1435            |        \
1436           st2 →taaf→ st2'
1438           the taaf can be empty (e.g. tunneling) but we ask it must not be the
1439           case when both st1 and st1' are labelled (we would be able to collapse
1440           labels otherwise)
1441         *)
1442        ∃st2'.
1443        ∃taa2 : trace_any_any_free … st2 st2'.
1444        (if taaf_non_empty … taa2 then True else (¬as_costed … st1 ∨ ¬as_costed … st1')) ∧
1445        sim_status_rel st1' st2' ∧
1446        label_rel … st1' st2'
1447      | cl_jump ⇒
1448        (* just like cl_other, but with a hypothesis more *)
1449        as_costed … st1' →
1450        ∃st2'.
1451        ∃taa2 : trace_any_any_free … st2 st2'.
1452        (if taaf_non_empty … taa2 then True else (¬as_costed … st1 ∨ ¬as_costed … st1')) ∧
1453        sim_status_rel st1' st2' ∧
1454        label_rel … st1' st2'
1455      ]
1456    ].
1460\paragraph{Main result: the 1-to-many forward simulation conditions
1461are sufficient to trace reconstruction}
1463Let us assume that a relation set is given such that the 1-to-many
1464forward simulation conditions are satisfied. Under this assumption we
1465can prove the following three trace reconstruction theorems by mutual
1466structural induction over the traces given in input between the
1467$s_1$ and $s_1'$ states.
1469In particular, the \texttt{status\_simulation\_produce\_tlr} theorem
1470applied to the \texttt{main} function of the program and equal
1471$s_{2_b}$ and $s_2$ states shows that, for every initial state in the
1472source code that induces a structured trace in the source code,
1473the compiled code produces a similar structured trace.
1476For every $s_1,s_1',s_{2_b},s_2$ s.t.
1477there is a $\texttt{TLR}~s_1~s_1'$ called $tlr_1$ and a
1478$\verb+TAA+~s_{2_b}~s_2$ and $s_1 \L s_{2_b}$ and
1479$s_1 \S s_2$, there exists $s_{2_m},s_2'$ s.t.
1480there is a $\texttt{TLR}~s_{2_b}~s_{2_m}$ called $tlr_2$ and
1481there is a $\verb+TAAF+~s_{2_m}~s_2'$ called $taaf$
1482s.t. if $taaf$ is non empty then $\lnot (\ell~s_{2_m})$,
1483and $tlr_1\approx tlr_2$
1484and $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1485$s_1' \R s_{2_m}$.
1488The theorem states that a \texttt{trace\_label\_return} in the source code
1489together with a precomputed preamble of silent states
1490(the \verb+TAA+) in the target code induces a
1491similar \texttt{trace\_label\_return} in the target code which can be
1492followed by a sequence of silent states. Note that the statement does not
1493require the produced \texttt{trace\_label\_return} to start with the
1494precomputed preamble, even if this is likely to be the case in concrete
1495implementations. The preamble in input is necessary for compositionality, e.g.
1496because the 1-to-many forward simulation conditions allow in the
1497case of function calls to execute a preamble of silent instructions just after
1498the function call.
1500Clearly similar results are also available for the other two types of structured
1501traces (in fact, they are all proved simultaneously by mutual induction).
1502% \begin{theorem}[\texttt{status\_simulation\_produce\_tll}]
1503% For every $s_1,s_1',s_{2_b},s_2$ s.t.
1504% there is a $\texttt{TLL}~b~s_1~s_1'$ called $tll_1$ and a
1505% $\verb+TAA+~s_{2_b}~s_2$ and $s_1 \L s_{2_b}$ and
1506% $s_1 \S s_2$, there exists $s_{2_m},s_2'$ s.t.
1507% \begin{itemize}
1508%  \item if $b$ (the trace ends with a return) then there exists $s_{2_m},s_2'$
1509%        and a trace $\texttt{TLL}~b~s_{2_b}~s_{2_m}$ called $tll_2$
1510%        and a $\texttt{TAAF}~s_{2_m}~s_2'$ called $taa_2$ s.t.
1511%        $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1512%        $s_1' \R s_{2_m}$ and
1513%        $tll_1\approx tll_2$ and
1514%        if $taa_2$ is non empty then $\lnot \ell~s_{2_m}$;
1515%  \item else there exists $s_2'$ and a
1516%        $\texttt{TLL}~b~s_{2_b}~s_2'$ called $tll_2$ such that
1517%        $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1518%        $tll_1\approx tll_2$.
1519% \end{itemize}
1520% \end{theorem}
1522% The statement is similar to the previous one: a source
1523% \texttt{trace\_label\_label} and a given target preamble of silent states
1524% in the target code induce a similar \texttt{trace\_label\_label} in the
1525% target code, possibly followed by a sequence of silent moves that become the
1526% preamble for the next \texttt{trace\_label\_label} translation.
1528% \begin{theorem}[\texttt{status\_simulation\_produce\_tal}]
1529% For every $s_1,s_1',s_2$ s.t.
1530% there is a $\texttt{TAL}~b~s_1~s_1'$ called $tal_1$ and
1531% $s_1 \S s_2$
1532% \begin{itemize}
1533%  \item if $b$ (the trace ends with a return) then there exists $s_{2_m},s_2'$
1534%    and a trace $\texttt{TAL}~b~s_2~s_{2_m}$ called $tal_2$ and a
1535%    $\texttt{TAAF}~s_{2_m}~s_2'$ called $taa_2$ s.t.
1536%    $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1537%    $s_1' \R s_{2_m}$ and
1538%    $tal_1 \approx tal_2$ and
1539%    if $taa_2$ is non empty then $\lnot \ell~s_{2_m}$;
1540%  \item else there exists $s_2'$ and a
1541%    $\texttt{TAL}~b~s_2~s_2'$ called $tal_2$ such that
1542%    either $s_1' \mathrel{{\S} \cap {\L}} s_2'$ and
1543%        $tal_1\approx tal_2$
1544%    or $s_1' \mathrel{{\S} \cap {\L}} s_2$ and
1545%    $\texttt{tal\_collapsable}~tal_1$ and $\lnot \ell~s_1$.
1546% \end{itemize}
1547% \end{theorem}
1549% The statement is also similar to the previous ones, but for the lack of
1550% the target code preamble.
1554For every $s_1,s_1',s_2$ s.t.
1555there is a $\texttt{trace\_label\_return}~s_1~s_1'$ called $tlr_1$ and
1556$s_1 (\L \cap \S) s_2$
1557there exists $s_{2_m},s_2'$ s.t.
1558there is a $\texttt{trace\_label\_return}~s_2~s_{2_m}$ called $tlr_2$ and
1559there is a $\texttt{trace\_any\_any\_free}~s_{2_m}~s_2'$ called $taaf$
1560s.t. if $taaf$ is non empty then $\lnot (\texttt{as\_costed}~s_{2_m})$,
1561and $\texttt{tlr\_rel}~tlr_1~tlr_2$
1562and $s_1' (\S \cap \L) s_2'$ and
1563$s_1' \R s_{2_m}$.
1569status_simulation_produce_tlr S1 S2 R
1570(* we start from this situation
1571     st1 →→→→tlr→→→→ st1'
1572      | \
1573      L  \---S--\
1574      |          \
1575   st2_lab →taa→ st2   (the taa preamble is in general either empty or given
1576                        by the preceding call)
1578   and we produce
1579     st1 →→→→tlr→→→→ st1'
1580             \\      /  \
1581             //     R    \-L,S-\
1582             \\     |           \
1583   st2_lab →tlr→ st2_mid →taaf→ st2'
1585  st1 st1' st2_lab st2
1586  (tlr1 : trace_label_return S1 st1 st1')
1587  (taa2_pre : trace_any_any S2 st2_lab st2)
1588  (sim_execute : status_simulation S1 S2 R)
1589  on tlr1 : R st1 st2 → label_rel … st1 st2_lab →
1590  ∃st2_mid.∃st2'.
1591  ∃tlr2 : trace_label_return S2 st2_lab st2_mid.
1592  ∃taa2 : trace_any_any_free … st2_mid st2'.
1593  (if taaf_non_empty … taa2 then ¬as_costed … st2_mid else True) ∧
1594  R st1' st2' ∧ ret_rel … R st1' st2_mid ∧ label_rel … st1' st2' ∧
1595  tlr_rel … tlr1 tlr2
1599\section{Conclusions and future works}
1601The labelling approach is a technique to implement compilers that induce on
1602the source code a non uniform cost model determined from the object code
1603produced. The cost model assigns a cost to each basic block of the program.
1604The main theorem of the approach says that there is an exact
1605correspondence between the sequence of basic blocks started in the source
1606and object code, and that no instruction in the source or object code is
1607executed outside a basic block. Thus the cost of object code execution
1608can be computed precisely on the source.
1610In this paper we scale the labelling approach to cover a programming language
1611with function calls. This introduces new difficulties only when the language
1612is unstructured, i.e. it allows function calls to return anywhere in the code,
1613destroying the hope of a static prediction of the cost of basic blocks.
1614We restore static predictability by introducing a new semantics for unstructured
1615programs that single outs well structured executions. The latter are represented
1616by structured traces, a generalisation of streams of observables that capture
1617several structural invariants of the execution, like well nesting of functions
1618or the fact that every basic block must start with a code emission statement.
1619We show that structured traces are sufficiently structured to statically compute
1620a precise cost model on the object code.
1622We introduce a similarity relation on structured traces that must hold between
1623source and target traces. When the relation holds for every program, we prove
1624that the cost model can be lifted from the object to the source code.
1626In order to prove that similarity holds, we present a generic proof of forward
1627simulation that is aimed at pulling apart as much as possible the part of the
1628simulation related to non-functional properties (preservation of structure)
1629from that related to functional properties. In particular, we reduce the
1630problem of preservation of structure to that of showing a 1-to-many
1631forward simulation that only adds a few additional proof obligations to those
1632of a traditional, function properties only, proof.
1634All results presented in the paper are part of a larger certification of a
1635C compiler which is based on the labelling approach. The certification, done
1636in Matita, is the main deliverable of the FET-Open Certified Complexity (CerCo).
1638The short term future work consists in the completion of the certification of
1639the CerCo compiler exploiting the main theorem of this paper.
1641\paragraph{Related works.}
1642CerCo is the first project that explicitly tries to induce a
1643precise cost model on the source code in order to establish non-functional
1644properties of programs on an high level language. Traditional certifications
1645of compilers, like~\cite{compcert2,piton}, only explicitly prove preservation
1646of the functional properties.
1648Usually forward simulations take the following form: for each transition
1649from $s_1$ to $s_2$ in the source code, there exists an equivalent sequence of
1650transitions in the target code of length $n$. The number $n$ of transition steps
1651in the target code can just be the witness of the existential statement.
1652An equivalent alternative when the proof of simulation is constructive consists
1653in providing an explicit function, called \emph{clock function} in the
1654literature~\cite{clockfunctions}, that computes $n$ from $s_1$. Every clock
1655function constitutes then a cost model for the source code, in the spirit of
1656what we are doing in CerCo. However, we believe our solution to be superior
1657in the following respects: 1) the machinery of the labelling approach is
1658insensible to the resource being measured. Indeed, any cost model computed on
1659the object code can be lifted to the source code (e.g. stack space used,
1660energy consumed, etc.). On the contrary, clock functions only talk about
1661number of transition steps. In order to extend the approach with clock functions
1662to other resources, additional functions must be introduced. Moreover, the
1663additional functions would be handled differently in the proof.
16642) the cost models induced by the labelling approach have a simple presentation.
1665In particular, they associate a number to each basic block. More complex
1666models can be induced when the approach is scaled to cover, for instance,
1667loop optimisations~\cite{loopoptimizations}, but the costs are still meant to
1668be easy to understand and manipulate in an interactive theorem prover or
1669in Frama-C.
1670On the contrary, a clock function is a complex function of the state $s_1$
1671which, as a function, is an opaque object that is difficult to reify as
1672source code in order to reason on it.
1677% \appendix
1678% \section{Notes for the reviewers}
1680% The results described in the paper are part of a larger formalization
1681% (the certification of the CerCo compiler). At the moment of the submission
1682% we need to single out from the CerCo formalization the results presented here.
1683% Before the 16-th of February we will submit an attachment that contains the
1684% minimal subset of the CerCo formalization that allows to prove those results.
1685% At that time it will also be possible to measure exactly the size of the
1686% formalization described here. At the moment a rough approximation suggests
1687% about 2700 lines of Matita code.
1689% We will also attach the development version of the interactive theorem
1690% prover Matita that compiles the submitted formalization. Another possibility
1691% is to backport the development to the last released version of the system
1692% to avoid having to re-compile Matita from scratch.
1694% The programming and certification style used in the formalization heavily
1695% exploit dependent types. Dependent types are used: 1) to impose invariants
1696% by construction on the data types and operations (e.g. a traces from a state
1697% $s_1$ to a state $s_2$ can be concatenad to a trace from a state
1698% $s_2'$ to a state $s_3$ only if $s_2$ is convertible with $s_2'$); 2)
1699% to state and prove the theorems by using the Russell methodology of
1700% 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.
1701% }, better known in the Coq world as ``\texttt{Program}'' and reimplemented in a simpler way in Matita using coercion propagations\footnote{Andrea Asperti, Wilmer Ricciotti, Claudio Sacerdoti Coen, Enrico Tassi: A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions. Logical Methods in Computer Science 8(1) (2012)}. However, no result presented depends
1702% mandatorily on dependent types: it should be easy to adapt the technique
1703% and results presented in the paper to HOL.
1705% Finally, Matita and Coq are based on minor variations of the Calculus of
1706% (Co)Inductive Constructions. These variations do not affect the CerCo
1707% formalization. Therefore a porting of the proofs and ideas to Coq would be
1708% rather straightforward.
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