source: Papers/fopara2013/fopara13.tex @ 3452

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43\title{Certified Complexity (CerCo)\thanks{The project CerCo acknowledges the
44financial support of the Future and Emerging Technologies (FET) programme within
45the Seventh Framework Programme for Research of the European Commission, under
46FET-Open grant number: 243881}}
49R.~M. Amadio$^{4}$ \and
51N.~Ayache$^{3,4}$ \and
53F.~Bobot$^{3,4}$ \and
55J.~P. Boender$^1$ \and
57B.~Campbell$^2$ \and
59I.~Garnier$^2$ \and
61A.~Madet$^4$ \and
63J.~McKinna$^2$ \and
65D.~P. Mulligan$^1$ \and
67M.~Piccolo$^1$ \and
69R.~Pollack$^2$ \and
71Y.~R\'egis-Gianas$^{3,4}$ \and
73C.~Sacerdoti Coen$^1$ \and
75I.~Stark$^2$ \and
78\institute{Dipartimento di Informatica - Scienza e Ingegneria, Universit\'a di
79Bologna \and
80LFCS, School of Informatics, University of Edinburgh
81\and INRIA (Team $\pi r^2$)
83Universit\`e Paris Diderot
92% Brian: I've changed "without loss of accuracy" because we discuss
93% important precision/complexity tradeoffs for advanced architectures
94% in the dependent labelling section.
96We provide an overview of the FET-Open Project CerCo
97(`Certified Complexity'). Our main achievement is
98the development of a technique for analysing non-functional
99properties of programs (time, space) at the source level, with little or no loss of accuracy,
100and with a small trusted code base. We developed a certified compiler that translates C source code to object code, producing a copy of the source code instrumented with cost annotations as an additional side-effect. These instrumentations expose at
101the source level---and track precisely---the actual (non-asymptotic)
102computational cost of the input program. Untrusted invariant generators and trusted theorem provers
103may then be used to compute and certify the parametric execution time of the
107% ---------------------------------------------------------------------------- %
108% SECTION                                                                      %
109% ---------------------------------------------------------------------------- %
112%\paragraph{Problem statement.}
113Programs can be specified with both
114functional constraints (what the program must do) and non-functional constraints (what time, space or other resources the program may use).  In the current
115state of the art, functional properties are verified
116by combining user annotations---preconditions, invariants, and so on---with a
117multitude of automated analyses---invariant generators, type systems, abstract
118interpretation, theorem proving, and so on---on the program's high-level source code.
119By contrast, many non-functional properties
120are verified using analyses on low-level object code,
121%\footnote{A notable
122%  exception is the explicit allocation of heap space in languages like C, which
123%  can be handled at the source level.}
124but these analyses may then need information
125about the high-level functional behaviour of the program that must then be reconstructed.
127The analysis of non-functional constraints on low-level object code presents several problems:
130It can be hard to deduce the high-level structure of the program in the presence of compiler optimisations.
131The object code produced by an optimising compiler may have radically different control flow to the original source code program.
133Techniques that operate on object code are not useful early in the development cycle of a program, yet problems with a program's design or implementation are cheaper to resolve earlier rather than later.
135Parametric cost analysis is very hard: how can we reflect a cost that depends on the execution state, for example the
136value of a register or a carry bit, to a cost that the user can understand
137looking at the source code?
139Performing functional analyses on object code makes it hard for the programmer to provide information about the program and its expected execution, leading to a loss of precision in the resulting analyses.
142\paragraph{Vision and approach.}
143 We want to reconcile functional and
144non-functional analyses: to share information and perform both at the same time
145on source code.
147What has previously prevented this approach is the lack of a uniform and precise
148cost model for high-level code: 1) each statement occurrence is compiled
149differently and optimisations may change control flow; 2) the cost of an object
150code instruction may depend on the runtime state of hardware components like
151pipelines and caches, all of which are not visible in the source code.
153To solve the issue, we envision a new generation of compilers able to keep track
154of program structure through compilation and optimisation, and to exploit that
155information to define a cost model for source code that is precise, non-uniform,
156and which accounts for runtime state. With such a source-level cost model we can
157reduce non-functional verification to the functional case and exploit the state
158of the art in automated high-level verification~\cite{survey}. The techniques
159currently used by the Worst Case Execution Time (WCET) community, who perform analyses on object code,
160are still available but can now be coupled with an additional source-level
161analysis. Where the approach produces precise cost models too complex to reason
162about, safe approximations can be used to trade complexity with precision.
163Finally, source code analysis can be used during the early stages of development, when
164components have been specified but not implemented: source code modularity means
165that it is enough to specify the non-functional behaviour of unimplemented
169We have developed what we refer to as \emph{the labelling approach} \cite{labelling}, a
170technique to implement compilers that induce cost models on source programs by
171very lightweight tracking of code changes through compilation. We have studied
172how to formally prove the correctness of compilers implementing this technique.
173We have implemented such a compiler from C to object binaries for the MCS-51
174micro-controller for predicting execution time and stack space usage,
175and verified it in an interactive theorem prover.  As we are targeting
176an embedded micro-controller we do not consider dynamic memory allocation.
178To demonstrate source-level verification of costs we have implemented
179a Frama-C plugin \cite{framac} that invokes the compiler on a source
180program and uses this to generate invariants on the high-level source
181that correctly model low-level costs. The plugin certifies that the
182program respects these costs by calling automated theorem provers, a
183new and innovative technique in the field of cost analysis. Finally,
184we have conducted several case studies, including showing that the
185plugin can automatically compute and certify the exact reaction time
186of Lustre~\cite{lustre} data flow programs compiled into C.
188\section{Project context and approach}
189Formal methods for the verification of functional properties of programs have
190now reached a level of maturity and automation that is facilitating a slow but
191increasing adoption in production environments. For safety critical code, it is
192becoming commonplace to combine rigorous software engineering methodologies and testing
193with static analysis, taking the strong points of each approach and mitigating
194their weaknesses. Of particular interest are open frameworks
195for the combination of different formal methods, where the programs can be
196progressively specified and are continuously enriched with new safety
197guarantees: every method contributes knowledge (e.g. new invariants) that
198becomes an assumption for later analysis.
200The outlook for verifying non-functional properties of programs (time spent,
201memory used, energy consumed) is bleaker.
202% and the future seems to be getting even worse.
203Most industries verify that real time systems meet their deadlines
204by simply performing many runs of the system and timing their execution,  computing the
205maximum time and adding an empirical safety margin, claiming the result to be a
206bound for the WCET of the program. Formal methods and software to statically
207analyse the WCET of programs exist, but they often produce bounds that are too
208pessimistic to be useful. Recent advancements in hardware architecture
209have been
210focused on the improvement of the average case performance, not the
211predictability of the worst case. Execution time is becoming increasingly
212dependent on execution history and the internal state of
213hardware components like pipelines and caches. Multi-core processors and non-uniform
214memory models are drastically reducing the possibility of performing
215static analysis in isolation, because programs are less and less time
216composable. Clock-precise hardware models are necessary for static analysis, and
217obtaining them is becoming harder due to the increased sophistication
218of hardware design.
220Despite the aforementioned problems, the need for reliable real time
221systems and programs is increasing, and there is increasing pressure
222from the research community towards the introduction of alternative
223hardware with more predictable behaviour, which would be more suitable
224for static analysis.  One example, being investigated by the Proartis
225project~\cite{proartis}, is to decouple execution time from execution
226history by introducing randomisation.
228In the CerCo project \cite{cerco} we do not try to address this problem, optimistically
229assuming that improvements in low-level timing analysis or architecture will make
230verification feasible in the longer term. Instead, the main objective of our work is
231to bring together the static analysis of functional and non-functional
232properties, which, according to the current state of the art, are completely
233independent activities with limited exchange of information: while the
234functional properties are verified on the source code, the analysis of
235non-functional properties is entirely performed on the object code to exploit
236clock-precise hardware models.
238\subsection{Current object-code methods}
240Analysis currently takes place on object code for two main reasons.
241First, there cannot be a uniform, precise cost model for source
242code instructions (or even basic blocks). During compilation, high level
243instructions are torn apart and reassembled in context-specific ways so that
244identifying a fragment of object code and a single high level instruction is
245infeasible. Even the control flow of the object and source code can be very
246different as a result of optimisations, for example aggressive loop
247optimisations may completely transform source level loops. Despite the lack of a uniform, compilation- and
248program-independent cost model on the source language, the literature on the
249analysis of non-asymptotic execution time on high level languages that assumes
250such a model is growing and gaining momentum. However, unless we provide a
251replacement for such cost models, this literature's future practical impact looks
252to be minimal. Some hope has been provided by the EmBounded project \cite{embounded}, which
253compositionally compiles high-level code to a byte code that is executed by an
254interpreter with guarantees on the maximal execution time spent for each byte code
255instruction. This provides a uniform model at the expense of the model's
256precision (each cost is a pessimistic upper bound) and the performance of the
257executed code (because the byte code is interpreted compositionally instead of
258performing a fully non-compositional compilation).
260The second reason to perform the analysis on the object code is that bounding
261the worst case execution time of small code fragments in isolation (e.g. loop
262bodies) and then adding up the bounds yields very poor estimates as no
263knowledge of the hardware state prior to executing the fragment can be assumed. By
264analysing longer runs the bound obtained becomes more precise because the lack
265of information about the initial state has a relatively small impact.
267To calculate the cost of an execution, value and control flow analyses
268are required to bound the number of times each basic block is
269executed.  Current state
270of the art WCET technology such as AbsInt's aiT analysis tools~\cite{absint} performs these analyses on the
271object code, where the logic of the program is harder to reconstruct
272and most information available at the source code level has been lost;
273see~\cite{stateart} for a survey. Imprecision in the analysis can lead to useless bounds. To
274augment precision, the tools ask the user to provide constraints on
275the object code control flow, usually in the form of bounds on the
276number of iterations of loops or linear inequalities on them. This
277requires the user to manually link the source and object code,
278translating his assumptions on the source code (which may be wrong) to
279object code constraints. The task is error prone and hard, especially
280in the presence of complex compiler optimisations.
282Traditional techniques for WCET that work on object code are also affected by
283another problem: they cannot be applied before the generation of the object
284code. Functional properties can be analysed in early development stages, while
285analysis of non-functional properties may come too late to avoid expensive
286changes to the program architecture.
288\subsection{CerCo's approach}
290In CerCo we propose a radically new approach to the problem: we reject the idea
291of a uniform cost model and we propose that the compiler, which knows how the
292code is translated, must return the cost model for basic blocks of high level
293instructions. It must do so by keeping track of the control flow modifications
294to reverse them and by interfacing with processor timing analysis.
296By embracing compilation, instead of avoiding it like EmBounded did, a CerCo
297compiler can both produce efficient code and return costs that are
298as precise as the processor timing analysis can be. Moreover, our costs can be
299parametric: the cost of a block can depend on actual program data, on a
300summary of the execution history, or on an approximated representation of the
301hardware state. For example, loop optimisations may assign a cost to a loop body
302that is a function of the number of iterations performed. As another example,
303the cost of a block may be a function of the vector of stalled pipeline states,
304which can be exposed in the source code and updated at each basic block exit. It
305is parametricity that allows one to analyse small code fragments without losing
306precision: in the analysis of the code fragment we do not have to ignore the
307initial hardware state. On the contrary, we can assume that we know exactly which
308state (or mode, as the WCET literature calls it) we are in.
310The CerCo approach has the potential to dramatically improve the state of the
311art. By performing control and data flow analyses on the source code, the error
312prone translation of invariants is completely avoided. Instead, this
313work is done at the source level using tools of the user's choice.
314Any available technique for the verification of functional properties
315can be immediately reused and multiple techniques can collaborate together to
316infer and certify cost invariants for the program.  There are no
317limitations on the types of loops or data structures involved. Parametric cost analysis
318becomes the default one, with non-parametric bounds used as a last resort when
319trading the complexity of the analysis with its precision. \emph{A priori}, no
320technique previously used in traditional WCET is lost: processor
321timing analyses can be used by the compiler on the object code, and the rest can be applied
322at the source code level.
323 Our approach can also work in the early
324stages of development by axiomatically attaching costs to unimplemented components.
327All software used to verify properties of programs must be as bug free as
328possible. The trusted code base for verification consists of the code that needs
329to be trusted to believe that the property holds. The trusted code base of
330state-of-the-art WCET tools is very large: one needs to trust the control flow
331analyser and the linear programming libraries it uses and also the formal models
332of the hardware under analysis. In CerCo we are moving the control flow analysis to the source
333code and we are introducing a non-standard compiler too. To reduce the trusted
334code base, we implemented a prototype and a static analyser in an interactive
335theorem prover, which was used to certify that the costs added to the source
336code are indeed those incurred by the hardware. Formal models of the
337hardware and of the high level source languages were also implemented in the
338interactive theorem prover. Control flow analysis on the source code has been
339obtained using invariant generators, tools to produce proof obligations from
340generated invariants and automatic theorem provers to verify the obligations. If
341the automatic provers are able to generate proof traces that can be
342independently checked, the only remaining component that enters the trusted code
343base is an off-the-shelf invariant generator which, in turn, can be proved
344correct using an interactive theorem prover. Therefore we achieve the double
345objective of allowing the use of more off-the-shelf components (e.g. provers and
346invariant generators) whilst reducing the trusted code base at the same time.
348%\paragraph{Summary of the CerCo objectives.} To summarize, the goal of CerCo is
349% to reconcile functional with non-functional analysis by performing them together
350% on the source code, sharing common knowledge about execution invariants. We want
351% to achieve the goal implementing a new generation of compilers that induce a
352% parametric, precise cost model for basic blocks on the source code. The compiler
353% should be certified using an interactive theorem prover to minimize the trusted
354% code base of the analysis. Once the cost model is induced, off-the-shelf tools
355% and techniques can be combined together to infer and prove parametric cost
356% bounds.
357%The long term benefits of the CerCo vision are expected to be:
358%1. the possibility to perform static analysis during early development stages
359%2.  parametric bounds made easier
360%3.  the application of off-the-shelf techniques currently unused for the
361% analysis of non-functional properties, like automated proving and type systems
362%4. simpler and safer interaction with the user, that is still asked for
363% knowledge, but on the source code, with the additional possibility of actually
364% verifying the provided knowledge
365%5. a reduced trusted code base
366%6. the increased accuracy of the bounds themselves.
368%The long term success of the project is hindered by the increased complexity of
369% the static prediction of the non-functional behaviour of modern hardware. In the
370% time frame of the European contribution we have focused on the general
371% methodology and on the difficulties related to the development and certification
372% of a cost model inducing compiler.
374\section{The typical CerCo workflow}
379char a[] = {3, 2, 7, 14};
380char threshold = 4;
382int count(char *p, int len) {
383  char j;
384  int found = 0;
385  for (j=0; j < len; j++) {
386    if (*p <= threshold)
387      found++;
388    p++;
389    }
390  return found;
393int main() {
394  return count(a,4);
398%  $\vcenter{\includegraphics[width=7.5cm]{interaction_diagram.pdf}}$
400    baseline={([yshift=-.5ex]current bounding box)},
401    element/.style={draw, text width=1.6cm, on chain, text badly centered},
402    >=stealth
403    ]
404{[start chain=going below, node distance=.5cm]
405\node [element] (cerco) {CerCo\\compiler};
406\node [element] (cost)  {CerCo\\cost plugin};
407{[densely dashed]
408\node [element] (ded)   {Deductive\\platform};
409\node [element] (check) {Proof\\checker};
412\coordinate [left=3.5cm of cerco] (left);
413{[every node/.style={above, text width=3.5cm, text badly centered,
414                     font=\scriptsize}]
415\draw [<-] ([yshift=-1ex]cerco.north west) coordinate (t) --
416    node {C source}
417    (t-|left);
418\draw [->] (cerco) -- (cost);
419\draw [->] ([yshift=1ex]cerco.south west) coordinate (t) --
420    node {C source+\color{red}{cost annotations}}
421    (t-|left) coordinate (cerco out);
422\draw [->] ([yshift=1ex]cost.south west) coordinate (t) --
423    node {C source+\color{red}{cost annotations}\\+\color{blue}{synthesized assertions}}
424    (t-|left) coordinate (out);
425{[densely dashed]
426\draw [<-] ([yshift=-1ex]ded.north west) coordinate (t) --
427    node {C source+\color{red}{cost annotations}\\+\color{blue}{complexity assertions}}
428    (t-|left) coordinate (ded in) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
429\draw [->] ([yshift=1ex]ded.south west) coordinate (t) --
430    node {complexity obligations}
431    (t-|left) coordinate (out);
432\draw [<-] ([yshift=-1ex]check.north west) coordinate (t) --
433    node {complexity proof}
434    (t-|left) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
435\draw [dash phase=2.5pt] (cerco out) .. controls +(-1, 0) and +(-1, 0) ..
436    (ded in);
438%% user:
439% head
440\draw (current bounding box.west) ++(-.2,.5)
441    circle (.2) ++(0,-.2) -- ++(0,-.1) coordinate (t);
442% arms
443\draw (t) -- +(-.3,-.2);
444\draw (t) -- +(.3,-.2);
445% body
446\draw (t) -- ++(0,-.4) coordinate (t);
447% legs
448\draw (t) -- +(-.2,-.6);
449\draw (t) -- +(.2,-.6);
452\caption{On the left: code to count the array's elements
453 that are less than or equal to the threshold. On the right: CerCo's interaction
454 diagram. CerCo's components are drawn solid.}
457We illustrate the workflow we envisage (on the right
458of~\autoref{test1}) on an example program (on the left
459of~\autoref{test1}).  The user writes the program and feeds it to the
460CerCo compiler, which outputs an instrumented version of the same
461program that updates global variables that record the elapsed
462execution time and the stack space usage.  The red lines in
463\autoref{itest1} introducing variables, functions and function calls
464starting with \lstinline'__cost' and \lstinline'__stack' are the instrumentation introduced by the
465compiler.  For example, the two calls at the start of
466\lstinline'count' say that 4 bytes of stack are required, and that it
467takes 111 cycles to reach the next cost annotation (in the loop body).
468The compiler measures these on the labelled object code that it generates.
470 The annotated program can then be enriched with complexity
471assertions in the style of Hoare logic, that are passed to a deductive
472platform (in our case Frama-C). We provide as a Frama-C cost plugin a
473simple automatic synthesiser for complexity assertions which can
474be overridden by the user to increase or decrease accuracy.  These are the blue
475comments starting with \lstinline'/*@' in \autoref{itest1}, written in
476Frama-C's specification language, ACSL.  From the
477assertions, a general purpose deductive platform produces proof
478obligations which in turn can be closed by automatic or interactive
479provers, ending in a proof certificate.
481% NB: if you try this example with the live CD you should increase the timeout
483Twelve proof obligations are generated from~\autoref{itest1} (to prove
484that the loop invariant holds after one execution if it holds before,
485to prove that the whole program execution takes at most 1358 cycles,
486etc.).  Note that the synthesised time bound for \lstinline'count',
487$178+214*(1+\text{\lstinline'len'})$ cycles, is parametric in the length of
488the array. The CVC3 prover
489closes all obligations within half a minute on routine commodity
490hardware.  A simpler non-parametric version can be solved in a few
495        moredelim=[is][\color{blue}]{$}{$},
496        moredelim=[is][\color{red}]{|}{|},
497        lineskip=-2pt}
500|int __cost = 33, __stack = 5, __stack_max = 5;|
501|void __cost_incr(int incr) { __cost += incr; }|
502|void __stack_incr(int incr) {
503  __stack += incr;
504  __stack_max = __stack_max < __stack ? __stack : __stack_max;
507char a[4] = {3, 2, 7, 14};  char threshold = 4;
509/*@ behaviour stack_cost:
510      ensures __stack_max <= __max(\old(__stack_max), 4+\old(__stack));
511      ensures __stack == \old(__stack);
512    behaviour time_cost:
513      ensures __cost <= \old(__cost)+(178+214*__max(1+\at(len,Pre), 0));
515int count(char *p, int len) {
516  char j;  int found = 0;
517  |__stack_incr(4);  __cost_incr(111);|
518  $__l: /* internal */$
519  /*@ for time_cost: loop invariant
520        __cost <= \at(__cost,__l)+
521                  214*(__max(\at((len-j)+1,__l), 0)-__max(1+(len-j), 0));
522      for stack_cost: loop invariant
523        __stack_max == \at(__stack_max,__l);
524      for stack_cost: loop invariant
525        __stack == \at(__stack,__l);
526      loop variant len-j;
527  */
528  for (j = 0; j < len; j++) {
529    |__cost_incr(78);|
530    if (*p <= threshold) { |__cost_incr(136);| found ++; }
531    else { |__cost_incr(114);| }
532    p ++;
533  }
534  |__cost_incr(67);  __stack_incr(-4);|
535  return found;
538/*@ behaviour stack_cost:
539      ensures __stack_max <= __max(\old(__stack_max), 6+\old(__stack));
540      ensures __stack == \old(__stack);
541    behaviour time_cost:
542      ensures __cost <= \old(__cost)+1358;
544int main(void) {
545  int t;
546  |__stack_incr(2);  __cost_incr(110);|
547  t = count(a,4);
548  |__stack_incr(-2);|
549  return t;
552\caption{The instrumented version of the program in \autoref{test1},
553 with instrumentation added by the CerCo compiler in red and cost invariants
554 added by the CerCo Frama-C plugin in blue. The \texttt{\_\_cost},
555 \texttt{\_\_stack} and \texttt{\_\_stack\_max} variables hold the elapsed time
556in clock cycles and the current and maximum stack usage. Their initial values
557hold the clock cycles spent in initialising the global data before calling
558\texttt{main} and the space required by global data (and thus unavailable for
559the stack).
563\section{Main scientific and technical results}
564First we describe the basic labelling approach and our compiler
565implementations that use it.  This is suitable for basic architectures
566with simple cost models.  Then we will discuss the dependent labelling
567extension which is suitable for more advanced processor architectures
568and compiler optimisations.  At the end of this section we will
569demonstrate automated high level reasoning about the source level
570costs provided by the compilers.
572% \emph{Dependent labelling~\cite{?}.} The basic labelling approach assumes a
573% bijective mapping between object code and source code O(1) blocks (called basic
574% blocks). This assumption is violated by many program optimizations (e.g. loop
575% peeling and loop unrolling). It also assumes the cost model computed on the
576% object code to be non parametric: every block must be assigned a cost that does
577% not depend on the state. This assumption is violated by stateful hardware like
578% pipelines, caches, branch prediction units. The dependent labelling approach is
579% an extension of the basic labelling approach that allows to handle parametric
580% cost models. We showed how the method allows to deal with loop optimizations and
581% pipelines, and we speculated about its applications to caches.
583% \emph{Techniques to exploit the induced cost model.} Every technique used for
584% the analysis of functional properties of programs can be adapted to analyse the
585% non-functional properties of the source code instrumented by compilers that
586% implement the labelling approach. In order to gain confidence in this claim, we
587% showed how to implement a cost invariant generator combining abstract
588% interpretation with separation logic ideas \cite{separation}. We integrated
589% everything in the Frama-C modular architecture, in order to compute proof
590% obligations from functional and cost invariants and to use automatic theorem
591% provers on them. This is an example of a new technique that is not currently
592% exploited in WCET analysis. It also shows how precise functional invariants
593% benefits the non-functional analysis too. Finally, we show how to fully
594% automatically analyse the reaction time of Lustre nodes that are first compiled
595% to C using a standard Lustre compiler and then processed by a C compiler that
596% implements the labelling approach.
598% \emph{The CerCo compiler.} This is a compiler from a large subset of the C
599% program to 8051/8052 object code,
600% integrating the labelling approach and a static analyser for 8051 executables.
601% The latter can be implemented easily and does not require dependent costs
602% because the 8051 microprocessor is a very simple one, with constant-cost
603% instruction. It was chosen to separate the issue of exact propagation of the
604% cost model from the orthogonal problem of the static analysis of object code
605% that may require approximations or dependent costs. The compiler comes in
606% several versions: some are prototypes implemented directly in OCaml, and they
607% implement both the basic and dependent labelling approaches; the final version
608% is extracted from a Matita certification and at the moment implements only the
609% basic approach.
611\subsection{The (basic) labelling approach}
612The labelling approach is the foundational insight that underlies all the developments in CerCo.
613It facilitates the tracking of basic block evolution during the compilation process in order to propagate the cost model from the
614object code to the source code without losing precision in the process.
615\paragraph{Problem statement.} Given a source program $P$, we want to obtain an
616instrumented source program $P'$,  written in the same programming language, and
617the object code $O$ such that: 1) $P'$ is obtained by inserting into $P$ some
618additional instructions to update global cost information like the amount of
619time spent during execution or the maximal stack space required; 2) $P$ and $P'$ 
620must have the same functional behaviour, i.e.\ they must produce that same output
621and intermediate observables; 3) $P$ and $O$ must have the same functional
622behaviour; 4) after execution and in interesting points during execution, the
623cost information computed by $P'$ must be an upper bound of the one spent by $O$ 
624to perform the corresponding operations (\emph{soundness} property); 5) the difference
625between the costs computed by $P'$ and the execution costs of $O$ must be
626bounded by a program-dependent constant (\emph{precision} property).
628\paragraph{The labelling software components.} We solve the problem in four
629stages \cite{labelling}, implemented by four software components that are used
630in sequence.
632The first component labels the source program $P$ by injecting label emission
633statements in appropriate positions to mark the beginning of basic blocks.
634These are the positions where the cost instrumentation will appear in the final output.
635% The
636% set of labels with their positions is called labelling.
637The syntax and semantics
638of the source programming language is augmented with label emission statements.
639The statement ``EMIT $\ell$'' behaves like a NOP instruction that does not affect the
640program state or control flow, but its execution is observable.
641% Therefore the observables of a run of a program becomes a stream
642% of label emissions: $\ell_1,\ldots,\ell_n$, called the program trace. We clarify the conditions
643% that the labelling must respect later.
644For the example in Section~\ref{sec:workflow} this is just the original C code
645with ``EMIT'' instructions added at every point a \lstinline'__cost_incr' call
646appears in the final code.
648The second component is a labelling preserving compiler. It can be obtained from
649an existing compiler by adding label emission statements to every intermediate
650language and by propagating label emission statements during compilation. The
651compiler is correct if it preserves both the functional behaviour of the program
652and the traces of observables, including the labels `emitted'.
653% We may also ask that the function that erases the cost
654% emission statements commute with compilation. This optional property grants that
655% the labelling does not interfere with the original compiler behaviour. A further
656% set of requirements will be added later.
658The third component analyses the labelled
659object code to compute the scope of each of its label emission statements,
660i.e.\ the instructions that may be executed after the statement and before
661a new label emission is encountered, and then computes the maximum cost of
662each.  Note that we only have enough information at this point to compute
663the cost of loop-free portions of code.  We will consider how to ensure that
664every loop is broken by a cost label shortly.
665%It is a tree and not a sequence because the scope
666%may contain a branching statement. In order to grant that such a finite tree
667%exists, the object code must not contain any loop that is not broken by a label
668%emission statement. This is the first requirement of a sound labelling. The
669%analyser fails if the labelling is unsound. For each scope, the analyser
670%computes an upper bound of the execution time required by the scope, using the
671%maximum of the costs of the two branches in case of a conditional statement.
672%Finally, the analyser computes the cost of a label by taking the maximum of the
673%costs of the scopes of every statement that emits that label.
675The fourth and final component replaces the labels in the labelled
676version of the source code produced at the start with the costs
677computed for each label's scope.  This yields the instrumented source
678code.  For the example, this is the code in \autoref{itest1}, except
679for the specifications in comments, which we consider in
682\paragraph{Correctness.} Requirements 1 and 2 hold because of the
683non-invasive labelling procedure.  Requirement 3 can be satisfied by
684implementing compilation correctly.  It is obvious that the value of
685the global cost variable of the instrumented source code is always
686equal to the sum of the costs of the labels emitted by the
687corresponding labelled code. Moreover, because the compiler preserves
688all traces, the sum of the costs of the labels emitted in the source
689and target labelled code are the same. Therefore, to satisfy the
690soundness requirement, we need to ensure that the time taken to
691execute the object code is equal to the sum of the costs of the labels
692emitted by the object code. We collect all the necessary conditions
693for this to happen in the definition of a \emph{sound} labelling: a)
694all loops must be broken by a cost emission statement; b) all program
695instructions must be in the scope of some cost emission statement.
696This ensures that every label's scope is a tree of instructions, with
697the cost being the most expensive path. To satisfy also the precision
698requirement, we must make the scopes flat sequences of instructions.
699We require a \emph{precise} labelling where every label is emitted at
700most once and both branches of each conditional jump start with a label
701emission statement.
703The correctness and precision of the labelling approach only rely on the
704correctness and precision of the object code labelling. The simplest
705way to achieve that is to impose correctness and precision
706requirements on the source code labelling produced at the start, and
707to demand that the compiler preserves these
708properties too. The latter requirement imposes serious limitations on the
709compilation strategy and optimisations: the compiler may not duplicate any code
710that contains label emission statements, like loop bodies. Therefore various
711loop optimisations like peeling or unrolling are prevented. Moreover, precision
712of the object code labelling is not sufficient \emph{per se} to obtain global
713precision: we implicitly assumed that a precise constant cost can be assigned
714to every instruction. This is not possible in
715the presence of stateful hardware whose state influences the cost of operations,
716like pipelines and caches. In Section~\ref{lab:deplabel} we will see an extension of the
717basic labelling approach which tackles these problems.
719In CerCo we have developed several cost preserving compilers based on
720the labelling approach. Excluding an initial certified compiler for a
721`while' language, all remaining compilers target realistic source
722languages---a pure higher order functional language and a large subset
723of C with pointers, \texttt{goto}s and all data structures---and real world
724target processors---MIPS and the Intel 8051/8052 processor
725family. Moreover, they achieve a level of optimisation that ranges
726from moderate (comparable to GCC level 1) to intermediate (including
727loop peeling and unrolling, hoisting and late constant propagation).
728We describe the C compilers in detail in the following section.
730Two compilation chains were implemented for a purely functional
731higher-order language~\cite{labelling2}. The two main changes required
732to deal with functional languages are: 1) because global variables and
733updates are not available, the instrumentation phase produces monadic
734code to `update' the global costs; 2) the requirements for a sound and
735precise labelling of the source code must be changed when the
736compilation is based on CPS translations.  In particular, we need to
737introduce labels emitted before a statement is executed and also
738labels emitted after a statement is executed. The latter capture code
739that is inserted by the CPS translation and that would escape all
740label scopes.
742% Brian: one of the reviewers pointed out that standard Prolog implementations
743% do have some global state that apparently survives backtracking and might be
744% used.  As we haven't experimented with this, I think it's best to elide it
745% entirely.
747% Phases 1, 2 and 3 can be applied as well to logic languages (e.g. Prolog).
748% However, the instrumentation phase cannot: in standard Prolog there is no notion
749% of (global) variable whose state is not retracted during backtracking.
750% Therefore, the cost of executing computations that are later backtracked would
751% not be correctly counted in. Any extension of logic languages with
752% non-backtrackable state could support our labelling approach.
754\subsection{The CerCo C compilers}
755We implemented two C compilers, one implemented directly in OCaml and
756the other implemented in the interactive theorem prover
757Matita~\cite{matita}.  The first acted as a prototype for the second,
758but also supported MIPS and acted as a testbed for more advanced
759features such as the dependent labelling approach
760in Section~\ref{lab:deplabel}.
762The second C compiler is the \emph{Trusted CerCo Compiler}, whose cost
763predictions are formally verified. The executable code
764is OCaml code extracted from the Matita implementation. The Trusted
765CerCo Compiler only targets
766the C language and the 8051/8052 family, and does not yet implement any
767advanced optimisations. Its user interface, however, is the same as the
768other version for interoperability purposes. In
769particular, the Frama-C CerCo plugin descibed in
770Section~\ref{sec:exploit} can work without recompilation
771with both of our C compilers.
773The 8051/8052 microprocessor is a very simple one, with constant-cost
774instructions. It was chosen to separate the issue of exact propagation of the
775cost model from the orthogonal problem of low-level timing analysis of object code
776that may require approximation or dependent costs.
778The (trusted) CerCo compiler implements the following optimisations: cast
779simplification, constant propagation in expressions, liveness analysis driven
780spilling of registers, dead code elimination, branch displacement, and tunnelling.
781The two latter optimisations are performed by our optimising assembler
782\cite{correctness}. The back-end of the compiler works on three address
783instructions, preferred to static single assignment code for the simplicity of
784the formal certification.
786The CerCo compiler is loosely based on the CompCert compiler \cite{compcert}, a
787recently developed certified compiler from C to the PowerPC, ARM and x86
788microprocessors. In contrast to CompCert, both the CerCo code and its
789certification are fully open source. Some data structures and language definitions for
790the front-end are directly taken from CompCert, while the back-end is a redesign
791of a compiler from Pascal to MIPS used by Fran\c{c}ois Pottier for a course at the
792\'{E}cole Polytechnique.
794The main differences in the CerCo compiler are:
796\item All the intermediate languages include label emitting instructions to
797implement the labelling approach, and the compiler preserves execution traces.
798\item Instead of targeting an assembly language with additional
799macro-instructions which are expanded before assembly, we directly
800produce object code in order to perform the timing analysis, using
801an integrated optimising assembler.
802\item In order to avoid the additional work of implementing a linker
803 and a loader, we do not support separate
804compilation and external calls. Adding them is orthogonal
805to the labelling approach and should not introduce extra problems.
806\item We target an 8-bit processor, in contrast to CompCert's 32-bit targets. This requires
807many changes and more compiler code, but it is not fundamentally more
808complex. The proof of correctness, however, becomes much harder.
809\item We target a microprocessor that has a non-uniform memory model, which is
810still often the case for microprocessors used in embedded systems and that is
811becoming common again in multi-core processors. Therefore the compiler has to
812keep track of data and it must move data between memory regions in the proper
813way. Moreover the size of pointers to different regions is not uniform. %An
814%additional difficulty was that the space available for the stack in internal
815%memory in the 8051 is tiny, allowing only a minor number of nested calls. To
816%support full recursion in order to test the CerCo tools also on recursive
817%programs, the compiler implements a stack in external memory.
820\subsection{Formal certification of the CerCo compiler}
821We have formally
822certified in the Matita interactive proof assistant that the cost models induced on the source code by the
823Trusted CerCo Compiler correctly and precisely
824predict the object code behaviour. There are two cost models, one for execution
825time and one for stack space consumption. We show the correctness of the prediction
826only for those programs that do not exhaust the available stack space, a
827property that---thanks to the stack cost model---we can statically analyse on the
828source code in sharp contrast to other certified compilers.  Other projects have
829already certified the preservation of functional semantics in similar compilers,
830so we have not attempted to directly repeat that work and assume
831functional correctness for most passes. In order to complete the
832proof for non-functional properties, we have introduced a new,
833structured, form of execution trace, with the
834related notions for forward similarity and the intensional
835consequences of forward similarity. We have also introduced a unified
836representation for back-end intermediate languages that was exploited to provide
837a uniform proof of forward similarity.
839The details on the proof techniques employed
841the proof sketch can be found in the CerCo deliverables and papers~\cite{cerco-deliverables}.
842In this section we will only hint at the correctness statement, which turned
843out to be more complex than expected.
845\paragraph{The correctness statement.}
846Real time programs are often reactive programs that loop forever responding to
847events (inputs) by performing some computation followed by some action (output)
848and continuing as before. For these programs the overall execution
849time does not make sense. The same is true for reactive programs that spend an
850unpredictable amount of time in I/O. Instead, what is interesting is the reaction time ---
851 the time spent between I/O events. Moreover, we are interested in
852predicting and ruling out crashes due to running out of space on certain
854Therefore we need a statement that talks about sub-runs of a
855program. A natural candidate is that the time predicted on the source
856code and spent on the object code by two corresponding sub-runs are the same.
857To make this statement formal we must identify the
858corresponding sub-runs and how to single out those that are meaningful.
859We introduce the notion of a \emph{measurable} sub-run of a run
860which does not exhaust the available stack before or during the
861sub-run, the number of function calls and returns
862in the sub-run is the same, the sub-run does not perform any I/O, and the sub-run
863starts with a label emission statement and ends with a return or another label
864emission statement. The stack usage is bounded using the stack usage model
865that is computed by the compiler.
867The statement that we formally proved is: for each C run with a measurable
868sub-run, there exists an object code run with a sub-run, with the same
869execution trace for both the prefix of the run and the sub-run itself,
870 and where the time spent by the
871object code in the sub-run is the same as the time predicted on the source code
872using the time cost model generated by the compiler.
875We briefly discuss the constraints for measurability. Not exhausting the stack
876space is necessary for a run to be meaningful, because the source semantics
877has no notion of running out of memory. Balancing
878function calls and returns is a requirement for precision: the labelling approach
879allows the scope of a label to extend after function calls to
880minimize the number of labels. (The scope excludes the called
881function's execution.) If the number of
882calls/returns is unbalanced, it means that there is a call we have not returned
883to that could be followed by additional instructions whose cost has already been
884taken in account.  The last condition on the start and end points of
885a run is also required to make the bound precise.  With these
886restrictions and the 8051's simple timing model we obtain \emph{exact}
887predictions.  If we relax these conditions then we obtain a corollary
888with an upper bound on the cost.
889Finally, I/O operations can be performed in the prefix of the run, but not in the
890measurable sub-run. Therefore we prove that we can predict reaction times, but
891not I/O times, as desired.
893\subsection{Dependent labelling}
895The core idea of the basic labelling approach is to establish a tight connection
896between basic blocks executed in the source and target languages. Once the
897connection is established, any cost model computed on the object code can be
898transferred to the source code, without affecting the code of the compiler or
899its proof. In particular, we can also transport cost models
900that associate to each label a \emph{function} from the hardware state
901to a natural number.
902However, a problem arises during the instrumentation phase that replaces label
903emission statements with increments of global cost variables. They are
904incremented by the result of applying the label's cost function
905to the hardware state at the time of execution of the block. However,
906the hardware state comprises both the functional state that affects the
907computation (the value of the registers and memory) and the non-functional
908state that does not (the pipeline and cache contents, for example).
909We can find corresponding information for the former in the source code state, but constructing the
910correspondence may be hard and lifting the cost model to work on the source code
911state is likely to produce cost expressions that are too complex to understand and reason about.
912Fortunately, in modern architectures the cost of executing single
913instructions is either independent of the functional state or the jitter---the
914difference between the worst and best case execution times---is small enough
915to be bounded without losing too much precision. Therefore we only consider
916dependencies on the `non-functional' parts of the state.
918The non-functional state is not directly related to the high level
919state and does not influence the functional properties. What can be
920done is to expose key aspects of the non-functional state in the
921source code. We present here the basic intuition in a simplified form:
922the technical details that allow us to handle the general case are
923more complex and can be found in~\cite{paolo}. We add to the source
924code an additional global variable that represents the non-functional
925state and another one that remembers the last few labels emitted. The
926state variable must be updated at every label emission statement,
927using an update function which is computed during the processor timing
929This update function assigns to each label a function from the
930recently emitted labels and old state to the new state. It is computed
931by composing the semantics of every instruction in a basic block
932restricted to the non-functional part of the state.
934Not all the details of the non-functional state needs to be exposed, and the
935technique works better when the part of state that is required can be summarised
936in a simple data structure. For example, to handle simple but realistic
937pipelines it is sufficient to remember a short integer that encodes the position
938of bubbles (stuck instructions) in the pipeline. In any case, it is not necessary
939for the user to understand the meaning of the state to reason over the properties
940of the
941program. Moreover, the user, or the invariant generator tools that
942analyse the instrumented source code produced by the compiler, can decide to
943trade precision of the analysis for simplicity by approximating the
944cost by safe bounds that do not depend on the processor state. Interestingly, the functional analysis of
945the code could determine which blocks are executed more frequently in order to
946use more aggressive approximations for those that are executed least.
948Dependent labelling can also be applied to allow the compiler to duplicate
949blocks that contain labels (e.g. in loop optimisations)~\cite{paolo}. The effect is to assign
950a different cost to the different occurrences of a duplicated label. For
951example, loop peeling turns a loop into the concatenation of a copy of the loop
952body for the first iteration and the conditional execution of the
953loop for successive iterations. Further optimisations will compile the two
954copies of the loop differently, with the first body usually
955taking more time.
957By introducing a variable that keeps track of the iteration number, we can
958associate to the label a cost that is a function of the iteration number. The
959same technique works for loop unrolling without modification: the function will
960assign one cost to the even iterations and another cost to the odd
961ones.  The optimisation code
962that duplicates the loop bodies must also modify the code to correctly propagate
963the update of the iteration numbers. The technical details are more complicated and
964can be found in the CerCo reports and publications. The implementation, however,
965is quite simple (and forms part of our OCaml version of the compiler)
966and the changes to a loop optimising compiler are minimal.
968\subsection{Techniques to exploit the induced cost model}
970We now turn our attention to synthesising high-level costs, such as
971the reaction time of a real-time program.  We consider as our starting point source level costs
972provided by basic labelling, in other words annotations
973on the source code which are constants that provide a sound and sufficiently
974precise upper bound on the cost of executing the blocks after compilation to
975object code.
977The principle that we have followed in designing the cost synthesis tools is
978that the synthesised bounds should be expressed and proved within a general
979purpose tool built to reason on the source code. In particular, we rely on the
980Frama-C tool to reason on C code and on the Coq proof-assistant to reason on
981higher-order functional programs.
982This principle entails that
983the inferred synthetic bounds are indeed correct as long as the general purpose
984tool is, and that there is no limitation on the class of programs that can be handled,
985for example by resorting to interactive proof.
987Of course, automation is desirable whenever possible. Within this framework,
988automation means writing programs that give hints to the general purpose tool.
989These hints may take the form, say, of loop invariants/variants, of predicates
990describing the structure of the heap, or of types in a light logic. If these
991hints are correct and sufficiently precise the general purpose tool will produce
992a proof automatically, otherwise, user interaction is required.
994\paragraph{The Cost plugin and its application to the Lustre compiler.}
995Frama-C \cite{framac} is a set of analysers for C programs with a
996specification language called ACSL. New analyses can be dynamically added
997via a plugin system. For instance, the Jessie plugin~\cite{jessie} allows deductive
998verification of C programs with respect to their specification in ACSL, with
999various provers as back-end tools.
1000We developed the CerCo Cost plugin for the Frama-C platform as a proof of
1001concept of an automatic environment exploiting the cost annotations produced by
1002the CerCo compiler. It consists of an OCaml program which essentially
1003uses the CerCo compiler to produce a related C program with cost annotations,
1004and applies some heuristics to produce a tentative bound on the cost of
1005executing the C functions of the program as a function of the value of their
1006parameters. The user can then call the Jessie plugin to discharge the
1007related proof obligations.
1008In the following we elaborate on the soundness of the framework and the
1009experiments we performed with the Cost tool on C programs, including some produced by a
1010Lustre compiler.
1012\paragraph{Soundness.} The soundness of the whole framework depends on the cost
1013annotations added by the CerCo compiler,
1014the verification conditions (VCs) generated by Jessie, and the
1015external provers discharging the VCs. Jessie can be used to verify the
1016synthesised bounds because our plugin generates them in ACSL format. Thus, even if the added synthetic costs are
1017incorrect (relatively to the cost annotations), the process as a whole is still
1018correct: indeed, Jessie will not validate incorrect costs and no conclusion can
1019be made about the WCET of the program in this case. In other terms, the
1020soundness does not depend on the cost plugin, which can in
1021principle produce any synthetic cost. However, in order to be able to actually
1022prove a WCET of a C function, we need to add correct annotations in a way that
1023Jessie and subsequent automatic provers have enough information to deduce their
1024validity. In practice this is not straightforward even for very simple programs
1025composed of branching and assignments (no loops and no recursion) because a fine
1026analysis of the VCs associated with branching may lead to a complexity blow up.
1028\paragraph{Experience with Lustre.} Lustre~\cite{lustre} is a data-flow language for programming
1029synchronous systems, with a compiler which targets C. We designed a
1030wrapper for supporting Lustre files.
1031The C function produced by the compiler is relatively simple loop-free
1032code which implements the step function of the
1033synchronous system and computing the WCET of the function amounts to obtaining a
1034bound on the reaction time of the system. We tested the Cost plugin and the
1035Lustre wrapper on the C programs generated by the Lustre compiler. For programs
1036consisting of a few hundred lines of code, the cost plugin computes a
1037WCET and Alt-Ergo is able to discharge all VCs automatically.
1039\paragraph{Handling C programs with simple loops.}
1040The cost annotations added by the CerCo compiler take the form of C instructions
1041that update a fresh global variable called the cost variable by a constant.
1042Synthesizing a WCET of a C function thus consists of statically resolving an
1043upper bound of the difference between the value of the cost variable before and
1044after the execution of the function, i.e. finding the instructions
1045that update the cost variable and establish the number of times they are passed
1046through during the flow of execution. To perform the analysis the plugin
1047assumes that there are no recursive functions in the program, and that
1048every loop is annotated with a variant. In the case of `for' loops the
1049variants are automatically inferred where a loop counter can be
1050syntactically detected.
1052The plugin computes a call-graph and proceeds to calculate bounds for
1053each function from the leaves up to the main function.
1054The computation of the cost of each function is performed by traversing its
1055control flow graph, where the cost of a node is the maximum of the
1056costs of the successors.
1057In the case of a loop with a body that has a constant cost for every step of the
1058loop, the cost is the product of the cost of the body and of the variant taken
1059at the start of the loop.
1060In the case of a loop with a body whose cost depends on the values of some free
1061variables, a fresh logic function $f$ is introduced to represent the cost of the
1062loop in the logic assertions. This logic function takes the variant as a first
1063parameter. The other parameters of $f$ are the free variables of the body of the
1064loop. An axiom is added to account for the fact that the cost is accumulated at each
1065step of the loop.
1066The cost of the function is directly added as post-condition of the function.
1068The user can also specify more precise variants and annotate functions
1069with their own cost specifications. The plugin will use these
1070instead of computing its own, allowing greater precision and the
1071ability to analyse programs which the variant generator does not
1074In addition to the loop-free Lustre code, this method was successfully
1075applied to a small range of cryptographic code.  See~\cite{labelling}
1076for more details.  The example in Section~\ref{sec:workflow} was also
1077produced using the plug-in.  The variant was calculated automatically
1078by noticing that \lstinline'j' is a loop counter with maximum value
1079\lstinline'len'.  The most expensive path through the loop body
1080($78+136 = 214$) is then multiplied by the number of iterations to
1081give the cost of the loop.
1083\paragraph{C programs with pointers.}
1084When it comes to verifying programs involving pointer-based data structures,
1085such as linked lists, trees, or graphs, the use of traditional first-order logic
1086to specify, and of SMT solvers to verify, shows some limitations. Separation
1087logic is an elegant alternative. It is a program logic
1088with a new notion of conjunction to express spatial heap separation. Bobot has
1089recently introduced automatically generated separation
1090predicates to simulate separation logic reasoning in the Jessie plugin where the specification language, the verification condition
1091generator, and the theorem provers were not designed with separation logic in
1092mind~\cite{bobot}. CerCo's plugin can exploit the separation predicates to automatically
1093reason on the cost of execution of simple heap manipulation programs such as an
1094in-place list reversal.
1096\section{Conclusions and future work}
1098All CerCo software and deliverables may be found on the project homepage~\cite{cerco}.
1100The results obtained so far are encouraging and provide evidence that
1101it is possible to perform static time and space analysis at the source level
1102without losing accuracy, reducing the trusted code base and reconciling the
1103study of functional and non-functional properties of programs. The
1104techniques introduced seem to be scalable, cover both imperative and
1105functional languages and are compatible with every compiler optimisation
1106considered by us so far.
1108To prove that compilers can keep track of optimisations
1109and induce a precise cost model on the source code, we targeted a simple
1110architecture that admits a cost model that is execution history independent.
1111The most important future work is dealing with hardware architectures
1112characterised by history-dependent stateful components, like caches and
1113pipelines. The main issue is to assign a parametric, dependent cost
1114to basic blocks that can be later transferred by the labelling approach to
1115the source code and represented in a meaningful way to the user. The dependent
1116labelling approach that we have studied seems a promising tool to achieve
1117this goal, but more work is required to provide good source level
1118approximations of the relevant processor state.
1120Other examples of future work are to improve the cost invariant
1121generator algorithms and the coverage of compiler optimisations, to combining
1122the labelling approach with the type and effect discipline of~\cite{typeffects}
1123to handle languages with implicit memory management, and to experiment with
1124our tools in the early phases of development. Larger case studies are also necessary
1125to evaluate the CerCo's prototype on realistic, industrial-scale programs.
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