source: Papers/fopara2013/fopara13.tex @ 3451

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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.
127This situation presents several problems: 1) it can be hard to infer this
128high-level structure in the presence of compiler optimisations; 2) techniques
129working on object code are not useful in early development, yet problems
130detected later are more expensive to tackle; 3) parametric cost analysis is very
131hard: how can we reflect a cost that depends on the execution state (e.g. the
132value of a register or a carry bit) to a cost that the user can understand
133looking at source code? 4) performing functional analysis on the object code
134makes it hard for the programmer to provide information, leading to less precision in
135the estimates.
137\paragraph{Vision and approach.}
138 We want to reconcile functional and
139non-functional analyses: to share information and perform both at the same time
140on source code.
142What has previously prevented this approach is the lack of a uniform and precise
143cost model for high-level code: 1) each statement occurrence is compiled
144differently and optimisations may change control flow; 2) the cost of an object
145code instruction may depend on the runtime state of hardware components like
146pipelines and caches, all of which are not visible in the source code.
148To solve the issue, we envision a new generation of compilers able to keep track
149of program structure through compilation and optimisation, and to exploit that
150information to define a cost model for source code that is precise, non-uniform,
151and which accounts for runtime state. With such a source-level cost model we can
152reduce non-functional verification to the functional case and exploit the state
153of the art in automated high-level verification~\cite{survey}. The techniques
154currently used by the Worst Case Execution Time (WCET) community, who perform analyses on object code,
155are still available but can now be coupled with an additional source-level
156analysis. Where the approach produces precise cost models too complex to reason
157about, safe approximations can be used to trade complexity with precision.
158Finally, source code analysis can be used during the early stages of development, when
159components have been specified but not implemented: source code modularity means
160that it is enough to specify the non-functional behaviour of unimplemented
164We have developed what we refer to as \emph{the labelling approach} \cite{labelling}, a
165technique to implement compilers that induce cost models on source programs by
166very lightweight tracking of code changes through compilation. We have studied
167how to formally prove the correctness of compilers implementing this technique.
168We have implemented such a compiler from C to object binaries for the MCS-51
169micro-controller for predicting execution time and stack space usage,
170and verified it in an interactive theorem prover.  As we are targeting
171an embedded micro-controller we do not consider dynamic memory allocation.
173To demonstrate source-level verification of costs we have implemented
174a Frama-C plugin \cite{framac} that invokes the compiler on a source
175program and uses this to generate invariants on the high-level source
176that correctly model low-level costs. The plugin certifies that the
177program respects these costs by calling automated theorem provers, a
178new and innovative technique in the field of cost analysis. Finally,
179we have conducted several case studies, including showing that the
180plugin can automatically compute and certify the exact reaction time
181of Lustre~\cite{lustre} data flow programs compiled into C.
183\section{Project context and approach}
184Formal methods for the verification of functional properties of programs have
185now reached a level of maturity and automation that is facilitating a slow but
186increasing adoption in production environments. For safety critical code, it is
187becoming commonplace to combine rigorous software engineering methodologies and testing
188with static analysis, taking the strong points of each approach and mitigating
189their weaknesses. Of particular interest are open frameworks
190for the combination of different formal methods, where the programs can be
191progressively specified and are continuously enriched with new safety
192guarantees: every method contributes knowledge (e.g. new invariants) that
193becomes an assumption for later analysis.
195The outlook for verifying non-functional properties of programs (time spent,
196memory used, energy consumed) is bleaker.
197% and the future seems to be getting even worse.
198Most industries verify that real time systems meet their deadlines
199by simply performing many runs of the system and timing their execution,  computing the
200maximum time and adding an empirical safety margin, claiming the result to be a
201bound for the WCET of the program. Formal methods and software to statically
202analyse the WCET of programs exist, but they often produce bounds that are too
203pessimistic to be useful. Recent advancements in hardware architecture
204have been
205focused on the improvement of the average case performance, not the
206predictability of the worst case. Execution time is becoming increasingly
207dependent on execution history and the internal state of
208hardware components like pipelines and caches. Multi-core processors and non-uniform
209memory models are drastically reducing the possibility of performing
210static analysis in isolation, because programs are less and less time
211composable. Clock-precise hardware models are necessary for static analysis, and
212obtaining them is becoming harder due to the increased sophistication
213of hardware design.
215Despite the aforementioned problems, the need for reliable real time
216systems and programs is increasing, and there is increasing pressure
217from the research community towards the introduction of alternative
218hardware with more predictable behaviour, which would be more suitable
219for static analysis.  One example, being investigated by the Proartis
220project~\cite{proartis}, is to decouple execution time from execution
221history by introducing randomisation.
223In the CerCo project \cite{cerco} we do not try to address this problem, optimistically
224assuming that improvements in low-level timing analysis or architecture will make
225verification feasible in the longer term. Instead, the main objective of our work is
226to bring together the static analysis of functional and non-functional
227properties, which, according to the current state of the art, are completely
228independent activities with limited exchange of information: while the
229functional properties are verified on the source code, the analysis of
230non-functional properties is entirely performed on the object code to exploit
231clock-precise hardware models.
233\subsection{Current object-code methods}
235Analysis currently takes place on object code for two main reasons.
236First, there cannot be a uniform, precise cost model for source
237code instructions (or even basic blocks). During compilation, high level
238instructions are torn apart and reassembled in context-specific ways so that
239identifying a fragment of object code and a single high level instruction is
240infeasible. Even the control flow of the object and source code can be very
241different as a result of optimisations, for example aggressive loop
242optimisations may completely transform source level loops. Despite the lack of a uniform, compilation- and
243program-independent cost model on the source language, the literature on the
244analysis of non-asymptotic execution time on high level languages that assumes
245such a model is growing and gaining momentum. However, unless we provide a
246replacement for such cost models, this literature's future practical impact looks
247to be minimal. Some hope has been provided by the EmBounded project \cite{embounded}, which
248compositionally compiles high-level code to a byte code that is executed by an
249interpreter with guarantees on the maximal execution time spent for each byte code
250instruction. This provides a uniform model at the expense of the model's
251precision (each cost is a pessimistic upper bound) and the performance of the
252executed code (because the byte code is interpreted compositionally instead of
253performing a fully non-compositional compilation).
255The second reason to perform the analysis on the object code is that bounding
256the worst case execution time of small code fragments in isolation (e.g. loop
257bodies) and then adding up the bounds yields very poor estimates as no
258knowledge of the hardware state prior to executing the fragment can be assumed. By
259analysing longer runs the bound obtained becomes more precise because the lack
260of information about the initial state has a relatively small impact.
262To calculate the cost of an execution, value and control flow analyses
263are required to bound the number of times each basic block is
264executed.  Current state
265of the art WCET technology such as AbsInt's aiT analysis tools~\cite{absint} performs these analyses on the
266object code, where the logic of the program is harder to reconstruct
267and most information available at the source code level has been lost;
268see~\cite{stateart} for a survey. Imprecision in the analysis can lead to useless bounds. To
269augment precision, the tools ask the user to provide constraints on
270the object code control flow, usually in the form of bounds on the
271number of iterations of loops or linear inequalities on them. This
272requires the user to manually link the source and object code,
273translating his assumptions on the source code (which may be wrong) to
274object code constraints. The task is error prone and hard, especially
275in the presence of complex compiler optimisations.
277Traditional techniques for WCET that work on object code are also affected by
278another problem: they cannot be applied before the generation of the object
279code. Functional properties can be analysed in early development stages, while
280analysis of non-functional properties may come too late to avoid expensive
281changes to the program architecture.
283\subsection{CerCo's approach}
285In CerCo we propose a radically new approach to the problem: we reject the idea
286of a uniform cost model and we propose that the compiler, which knows how the
287code is translated, must return the cost model for basic blocks of high level
288instructions. It must do so by keeping track of the control flow modifications
289to reverse them and by interfacing with processor timing analysis.
291By embracing compilation, instead of avoiding it like EmBounded did, a CerCo
292compiler can both produce efficient code and return costs that are
293as precise as the processor timing analysis can be. Moreover, our costs can be
294parametric: the cost of a block can depend on actual program data, on a
295summary of the execution history, or on an approximated representation of the
296hardware state. For example, loop optimisations may assign a cost to a loop body
297that is a function of the number of iterations performed. As another example,
298the cost of a block may be a function of the vector of stalled pipeline states,
299which can be exposed in the source code and updated at each basic block exit. It
300is parametricity that allows one to analyse small code fragments without losing
301precision: in the analysis of the code fragment we do not have to ignore the
302initial hardware state. On the contrary, we can assume that we know exactly which
303state (or mode, as the WCET literature calls it) we are in.
305The CerCo approach has the potential to dramatically improve the state of the
306art. By performing control and data flow analyses on the source code, the error
307prone translation of invariants is completely avoided. Instead, this
308work is done at the source level using tools of the user's choice.
309Any available technique for the verification of functional properties
310can be immediately reused and multiple techniques can collaborate together to
311infer and certify cost invariants for the program.  There are no
312limitations on the types of loops or data structures involved. Parametric cost analysis
313becomes the default one, with non-parametric bounds used as a last resort when
314trading the complexity of the analysis with its precision. \emph{A priori}, no
315technique previously used in traditional WCET is lost: processor
316timing analyses can be used by the compiler on the object code, and the rest can be applied
317at the source code level.
318 Our approach can also work in the early
319stages of development by axiomatically attaching costs to unimplemented components.
322All software used to verify properties of programs must be as bug free as
323possible. The trusted code base for verification consists of the code that needs
324to be trusted to believe that the property holds. The trusted code base of
325state-of-the-art WCET tools is very large: one needs to trust the control flow
326analyser and the linear programming libraries it uses and also the formal models
327of the hardware under analysis. In CerCo we are moving the control flow analysis to the source
328code and we are introducing a non-standard compiler too. To reduce the trusted
329code base, we implemented a prototype and a static analyser in an interactive
330theorem prover, which was used to certify that the costs added to the source
331code are indeed those incurred by the hardware. Formal models of the
332hardware and of the high level source languages were also implemented in the
333interactive theorem prover. Control flow analysis on the source code has been
334obtained using invariant generators, tools to produce proof obligations from
335generated invariants and automatic theorem provers to verify the obligations. If
336the automatic provers are able to generate proof traces that can be
337independently checked, the only remaining component that enters the trusted code
338base is an off-the-shelf invariant generator which, in turn, can be proved
339correct using an interactive theorem prover. Therefore we achieve the double
340objective of allowing the use of more off-the-shelf components (e.g. provers and
341invariant generators) whilst reducing the trusted code base at the same time.
343%\paragraph{Summary of the CerCo objectives.} To summarize, the goal of CerCo is
344% to reconcile functional with non-functional analysis by performing them together
345% on the source code, sharing common knowledge about execution invariants. We want
346% to achieve the goal implementing a new generation of compilers that induce a
347% parametric, precise cost model for basic blocks on the source code. The compiler
348% should be certified using an interactive theorem prover to minimize the trusted
349% code base of the analysis. Once the cost model is induced, off-the-shelf tools
350% and techniques can be combined together to infer and prove parametric cost
351% bounds.
352%The long term benefits of the CerCo vision are expected to be:
353%1. the possibility to perform static analysis during early development stages
354%2.  parametric bounds made easier
355%3.  the application of off-the-shelf techniques currently unused for the
356% analysis of non-functional properties, like automated proving and type systems
357%4. simpler and safer interaction with the user, that is still asked for
358% knowledge, but on the source code, with the additional possibility of actually
359% verifying the provided knowledge
360%5. a reduced trusted code base
361%6. the increased accuracy of the bounds themselves.
363%The long term success of the project is hindered by the increased complexity of
364% the static prediction of the non-functional behaviour of modern hardware. In the
365% time frame of the European contribution we have focused on the general
366% methodology and on the difficulties related to the development and certification
367% of a cost model inducing compiler.
369\section{The typical CerCo workflow}
374char a[] = {3, 2, 7, 14};
375char threshold = 4;
377int count(char *p, int len) {
378  char j;
379  int found = 0;
380  for (j=0; j < len; j++) {
381    if (*p <= threshold)
382      found++;
383    p++;
384    }
385  return found;
388int main() {
389  return count(a,4);
393%  $\vcenter{\includegraphics[width=7.5cm]{interaction_diagram.pdf}}$
395    baseline={([yshift=-.5ex]current bounding box)},
396    element/.style={draw, text width=1.6cm, on chain, text badly centered},
397    >=stealth
398    ]
399{[start chain=going below, node distance=.5cm]
400\node [element] (cerco) {CerCo\\compiler};
401\node [element] (cost)  {CerCo\\cost plugin};
402{[densely dashed]
403\node [element] (ded)   {Deductive\\platform};
404\node [element] (check) {Proof\\checker};
407\coordinate [left=3.5cm of cerco] (left);
408{[every node/.style={above, text width=3.5cm, text badly centered,
409                     font=\scriptsize}]
410\draw [<-] ([yshift=-1ex]cerco.north west) coordinate (t) --
411    node {C source}
412    (t-|left);
413\draw [->] (cerco) -- (cost);
414\draw [->] ([yshift=1ex]cerco.south west) coordinate (t) --
415    node {C source+\color{red}{cost annotations}}
416    (t-|left) coordinate (cerco out);
417\draw [->] ([yshift=1ex]cost.south west) coordinate (t) --
418    node {C source+\color{red}{cost annotations}\\+\color{blue}{synthesized assertions}}
419    (t-|left) coordinate (out);
420{[densely dashed]
421\draw [<-] ([yshift=-1ex]ded.north west) coordinate (t) --
422    node {C source+\color{red}{cost annotations}\\+\color{blue}{complexity assertions}}
423    (t-|left) coordinate (ded in) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
424\draw [->] ([yshift=1ex]ded.south west) coordinate (t) --
425    node {complexity obligations}
426    (t-|left) coordinate (out);
427\draw [<-] ([yshift=-1ex]check.north west) coordinate (t) --
428    node {complexity proof}
429    (t-|left) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
430\draw [dash phase=2.5pt] (cerco out) .. controls +(-1, 0) and +(-1, 0) ..
431    (ded in);
433%% user:
434% head
435\draw (current bounding box.west) ++(-.2,.5)
436    circle (.2) ++(0,-.2) -- ++(0,-.1) coordinate (t);
437% arms
438\draw (t) -- +(-.3,-.2);
439\draw (t) -- +(.3,-.2);
440% body
441\draw (t) -- ++(0,-.4) coordinate (t);
442% legs
443\draw (t) -- +(-.2,-.6);
444\draw (t) -- +(.2,-.6);
447\caption{On the left: code to count the array's elements
448 that are less than or equal to the threshold. On the right: CerCo's interaction
449 diagram. CerCo's components are drawn solid.}
452We illustrate the workflow we envisage (on the right
453of~\autoref{test1}) on an example program (on the left
454of~\autoref{test1}).  The user writes the program and feeds it to the
455CerCo compiler, which outputs an instrumented version of the same
456program that updates global variables that record the elapsed
457execution time and the stack space usage.  The red lines in
458\autoref{itest1} introducing variables, functions and function calls
459starting with \lstinline'__cost' and \lstinline'__stack' are the instrumentation introduced by the
460compiler.  For example, the two calls at the start of
461\lstinline'count' say that 4 bytes of stack are required, and that it
462takes 111 cycles to reach the next cost annotation (in the loop body).
463The compiler measures these on the labelled object code that it generates.
465 The annotated program can then be enriched with complexity
466assertions in the style of Hoare logic, that are passed to a deductive
467platform (in our case Frama-C). We provide as a Frama-C cost plugin a
468simple automatic synthesiser for complexity assertions which can
469be overridden by the user to increase or decrease accuracy.  These are the blue
470comments starting with \lstinline'/*@' in \autoref{itest1}, written in
471Frama-C's specification language, ACSL.  From the
472assertions, a general purpose deductive platform produces proof
473obligations which in turn can be closed by automatic or interactive
474provers, ending in a proof certificate.
476% NB: if you try this example with the live CD you should increase the timeout
478Twelve proof obligations are generated from~\autoref{itest1} (to prove
479that the loop invariant holds after one execution if it holds before,
480to prove that the whole program execution takes at most 1358 cycles,
481etc.).  Note that the synthesised time bound for \lstinline'count',
482$178+214*(1+\text{\lstinline'len'})$ cycles, is parametric in the length of
483the array. The CVC3 prover
484closes all obligations within half a minute on routine commodity
485hardware.  A simpler non-parametric version can be solved in a few
490        moredelim=[is][\color{blue}]{$}{$},
491        moredelim=[is][\color{red}]{|}{|},
492        lineskip=-2pt}
495|int __cost = 33, __stack = 5, __stack_max = 5;|
496|void __cost_incr(int incr) { __cost += incr; }|
497|void __stack_incr(int incr) {
498  __stack += incr;
499  __stack_max = __stack_max < __stack ? __stack : __stack_max;
502char a[4] = {3, 2, 7, 14};  char threshold = 4;
504/*@ behaviour stack_cost:
505      ensures __stack_max <= __max(\old(__stack_max), 4+\old(__stack));
506      ensures __stack == \old(__stack);
507    behaviour time_cost:
508      ensures __cost <= \old(__cost)+(178+214*__max(1+\at(len,Pre), 0));
510int count(char *p, int len) {
511  char j;  int found = 0;
512  |__stack_incr(4);  __cost_incr(111);|
513  $__l: /* internal */$
514  /*@ for time_cost: loop invariant
515        __cost <= \at(__cost,__l)+
516                  214*(__max(\at((len-j)+1,__l), 0)-__max(1+(len-j), 0));
517      for stack_cost: loop invariant
518        __stack_max == \at(__stack_max,__l);
519      for stack_cost: loop invariant
520        __stack == \at(__stack,__l);
521      loop variant len-j;
522  */
523  for (j = 0; j < len; j++) {
524    |__cost_incr(78);|
525    if (*p <= threshold) { |__cost_incr(136);| found ++; }
526    else { |__cost_incr(114);| }
527    p ++;
528  }
529  |__cost_incr(67);  __stack_incr(-4);|
530  return found;
533/*@ behaviour stack_cost:
534      ensures __stack_max <= __max(\old(__stack_max), 6+\old(__stack));
535      ensures __stack == \old(__stack);
536    behaviour time_cost:
537      ensures __cost <= \old(__cost)+1358;
539int main(void) {
540  int t;
541  |__stack_incr(2);  __cost_incr(110);|
542  t = count(a,4);
543  |__stack_incr(-2);|
544  return t;
547\caption{The instrumented version of the program in \autoref{test1},
548 with instrumentation added by the CerCo compiler in red and cost invariants
549 added by the CerCo Frama-C plugin in blue. The \texttt{\_\_cost},
550 \texttt{\_\_stack} and \texttt{\_\_stack\_max} variables hold the elapsed time
551in clock cycles and the current and maximum stack usage. Their initial values
552hold the clock cycles spent in initialising the global data before calling
553\texttt{main} and the space required by global data (and thus unavailable for
554the stack).
558\section{Main scientific and technical results}
559First we describe the basic labelling approach and our compiler
560implementations that use it.  This is suitable for basic architectures
561with simple cost models.  Then we will discuss the dependent labelling
562extension which is suitable for more advanced processor architectures
563and compiler optimisations.  At the end of this section we will
564demonstrate automated high level reasoning about the source level
565costs provided by the compilers.
567% \emph{Dependent labelling~\cite{?}.} The basic labelling approach assumes a
568% bijective mapping between object code and source code O(1) blocks (called basic
569% blocks). This assumption is violated by many program optimizations (e.g. loop
570% peeling and loop unrolling). It also assumes the cost model computed on the
571% object code to be non parametric: every block must be assigned a cost that does
572% not depend on the state. This assumption is violated by stateful hardware like
573% pipelines, caches, branch prediction units. The dependent labelling approach is
574% an extension of the basic labelling approach that allows to handle parametric
575% cost models. We showed how the method allows to deal with loop optimizations and
576% pipelines, and we speculated about its applications to caches.
578% \emph{Techniques to exploit the induced cost model.} Every technique used for
579% the analysis of functional properties of programs can be adapted to analyse the
580% non-functional properties of the source code instrumented by compilers that
581% implement the labelling approach. In order to gain confidence in this claim, we
582% showed how to implement a cost invariant generator combining abstract
583% interpretation with separation logic ideas \cite{separation}. We integrated
584% everything in the Frama-C modular architecture, in order to compute proof
585% obligations from functional and cost invariants and to use automatic theorem
586% provers on them. This is an example of a new technique that is not currently
587% exploited in WCET analysis. It also shows how precise functional invariants
588% benefits the non-functional analysis too. Finally, we show how to fully
589% automatically analyse the reaction time of Lustre nodes that are first compiled
590% to C using a standard Lustre compiler and then processed by a C compiler that
591% implements the labelling approach.
593% \emph{The CerCo compiler.} This is a compiler from a large subset of the C
594% program to 8051/8052 object code,
595% integrating the labelling approach and a static analyser for 8051 executables.
596% The latter can be implemented easily and does not require dependent costs
597% because the 8051 microprocessor is a very simple one, with constant-cost
598% instruction. It was chosen to separate the issue of exact propagation of the
599% cost model from the orthogonal problem of the static analysis of object code
600% that may require approximations or dependent costs. The compiler comes in
601% several versions: some are prototypes implemented directly in OCaml, and they
602% implement both the basic and dependent labelling approaches; the final version
603% is extracted from a Matita certification and at the moment implements only the
604% basic approach.
606\subsection{The (basic) labelling approach}
607The labelling approach is the foundational insight that underlies all the developments in CerCo.
608It facilitates the tracking of basic block evolution during the compilation process in order to propagate the cost model from the
609object code to the source code without losing precision in the process.
610\paragraph{Problem statement.} Given a source program $P$, we want to obtain an
611instrumented source program $P'$,  written in the same programming language, and
612the object code $O$ such that: 1) $P'$ is obtained by inserting into $P$ some
613additional instructions to update global cost information like the amount of
614time spent during execution or the maximal stack space required; 2) $P$ and $P'$ 
615must have the same functional behaviour, i.e.\ they must produce that same output
616and intermediate observables; 3) $P$ and $O$ must have the same functional
617behaviour; 4) after execution and in interesting points during execution, the
618cost information computed by $P'$ must be an upper bound of the one spent by $O$ 
619to perform the corresponding operations (\emph{soundness} property); 5) the difference
620between the costs computed by $P'$ and the execution costs of $O$ must be
621bounded by a program-dependent constant (\emph{precision} property).
623\paragraph{The labelling software components.} We solve the problem in four
624stages \cite{labelling}, implemented by four software components that are used
625in sequence.
627The first component labels the source program $P$ by injecting label emission
628statements in appropriate positions to mark the beginning of basic blocks.
629These are the positions where the cost instrumentation will appear in the final output.
630% The
631% set of labels with their positions is called labelling.
632The syntax and semantics
633of the source programming language is augmented with label emission statements.
634The statement ``EMIT $\ell$'' behaves like a NOP instruction that does not affect the
635program state or control flow, but its execution is observable.
636% Therefore the observables of a run of a program becomes a stream
637% of label emissions: $\ell_1,\ldots,\ell_n$, called the program trace. We clarify the conditions
638% that the labelling must respect later.
639For the example in Section~\ref{sec:workflow} this is just the original C code
640with ``EMIT'' instructions added at every point a \lstinline'__cost_incr' call
641appears in the final code.
643The second component is a labelling preserving compiler. It can be obtained from
644an existing compiler by adding label emission statements to every intermediate
645language and by propagating label emission statements during compilation. The
646compiler is correct if it preserves both the functional behaviour of the program
647and the traces of observables, including the labels `emitted'.
648% We may also ask that the function that erases the cost
649% emission statements commute with compilation. This optional property grants that
650% the labelling does not interfere with the original compiler behaviour. A further
651% set of requirements will be added later.
653The third component analyses the labelled
654object code to compute the scope of each of its label emission statements,
655i.e.\ the instructions that may be executed after the statement and before
656a new label emission is encountered, and then computes the maximum cost of
657each.  Note that we only have enough information at this point to compute
658the cost of loop-free portions of code.  We will consider how to ensure that
659every loop is broken by a cost label shortly.
660%It is a tree and not a sequence because the scope
661%may contain a branching statement. In order to grant that such a finite tree
662%exists, the object code must not contain any loop that is not broken by a label
663%emission statement. This is the first requirement of a sound labelling. The
664%analyser fails if the labelling is unsound. For each scope, the analyser
665%computes an upper bound of the execution time required by the scope, using the
666%maximum of the costs of the two branches in case of a conditional statement.
667%Finally, the analyser computes the cost of a label by taking the maximum of the
668%costs of the scopes of every statement that emits that label.
670The fourth and final component replaces the labels in the labelled
671version of the source code produced at the start with the costs
672computed for each label's scope.  This yields the instrumented source
673code.  For the example, this is the code in \autoref{itest1}, except
674for the specifications in comments, which we consider in
677\paragraph{Correctness.} Requirements 1 and 2 hold because of the
678non-invasive labelling procedure.  Requirement 3 can be satisfied by
679implementing compilation correctly.  It is obvious that the value of
680the global cost variable of the instrumented source code is always
681equal to the sum of the costs of the labels emitted by the
682corresponding labelled code. Moreover, because the compiler preserves
683all traces, the sum of the costs of the labels emitted in the source
684and target labelled code are the same. Therefore, to satisfy the
685soundness requirement, we need to ensure that the time taken to
686execute the object code is equal to the sum of the costs of the labels
687emitted by the object code. We collect all the necessary conditions
688for this to happen in the definition of a \emph{sound} labelling: a)
689all loops must be broken by a cost emission statement; b) all program
690instructions must be in the scope of some cost emission statement.
691This ensures that every label's scope is a tree of instructions, with
692the cost being the most expensive path. To satisfy also the precision
693requirement, we must make the scopes flat sequences of instructions.
694We require a \emph{precise} labelling where every label is emitted at
695most once and both branches of each conditional jump start with a label
696emission statement.
698The correctness and precision of the labelling approach only rely on the
699correctness and precision of the object code labelling. The simplest
700way to achieve that is to impose correctness and precision
701requirements on the source code labelling produced at the start, and
702to demand that the compiler preserves these
703properties too. The latter requirement imposes serious limitations on the
704compilation strategy and optimisations: the compiler may not duplicate any code
705that contains label emission statements, like loop bodies. Therefore various
706loop optimisations like peeling or unrolling are prevented. Moreover, precision
707of the object code labelling is not sufficient \emph{per se} to obtain global
708precision: we implicitly assumed that a precise constant cost can be assigned
709to every instruction. This is not possible in
710the presence of stateful hardware whose state influences the cost of operations,
711like pipelines and caches. In Section~\ref{lab:deplabel} we will see an extension of the
712basic labelling approach which tackles these problems.
714In CerCo we have developed several cost preserving compilers based on
715the labelling approach. Excluding an initial certified compiler for a
716`while' language, all remaining compilers target realistic source
717languages---a pure higher order functional language and a large subset
718of C with pointers, \texttt{goto}s and all data structures---and real world
719target processors---MIPS and the Intel 8051/8052 processor
720family. Moreover, they achieve a level of optimisation that ranges
721from moderate (comparable to GCC level 1) to intermediate (including
722loop peeling and unrolling, hoisting and late constant propagation).
723We describe the C compilers in detail in the following section.
725Two compilation chains were implemented for a purely functional
726higher-order language~\cite{labelling2}. The two main changes required
727to deal with functional languages are: 1) because global variables and
728updates are not available, the instrumentation phase produces monadic
729code to `update' the global costs; 2) the requirements for a sound and
730precise labelling of the source code must be changed when the
731compilation is based on CPS translations.  In particular, we need to
732introduce labels emitted before a statement is executed and also
733labels emitted after a statement is executed. The latter capture code
734that is inserted by the CPS translation and that would escape all
735label scopes.
737% Brian: one of the reviewers pointed out that standard Prolog implementations
738% do have some global state that apparently survives backtracking and might be
739% used.  As we haven't experimented with this, I think it's best to elide it
740% entirely.
742% Phases 1, 2 and 3 can be applied as well to logic languages (e.g. Prolog).
743% However, the instrumentation phase cannot: in standard Prolog there is no notion
744% of (global) variable whose state is not retracted during backtracking.
745% Therefore, the cost of executing computations that are later backtracked would
746% not be correctly counted in. Any extension of logic languages with
747% non-backtrackable state could support our labelling approach.
749\subsection{The CerCo C compilers}
750We implemented two C compilers, one implemented directly in OCaml and
751the other implemented in the interactive theorem prover
752Matita~\cite{matita}.  The first acted as a prototype for the second,
753but also supported MIPS and acted as a testbed for more advanced
754features such as the dependent labelling approach
755in Section~\ref{lab:deplabel}.
757The second C compiler is the \emph{Trusted CerCo Compiler}, whose cost
758predictions are formally verified. The executable code
759is OCaml code extracted from the Matita implementation. The Trusted
760CerCo Compiler only targets
761the C language and the 8051/8052 family, and does not yet implement any
762advanced optimisations. Its user interface, however, is the same as the
763other version for interoperability purposes. In
764particular, the Frama-C CerCo plugin descibed in
765Section~\ref{sec:exploit} can work without recompilation
766with both of our C compilers.
768The 8051/8052 microprocessor is a very simple one, with constant-cost
769instructions. It was chosen to separate the issue of exact propagation of the
770cost model from the orthogonal problem of low-level timing analysis of object code
771that may require approximation or dependent costs.
773The (trusted) CerCo compiler implements the following optimisations: cast
774simplification, constant propagation in expressions, liveness analysis driven
775spilling of registers, dead code elimination, branch displacement, and tunnelling.
776The two latter optimisations are performed by our optimising assembler
777\cite{correctness}. The back-end of the compiler works on three address
778instructions, preferred to static single assignment code for the simplicity of
779the formal certification.
781The CerCo compiler is loosely based on the CompCert compiler \cite{compcert}, a
782recently developed certified compiler from C to the PowerPC, ARM and x86
783microprocessors. In contrast to CompCert, both the CerCo code and its
784certification are fully open source. Some data structures and language definitions for
785the front-end are directly taken from CompCert, while the back-end is a redesign
786of a compiler from Pascal to MIPS used by Fran\c{c}ois Pottier for a course at the
787\'{E}cole Polytechnique.
789The main differences in the CerCo compiler are:
791\item All the intermediate languages include label emitting instructions to
792implement the labelling approach, and the compiler preserves execution traces.
793\item Instead of targeting an assembly language with additional
794macro-instructions which are expanded before assembly, we directly
795produce object code in order to perform the timing analysis, using
796an integrated optimising assembler.
797\item In order to avoid the additional work of implementing a linker
798 and a loader, we do not support separate
799compilation and external calls. Adding them is orthogonal
800to the labelling approach and should not introduce extra problems.
801\item We target an 8-bit processor, in contrast to CompCert's 32-bit targets. This requires
802many changes and more compiler code, but it is not fundamentally more
803complex. The proof of correctness, however, becomes much harder.
804\item We target a microprocessor that has a non-uniform memory model, which is
805still often the case for microprocessors used in embedded systems and that is
806becoming common again in multi-core processors. Therefore the compiler has to
807keep track of data and it must move data between memory regions in the proper
808way. Moreover the size of pointers to different regions is not uniform. %An
809%additional difficulty was that the space available for the stack in internal
810%memory in the 8051 is tiny, allowing only a minor number of nested calls. To
811%support full recursion in order to test the CerCo tools also on recursive
812%programs, the compiler implements a stack in external memory.
815\subsection{Formal certification of the CerCo compiler}
816We have formally
817certified in the Matita interactive proof assistant that the cost models induced on the source code by the
818Trusted CerCo Compiler correctly and precisely
819predict the object code behaviour. There are two cost models, one for execution
820time and one for stack space consumption. We show the correctness of the prediction
821only for those programs that do not exhaust the available stack space, a
822property that---thanks to the stack cost model---we can statically analyse on the
823source code in sharp contrast to other certified compilers.  Other projects have
824already certified the preservation of functional semantics in similar compilers,
825so we have not attempted to directly repeat that work and assume
826functional correctness for most passes. In order to complete the
827proof for non-functional properties, we have introduced a new,
828structured, form of execution trace, with the
829related notions for forward similarity and the intensional
830consequences of forward similarity. We have also introduced a unified
831representation for back-end intermediate languages that was exploited to provide
832a uniform proof of forward similarity.
834The details on the proof techniques employed
836the proof sketch can be found in the CerCo deliverables and papers~\cite{cerco-deliverables}.
837In this section we will only hint at the correctness statement, which turned
838out to be more complex than expected.
840\paragraph{The correctness statement.}
841Real time programs are often reactive programs that loop forever responding to
842events (inputs) by performing some computation followed by some action (output)
843and continuing as before. For these programs the overall execution
844time does not make sense. The same is true for reactive programs that spend an
845unpredictable amount of time in I/O. Instead, what is interesting is the reaction time ---
846 the time spent between I/O events. Moreover, we are interested in
847predicting and ruling out crashes due to running out of space on certain
849Therefore we need a statement that talks about sub-runs of a
850program. A natural candidate is that the time predicted on the source
851code and spent on the object code by two corresponding sub-runs are the same.
852To make this statement formal we must identify the
853corresponding sub-runs and how to single out those that are meaningful.
854We introduce the notion of a \emph{measurable} sub-run of a run
855which does not exhaust the available stack before or during the
856sub-run, the number of function calls and returns
857in the sub-run is the same, the sub-run does not perform any I/O, and the sub-run
858starts with a label emission statement and ends with a return or another label
859emission statement. The stack usage is bounded using the stack usage model
860that is computed by the compiler.
862The statement that we formally proved is: for each C run with a measurable
863sub-run, there exists an object code run with a sub-run, with the same
864execution trace for both the prefix of the run and the sub-run itself,
865 and where the time spent by the
866object code in the sub-run is the same as the time predicted on the source code
867using the time cost model generated by the compiler.
870We briefly discuss the constraints for measurability. Not exhausting the stack
871space is necessary for a run to be meaningful, because the source semantics
872has no notion of running out of memory. Balancing
873function calls and returns is a requirement for precision: the labelling approach
874allows the scope of a label to extend after function calls to
875minimize the number of labels. (The scope excludes the called
876function's execution.) If the number of
877calls/returns is unbalanced, it means that there is a call we have not returned
878to that could be followed by additional instructions whose cost has already been
879taken in account.  The last condition on the start and end points of
880a run is also required to make the bound precise.  With these
881restrictions and the 8051's simple timing model we obtain \emph{exact}
882predictions.  If we relax these conditions then we obtain a corollary
883with an upper bound on the cost.
884Finally, I/O operations can be performed in the prefix of the run, but not in the
885measurable sub-run. Therefore we prove that we can predict reaction times, but
886not I/O times, as desired.
888\subsection{Dependent labelling}
890The core idea of the basic labelling approach is to establish a tight connection
891between basic blocks executed in the source and target languages. Once the
892connection is established, any cost model computed on the object code can be
893transferred to the source code, without affecting the code of the compiler or
894its proof. In particular, we can also transport cost models
895that associate to each label a \emph{function} from the hardware state
896to a natural number.
897However, a problem arises during the instrumentation phase that replaces label
898emission statements with increments of global cost variables. They are
899incremented by the result of applying the label's cost function
900to the hardware state at the time of execution of the block. However,
901the hardware state comprises both the functional state that affects the
902computation (the value of the registers and memory) and the non-functional
903state that does not (the pipeline and cache contents, for example).
904We can find corresponding information for the former in the source code state, but constructing the
905correspondence may be hard and lifting the cost model to work on the source code
906state is likely to produce cost expressions that are too complex to understand and reason about.
907Fortunately, in modern architectures the cost of executing single
908instructions is either independent of the functional state or the jitter---the
909difference between the worst and best case execution times---is small enough
910to be bounded without losing too much precision. Therefore we only consider
911dependencies on the `non-functional' parts of the state.
913The non-functional state is not directly related to the high level
914state and does not influence the functional properties. What can be
915done is to expose key aspects of the non-functional state in the
916source code. We present here the basic intuition in a simplified form:
917the technical details that allow us to handle the general case are
918more complex and can be found in~\cite{paolo}. We add to the source
919code an additional global variable that represents the non-functional
920state and another one that remembers the last few labels emitted. The
921state variable must be updated at every label emission statement,
922using an update function which is computed during the processor timing
924This update function assigns to each label a function from the
925recently emitted labels and old state to the new state. It is computed
926by composing the semantics of every instruction in a basic block
927restricted to the non-functional part of the state.
929Not all the details of the non-functional state needs to be exposed, and the
930technique works better when the part of state that is required can be summarised
931in a simple data structure. For example, to handle simple but realistic
932pipelines it is sufficient to remember a short integer that encodes the position
933of bubbles (stuck instructions) in the pipeline. In any case, it is not necessary
934for the user to understand the meaning of the state to reason over the properties
935of the
936program. Moreover, the user, or the invariant generator tools that
937analyse the instrumented source code produced by the compiler, can decide to
938trade precision of the analysis for simplicity by approximating the
939cost by safe bounds that do not depend on the processor state. Interestingly, the functional analysis of
940the code could determine which blocks are executed more frequently in order to
941use more aggressive approximations for those that are executed least.
943Dependent labelling can also be applied to allow the compiler to duplicate
944blocks that contain labels (e.g. in loop optimisations)~\cite{paolo}. The effect is to assign
945a different cost to the different occurrences of a duplicated label. For
946example, loop peeling turns a loop into the concatenation of a copy of the loop
947body for the first iteration and the conditional execution of the
948loop for successive iterations. Further optimisations will compile the two
949copies of the loop differently, with the first body usually
950taking more time.
952By introducing a variable that keeps track of the iteration number, we can
953associate to the label a cost that is a function of the iteration number. The
954same technique works for loop unrolling without modification: the function will
955assign one cost to the even iterations and another cost to the odd
956ones.  The optimisation code
957that duplicates the loop bodies must also modify the code to correctly propagate
958the update of the iteration numbers. The technical details are more complicated and
959can be found in the CerCo reports and publications. The implementation, however,
960is quite simple (and forms part of our OCaml version of the compiler)
961and the changes to a loop optimising compiler are minimal.
963\subsection{Techniques to exploit the induced cost model}
965We now turn our attention to synthesising high-level costs, such as
966the reaction time of a real-time program.  We consider as our starting point source level costs
967provided by basic labelling, in other words annotations
968on the source code which are constants that provide a sound and sufficiently
969precise upper bound on the cost of executing the blocks after compilation to
970object code.
972The principle that we have followed in designing the cost synthesis tools is
973that the synthesised bounds should be expressed and proved within a general
974purpose tool built to reason on the source code. In particular, we rely on the
975Frama-C tool to reason on C code and on the Coq proof-assistant to reason on
976higher-order functional programs.
977This principle entails that
978the inferred synthetic bounds are indeed correct as long as the general purpose
979tool is, and that there is no limitation on the class of programs that can be handled,
980for example by resorting to interactive proof.
982Of course, automation is desirable whenever possible. Within this framework,
983automation means writing programs that give hints to the general purpose tool.
984These hints may take the form, say, of loop invariants/variants, of predicates
985describing the structure of the heap, or of types in a light logic. If these
986hints are correct and sufficiently precise the general purpose tool will produce
987a proof automatically, otherwise, user interaction is required.
989\paragraph{The Cost plugin and its application to the Lustre compiler.}
990Frama-C \cite{framac} is a set of analysers for C programs with a
991specification language called ACSL. New analyses can be dynamically added
992via a plugin system. For instance, the Jessie plugin~\cite{jessie} allows deductive
993verification of C programs with respect to their specification in ACSL, with
994various provers as back-end tools.
995We developed the CerCo Cost plugin for the Frama-C platform as a proof of
996concept of an automatic environment exploiting the cost annotations produced by
997the CerCo compiler. It consists of an OCaml program which essentially
998uses the CerCo compiler to produce a related C program with cost annotations,
999and applies some heuristics to produce a tentative bound on the cost of
1000executing the C functions of the program as a function of the value of their
1001parameters. The user can then call the Jessie plugin to discharge the
1002related proof obligations.
1003In the following we elaborate on the soundness of the framework and the
1004experiments we performed with the Cost tool on C programs, including some produced by a
1005Lustre compiler.
1007\paragraph{Soundness.} The soundness of the whole framework depends on the cost
1008annotations added by the CerCo compiler,
1009the verification conditions (VCs) generated by Jessie, and the
1010external provers discharging the VCs. Jessie can be used to verify the
1011synthesised bounds because our plugin generates them in ACSL format. Thus, even if the added synthetic costs are
1012incorrect (relatively to the cost annotations), the process as a whole is still
1013correct: indeed, Jessie will not validate incorrect costs and no conclusion can
1014be made about the WCET of the program in this case. In other terms, the
1015soundness does not depend on the cost plugin, which can in
1016principle produce any synthetic cost. However, in order to be able to actually
1017prove a WCET of a C function, we need to add correct annotations in a way that
1018Jessie and subsequent automatic provers have enough information to deduce their
1019validity. In practice this is not straightforward even for very simple programs
1020composed of branching and assignments (no loops and no recursion) because a fine
1021analysis of the VCs associated with branching may lead to a complexity blow up.
1023\paragraph{Experience with Lustre.} Lustre~\cite{lustre} is a data-flow language for programming
1024synchronous systems, with a compiler which targets C. We designed a
1025wrapper for supporting Lustre files.
1026The C function produced by the compiler is relatively simple loop-free
1027code which implements the step function of the
1028synchronous system and computing the WCET of the function amounts to obtaining a
1029bound on the reaction time of the system. We tested the Cost plugin and the
1030Lustre wrapper on the C programs generated by the Lustre compiler. For programs
1031consisting of a few hundred lines of code, the cost plugin computes a
1032WCET and Alt-Ergo is able to discharge all VCs automatically.
1034\paragraph{Handling C programs with simple loops.}
1035The cost annotations added by the CerCo compiler take the form of C instructions
1036that update a fresh global variable called the cost variable by a constant.
1037Synthesizing a WCET of a C function thus consists of statically resolving an
1038upper bound of the difference between the value of the cost variable before and
1039after the execution of the function, i.e. finding the instructions
1040that update the cost variable and establish the number of times they are passed
1041through during the flow of execution. To perform the analysis the plugin
1042assumes that there are no recursive functions in the program, and that
1043every loop is annotated with a variant. In the case of `for' loops the
1044variants are automatically inferred where a loop counter can be
1045syntactically detected.
1047The plugin computes a call-graph and proceeds to calculate bounds for
1048each function from the leaves up to the main function.
1049The computation of the cost of each function is performed by traversing its
1050control flow graph, where the cost of a node is the maximum of the
1051costs of the successors.
1052In the case of a loop with a body that has a constant cost for every step of the
1053loop, the cost is the product of the cost of the body and of the variant taken
1054at the start of the loop.
1055In the case of a loop with a body whose cost depends on the values of some free
1056variables, a fresh logic function $f$ is introduced to represent the cost of the
1057loop in the logic assertions. This logic function takes the variant as a first
1058parameter. The other parameters of $f$ are the free variables of the body of the
1059loop. An axiom is added to account for the fact that the cost is accumulated at each
1060step of the loop.
1061The cost of the function is directly added as post-condition of the function.
1063The user can also specify more precise variants and annotate functions
1064with their own cost specifications. The plugin will use these
1065instead of computing its own, allowing greater precision and the
1066ability to analyse programs which the variant generator does not
1069In addition to the loop-free Lustre code, this method was successfully
1070applied to a small range of cryptographic code.  See~\cite{labelling}
1071for more details.  The example in Section~\ref{sec:workflow} was also
1072produced using the plug-in.  The variant was calculated automatically
1073by noticing that \lstinline'j' is a loop counter with maximum value
1074\lstinline'len'.  The most expensive path through the loop body
1075($78+136 = 214$) is then multiplied by the number of iterations to
1076give the cost of the loop.
1078\paragraph{C programs with pointers.}
1079When it comes to verifying programs involving pointer-based data structures,
1080such as linked lists, trees, or graphs, the use of traditional first-order logic
1081to specify, and of SMT solvers to verify, shows some limitations. Separation
1082logic is an elegant alternative. It is a program logic
1083with a new notion of conjunction to express spatial heap separation. Bobot has
1084recently introduced automatically generated separation
1085predicates to simulate separation logic reasoning in the Jessie plugin where the specification language, the verification condition
1086generator, and the theorem provers were not designed with separation logic in
1087mind~\cite{bobot}. CerCo's plugin can exploit the separation predicates to automatically
1088reason on the cost of execution of simple heap manipulation programs such as an
1089in-place list reversal.
1091\section{Conclusions and future work}
1093All CerCo software and deliverables may be found on the project homepage~\cite{cerco}.
1095The results obtained so far are encouraging and provide evidence that
1096it is possible to perform static time and space analysis at the source level
1097without losing accuracy, reducing the trusted code base and reconciling the
1098study of functional and non-functional properties of programs. The
1099techniques introduced seem to be scalable, cover both imperative and
1100functional languages and are compatible with every compiler optimisation
1101considered by us so far.
1103To prove that compilers can keep track of optimisations
1104and induce a precise cost model on the source code, we targeted a simple
1105architecture that admits a cost model that is execution history independent.
1106The most important future work is dealing with hardware architectures
1107characterised by history-dependent stateful components, like caches and
1108pipelines. The main issue is to assign a parametric, dependent cost
1109to basic blocks that can be later transferred by the labelling approach to
1110the source code and represented in a meaningful way to the user. The dependent
1111labelling approach that we have studied seems a promising tool to achieve
1112this goal, but more work is required to provide good source level
1113approximations of the relevant processor state.
1115Other examples of future work are to improve the cost invariant
1116generator algorithms and the coverage of compiler optimisations, to combining
1117the labelling approach with the type and effect discipline of~\cite{typeffects}
1118to handle languages with implicit memory management, and to experiment with
1119our tools in the early phases of development. Larger case studies are also necessary
1120to evaluate the CerCo's prototype on realistic, industrial-scale programs.
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