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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 a small trusted code base. We developed a C compiler, verified in Matita, that translates source code to object code, producing a copy of the source code embellished with cost annotations as an additional side-effect. These annotations expose and track precisely the actual (non-asymptotic)
101computational cost of the input program at
102the source level. 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.
126This analysis on low-level object code has several problems:
129It can be hard to deduce the high-level structure of the program after compiler optimisations.
130The object code produced by an optimising compiler may have radically different control flow to the original source code program.
132Techniques that operate on object code are not useful early in the development process of a program, yet problems with a program's design or implementation are cheaper to resolve earlier in the process, rather than later.
134Parametric cost analysis is very hard: how can we reflect a cost that depends on the execution state, for example the
135value of a register or a carry bit, to a cost that the user can understand
136looking at the source code?
138Performing 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.
140\paragraph{Vision and approach.}
141We wish to reconcile functional and
142non-functional analyses: to share information and perform both at the same time
143on high-level source code.
145What has previously prevented this approach is the lack of a uniform and precise
146cost model for high-level code as each statement occurrence is compiled
147differently, optimisations may change control flow, and the cost of an object
148code instruction may depend on the runtime state of hardware components like
149pipelines and caches, all of which are not visible in the source code.
151We envision a new generation of compilers that track program structure through compilation and optimisation and exploit this
152information to define a precise, non-uniform cost model for source code that accounts for runtime state. With such a cost model we can
153reduce non-functional verification to the functional case and exploit the state
154of the art in automated high-level verification~\cite{survey}. The techniques
155currently used by the Worst Case Execution Time (WCET) community, who perform analyses on object code,
156are still available but can be coupled with additional source-level
157analyses. Where our approach produces overly complex cost models, safe approximations can be used to trade complexity with precision.
158Finally, source code analysis can be used early in the development process, when
159components have been specified but not implemented, as modularity means
160that it is enough to specify the non-functional behaviour of missing
163We have developed \emph{the labelling approach}~\cite{labelling}, a
164technique to implement compilers that induce cost models on source programs by
165very lightweight tracking of code changes through compilation. We have studied
166how to formally prove the correctness of compilers implementing this technique, and
167have implemented such a compiler from C to object binaries for the MCS-51
168microcontroller for predicting execution time and stack space usage,
169verifying it in an interactive theorem prover.  As we are targeting
170an embedded microcontroller we do not consider dynamic memory allocation.
172To demonstrate source-level verification of costs we have implemented
173a Frama-C plugin~\cite{framac} that invokes the compiler on a source
174program and uses it to generate invariants on the high-level source
175that correctly model low-level costs. The plugin certifies that the
176program respects these costs by calling automated theorem provers, a
177new and innovative technique in the field of cost analysis. Finally,
178we have conducted several case studies, including showing that the
179plugin can automatically compute and certify the exact reaction time
180of Lustre~\cite{lustre} data flow programs compiled into C.
182\section{Project context and approach}
183Formal methods for the verification of functional properties of programs are now mature enough to find increasing application in production environments. For safety critical code, it is commonplace to combine rigorous engineering methodologies and testing
184with static analyses, taking the strengths of each and mitigating
185their weaknesses. Of particular interest are open frameworks
186for the combination of different formal methods, where the programs can be
187progressively specified and enriched with new safety
188guarantees: every method contributes knowledge (e.g. new invariants) that
189becomes an assumption for later analysis.
191The outlook for verifying non-functional properties of programs (time spent,
192memory used, energy consumed) is bleaker.
193% and the future seems to be getting even worse.
194Most industries verify that real time systems meet their deadlines
195by simply performing many runs of the system and timing their execution,  computing the
196maximum time and adding an empirical safety margin, claiming the result to be a
197bound for the WCET of the program. Formal methods and software to statically
198analyse the WCET of programs exist, but they often produce bounds that are too
199pessimistic to be useful. Recent advancements in hardware architecture
200have been
201focused on the improvement of the average case performance, not the
202predictability of the worst case. Execution time is becoming increasingly
203dependent on execution history and the internal state of
204hardware components like pipelines and caches. Multi-core processors and non-uniform
205memory models are drastically reducing the possibility of performing
206static analysis in isolation, because programs are less and less time
207composable. Clock-precise hardware models are necessary for static analysis, and
208obtaining them is becoming harder due to the increased sophistication
209of hardware design.
211Despite these problems, the need for reliable real time
212systems and programs is increasing, and there is a push
213from the research community towards the introduction of
214hardware with more predictable behaviour, which would be more suitable
215for static analysis.  One example, exemplified by the Proartis
216project~\cite{proartis}, is to decouple execution time from execution
217history by introducing randomisation.
219In CerCo~\cite{cerco} we do not address this problem, optimistically
220assuming that improvements in low-level timing analysis or architecture will make
221verification feasible in the longer term. Instead, the main objective of our work is
222to bring together the static analysis of functional and non-functional
223properties, which in the current state of the art are
224independent activities with limited exchange of information: while the
225functional properties are verified on the source code, the analysis of
226non-functional properties is performed on object code to exploit
227clock-precise hardware models.
229\subsection{Current object-code methods}
231Analysis currently takes place on object code for two main reasons.
232First, there cannot be a uniform, precise cost model for source
233code instructions (or even basic blocks). During compilation, high level
234instructions are disassembled and reassembled in context-specific ways so that
235identifying a fragment of object code and a single high level instruction is
236infeasible. Even the control flow of the object and source code can be very
237different as a result of optimisations, for example aggressive loop
238optimisations may completely transform source level loops. Despite the lack of a uniform, compilation- and
239program-independent cost model on the source language, the literature on the
240analysis of non-asymptotic execution time on high level languages assuming
241such a model is growing and gaining momentum. However, unless we provide a
242replacement for such cost models, this literature's future practical impact looks
243to be minimal. Some hope has been provided by the EmBounded project \cite{embounded}, which
244compositionally compiles high-level code to a byte code that is executed by an
245interpreter with guarantees on the maximal execution time spent for each byte code
246instruction. This provides a uniform model at the expense of the model's
247precision (each cost is a pessimistic upper bound) and the performance of the
248executed code (because the byte code is interpreted compositionally instead of
249performing a fully non-compositional compilation).
251The second reason to perform the analysis on the object code is that bounding
252the worst case execution time of small code fragments in isolation (e.g. loop
253bodies) and then adding up the bounds yields very poor estimates as no
254knowledge of the hardware state prior to executing the fragment can be assumed. By
255analysing longer runs the bound obtained becomes more precise because the lack
256of information about the initial state has a relatively small impact.
258To calculate the cost of an execution, value and control flow analyses
259are required to bound the number of times each basic block is
260executed.  Currently, state
261of the art WCET analysis tools, such as AbsInt's aiT toolset~\cite{absint}, perform these analyses on
262object code, where the logic of the program is harder to reconstruct
263and most information available at the source code level has been lost;
264see~\cite{stateart} for a survey. Imprecision in the analysis can lead to useless bounds. To
265augment precision, the tools ask the user to provide constraints on
266the object code control flow, usually in the form of bounds on the
267number of iterations of loops or linear inequalities on them. This
268requires the user to manually link the source and object code,
269translating his assumptions on the source code (which may be wrong) to
270object code constraints. The task is error prone and hard, especially
271in the presence of complex compiler optimisations.
273Traditional techniques for WCET that work on object code are also affected by
274another problem: they cannot be applied before the generation of the object
275code. Functional properties can be analysed in early development stages, while
276analysis of non-functional properties may come too late to avoid expensive
277changes to the program architecture.
279\subsection{CerCo's approach}
281In CerCo we propose a new approach to the problem: we abandon the idea
282of a uniform cost model and we propose that the compiler, which knows how the
283code is translated, must return the cost model for basic blocks of high level
284instructions. It must do so by keeping track of the control flow modifications
285to reverse them and by interfacing with processor timing analysis.
287By embracing the knowledge implicit in the compilation process, instead of avoiding it like EmBounded did, a CerCo
288compiler can both produce efficient code and return costs that are
289as precise as the processor timing analysis can be. Moreover, our costs can be
290parametric: the cost of a block can depend on actual program data, on a
291summary of the execution history, or on an approximated representation of the
292hardware state. For example, loop optimisations may assign a cost to a loop body
293that is a function of the number of iterations performed. As another example,
294the cost of a block may be a function of the vector of stalled pipeline states,
295which can be exposed in the source code and updated at each basic block exit. It
296is parametricity that allows one to analyse small code fragments without losing
297precision. In the analysis of the code fragment we do not have to ignore the
298initial hardware state, rather, we may assume that we know exactly which
299state (or mode, as the WCET literature calls it) we are in.
301The CerCo approach has the potential to substantially improve upon the current state of the
302art. By performing control and data flow analyses on the source code, the error
303prone translation of invariants is avoided. Instead, this
304work is done at the source level using tools of the user's choice.
305Any available technique for the verification of functional properties
306can be immediately reused and multiple techniques can collaborate together to
307infer and certify cost invariants for the program.  There are no
308limitations on the types of loops or data structures involved. Parametric cost analysis
309becomes the default one, with non-parametric bounds used as a last resort when the user
310decides to trade the complexity of the analysis with its precision. \emph{A priori}, no
311technique previously used in traditional WCET is lost: processor
312timing analyses can be used by the compiler on the object code, and the rest can be applied
313at the source code level.
314 Our approach can also work in the early
315stages of development by axiomatically attaching costs to unimplemented components.
318We take as granted the position that all software used to verify properties of programs must be as bug free as
319possible. The trusted code base for verification consists of the code that needs
320to be trusted to believe that the property holds. The trusted code base of
321state-of-the-art WCET tools is very large: one needs to trust the control flow
322analyser, the linear programming libraries used, and also the formal models
323of the hardware under analysis, for example. In CerCo we are moving the control flow analysis to the source
324code and we are introducing a non-standard compiler too. To reduce the trusted
325code base, we implemented a prototype and a static analyser in an interactive
326theorem prover, which was used to certify that the costs added to the source
327code are indeed those incurred by the hardware. Formal models of the
328hardware and of the high level source languages were also implemented in the
329interactive theorem prover. Control flow analysis on the source code has been
330obtained using invariant generators, tools to produce proof obligations from
331generated invariants and automatic theorem provers to verify the obligations. If
332these tools are able to generate proof traces that can be
333independently checked, the only remaining component that enters the trusted code
334base is an off-the-shelf invariant generator which, in turn, can be proved
335correct using an interactive theorem prover. Therefore we achieve the double
336objective of allowing the use of more off-the-shelf components (e.g. provers and
337invariant generators) whilst reducing the trusted code base at the same time.
339%\paragraph{Summary of the CerCo objectives.} To summarize, the goal of CerCo is
340% to reconcile functional with non-functional analysis by performing them together
341% on the source code, sharing common knowledge about execution invariants. We want
342% to achieve the goal implementing a new generation of compilers that induce a
343% parametric, precise cost model for basic blocks on the source code. The compiler
344% should be certified using an interactive theorem prover to minimize the trusted
345% code base of the analysis. Once the cost model is induced, off-the-shelf tools
346% and techniques can be combined together to infer and prove parametric cost
347% bounds.
348%The long term benefits of the CerCo vision are expected to be:
349%1. the possibility to perform static analysis during early development stages
350%2.  parametric bounds made easier
351%3.  the application of off-the-shelf techniques currently unused for the
352% analysis of non-functional properties, like automated proving and type systems
353%4. simpler and safer interaction with the user, that is still asked for
354% knowledge, but on the source code, with the additional possibility of actually
355% verifying the provided knowledge
356%5. a reduced trusted code base
357%6. the increased accuracy of the bounds themselves.
359%The long term success of the project is hindered by the increased complexity of
360% the static prediction of the non-functional behaviour of modern hardware. In the
361% time frame of the European contribution we have focused on the general
362% methodology and on the difficulties related to the development and certification
363% of a cost model inducing compiler.
365\section{The typical CerCo workflow}
370char a[] = {3, 2, 7, 14};
371char threshold = 4;
373int count(char *p, int len) {
374  char j;
375  int found = 0;
376  for (j=0; j < len; j++) {
377    if (*p <= threshold)
378      found++;
379    p++;
380    }
381  return found;
384int main() {
385  return count(a,4);
389%  $\vcenter{\includegraphics[width=7.5cm]{interaction_diagram.pdf}}$
391    baseline={([yshift=-.5ex]current bounding box)},
392    element/.style={draw, text width=1.6cm, on chain, text badly centered},
393    >=stealth
394    ]
395{[start chain=going below, node distance=.5cm]
396\node [element] (cerco) {CerCo\\compiler};
397\node [element] (cost)  {CerCo\\cost plugin};
398{[densely dashed]
399\node [element] (ded)   {Deductive\\platform};
400\node [element] (check) {Proof\\checker};
403\coordinate [left=3.5cm of cerco] (left);
404{[every node/.style={above, text width=3.5cm, text badly centered,
405                     font=\scriptsize}]
406\draw [<-] ([yshift=-1ex]cerco.north west) coordinate (t) --
407    node {C source}
408    (t-|left);
409\draw [->] (cerco) -- (cost);
410\draw [->] ([yshift=1ex]cerco.south west) coordinate (t) --
411    node {C source+\color{red}{cost annotations}}
412    (t-|left) coordinate (cerco out);
413\draw [->] ([yshift=1ex]cost.south west) coordinate (t) --
414    node {C source+\color{red}{cost annotations}\\+\color{blue}{synthesized assertions}}
415    (t-|left) coordinate (out);
416{[densely dashed]
417\draw [<-] ([yshift=-1ex]ded.north west) coordinate (t) --
418    node {C source+\color{red}{cost annotations}\\+\color{blue}{complexity assertions}}
419    (t-|left) coordinate (ded in) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
420\draw [->] ([yshift=1ex]ded.south west) coordinate (t) --
421    node {complexity obligations}
422    (t-|left) coordinate (out);
423\draw [<-] ([yshift=-1ex]check.north west) coordinate (t) --
424    node {complexity proof}
425    (t-|left) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
426\draw [dash phase=2.5pt] (cerco out) .. controls +(-1, 0) and +(-1, 0) ..
427    (ded in);
429%% user:
430% head
431\draw (current bounding box.west) ++(-.2,.5)
432    circle (.2) ++(0,-.2) -- ++(0,-.1) coordinate (t);
433% arms
434\draw (t) -- +(-.3,-.2);
435\draw (t) -- +(.3,-.2);
436% body
437\draw (t) -- ++(0,-.4) coordinate (t);
438% legs
439\draw (t) -- +(-.2,-.6);
440\draw (t) -- +(.2,-.6);
443\caption{On the left: C code to count the number of elements in an array
444 that are less than or equal to a given threshold. On the right: CerCo's interaction
445 diagram. Components provided by CerCo are drawn with a solid border.}
448We illustrate the workflow we envisage (on the right
449of~\autoref{test1}) on an example program (on the left
450of~\autoref{test1}).  The user writes the program and feeds it to the
451CerCo compiler, which outputs an instrumented version of the same
452program that updates global variables that record the elapsed
453execution time and the stack space usage.  The red lines in
454\autoref{itest1} introducing variables, functions and function calls
455starting with \lstinline'__cost' and \lstinline'__stack' are the instrumentation introduced by the
456compiler.  For example, the two calls at the start of
457\lstinline'count' say that 4 bytes of stack are required, and that it
458takes 111 cycles to reach the next cost annotation (in the loop body).
459The compiler measures these on the labelled object code that it generates.
461 The annotated program can then be enriched with complexity
462assertions in the style of Hoare logic, that are passed to a deductive
463platform (in our case Frama-C). We provide as a Frama-C cost plugin a
464simple automatic synthesiser for complexity assertions which can
465be overridden by the user to increase or decrease accuracy.  These are the blue
466comments starting with \lstinline'/*@' in \autoref{itest1}, written in
467Frama-C's specification language, ACSL.  From the
468assertions, a general purpose deductive platform produces proof
469obligations which in turn can be closed by automatic or interactive
470provers, ending in a proof certificate.
472% NB: if you try this example with the live CD you should increase the timeout
474Twelve proof obligations are generated from~\autoref{itest1} (to prove
475that the loop invariant holds after one execution if it holds before,
476to prove that the whole program execution takes at most 1358 cycles, and so on).  Note that the synthesised time bound for \lstinline'count',
477$178+214*(1+\text{\lstinline'len'})$ cycles, is parametric in the length of
478the array. The CVC3 prover
479closes all obligations within half a minute on routine commodity
480hardware.  A simpler non-parametric version can be solved in a few
485        moredelim=[is][\color{blue}]{$}{$},
486        moredelim=[is][\color{red}]{|}{|},
487        lineskip=-2pt}
490|int __cost = 33, __stack = 5, __stack_max = 5;|
491|void __cost_incr(int incr) { __cost += incr; }|
492|void __stack_incr(int incr) {
493  __stack += incr;
494  __stack_max = __stack_max < __stack ? __stack : __stack_max;
497char a[4] = {3, 2, 7, 14};  char threshold = 4;
499/*@ behaviour stack_cost:
500      ensures __stack_max <= __max(\old(__stack_max), 4+\old(__stack));
501      ensures __stack == \old(__stack);
502    behaviour time_cost:
503      ensures __cost <= \old(__cost)+(178+214*__max(1+\at(len,Pre), 0));
505int count(char *p, int len) {
506  char j;  int found = 0;
507  |__stack_incr(4);  __cost_incr(111);|
508  $__l: /* internal */$
509  /*@ for time_cost: loop invariant
510        __cost <= \at(__cost,__l)+
511                  214*(__max(\at((len-j)+1,__l), 0)-__max(1+(len-j), 0));
512      for stack_cost: loop invariant
513        __stack_max == \at(__stack_max,__l);
514      for stack_cost: loop invariant
515        __stack == \at(__stack,__l);
516      loop variant len-j;
517  */
518  for (j = 0; j < len; j++) {
519    |__cost_incr(78);|
520    if (*p <= threshold) { |__cost_incr(136);| found ++; }
521    else { |__cost_incr(114);| }
522    p ++;
523  }
524  |__cost_incr(67);  __stack_incr(-4);|
525  return found;
528/*@ behaviour stack_cost:
529      ensures __stack_max <= __max(\old(__stack_max), 6+\old(__stack));
530      ensures __stack == \old(__stack);
531    behaviour time_cost:
532      ensures __cost <= \old(__cost)+1358;
534int main(void) {
535  int t;
536  |__stack_incr(2);  __cost_incr(110);|
537  t = count(a,4);
538  |__stack_incr(-2);|
539  return t;
542\caption{The instrumented version of the program in \autoref{test1},
543 with instrumentation added by the CerCo compiler in red and cost invariants
544 added by the CerCo Frama-C plugin in blue. The \texttt{\_\_cost},
545 \texttt{\_\_stack} and \texttt{\_\_stack\_max} variables hold the elapsed time
546in clock cycles and the current and maximum stack usage. Their initial values
547hold the clock cycles spent in initialising the global data before calling
548\texttt{main} and the space required by global data (and thus unavailable for
549the stack).
553\section{Main scientific and technical results}
554First we describe the basic labelling approach and our compiler
555implementations that use it.  This is suitable for basic architectures
556with simple cost models.  Then we will discuss the dependent labelling
557extension which is suitable for more advanced processor architectures
558and compiler optimisations.  At the end of this section we will
559demonstrate automated high level reasoning about the source level
560costs provided by the compilers.
562% \emph{Dependent labelling~\cite{?}.} The basic labelling approach assumes a
563% bijective mapping between object code and source code O(1) blocks (called basic
564% blocks). This assumption is violated by many program optimizations (e.g. loop
565% peeling and loop unrolling). It also assumes the cost model computed on the
566% object code to be non parametric: every block must be assigned a cost that does
567% not depend on the state. This assumption is violated by stateful hardware like
568% pipelines, caches, branch prediction units. The dependent labelling approach is
569% an extension of the basic labelling approach that allows to handle parametric
570% cost models. We showed how the method allows to deal with loop optimizations and
571% pipelines, and we speculated about its applications to caches.
573% \emph{Techniques to exploit the induced cost model.} Every technique used for
574% the analysis of functional properties of programs can be adapted to analyse the
575% non-functional properties of the source code instrumented by compilers that
576% implement the labelling approach. In order to gain confidence in this claim, we
577% showed how to implement a cost invariant generator combining abstract
578% interpretation with separation logic ideas \cite{separation}. We integrated
579% everything in the Frama-C modular architecture, in order to compute proof
580% obligations from functional and cost invariants and to use automatic theorem
581% provers on them. This is an example of a new technique that is not currently
582% exploited in WCET analysis. It also shows how precise functional invariants
583% benefits the non-functional analysis too. Finally, we show how to fully
584% automatically analyse the reaction time of Lustre nodes that are first compiled
585% to C using a standard Lustre compiler and then processed by a C compiler that
586% implements the labelling approach.
588% \emph{The CerCo compiler.} This is a compiler from a large subset of the C
589% program to 8051/8052 object code,
590% integrating the labelling approach and a static analyser for 8051 executables.
591% The latter can be implemented easily and does not require dependent costs
592% because the 8051 microprocessor is a very simple one, with constant-cost
593% instruction. It was chosen to separate the issue of exact propagation of the
594% cost model from the orthogonal problem of the static analysis of object code
595% that may require approximations or dependent costs. The compiler comes in
596% several versions: some are prototypes implemented directly in OCaml, and they
597% implement both the basic and dependent labelling approaches; the final version
598% is extracted from a Matita certification and at the moment implements only the
599% basic approach.
601\subsection{The (basic) labelling approach}
602The labelling approach is the foundational insight that underlies all the developments in CerCo.
603It facilitates the tracking of basic block evolution during the compilation process in order to propagate the cost model from the
604object code to the source code without losing precision in the process.
605\paragraph{Problem statement.} Given a source program $P$, we want to obtain an
606instrumented source program $P'$,  written in the same programming language, and
607the object code $O$ such that: 1) $P'$ is obtained by inserting into $P$ some
608additional instructions to update global cost information like the amount of
609time spent during execution or the maximal stack space required; 2) $P$ and $P'$ 
610must have the same functional behaviour, i.e.\ they must produce that same output
611and intermediate observables; 3) $P$ and $O$ must have the same functional
612behaviour; 4) after execution and in interesting points during execution, the
613cost information computed by $P'$ must be an upper bound of the one spent by $O$ 
614to perform the corresponding operations (\emph{soundness} property); 5) the difference
615between the costs computed by $P'$ and the execution costs of $O$ must be
616bounded by a program-dependent constant (\emph{precision} property).
618\paragraph{The labelling software components.} We solve the problem in four
619stages \cite{labelling}, implemented by four software components that are used
620in sequence.
622The first component labels the source program $P$ by injecting label emission
623statements in appropriate positions to mark the beginning of basic blocks.
624These are the positions where the cost instrumentation will appear in the final output.
625% The
626% set of labels with their positions is called labelling.
627The syntax and semantics
628of the source programming language is augmented with label emission statements.
629The statement ``EMIT $\ell$'' behaves like a NOP instruction that does not affect the
630program state or control flow, but its execution is observable.
631% Therefore the observables of a run of a program becomes a stream
632% of label emissions: $\ell_1,\ldots,\ell_n$, called the program trace. We clarify the conditions
633% that the labelling must respect later.
634For the example in Section~\ref{sec:workflow} this is just the original C code
635with ``EMIT'' instructions added at every point a \lstinline'__cost_incr' call
636appears in the final code.
638The second component is a labelling preserving compiler. It can be obtained from
639an existing compiler by adding label emission statements to every intermediate
640language and by propagating label emission statements during compilation. The
641compiler is correct if it preserves both the functional behaviour of the program
642and the traces of observables, including the labels `emitted'.
643% We may also ask that the function that erases the cost
644% emission statements commute with compilation. This optional property grants that
645% the labelling does not interfere with the original compiler behaviour. A further
646% set of requirements will be added later.
648The third component analyses the labelled
649object code to compute the scope of each of its label emission statements,
650i.e.\ the instructions that may be executed after the statement and before
651a new label emission is encountered, and then computes the maximum cost of
652each.  Note that we only have enough information at this point to compute
653the cost of loop-free portions of code.  We will consider how to ensure that
654every loop is broken by a cost label shortly.
655%It is a tree and not a sequence because the scope
656%may contain a branching statement. In order to grant that such a finite tree
657%exists, the object code must not contain any loop that is not broken by a label
658%emission statement. This is the first requirement of a sound labelling. The
659%analyser fails if the labelling is unsound. For each scope, the analyser
660%computes an upper bound of the execution time required by the scope, using the
661%maximum of the costs of the two branches in case of a conditional statement.
662%Finally, the analyser computes the cost of a label by taking the maximum of the
663%costs of the scopes of every statement that emits that label.
665The fourth and final component replaces the labels in the labelled
666version of the source code produced at the start with the costs
667computed for each label's scope.  This yields the instrumented source
668code.  For the example, this is the code in \autoref{itest1}, except
669for the specifications in comments, which we consider in
672\paragraph{Correctness.} Requirements 1 and 2 hold because of the
673non-invasive labelling procedure.  Requirement 3 can be satisfied by
674implementing compilation correctly.  It is obvious that the value of
675the global cost variable of the instrumented source code is always
676equal to the sum of the costs of the labels emitted by the
677corresponding labelled code. Moreover, because the compiler preserves
678all traces, the sum of the costs of the labels emitted in the source
679and target labelled code are the same. Therefore, to satisfy the
680soundness requirement, we need to ensure that the time taken to
681execute the object code is equal to the sum of the costs of the labels
682emitted by the object code. We collect all the necessary conditions
683for this to happen in the definition of a \emph{sound} labelling: a)
684all loops must be broken by a cost emission statement; b) all program
685instructions must be in the scope of some cost emission statement.
686This ensures that every label's scope is a tree of instructions, with
687the cost being the most expensive path. To satisfy also the precision
688requirement, we must make the scopes flat sequences of instructions.
689We require a \emph{precise} labelling where every label is emitted at
690most once and both branches of each conditional jump start with a label
691emission statement.
693The correctness and precision of the labelling approach only rely on the
694correctness and precision of the object code labelling. The simplest
695way to achieve that is to impose correctness and precision
696requirements on the source code labelling produced at the start, and
697to demand that the compiler preserves these
698properties too. The latter requirement imposes serious limitations on the
699compilation strategy and optimisations: the compiler may not duplicate any code
700that contains label emission statements, like loop bodies. Therefore various
701loop optimisations like peeling or unrolling are prevented. Moreover, precision
702of the object code labelling is not sufficient \emph{per se} to obtain global
703precision: we implicitly assumed that a precise constant cost can be assigned
704to every instruction. This is not possible in
705the presence of stateful hardware whose state influences the cost of operations,
706like pipelines and caches. In Section~\ref{lab:deplabel} we will see an extension of the
707basic labelling approach which tackles these problems.
709In CerCo we have developed several cost preserving compilers based on
710the labelling approach. Excluding an initial certified compiler for a
711`while' language, all remaining compilers target realistic source
712languages---a pure higher order functional language and a large subset
713of C with pointers, \texttt{goto}s and all data structures---and real world
714target processors---MIPS and the Intel 8051/8052 processor
715family. Moreover, they achieve a level of optimisation that ranges
716from moderate (comparable to GCC level 1) to intermediate (including
717loop peeling and unrolling, hoisting and late constant propagation).
718We describe the C compilers in detail in the following section.
720Two compilation chains were implemented for a purely functional
721higher-order language~\cite{labelling2}. The two main changes required
722to deal with functional languages are: 1) because global variables and
723updates are not available, the instrumentation phase produces monadic
724code to `update' the global costs; 2) the requirements for a sound and
725precise labelling of the source code must be changed when the
726compilation is based on CPS translations.  In particular, we need to
727introduce labels emitted before a statement is executed and also
728labels emitted after a statement is executed. The latter capture code
729that is inserted by the CPS translation and that would escape all
730label scopes.
732% Brian: one of the reviewers pointed out that standard Prolog implementations
733% do have some global state that apparently survives backtracking and might be
734% used.  As we haven't experimented with this, I think it's best to elide it
735% entirely.
737% Phases 1, 2 and 3 can be applied as well to logic languages (e.g. Prolog).
738% However, the instrumentation phase cannot: in standard Prolog there is no notion
739% of (global) variable whose state is not retracted during backtracking.
740% Therefore, the cost of executing computations that are later backtracked would
741% not be correctly counted in. Any extension of logic languages with
742% non-backtrackable state could support our labelling approach.
744\subsection{The CerCo C compilers}
745We implemented two C compilers, one implemented directly in OCaml and
746the other implemented in Matita, an interactive theorem prover~\cite{matita}.  The first acted as a prototype for the second,
747but also supported MIPS and acted as a testbed for more advanced
748features such as the dependent labelling approach
749in Section~\ref{lab:deplabel}.
751The second C compiler is the \emph{Trusted CerCo Compiler}, whose cost
752predictions are formally verified. The executable code
753is OCaml code extracted from the Matita implementation. The Trusted
754CerCo Compiler only targets
755the C language and the 8051/8052 family, and does not yet implement any
756advanced optimisations. Its user interface, however, is the same as the
757other version for interoperability purposes. In
758particular, the Frama-C CerCo plugin descibed in
759Section~\ref{sec:exploit} can work without recompilation
760with both of our C compilers.
762The 8051/8052 microprocessor is a very simple one, with constant-cost
763instructions. It was chosen to separate the issue of exact propagation of the
764cost model from the orthogonal problem of low-level timing analysis of object code
765that may require approximation or dependent costs.
767The (trusted) CerCo compiler implements the following optimisations: cast
768simplification, constant propagation in expressions, liveness analysis driven
769spilling of registers, dead code elimination, branch displacement, and tunnelling.
770The two latter optimisations are performed by our optimising assembler
771\cite{correctness}. The back-end of the compiler works on three address
772instructions, preferred to static single assignment code for the simplicity of
773the formal certification.
775The CerCo compiler is loosely based on the CompCert compiler \cite{compcert}, a
776recently developed certified compiler from C to the PowerPC, ARM and x86
777microprocessors. In contrast to CompCert, both the CerCo code and its
778certification are fully open source. Some data structures and language definitions for
779the front-end are directly taken from CompCert, while the back-end is a redesign
780of a compiler from Pascal to MIPS used by Fran\c{c}ois Pottier for a course at the
781\'{E}cole Polytechnique.
782The main differences in the CerCo compiler are:
784\item All the intermediate languages include label emitting instructions to
785implement the labelling approach, and the compiler preserves execution traces.
786\item Instead of targeting an assembly language with additional
787macro-instructions which are expanded before assembly, we directly
788produce object code in order to perform the timing analysis, using
789an integrated optimising assembler.
790\item In order to avoid the additional work of implementing a linker
791 and a loader, we do not support separate
792compilation and external calls. Adding them is orthogonal
793to the labelling approach and should not introduce extra problems.
794\item We target an 8-bit processor, in contrast to CompCert's 32-bit targets. This requires
795many changes and more compiler code, but it is not fundamentally more
796complex. The proof of correctness, however, becomes much harder.
797\item We target a microprocessor that has a non-uniform memory model, which is
798still often the case for microprocessors used in embedded systems and that is
799becoming common again in multi-core processors. Therefore the compiler has to
800keep track of data and it must move data between memory regions in the proper
801way. Moreover the size of pointers to different regions is not uniform. %An
802%additional difficulty was that the space available for the stack in internal
803%memory in the 8051 is tiny, allowing only a minor number of nested calls. To
804%support full recursion in order to test the CerCo tools also on recursive
805%programs, the compiler implements a stack in external memory.
808\subsection{Formal certification of the CerCo compiler}
809We have formally
810certified in Matita that the cost models induced on the source code by the
811Trusted CerCo Compiler correctly and precisely
812predict the object code behaviour. There are two cost models, one for execution
813time and one for stack space consumption. We show the correctness of the prediction
814only for those programs that do not exhaust the available stack space, a
815property that---thanks to the stack cost model---we can statically analyse on the
816source code in sharp contrast to other certified compilers.  Other projects have
817already certified the preservation of functional semantics in similar compilers,
818so we have not attempted to directly repeat that work and assume
819functional correctness for most passes. In order to complete the
820proof for non-functional properties, we have introduced a new,
821structured, form of execution trace, with the
822related notions for forward similarity and the intensional
823consequences of forward similarity. We have also introduced a unified
824representation for back-end intermediate languages that was exploited to provide
825a uniform proof of forward similarity.
827The details on the proof techniques employed
829the proof sketch can be found in the CerCo deliverables and papers~\cite{cerco-deliverables}.
830In this section we will only hint at the correctness statement, which turned
831out to be more complex than expected.
833\paragraph{The correctness statement.}
834Real time programs are often reactive programs that loop forever responding to
835events (inputs) by performing some computation followed by some action (output)
836and continuing as before. For these programs the overall execution
837time does not make sense. The same is true for reactive programs that spend an
838unpredictable amount of time in I/O. Instead, what is interesting is the reaction time ---
839 the time spent between I/O events. Moreover, we are interested in
840predicting and ruling out crashes due to running out of space on certain
842Therefore we need a statement that talks about sub-runs of a
843program. A natural candidate is that the time predicted on the source
844code and spent on the object code by two corresponding sub-runs are the same.
845To make this statement formal we must identify the
846corresponding sub-runs and how to single out those that are meaningful.
847We introduce the notion of a \emph{measurable} sub-run of a run
848which does not exhaust the available stack before or during the
849sub-run, the number of function calls and returns
850in the sub-run is the same, the sub-run does not perform any I/O, and the sub-run
851starts with a label emission statement and ends with a return or another label
852emission statement. The stack usage is bounded using the stack usage model
853that is computed by the compiler.
855The statement that we formally proved is: for each C run with a measurable
856sub-run, there exists an object code run with a sub-run, with the same
857execution trace for both the prefix of the run and the sub-run itself,
858 and where the time spent by the
859object code in the sub-run is the same as the time predicted on the source code
860using the time cost model generated by the compiler.
863We briefly discuss the constraints for measurability. Not exhausting the stack
864space is necessary for a run to be meaningful, because the source semantics
865has no notion of running out of memory. Balancing
866function calls and returns is a requirement for precision: the labelling approach
867allows the scope of a label to extend after function calls to
868minimize the number of labels. (The scope excludes the called
869function's execution.) If the number of
870calls/returns is unbalanced, it means that there is a call we have not returned
871to that could be followed by additional instructions whose cost has already been
872taken in account.  The last condition on the start and end points of
873a run is also required to make the bound precise.  With these
874restrictions and the 8051's simple timing model we obtain \emph{exact}
875predictions.  If we relax these conditions then we obtain a corollary
876with an upper bound on the cost.
877Finally, I/O operations can be performed in the prefix of the run, but not in the
878measurable sub-run. Therefore we prove that we can predict reaction times, but
879not I/O times, as desired.
881\subsection{Dependent labelling}
883The core idea of the basic labelling approach is to establish a tight connection
884between basic blocks executed in the source and target languages. Once the
885connection is established, any cost model computed on the object code can be
886transferred to the source code, without affecting the code of the compiler or
887its proof. In particular, we can also transport cost models
888that associate to each label a \emph{function} from the hardware state
889to a natural number.
890However, a problem arises during the instrumentation phase that replaces label
891emission statements with increments of global cost variables. They are
892incremented by the result of applying the label's cost function
893to the hardware state at the time of execution of the block. However,
894the hardware state comprises both the functional state that affects the
895computation (the value of the registers and memory) and the non-functional
896state that does not (the pipeline and cache contents, for example).
897We can find corresponding information for the former in the source code state, but constructing the
898correspondence may be hard and lifting the cost model to work on the source code
899state is likely to produce cost expressions that are too complex to understand and reason about.
900Fortunately, in modern architectures the cost of executing single
901instructions is either independent of the functional state or the jitter---the
902difference between the worst and best case execution times---is small enough
903to be bounded without losing too much precision. Therefore we only consider
904dependencies on the `non-functional' parts of the state.
906The non-functional state is not directly related to the high level
907state and does not influence the functional properties. What can be
908done is to expose key aspects of the non-functional state in the
909source code. We present here the basic intuition in a simplified form:
910the technical details that allow us to handle the general case are
911more complex and can be found in~\cite{paolo}. We add to the source
912code an additional global variable that represents the non-functional
913state and another one that remembers the last few labels emitted. The
914state variable must be updated at every label emission statement,
915using an update function which is computed during the processor timing
917This update function assigns to each label a function from the
918recently emitted labels and old state to the new state. It is computed
919by composing the semantics of every instruction in a basic block
920restricted to the non-functional part of the state.
922Not all the details of the non-functional state needs to be exposed, and the
923technique works better when the part of state that is required can be summarised
924in a simple data structure. For example, to handle simple but realistic
925pipelines it is sufficient to remember a short integer that encodes the position
926of bubbles (stuck instructions) in the pipeline. In any case, it is not necessary
927for the user to understand the meaning of the state to reason over the properties
928of the
929program. Moreover, the user, or the invariant generator tools that
930analyse the instrumented source code produced by the compiler, can decide to
931trade precision of the analysis for simplicity by approximating the
932cost by safe bounds that do not depend on the processor state. Interestingly, the functional analysis of
933the code could determine which blocks are executed more frequently in order to
934use more aggressive approximations for those that are executed least.
936Dependent labelling can also be applied to allow the compiler to duplicate
937blocks that contain labels (e.g. in loop optimisations)~\cite{paolo}. The effect is to assign
938a different cost to the different occurrences of a duplicated label. For
939example, loop peeling turns a loop into the concatenation of a copy of the loop
940body for the first iteration and the conditional execution of the
941loop for successive iterations. Further optimisations will compile the two
942copies of the loop differently, with the first body usually
943taking more time.
945By introducing a variable that keeps track of the iteration number, we can
946associate to the label a cost that is a function of the iteration number. The
947same technique works for loop unrolling without modification: the function will
948assign one cost to the even iterations and another cost to the odd
949ones.  The optimisation code
950that duplicates the loop bodies must also modify the code to correctly propagate
951the update of the iteration numbers. The technical details are more complicated and
952can be found in the CerCo reports and publications. The implementation, however,
953is quite simple (and forms part of our OCaml version of the compiler)
954and the changes to a loop optimising compiler are minimal.
956\subsection{Techniques to exploit the induced cost model}
958We now turn our attention to synthesising high-level costs, such as
959the reaction time of a real-time program.  We consider as our starting point source level costs
960provided by basic labelling, in other words annotations
961on the source code which are constants that provide a sound and sufficiently
962precise upper bound on the cost of executing the blocks after compilation to
963object code.
965The principle that we have followed in designing the cost synthesis tools is
966that the synthesised bounds should be expressed and proved within a general
967purpose tool built to reason on the source code. In particular, we rely on the
968Frama-C tool to reason on C code and on the Coq proof-assistant to reason on
969higher-order functional programs.
970This principle entails that
971the inferred synthetic bounds are indeed correct as long as the general purpose
972tool is, and that there is no limitation on the class of programs that can be handled,
973for example by resorting to interactive proof.
975Of course, automation is desirable whenever possible. Within this framework,
976automation means writing programs that give hints to the general purpose tool.
977These hints may take the form, say, of loop invariants/variants, of predicates
978describing the structure of the heap, or of types in a light logic. If these
979hints are correct and sufficiently precise the general purpose tool will produce
980a proof automatically, otherwise, user interaction is required.
982\paragraph{The Cost plugin and its application to the Lustre compiler.}
983Frama-C \cite{framac} is a set of analysers for C programs with a
984specification language, ACSL. New analyses can be added dynamically
985via plugins. For instance, the Jessie plugin~\cite{jessie} allows deductive
986verification of C programs with respect to their specification in ACSL, with
987various provers as back-end tools.
988We developed the CerCo Cost plugin for the Frama-C platform as a proof of
989concept of an automatic environment exploiting the cost annotations produced by
990the CerCo compiler. It consists of an OCaml program which essentially
991uses the CerCo compiler to produce a related C program with cost annotations,
992and applies some heuristics to produce a tentative bound on the cost of
993executing the C functions of the program as a function of the value of their
994parameters. The user can then call the Jessie plugin to discharge the
995related proof obligations.
996In the following we elaborate on the soundness of the framework and the
997experiments we performed with the Cost tool on C programs, including some produced by a
998Lustre compiler.
1000\paragraph{Soundness.} The soundness of the whole framework depends on the cost
1001annotations added by the CerCo compiler,
1002the verification conditions (VCs) generated by Jessie, and the
1003external provers discharging the VCs. Jessie can be used to verify the
1004synthesised bounds because our plugin generates them in ACSL format. Thus, even if the added synthetic costs are
1005incorrect (relatively to the cost annotations), the process as a whole is still
1006correct: indeed, Jessie will not validate incorrect costs and no conclusion can
1007be made about the WCET of the program in this case. In other terms, the
1008soundness does not depend on the cost plugin, which can in
1009principle produce any synthetic cost. However, in order to be able to actually
1010prove the WCET of a C function, we need to add correct annotations in a way that
1011Jessie and subsequent automatic provers have enough information to deduce their
1012validity. In practice this is not straightforward even for very simple programs
1013composed of branching and assignments (no loops and no recursion) because a fine
1014analysis of the VCs associated with branching may lead to a complexity blow up.
1016\paragraph{Experience with Lustre.} Lustre~\cite{lustre} is a data-flow language for programming
1017synchronous systems, with a compiler which targets C. We designed a
1018wrapper for supporting Lustre files.
1019The C function produced by the compiler is relatively simple loop-free
1020code which implements the step function of the
1021synchronous system and computing the WCET of the function amounts to obtaining a
1022bound on the reaction time of the system. We tested the Cost plugin and the
1023Lustre wrapper on the C programs generated by the Lustre compiler. For programs
1024consisting of a few hundred lines of code, the cost plugin computes a
1025WCET and Alt-Ergo is able to discharge all VCs automatically.
1027\paragraph{Handling C programs with simple loops.}
1028The cost annotations added by the CerCo compiler take the form of C instructions
1029that update a fresh global variable called the cost variable by a constant.
1030Synthesizing the WCET of a C function thus consists of statically resolving an
1031upper bound of the difference between the value of the cost variable before and
1032after the execution of the function, i.e. finding the instructions
1033that update the cost variable and establish the number of times they are passed
1034through during the flow of execution. To perform the analysis the plugin
1035assumes that there are no recursive functions in the program, and that
1036every loop is annotated with a variant. In the case of `for' loops the
1037variants are automatically inferred where a loop counter can be
1038syntactically detected.
1040The plugin computes a call-graph and proceeds to calculate bounds for
1041each function from the leaves up to the main function.
1042The computation of the cost of each function is performed by traversing its
1043control flow graph, where the cost of a node is the maximum of the
1044costs of the successors.
1045In the case of a loop with a body that has a constant cost for every step of the
1046loop, the cost is the product of the cost of the body and of the variant taken
1047at the start of the loop.
1048In the case of a loop with a body whose cost depends on the values of some free
1049variables, a fresh logic function $f$ is introduced to represent the cost of the
1050loop in the logic assertions. This logic function takes the variant as a first
1051parameter. The other parameters of $f$ are the free variables of the body of the
1052loop. An axiom is added to account for the fact that the cost is accumulated at each
1053step of the loop.
1054The cost of the function is added as post-condition of the function.
1056The user can also specify more precise variants and annotate functions
1057with their own cost specifications. The plugin will use these
1058instead of computing its own, allowing greater precision and the
1059ability to analyse programs which the variant generator does not
1062In addition to the loop-free Lustre code, this method was successfully
1063applied to a small range of cryptographic code.  See~\cite{labelling}
1064for more details.  The example in Section~\ref{sec:workflow} was also
1065produced using the plug-in.  The variant was calculated automatically
1066by noticing that \lstinline'j' is a loop counter with maximum value
1067\lstinline'len'.  The most expensive path through the loop body
1068($78+136 = 214$) is then multiplied by the number of iterations to
1069give the cost of the loop.
1071\paragraph{C programs with pointers.}
1072Using first-order logic and SMT solvers to specify and verify programs involving pointer-based
1073data structures such as linked-lists or graphs shows some limitations.
1074Separation logic, a program logic
1075with a new notion of conjunction to express spatial heap separation, is an elegant alternative.
1076Bobot has
1077recently introduced automatically generated separation
1078predicates to simulate separation logic reasoning in the Jessie plugin where the specification language, the verification condition
1079generator, and the theorem provers were not designed with separation logic in
1080mind~\cite{bobot}. CerCo's plugin can exploit these predicates to automatically
1081reason about the cost of execution of simple heap manipulation programs such as an
1082in-place list reversal.
1084\section{Conclusions and future work}
1086All CerCo software and deliverables may be found on the project homepage~\cite{cerco}.
1088The results obtained so far are encouraging and provide evidence that
1089it is possible to perform static time and space analysis at the source level
1090without losing accuracy, reducing the trusted code base and reconciling the
1091study of functional and non-functional properties of programs. The
1092techniques introduced seem to be scalable, cover both imperative and
1093functional languages and are compatible with every compiler optimisation
1094considered by us so far.
1096To prove that compilers can keep track of optimisations
1097and induce a precise cost model on the source code, we targeted a simple
1098architecture that admits a cost model that is execution history independent.
1099The most important future work is dealing with hardware architectures
1100characterised by history-dependent stateful components, like caches and
1101pipelines. The main issue is to assign a parametric, dependent cost
1102to basic blocks that can be later transferred by the labelling approach to
1103the source code and represented in a meaningful way to the user. The dependent
1104labelling approach that we have studied seems a promising tool to achieve
1105this goal, but more work is required to provide good source level
1106approximations of the relevant processor state.
1108Other examples of future work are to improve the cost invariant
1109generator algorithms and the coverage of compiler optimisations, to combining
1110the labelling approach with the type and effect discipline of~\cite{typeffects}
1111to handle languages with implicit memory management, and to experiment with
1112our tools in the early phases of development. Larger case studies are also necessary
1113to evaluate the CerCo's prototype on realistic, industrial-scale programs.
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