source: Papers/fopara2013/fopara13.tex @ 3453

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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 embellished with cost annotations as an additional side-effect. These annotations 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.
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 sufficiently mature to find increasing application in production environments. For safety critical code, it is
184becoming commonplace to combine rigorous software engineering methodologies and testing
185with static analysis, taking the strong points of each approach and mitigating
186their weaknesses. Of particular interest are open frameworks
187for the combination of different formal methods, where the programs can be
188progressively specified and are continuously enriched with new safety
189guarantees: every method contributes knowledge (e.g. new invariants) that
190becomes an assumption for later analysis.
192The outlook for verifying non-functional properties of programs (time spent,
193memory used, energy consumed) is bleaker.
194% and the future seems to be getting even worse.
195Most industries verify that real time systems meet their deadlines
196by simply performing many runs of the system and timing their execution,  computing the
197maximum time and adding an empirical safety margin, claiming the result to be a
198bound for the WCET of the program. Formal methods and software to statically
199analyse the WCET of programs exist, but they often produce bounds that are too
200pessimistic to be useful. Recent advancements in hardware architecture
201have been
202focused on the improvement of the average case performance, not the
203predictability of the worst case. Execution time is becoming increasingly
204dependent on execution history and the internal state of
205hardware components like pipelines and caches. Multi-core processors and non-uniform
206memory models are drastically reducing the possibility of performing
207static analysis in isolation, because programs are less and less time
208composable. Clock-precise hardware models are necessary for static analysis, and
209obtaining them is becoming harder due to the increased sophistication
210of hardware design.
212Despite the aforementioned problems, the need for reliable real time
213systems and programs is increasing, and there is increasing pressure
214from the research community towards the introduction of alternative
215hardware with more predictable behaviour, which would be more suitable
216for static analysis.  One example, being investigated by the Proartis
217project~\cite{proartis}, is to decouple execution time from execution
218history by introducing randomisation.
220In the CerCo project \cite{cerco} we do not try to address this problem, optimistically
221assuming that improvements in low-level timing analysis or architecture will make
222verification feasible in the longer term. Instead, the main objective of our work is
223to bring together the static analysis of functional and non-functional
224properties, which, according to the current state of the art, are completely
225independent activities with limited exchange of information: while the
226functional properties are verified on the source code, the analysis of
227non-functional properties is entirely performed on the object code to exploit
228clock-precise hardware models.
230\subsection{Current object-code methods}
232Analysis currently takes place on object code for two main reasons.
233First, there cannot be a uniform, precise cost model for source
234code instructions (or even basic blocks). During compilation, high level
235instructions are torn apart and reassembled in context-specific ways so that
236identifying a fragment of object code and a single high level instruction is
237infeasible. Even the control flow of the object and source code can be very
238different as a result of optimisations, for example aggressive loop
239optimisations may completely transform source level loops. Despite the lack of a uniform, compilation- and
240program-independent cost model on the source language, the literature on the
241analysis of non-asymptotic execution time on high level languages that assumes
242such a model is growing and gaining momentum. However, unless we provide a
243replacement for such cost models, this literature's future practical impact looks
244to be minimal. Some hope has been provided by the EmBounded project \cite{embounded}, which
245compositionally compiles high-level code to a byte code that is executed by an
246interpreter with guarantees on the maximal execution time spent for each byte code
247instruction. This provides a uniform model at the expense of the model's
248precision (each cost is a pessimistic upper bound) and the performance of the
249executed code (because the byte code is interpreted compositionally instead of
250performing a fully non-compositional compilation).
252The second reason to perform the analysis on the object code is that bounding
253the worst case execution time of small code fragments in isolation (e.g. loop
254bodies) and then adding up the bounds yields very poor estimates as no
255knowledge of the hardware state prior to executing the fragment can be assumed. By
256analysing longer runs the bound obtained becomes more precise because the lack
257of information about the initial state has a relatively small impact.
259To calculate the cost of an execution, value and control flow analyses
260are required to bound the number of times each basic block is
261executed.  Current state
262of the art WCET technology such as AbsInt's aiT analysis tools~\cite{absint} performs these analyses on the
263object code, where the logic of the program is harder to reconstruct
264and most information available at the source code level has been lost;
265see~\cite{stateart} for a survey. Imprecision in the analysis can lead to useless bounds. To
266augment precision, the tools ask the user to provide constraints on
267the object code control flow, usually in the form of bounds on the
268number of iterations of loops or linear inequalities on them. This
269requires the user to manually link the source and object code,
270translating his assumptions on the source code (which may be wrong) to
271object code constraints. The task is error prone and hard, especially
272in the presence of complex compiler optimisations.
274Traditional techniques for WCET that work on object code are also affected by
275another problem: they cannot be applied before the generation of the object
276code. Functional properties can be analysed in early development stages, while
277analysis of non-functional properties may come too late to avoid expensive
278changes to the program architecture.
280\subsection{CerCo's approach}
282In CerCo we propose a radically new approach to the problem: we reject the idea
283of a uniform cost model and we propose that the compiler, which knows how the
284code is translated, must return the cost model for basic blocks of high level
285instructions. It must do so by keeping track of the control flow modifications
286to reverse them and by interfacing with processor timing analysis.
288By embracing compilation, instead of avoiding it like EmBounded did, a CerCo
289compiler can both produce efficient code and return costs that are
290as precise as the processor timing analysis can be. Moreover, our costs can be
291parametric: the cost of a block can depend on actual program data, on a
292summary of the execution history, or on an approximated representation of the
293hardware state. For example, loop optimisations may assign a cost to a loop body
294that is a function of the number of iterations performed. As another example,
295the cost of a block may be a function of the vector of stalled pipeline states,
296which can be exposed in the source code and updated at each basic block exit. It
297is parametricity that allows one to analyse small code fragments without losing
298precision: in the analysis of the code fragment we do not have to ignore the
299initial hardware state. On the contrary, we can assume that we know exactly which
300state (or mode, as the WCET literature calls it) we are in.
302The CerCo approach has the potential to dramatically improve the state of the
303art. By performing control and data flow analyses on the source code, the error
304prone translation of invariants is completely avoided. Instead, this
305work is done at the source level using tools of the user's choice.
306Any available technique for the verification of functional properties
307can be immediately reused and multiple techniques can collaborate together to
308infer and certify cost invariants for the program.  There are no
309limitations on the types of loops or data structures involved. Parametric cost analysis
310becomes the default one, with non-parametric bounds used as a last resort when
311trading the complexity of the analysis with its precision. \emph{A priori}, no
312technique previously used in traditional WCET is lost: processor
313timing analyses can be used by the compiler on the object code, and the rest can be applied
314at the source code level.
315 Our approach can also work in the early
316stages of development by axiomatically attaching costs to unimplemented components.
319All software used to verify properties of programs must be as bug free as
320possible. The trusted code base for verification consists of the code that needs
321to be trusted to believe that the property holds. The trusted code base of
322state-of-the-art WCET tools is very large: one needs to trust the control flow
323analyser and the linear programming libraries it uses and also the formal models
324of the hardware under analysis. In CerCo we are moving the control flow analysis to the source
325code and we are introducing a non-standard compiler too. To reduce the trusted
326code base, we implemented a prototype and a static analyser in an interactive
327theorem prover, which was used to certify that the costs added to the source
328code are indeed those incurred by the hardware. Formal models of the
329hardware and of the high level source languages were also implemented in the
330interactive theorem prover. Control flow analysis on the source code has been
331obtained using invariant generators, tools to produce proof obligations from
332generated invariants and automatic theorem provers to verify the obligations. If
333the automatic provers are able to generate proof traces that can be
334independently checked, the only remaining component that enters the trusted code
335base is an off-the-shelf invariant generator which, in turn, can be proved
336correct using an interactive theorem prover. Therefore we achieve the double
337objective of allowing the use of more off-the-shelf components (e.g. provers and
338invariant generators) whilst reducing the trusted code base at the same time.
340%\paragraph{Summary of the CerCo objectives.} To summarize, the goal of CerCo is
341% to reconcile functional with non-functional analysis by performing them together
342% on the source code, sharing common knowledge about execution invariants. We want
343% to achieve the goal implementing a new generation of compilers that induce a
344% parametric, precise cost model for basic blocks on the source code. The compiler
345% should be certified using an interactive theorem prover to minimize the trusted
346% code base of the analysis. Once the cost model is induced, off-the-shelf tools
347% and techniques can be combined together to infer and prove parametric cost
348% bounds.
349%The long term benefits of the CerCo vision are expected to be:
350%1. the possibility to perform static analysis during early development stages
351%2.  parametric bounds made easier
352%3.  the application of off-the-shelf techniques currently unused for the
353% analysis of non-functional properties, like automated proving and type systems
354%4. simpler and safer interaction with the user, that is still asked for
355% knowledge, but on the source code, with the additional possibility of actually
356% verifying the provided knowledge
357%5. a reduced trusted code base
358%6. the increased accuracy of the bounds themselves.
360%The long term success of the project is hindered by the increased complexity of
361% the static prediction of the non-functional behaviour of modern hardware. In the
362% time frame of the European contribution we have focused on the general
363% methodology and on the difficulties related to the development and certification
364% of a cost model inducing compiler.
366\section{The typical CerCo workflow}
371char a[] = {3, 2, 7, 14};
372char threshold = 4;
374int count(char *p, int len) {
375  char j;
376  int found = 0;
377  for (j=0; j < len; j++) {
378    if (*p <= threshold)
379      found++;
380    p++;
381    }
382  return found;
385int main() {
386  return count(a,4);
390%  $\vcenter{\includegraphics[width=7.5cm]{interaction_diagram.pdf}}$
392    baseline={([yshift=-.5ex]current bounding box)},
393    element/.style={draw, text width=1.6cm, on chain, text badly centered},
394    >=stealth
395    ]
396{[start chain=going below, node distance=.5cm]
397\node [element] (cerco) {CerCo\\compiler};
398\node [element] (cost)  {CerCo\\cost plugin};
399{[densely dashed]
400\node [element] (ded)   {Deductive\\platform};
401\node [element] (check) {Proof\\checker};
404\coordinate [left=3.5cm of cerco] (left);
405{[every node/.style={above, text width=3.5cm, text badly centered,
406                     font=\scriptsize}]
407\draw [<-] ([yshift=-1ex]cerco.north west) coordinate (t) --
408    node {C source}
409    (t-|left);
410\draw [->] (cerco) -- (cost);
411\draw [->] ([yshift=1ex]cerco.south west) coordinate (t) --
412    node {C source+\color{red}{cost annotations}}
413    (t-|left) coordinate (cerco out);
414\draw [->] ([yshift=1ex]cost.south west) coordinate (t) --
415    node {C source+\color{red}{cost annotations}\\+\color{blue}{synthesized assertions}}
416    (t-|left) coordinate (out);
417{[densely dashed]
418\draw [<-] ([yshift=-1ex]ded.north west) coordinate (t) --
419    node {C source+\color{red}{cost annotations}\\+\color{blue}{complexity assertions}}
420    (t-|left) coordinate (ded in) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
421\draw [->] ([yshift=1ex]ded.south west) coordinate (t) --
422    node {complexity obligations}
423    (t-|left) coordinate (out);
424\draw [<-] ([yshift=-1ex]check.north west) coordinate (t) --
425    node {complexity proof}
426    (t-|left) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
427\draw [dash phase=2.5pt] (cerco out) .. controls +(-1, 0) and +(-1, 0) ..
428    (ded in);
430%% user:
431% head
432\draw (current bounding box.west) ++(-.2,.5)
433    circle (.2) ++(0,-.2) -- ++(0,-.1) coordinate (t);
434% arms
435\draw (t) -- +(-.3,-.2);
436\draw (t) -- +(.3,-.2);
437% body
438\draw (t) -- ++(0,-.4) coordinate (t);
439% legs
440\draw (t) -- +(-.2,-.6);
441\draw (t) -- +(.2,-.6);
444\caption{On the left: code to count the array's elements
445 that are less than or equal to the threshold. On the right: CerCo's interaction
446 diagram. CerCo's components are drawn solid.}
449We illustrate the workflow we envisage (on the right
450of~\autoref{test1}) on an example program (on the left
451of~\autoref{test1}).  The user writes the program and feeds it to the
452CerCo compiler, which outputs an instrumented version of the same
453program that updates global variables that record the elapsed
454execution time and the stack space usage.  The red lines in
455\autoref{itest1} introducing variables, functions and function calls
456starting with \lstinline'__cost' and \lstinline'__stack' are the instrumentation introduced by the
457compiler.  For example, the two calls at the start of
458\lstinline'count' say that 4 bytes of stack are required, and that it
459takes 111 cycles to reach the next cost annotation (in the loop body).
460The compiler measures these on the labelled object code that it generates.
462 The annotated program can then be enriched with complexity
463assertions in the style of Hoare logic, that are passed to a deductive
464platform (in our case Frama-C). We provide as a Frama-C cost plugin a
465simple automatic synthesiser for complexity assertions which can
466be overridden by the user to increase or decrease accuracy.  These are the blue
467comments starting with \lstinline'/*@' in \autoref{itest1}, written in
468Frama-C's specification language, ACSL.  From the
469assertions, a general purpose deductive platform produces proof
470obligations which in turn can be closed by automatic or interactive
471provers, ending in a proof certificate.
473% NB: if you try this example with the live CD you should increase the timeout
475Twelve proof obligations are generated from~\autoref{itest1} (to prove
476that the loop invariant holds after one execution if it holds before,
477to prove that the whole program execution takes at most 1358 cycles,
478etc.).  Note that the synthesised time bound for \lstinline'count',
479$178+214*(1+\text{\lstinline'len'})$ cycles, is parametric in the length of
480the array. The CVC3 prover
481closes all obligations within half a minute on routine commodity
482hardware.  A simpler non-parametric version can be solved in a few
487        moredelim=[is][\color{blue}]{$}{$},
488        moredelim=[is][\color{red}]{|}{|},
489        lineskip=-2pt}
492|int __cost = 33, __stack = 5, __stack_max = 5;|
493|void __cost_incr(int incr) { __cost += incr; }|
494|void __stack_incr(int incr) {
495  __stack += incr;
496  __stack_max = __stack_max < __stack ? __stack : __stack_max;
499char a[4] = {3, 2, 7, 14};  char threshold = 4;
501/*@ behaviour stack_cost:
502      ensures __stack_max <= __max(\old(__stack_max), 4+\old(__stack));
503      ensures __stack == \old(__stack);
504    behaviour time_cost:
505      ensures __cost <= \old(__cost)+(178+214*__max(1+\at(len,Pre), 0));
507int count(char *p, int len) {
508  char j;  int found = 0;
509  |__stack_incr(4);  __cost_incr(111);|
510  $__l: /* internal */$
511  /*@ for time_cost: loop invariant
512        __cost <= \at(__cost,__l)+
513                  214*(__max(\at((len-j)+1,__l), 0)-__max(1+(len-j), 0));
514      for stack_cost: loop invariant
515        __stack_max == \at(__stack_max,__l);
516      for stack_cost: loop invariant
517        __stack == \at(__stack,__l);
518      loop variant len-j;
519  */
520  for (j = 0; j < len; j++) {
521    |__cost_incr(78);|
522    if (*p <= threshold) { |__cost_incr(136);| found ++; }
523    else { |__cost_incr(114);| }
524    p ++;
525  }
526  |__cost_incr(67);  __stack_incr(-4);|
527  return found;
530/*@ behaviour stack_cost:
531      ensures __stack_max <= __max(\old(__stack_max), 6+\old(__stack));
532      ensures __stack == \old(__stack);
533    behaviour time_cost:
534      ensures __cost <= \old(__cost)+1358;
536int main(void) {
537  int t;
538  |__stack_incr(2);  __cost_incr(110);|
539  t = count(a,4);
540  |__stack_incr(-2);|
541  return t;
544\caption{The instrumented version of the program in \autoref{test1},
545 with instrumentation added by the CerCo compiler in red and cost invariants
546 added by the CerCo Frama-C plugin in blue. The \texttt{\_\_cost},
547 \texttt{\_\_stack} and \texttt{\_\_stack\_max} variables hold the elapsed time
548in clock cycles and the current and maximum stack usage. Their initial values
549hold the clock cycles spent in initialising the global data before calling
550\texttt{main} and the space required by global data (and thus unavailable for
551the stack).
555\section{Main scientific and technical results}
556First we describe the basic labelling approach and our compiler
557implementations that use it.  This is suitable for basic architectures
558with simple cost models.  Then we will discuss the dependent labelling
559extension which is suitable for more advanced processor architectures
560and compiler optimisations.  At the end of this section we will
561demonstrate automated high level reasoning about the source level
562costs provided by the compilers.
564% \emph{Dependent labelling~\cite{?}.} The basic labelling approach assumes a
565% bijective mapping between object code and source code O(1) blocks (called basic
566% blocks). This assumption is violated by many program optimizations (e.g. loop
567% peeling and loop unrolling). It also assumes the cost model computed on the
568% object code to be non parametric: every block must be assigned a cost that does
569% not depend on the state. This assumption is violated by stateful hardware like
570% pipelines, caches, branch prediction units. The dependent labelling approach is
571% an extension of the basic labelling approach that allows to handle parametric
572% cost models. We showed how the method allows to deal with loop optimizations and
573% pipelines, and we speculated about its applications to caches.
575% \emph{Techniques to exploit the induced cost model.} Every technique used for
576% the analysis of functional properties of programs can be adapted to analyse the
577% non-functional properties of the source code instrumented by compilers that
578% implement the labelling approach. In order to gain confidence in this claim, we
579% showed how to implement a cost invariant generator combining abstract
580% interpretation with separation logic ideas \cite{separation}. We integrated
581% everything in the Frama-C modular architecture, in order to compute proof
582% obligations from functional and cost invariants and to use automatic theorem
583% provers on them. This is an example of a new technique that is not currently
584% exploited in WCET analysis. It also shows how precise functional invariants
585% benefits the non-functional analysis too. Finally, we show how to fully
586% automatically analyse the reaction time of Lustre nodes that are first compiled
587% to C using a standard Lustre compiler and then processed by a C compiler that
588% implements the labelling approach.
590% \emph{The CerCo compiler.} This is a compiler from a large subset of the C
591% program to 8051/8052 object code,
592% integrating the labelling approach and a static analyser for 8051 executables.
593% The latter can be implemented easily and does not require dependent costs
594% because the 8051 microprocessor is a very simple one, with constant-cost
595% instruction. It was chosen to separate the issue of exact propagation of the
596% cost model from the orthogonal problem of the static analysis of object code
597% that may require approximations or dependent costs. The compiler comes in
598% several versions: some are prototypes implemented directly in OCaml, and they
599% implement both the basic and dependent labelling approaches; the final version
600% is extracted from a Matita certification and at the moment implements only the
601% basic approach.
603\subsection{The (basic) labelling approach}
604The labelling approach is the foundational insight that underlies all the developments in CerCo.
605It facilitates the tracking of basic block evolution during the compilation process in order to propagate the cost model from the
606object code to the source code without losing precision in the process.
607\paragraph{Problem statement.} Given a source program $P$, we want to obtain an
608instrumented source program $P'$,  written in the same programming language, and
609the object code $O$ such that: 1) $P'$ is obtained by inserting into $P$ some
610additional instructions to update global cost information like the amount of
611time spent during execution or the maximal stack space required; 2) $P$ and $P'$ 
612must have the same functional behaviour, i.e.\ they must produce that same output
613and intermediate observables; 3) $P$ and $O$ must have the same functional
614behaviour; 4) after execution and in interesting points during execution, the
615cost information computed by $P'$ must be an upper bound of the one spent by $O$ 
616to perform the corresponding operations (\emph{soundness} property); 5) the difference
617between the costs computed by $P'$ and the execution costs of $O$ must be
618bounded by a program-dependent constant (\emph{precision} property).
620\paragraph{The labelling software components.} We solve the problem in four
621stages \cite{labelling}, implemented by four software components that are used
622in sequence.
624The first component labels the source program $P$ by injecting label emission
625statements in appropriate positions to mark the beginning of basic blocks.
626These are the positions where the cost instrumentation will appear in the final output.
627% The
628% set of labels with their positions is called labelling.
629The syntax and semantics
630of the source programming language is augmented with label emission statements.
631The statement ``EMIT $\ell$'' behaves like a NOP instruction that does not affect the
632program state or control flow, but its execution is observable.
633% Therefore the observables of a run of a program becomes a stream
634% of label emissions: $\ell_1,\ldots,\ell_n$, called the program trace. We clarify the conditions
635% that the labelling must respect later.
636For the example in Section~\ref{sec:workflow} this is just the original C code
637with ``EMIT'' instructions added at every point a \lstinline'__cost_incr' call
638appears in the final code.
640The second component is a labelling preserving compiler. It can be obtained from
641an existing compiler by adding label emission statements to every intermediate
642language and by propagating label emission statements during compilation. The
643compiler is correct if it preserves both the functional behaviour of the program
644and the traces of observables, including the labels `emitted'.
645% We may also ask that the function that erases the cost
646% emission statements commute with compilation. This optional property grants that
647% the labelling does not interfere with the original compiler behaviour. A further
648% set of requirements will be added later.
650The third component analyses the labelled
651object code to compute the scope of each of its label emission statements,
652i.e.\ the instructions that may be executed after the statement and before
653a new label emission is encountered, and then computes the maximum cost of
654each.  Note that we only have enough information at this point to compute
655the cost of loop-free portions of code.  We will consider how to ensure that
656every loop is broken by a cost label shortly.
657%It is a tree and not a sequence because the scope
658%may contain a branching statement. In order to grant that such a finite tree
659%exists, the object code must not contain any loop that is not broken by a label
660%emission statement. This is the first requirement of a sound labelling. The
661%analyser fails if the labelling is unsound. For each scope, the analyser
662%computes an upper bound of the execution time required by the scope, using the
663%maximum of the costs of the two branches in case of a conditional statement.
664%Finally, the analyser computes the cost of a label by taking the maximum of the
665%costs of the scopes of every statement that emits that label.
667The fourth and final component replaces the labels in the labelled
668version of the source code produced at the start with the costs
669computed for each label's scope.  This yields the instrumented source
670code.  For the example, this is the code in \autoref{itest1}, except
671for the specifications in comments, which we consider in
674\paragraph{Correctness.} Requirements 1 and 2 hold because of the
675non-invasive labelling procedure.  Requirement 3 can be satisfied by
676implementing compilation correctly.  It is obvious that the value of
677the global cost variable of the instrumented source code is always
678equal to the sum of the costs of the labels emitted by the
679corresponding labelled code. Moreover, because the compiler preserves
680all traces, the sum of the costs of the labels emitted in the source
681and target labelled code are the same. Therefore, to satisfy the
682soundness requirement, we need to ensure that the time taken to
683execute the object code is equal to the sum of the costs of the labels
684emitted by the object code. We collect all the necessary conditions
685for this to happen in the definition of a \emph{sound} labelling: a)
686all loops must be broken by a cost emission statement; b) all program
687instructions must be in the scope of some cost emission statement.
688This ensures that every label's scope is a tree of instructions, with
689the cost being the most expensive path. To satisfy also the precision
690requirement, we must make the scopes flat sequences of instructions.
691We require a \emph{precise} labelling where every label is emitted at
692most once and both branches of each conditional jump start with a label
693emission statement.
695The correctness and precision of the labelling approach only rely on the
696correctness and precision of the object code labelling. The simplest
697way to achieve that is to impose correctness and precision
698requirements on the source code labelling produced at the start, and
699to demand that the compiler preserves these
700properties too. The latter requirement imposes serious limitations on the
701compilation strategy and optimisations: the compiler may not duplicate any code
702that contains label emission statements, like loop bodies. Therefore various
703loop optimisations like peeling or unrolling are prevented. Moreover, precision
704of the object code labelling is not sufficient \emph{per se} to obtain global
705precision: we implicitly assumed that a precise constant cost can be assigned
706to every instruction. This is not possible in
707the presence of stateful hardware whose state influences the cost of operations,
708like pipelines and caches. In Section~\ref{lab:deplabel} we will see an extension of the
709basic labelling approach which tackles these problems.
711In CerCo we have developed several cost preserving compilers based on
712the labelling approach. Excluding an initial certified compiler for a
713`while' language, all remaining compilers target realistic source
714languages---a pure higher order functional language and a large subset
715of C with pointers, \texttt{goto}s and all data structures---and real world
716target processors---MIPS and the Intel 8051/8052 processor
717family. Moreover, they achieve a level of optimisation that ranges
718from moderate (comparable to GCC level 1) to intermediate (including
719loop peeling and unrolling, hoisting and late constant propagation).
720We describe the C compilers in detail in the following section.
722Two compilation chains were implemented for a purely functional
723higher-order language~\cite{labelling2}. The two main changes required
724to deal with functional languages are: 1) because global variables and
725updates are not available, the instrumentation phase produces monadic
726code to `update' the global costs; 2) the requirements for a sound and
727precise labelling of the source code must be changed when the
728compilation is based on CPS translations.  In particular, we need to
729introduce labels emitted before a statement is executed and also
730labels emitted after a statement is executed. The latter capture code
731that is inserted by the CPS translation and that would escape all
732label scopes.
734% Brian: one of the reviewers pointed out that standard Prolog implementations
735% do have some global state that apparently survives backtracking and might be
736% used.  As we haven't experimented with this, I think it's best to elide it
737% entirely.
739% Phases 1, 2 and 3 can be applied as well to logic languages (e.g. Prolog).
740% However, the instrumentation phase cannot: in standard Prolog there is no notion
741% of (global) variable whose state is not retracted during backtracking.
742% Therefore, the cost of executing computations that are later backtracked would
743% not be correctly counted in. Any extension of logic languages with
744% non-backtrackable state could support our labelling approach.
746\subsection{The CerCo C compilers}
747We implemented two C compilers, one implemented directly in OCaml and
748the other implemented in the interactive theorem prover
749Matita~\cite{matita}.  The first acted as a prototype for the second,
750but also supported MIPS and acted as a testbed for more advanced
751features such as the dependent labelling approach
752in Section~\ref{lab:deplabel}.
754The second C compiler is the \emph{Trusted CerCo Compiler}, whose cost
755predictions are formally verified. The executable code
756is OCaml code extracted from the Matita implementation. The Trusted
757CerCo Compiler only targets
758the C language and the 8051/8052 family, and does not yet implement any
759advanced optimisations. Its user interface, however, is the same as the
760other version for interoperability purposes. In
761particular, the Frama-C CerCo plugin descibed in
762Section~\ref{sec:exploit} can work without recompilation
763with both of our C compilers.
765The 8051/8052 microprocessor is a very simple one, with constant-cost
766instructions. It was chosen to separate the issue of exact propagation of the
767cost model from the orthogonal problem of low-level timing analysis of object code
768that may require approximation or dependent costs.
770The (trusted) CerCo compiler implements the following optimisations: cast
771simplification, constant propagation in expressions, liveness analysis driven
772spilling of registers, dead code elimination, branch displacement, and tunnelling.
773The two latter optimisations are performed by our optimising assembler
774\cite{correctness}. The back-end of the compiler works on three address
775instructions, preferred to static single assignment code for the simplicity of
776the formal certification.
778The CerCo compiler is loosely based on the CompCert compiler \cite{compcert}, a
779recently developed certified compiler from C to the PowerPC, ARM and x86
780microprocessors. In contrast to CompCert, both the CerCo code and its
781certification are fully open source. Some data structures and language definitions for
782the front-end are directly taken from CompCert, while the back-end is a redesign
783of a compiler from Pascal to MIPS used by Fran\c{c}ois Pottier for a course at the
784\'{E}cole Polytechnique.
786The main differences in the CerCo compiler are:
788\item All the intermediate languages include label emitting instructions to
789implement the labelling approach, and the compiler preserves execution traces.
790\item Instead of targeting an assembly language with additional
791macro-instructions which are expanded before assembly, we directly
792produce object code in order to perform the timing analysis, using
793an integrated optimising assembler.
794\item In order to avoid the additional work of implementing a linker
795 and a loader, we do not support separate
796compilation and external calls. Adding them is orthogonal
797to the labelling approach and should not introduce extra problems.
798\item We target an 8-bit processor, in contrast to CompCert's 32-bit targets. This requires
799many changes and more compiler code, but it is not fundamentally more
800complex. The proof of correctness, however, becomes much harder.
801\item We target a microprocessor that has a non-uniform memory model, which is
802still often the case for microprocessors used in embedded systems and that is
803becoming common again in multi-core processors. Therefore the compiler has to
804keep track of data and it must move data between memory regions in the proper
805way. Moreover the size of pointers to different regions is not uniform. %An
806%additional difficulty was that the space available for the stack in internal
807%memory in the 8051 is tiny, allowing only a minor number of nested calls. To
808%support full recursion in order to test the CerCo tools also on recursive
809%programs, the compiler implements a stack in external memory.
812\subsection{Formal certification of the CerCo compiler}
813We have formally
814certified in the Matita interactive proof assistant that the cost models induced on the source code by the
815Trusted CerCo Compiler correctly and precisely
816predict the object code behaviour. There are two cost models, one for execution
817time and one for stack space consumption. We show the correctness of the prediction
818only for those programs that do not exhaust the available stack space, a
819property that---thanks to the stack cost model---we can statically analyse on the
820source code in sharp contrast to other certified compilers.  Other projects have
821already certified the preservation of functional semantics in similar compilers,
822so we have not attempted to directly repeat that work and assume
823functional correctness for most passes. In order to complete the
824proof for non-functional properties, we have introduced a new,
825structured, form of execution trace, with the
826related notions for forward similarity and the intensional
827consequences of forward similarity. We have also introduced a unified
828representation for back-end intermediate languages that was exploited to provide
829a uniform proof of forward similarity.
831The details on the proof techniques employed
833the proof sketch can be found in the CerCo deliverables and papers~\cite{cerco-deliverables}.
834In this section we will only hint at the correctness statement, which turned
835out to be more complex than expected.
837\paragraph{The correctness statement.}
838Real time programs are often reactive programs that loop forever responding to
839events (inputs) by performing some computation followed by some action (output)
840and continuing as before. For these programs the overall execution
841time does not make sense. The same is true for reactive programs that spend an
842unpredictable amount of time in I/O. Instead, what is interesting is the reaction time ---
843 the time spent between I/O events. Moreover, we are interested in
844predicting and ruling out crashes due to running out of space on certain
846Therefore we need a statement that talks about sub-runs of a
847program. A natural candidate is that the time predicted on the source
848code and spent on the object code by two corresponding sub-runs are the same.
849To make this statement formal we must identify the
850corresponding sub-runs and how to single out those that are meaningful.
851We introduce the notion of a \emph{measurable} sub-run of a run
852which does not exhaust the available stack before or during the
853sub-run, the number of function calls and returns
854in the sub-run is the same, the sub-run does not perform any I/O, and the sub-run
855starts with a label emission statement and ends with a return or another label
856emission statement. The stack usage is bounded using the stack usage model
857that is computed by the compiler.
859The statement that we formally proved is: for each C run with a measurable
860sub-run, there exists an object code run with a sub-run, with the same
861execution trace for both the prefix of the run and the sub-run itself,
862 and where the time spent by the
863object code in the sub-run is the same as the time predicted on the source code
864using the time cost model generated by the compiler.
867We briefly discuss the constraints for measurability. Not exhausting the stack
868space is necessary for a run to be meaningful, because the source semantics
869has no notion of running out of memory. Balancing
870function calls and returns is a requirement for precision: the labelling approach
871allows the scope of a label to extend after function calls to
872minimize the number of labels. (The scope excludes the called
873function's execution.) If the number of
874calls/returns is unbalanced, it means that there is a call we have not returned
875to that could be followed by additional instructions whose cost has already been
876taken in account.  The last condition on the start and end points of
877a run is also required to make the bound precise.  With these
878restrictions and the 8051's simple timing model we obtain \emph{exact}
879predictions.  If we relax these conditions then we obtain a corollary
880with an upper bound on the cost.
881Finally, I/O operations can be performed in the prefix of the run, but not in the
882measurable sub-run. Therefore we prove that we can predict reaction times, but
883not I/O times, as desired.
885\subsection{Dependent labelling}
887The core idea of the basic labelling approach is to establish a tight connection
888between basic blocks executed in the source and target languages. Once the
889connection is established, any cost model computed on the object code can be
890transferred to the source code, without affecting the code of the compiler or
891its proof. In particular, we can also transport cost models
892that associate to each label a \emph{function} from the hardware state
893to a natural number.
894However, a problem arises during the instrumentation phase that replaces label
895emission statements with increments of global cost variables. They are
896incremented by the result of applying the label's cost function
897to the hardware state at the time of execution of the block. However,
898the hardware state comprises both the functional state that affects the
899computation (the value of the registers and memory) and the non-functional
900state that does not (the pipeline and cache contents, for example).
901We can find corresponding information for the former in the source code state, but constructing the
902correspondence may be hard and lifting the cost model to work on the source code
903state is likely to produce cost expressions that are too complex to understand and reason about.
904Fortunately, in modern architectures the cost of executing single
905instructions is either independent of the functional state or the jitter---the
906difference between the worst and best case execution times---is small enough
907to be bounded without losing too much precision. Therefore we only consider
908dependencies on the `non-functional' parts of the state.
910The non-functional state is not directly related to the high level
911state and does not influence the functional properties. What can be
912done is to expose key aspects of the non-functional state in the
913source code. We present here the basic intuition in a simplified form:
914the technical details that allow us to handle the general case are
915more complex and can be found in~\cite{paolo}. We add to the source
916code an additional global variable that represents the non-functional
917state and another one that remembers the last few labels emitted. The
918state variable must be updated at every label emission statement,
919using an update function which is computed during the processor timing
921This update function assigns to each label a function from the
922recently emitted labels and old state to the new state. It is computed
923by composing the semantics of every instruction in a basic block
924restricted to the non-functional part of the state.
926Not all the details of the non-functional state needs to be exposed, and the
927technique works better when the part of state that is required can be summarised
928in a simple data structure. For example, to handle simple but realistic
929pipelines it is sufficient to remember a short integer that encodes the position
930of bubbles (stuck instructions) in the pipeline. In any case, it is not necessary
931for the user to understand the meaning of the state to reason over the properties
932of the
933program. Moreover, the user, or the invariant generator tools that
934analyse the instrumented source code produced by the compiler, can decide to
935trade precision of the analysis for simplicity by approximating the
936cost by safe bounds that do not depend on the processor state. Interestingly, the functional analysis of
937the code could determine which blocks are executed more frequently in order to
938use more aggressive approximations for those that are executed least.
940Dependent labelling can also be applied to allow the compiler to duplicate
941blocks that contain labels (e.g. in loop optimisations)~\cite{paolo}. The effect is to assign
942a different cost to the different occurrences of a duplicated label. For
943example, loop peeling turns a loop into the concatenation of a copy of the loop
944body for the first iteration and the conditional execution of the
945loop for successive iterations. Further optimisations will compile the two
946copies of the loop differently, with the first body usually
947taking more time.
949By introducing a variable that keeps track of the iteration number, we can
950associate to the label a cost that is a function of the iteration number. The
951same technique works for loop unrolling without modification: the function will
952assign one cost to the even iterations and another cost to the odd
953ones.  The optimisation code
954that duplicates the loop bodies must also modify the code to correctly propagate
955the update of the iteration numbers. The technical details are more complicated and
956can be found in the CerCo reports and publications. The implementation, however,
957is quite simple (and forms part of our OCaml version of the compiler)
958and the changes to a loop optimising compiler are minimal.
960\subsection{Techniques to exploit the induced cost model}
962We now turn our attention to synthesising high-level costs, such as
963the reaction time of a real-time program.  We consider as our starting point source level costs
964provided by basic labelling, in other words annotations
965on the source code which are constants that provide a sound and sufficiently
966precise upper bound on the cost of executing the blocks after compilation to
967object code.
969The principle that we have followed in designing the cost synthesis tools is
970that the synthesised bounds should be expressed and proved within a general
971purpose tool built to reason on the source code. In particular, we rely on the
972Frama-C tool to reason on C code and on the Coq proof-assistant to reason on
973higher-order functional programs.
974This principle entails that
975the inferred synthetic bounds are indeed correct as long as the general purpose
976tool is, and that there is no limitation on the class of programs that can be handled,
977for example by resorting to interactive proof.
979Of course, automation is desirable whenever possible. Within this framework,
980automation means writing programs that give hints to the general purpose tool.
981These hints may take the form, say, of loop invariants/variants, of predicates
982describing the structure of the heap, or of types in a light logic. If these
983hints are correct and sufficiently precise the general purpose tool will produce
984a proof automatically, otherwise, user interaction is required.
986\paragraph{The Cost plugin and its application to the Lustre compiler.}
987Frama-C \cite{framac} is a set of analysers for C programs with a
988specification language called ACSL. New analyses can be dynamically added
989via a plugin system. For instance, the Jessie plugin~\cite{jessie} allows deductive
990verification of C programs with respect to their specification in ACSL, with
991various provers as back-end tools.
992We developed the CerCo Cost plugin for the Frama-C platform as a proof of
993concept of an automatic environment exploiting the cost annotations produced by
994the CerCo compiler. It consists of an OCaml program which essentially
995uses the CerCo compiler to produce a related C program with cost annotations,
996and applies some heuristics to produce a tentative bound on the cost of
997executing the C functions of the program as a function of the value of their
998parameters. The user can then call the Jessie plugin to discharge the
999related proof obligations.
1000In the following we elaborate on the soundness of the framework and the
1001experiments we performed with the Cost tool on C programs, including some produced by a
1002Lustre compiler.
1004\paragraph{Soundness.} The soundness of the whole framework depends on the cost
1005annotations added by the CerCo compiler,
1006the verification conditions (VCs) generated by Jessie, and the
1007external provers discharging the VCs. Jessie can be used to verify the
1008synthesised bounds because our plugin generates them in ACSL format. Thus, even if the added synthetic costs are
1009incorrect (relatively to the cost annotations), the process as a whole is still
1010correct: indeed, Jessie will not validate incorrect costs and no conclusion can
1011be made about the WCET of the program in this case. In other terms, the
1012soundness does not depend on the cost plugin, which can in
1013principle produce any synthetic cost. However, in order to be able to actually
1014prove a WCET of a C function, we need to add correct annotations in a way that
1015Jessie and subsequent automatic provers have enough information to deduce their
1016validity. In practice this is not straightforward even for very simple programs
1017composed of branching and assignments (no loops and no recursion) because a fine
1018analysis of the VCs associated with branching may lead to a complexity blow up.
1020\paragraph{Experience with Lustre.} Lustre~\cite{lustre} is a data-flow language for programming
1021synchronous systems, with a compiler which targets C. We designed a
1022wrapper for supporting Lustre files.
1023The C function produced by the compiler is relatively simple loop-free
1024code which implements the step function of the
1025synchronous system and computing the WCET of the function amounts to obtaining a
1026bound on the reaction time of the system. We tested the Cost plugin and the
1027Lustre wrapper on the C programs generated by the Lustre compiler. For programs
1028consisting of a few hundred lines of code, the cost plugin computes a
1029WCET and Alt-Ergo is able to discharge all VCs automatically.
1031\paragraph{Handling C programs with simple loops.}
1032The cost annotations added by the CerCo compiler take the form of C instructions
1033that update a fresh global variable called the cost variable by a constant.
1034Synthesizing a WCET of a C function thus consists of statically resolving an
1035upper bound of the difference between the value of the cost variable before and
1036after the execution of the function, i.e. finding the instructions
1037that update the cost variable and establish the number of times they are passed
1038through during the flow of execution. To perform the analysis the plugin
1039assumes that there are no recursive functions in the program, and that
1040every loop is annotated with a variant. In the case of `for' loops the
1041variants are automatically inferred where a loop counter can be
1042syntactically detected.
1044The plugin computes a call-graph and proceeds to calculate bounds for
1045each function from the leaves up to the main function.
1046The computation of the cost of each function is performed by traversing its
1047control flow graph, where the cost of a node is the maximum of the
1048costs of the successors.
1049In the case of a loop with a body that has a constant cost for every step of the
1050loop, the cost is the product of the cost of the body and of the variant taken
1051at the start of the loop.
1052In the case of a loop with a body whose cost depends on the values of some free
1053variables, a fresh logic function $f$ is introduced to represent the cost of the
1054loop in the logic assertions. This logic function takes the variant as a first
1055parameter. The other parameters of $f$ are the free variables of the body of the
1056loop. An axiom is added to account for the fact that the cost is accumulated at each
1057step of the loop.
1058The cost of the function is directly added as post-condition of the function.
1060The user can also specify more precise variants and annotate functions
1061with their own cost specifications. The plugin will use these
1062instead of computing its own, allowing greater precision and the
1063ability to analyse programs which the variant generator does not
1066In addition to the loop-free Lustre code, this method was successfully
1067applied to a small range of cryptographic code.  See~\cite{labelling}
1068for more details.  The example in Section~\ref{sec:workflow} was also
1069produced using the plug-in.  The variant was calculated automatically
1070by noticing that \lstinline'j' is a loop counter with maximum value
1071\lstinline'len'.  The most expensive path through the loop body
1072($78+136 = 214$) is then multiplied by the number of iterations to
1073give the cost of the loop.
1075\paragraph{C programs with pointers.}
1076When it comes to verifying programs involving pointer-based data structures,
1077such as linked lists, trees, or graphs, the use of traditional first-order logic
1078to specify, and of SMT solvers to verify, shows some limitations. Separation
1079logic is an elegant alternative. It is a program logic
1080with a new notion of conjunction to express spatial heap separation. Bobot has
1081recently introduced automatically generated separation
1082predicates to simulate separation logic reasoning in the Jessie plugin where the specification language, the verification condition
1083generator, and the theorem provers were not designed with separation logic in
1084mind~\cite{bobot}. CerCo's plugin can exploit the separation predicates to automatically
1085reason on the cost of execution of simple heap manipulation programs such as an
1086in-place list reversal.
1088\section{Conclusions and future work}
1090All CerCo software and deliverables may be found on the project homepage~\cite{cerco}.
1092The results obtained so far are encouraging and provide evidence that
1093it is possible to perform static time and space analysis at the source level
1094without losing accuracy, reducing the trusted code base and reconciling the
1095study of functional and non-functional properties of programs. The
1096techniques introduced seem to be scalable, cover both imperative and
1097functional languages and are compatible with every compiler optimisation
1098considered by us so far.
1100To prove that compilers can keep track of optimisations
1101and induce a precise cost model on the source code, we targeted a simple
1102architecture that admits a cost model that is execution history independent.
1103The most important future work is dealing with hardware architectures
1104characterised by history-dependent stateful components, like caches and
1105pipelines. The main issue is to assign a parametric, dependent cost
1106to basic blocks that can be later transferred by the labelling approach to
1107the source code and represented in a meaningful way to the user. The dependent
1108labelling approach that we have studied seems a promising tool to achieve
1109this goal, but more work is required to provide good source level
1110approximations of the relevant processor state.
1112Other examples of future work are to improve the cost invariant
1113generator algorithms and the coverage of compiler optimisations, to combining
1114the labelling approach with the type and effect discipline of~\cite{typeffects}
1115to handle languages with implicit memory management, and to experiment with
1116our tools in the early phases of development. Larger case studies are also necessary
1117to evaluate the CerCo's prototype on realistic, industrial-scale programs.
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