source: Papers/fopara2013/fopara13.tex @ 3420

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Parametric example for FOPARA.

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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 $\pi$r2 )
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.
96This paper provides an overview of the FET-Open Project CerCo
97(`Certified Complexity'). The project's 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. The main software component
101developed is a certified compiler producing cost annotations. The compiler
102translates source code to object code and produces an instrumented copy of the
103source code. This instrumentation exposes at
104the source level---and tracks precisely---the actual (non-asymptotic)
105computational cost of the input program. Untrusted invariant generators and trusted theorem provers
106can then be used to compute and certify the parametric execution time of the
110% ---------------------------------------------------------------------------- %
111% SECTION                                                                      %
112% ---------------------------------------------------------------------------- %
115\paragraph{Problem statement.} Computer programs can be specified with both
116functional constraints (what a program must do) and non-functional constraints
117(e.g. what resources---time, space, etc.---the program may use).  In the current
118state of the art, functional properties are verified for high-level source code
119by combining user annotations (e.g. preconditions and invariants) with a
120multitude of automated analyses (invariant generators, type systems, abstract
121interpretation, theorem proving, and so on). By contrast, non-functional properties
122are generally checked on low-level object code, but also demand information
123about high-level functional behaviour that must somehow be recreated.
125This situation presents several problems: 1) it can be hard to infer this
126high-level structure in the presence of compiler optimisations; 2) techniques
127working on object code are not useful in early development, yet problems
128detected later are more expensive to tackle; 3) parametric cost analysis is very
129hard: how can we reflect a cost that depends on the execution state (e.g. the
130value of a register or a carry bit) to a cost that the user can understand
131looking at source code? 4) functional analysis performed only on object code
132makes any contribution from the programmer hard, leading to less precision in
133the estimates.
135\paragraph{Vision and approach.} We want to reconcile functional and
136non-functional analyses: to share information and perform both at the same time
137on source code.
139What has previously prevented this approach is the lack of a uniform and precise
140cost model for high-level code: 1) each statement occurrence is compiled
141differently and optimisations may change control flow; 2) the cost of an object
142code instruction may depend on the runtime state of hardware components like
143pipelines and caches, all of which are not visible in the source code.
145To solve the issue, we envision a new generation of compilers able to keep track
146of program structure through compilation and optimisation, and to exploit that
147information to define a cost model for source code that is precise, non-uniform,
148and accounts for runtime state. With such a source-level cost model we can
149reduce non-functional verification to the functional case and exploit the state
150of the art in automated high-level verification~\cite{survey}. The techniques
151currently used by the Worst Case Execution Time (WCET) community, which perform the analysis on object code,
152are still available but can now be coupled with an additional source-level
153analysis. Where the approach produces precise cost models too complex to reason
154about, safe approximations can be used to trade complexity with precision.
155Finally, source code analysis can be made during early development stages, when
156components have been specified but not implemented: source code modularity means
157that it is enough to specify the non-functional behaviour of unimplemented
160\paragraph{Contributions.} We have developed what we refer to as \emph{the labeling approach} \cite{labeling}, a
161technique to implement compilers that induce cost models on source programs by
162very lightweight tracking of code changes through compilation. We have studied
163how to formally prove the correctness of compilers implementing this technique.
164We have implemented such a compiler from C to object binaries for the 8051
165micro-controller, and verified it in an interactive theorem prover. We have
166implemented a Frama-C plugin \cite{framac} that invokes the compiler on a source
167program and uses this to generate invariants on the high-level source that
168correctly model low-level costs. Finally, the plugin certifies that the program
169respects these costs by calling automated theorem provers, a new and innovative
170technique in the field of cost analysis. As a case study, we show how the
171plugin can automatically compute and certify the exact reaction time of Lustre
172\cite{lustre} data flow programs compiled into C.
174\section{Project context and objectives}
175Formal methods for verification of functional properties of programs have
176now reached a level of maturity and automation that is facilitating a slow but
177increasing adoption in production environments. For safety critical code, it is
178becoming commonplace to combine rigorous software engineering methodologies and testing
179with static analysis, taking the strong points of each approach and mitigating
180their weaknesses. Of particular interest are open frameworks
181for the combination of different formal methods, where the programs can be
182progressively specified and are continuously enriched with new safety
183guarantees: every method contributes knowledge (e.g. new invariants) that
184becomes an assumption for later analysis.
186The scenario for the verification of non-functional properties (time spent,
187memory used, energy consumed) is bleaker and the future seems to be getting even
188worse. Most industries verify that real time systems meet their deadlines
189by simply performing many runs of the system and timing their execution,  computing the
190maximum time and adding an empirical safety margin, claiming the result to be a
191bound for the WCET of the program. Formal methods and software to statically
192analyse the WCET of programs exist, but they often produce bounds that are too
193pessimistic to be useful. Recent advancements in hardware architectures are all
194focused on the improvement of the average case performance, not the
195predictability of the worst case. Execution time is becoming increasingly
196dependent on execution history and the internal state of
197hardware components like pipelines and caches. Multi-core processors and non-uniform
198memory models are drastically reducing the possibility of performing
199static analysis in isolation, because programs are less and less time
200composable. Clock-precise hardware models are necessary for static analysis, and
201obtaining them is becoming harder as a consequence of the increased sophistication
202of hardware design.
204Despite the aforementioned problems, the need for reliable real time systems and programs is
205increasing, and there is increasing pressure from the research community towards
206the differentiation of hardware. The aim is to introduce alternative
207hardware with more predictable behaviour and hence more suitability for static
208analyses, for example, the decoupling of execution time from execution history
209by introducing randomisation \cite{proartis}.
211In the CerCo project \cite{cerco} we do not try to address this problem, optimistically
212assuming that static analysis of non-functional properties of programs will
213become feasible in the longer term. The main objective of our work is
214instead to bring together static analysis of functional and non-functional
215properties, which, according to the current state of the art, are completely
216independent activities with limited exchange of information: while the
217functional properties are verified on the source code, the analysis of
218non-functional properties is entirely performed on the object code to exploit
219clock-precise hardware models.
221Analysis currently takes place on object code for two main reasons.
222First, there is a lack of a uniform, precise cost model for source
223code instructions (or even basic blocks). During compilation, high level
224instructions are torn apart and reassembled in context-specific ways so that
225identifying a fragment of object code and a single high level instruction is
226infeasible. Even the control flow of the object and source code can be very
227different as a result of optimisations, for example aggressive loop
228optimisations may completely transform source level loops. Despite the lack of a uniform, compilation- and
229program-independent cost model on the source language, the literature on the
230analysis of non-asymptotic execution time on high level languages that assumes
231such a model is growing and gaining momentum. However, unless we can provide a
232replacement for such cost models, this literature's future practical impact looks
233to be minimal. Some hope has been provided by the EmBounded project \cite{embounded}, which
234compositionally compiles high-level code to a byte code that is executed by an
235emulator with guarantees on the maximal execution time spent for each byte code
236instruction. The approach provides a uniform model at the price of the model's
237precision (each cost is a pessimistic upper bound) and performance of the
238executed code (because the byte code is emulated  compositionally instead of
239performing a fully non-compositional compilation).
241The second reason to perform the analysis on the object code is that bounding
242the worst case execution time of small code fragments in isolation (e.g. loop
243bodies) and then adding up the bounds yields very poor estimates because no
244knowledge on the hardware state can be assumed when executing the fragment. By
245analysing longer runs the bound obtained becomes more precise because the lack
246of knowledge on the initial state has less of an effect on longer computations.
248In CerCo we propose a radically new approach to the problem: we reject the idea
249of a uniform cost model and we propose that the compiler, which knows how the
250code is translated, must return the cost model for basic blocks of high level
251instructions. It must do so by keeping track of the control flow modifications
252to reverse them and by interfacing with static analysers.
254By embracing compilation, instead of avoiding it like EmBounded did, a CerCo
255compiler can at the same time produce efficient code and return costs that are
256as precise as the static analysis can be. Moreover, we allow our costs to be
257parametric: the cost of a block can depend on actual program data or on a
258summary of the execution history or on an approximated representation of the
259hardware state. For example, loop optimizations may assign to a loop body a cost
260that is a function of the number of iterations performed. As another example,
261the cost of a block may be a function of the vector of stalled pipeline states,
262which can be exposed in the source code and updated at each basic block exit. It
263is parametricity that allows one to analyse small code fragments without losing
264precision: in the analysis of the code fragment we do not have to ignore the
265initial hardware state. On the contrary, we can assume that we know exactly which
266state (or mode, as the WCET literature calls it) we are in.
268The cost of an execution is the sum of the cost of basic blocks multiplied by
269the number of times they are executed, which is a functional property of the
270program. Therefore, in order to perform (parametric) time analysis of programs,
271it is necessary to combine a cost model with control and data flow analysis.
272Current state of the art WCET technology
273\cite{stateart} performs the analysis on the object code, where the logic of the
274program is harder to reconstruct and most information available at the source
275code level has been lost. Imprecision in the analysis leads to useless bounds. To
276augment precision, the tools ask the user to provide constraints on the object
277code control flow, usually in the form of bounds on the number of iterations of
278loops or linear inequalities on them. This requires the user to manually link
279the source and object code, translating his assumptions on the source code
280(which may be wrong) to object code constraints. The task is error prone and
281hard, especially in the presence of complex compiler optimisations.
283The CerCo approach has the potential to dramatically improve the state of the
284art. By performing control and data flow analyses on the source code, the error
285prone translation of invariants is completely avoided. It is in fact performed
286by the compiler itself when it induces the cost model on the source language.
287Moreover, any available technique for the verification of functional properties
288can be immediately reused and multiple techniques can collaborate together to
289infer and certify cost invariants for the program. Parametric cost analysis
290becomes the default one, with non parametric bounds used as last resorts when
291trading the complexity of the analysis with its precision. \emph{A priori}, no
292technique previously used in traditional WCET is lost: they can just be applied
293at the source code level.
295Traditional techniques for WCET that work on object code are also affected by
296another problem: they cannot be applied before the generation of the object
297code. Functional properties can be analysed in early development stages, while
298analysis of non-functional properties may come too late to avoid expensive
299changes to the program architecture. Our approach already works in early
300development stages by axiomatically attaching costs to unimplemented components.
302All software used to verify properties of programs must be as bug free as
303possible. The trusted code base for verification is made by the code that needs
304to be trusted to believe that the property holds. The trusted code base of
305state-of-the-art WCET tools is very large: one needs to trust the control flow
306analyser and the linear programming libraries it uses and also the formal models
307of the hardware. In CerCo we are moving the control flow analysis to the source
308code and we are introducing a non-standard compiler too. To reduce the trusted
309code base, we implemented a prototype and a static analyser in an interactive
310theorem prover, which was used to certify that the cost computed on the source
311code is indeed the one actually spent by the hardware. Formal models of the
312hardware and of the high level source languages were also implemented in the
313interactive theorem prover. Control flow analysis on the source code has been
314obtained using invariant generators, tools to produce proof obligations from
315generated invariants and automatic theorem provers to verify the obligations. If
316the automatic provers are able to generate proof traces that can be
317independently checked, the only remaining component that enters the trusted code
318base is an off-the-shelf invariant generator which, in turn, can be proved
319correct using an interactive theorem prover. Therefore we achieve the double
320objective of allowing the use of more off-the-shelf components (e.g. provers and
321invariant generators) whilst reducing the trusted code base at the same time.
323%\paragraph{Summary of the CerCo objectives.} To summarize, the goal of CerCo is
324% to reconcile functional with non-functional analysis by performing them together
325% on the source code, sharing common knowledge about execution invariants. We want
326% to achieve the goal implementing a new generation of compilers that induce a
327% parametric, precise cost model for basic blocks on the source code. The compiler
328% should be certified using an interactive theorem prover to minimize the trusted
329% code base of the analysis. Once the cost model is induced, off-the-shelf tools
330% and techniques can be combined together to infer and prove parametric cost
331% bounds.
332%The long term benefits of the CerCo vision are expected to be:
333%1. the possibility to perform static analysis during early development stages
334%2.  parametric bounds made easier
335%3.  the application of off-the-shelf techniques currently unused for the
336% analysis of non-functional properties, like automated proving and type systems
337%4. simpler and safer interaction with the user, that is still asked for
338% knowledge, but on the source code, with the additional possibility of actually
339% verifying the provided knowledge
340%5. a reduced trusted code base
341%6. the increased accuracy of the bounds themselves.
343%The long term success of the project is hindered by the increased complexity of
344% the static prediction of the non-functional behaviour of modern hardware. In the
345% time frame of the European contribution we have focused on the general
346% methodology and on the difficulties related to the development and certification
347% of a cost model inducing compiler.
349\section{The typical CerCo workflow}
353char a[] = {3, 2, 7, 14};
354char threshold = 4;
356int count(char *p, int len) {
357  char j;
358  int found = 0;
359  for (j=0; j < len; j++) {
360    if (*p <= threshold)
361      found++;
362    p++;
363    }
364  return found;
367int main() {
368  return count(a,4);
372%  $\vcenter{\includegraphics[width=7.5cm]{interaction_diagram.pdf}}$
374    baseline={([yshift=-.5ex]current bounding box)},
375    element/.style={draw, text width=1.6cm, on chain, text badly centered},
376    >=stealth
377    ]
378{[start chain=going below, node distance=.5cm]
379\node [element] (cerco) {CerCo\\compiler};
380\node [element] (cost)  {CerCo\\cost plugin};
381{[densely dashed]
382\node [element] (ded)   {Deductive\\platform};
383\node [element] (check) {Proof\\checker};
386\coordinate [left=3.5cm of cerco] (left);
387{[every node/.style={above, text width=3.5cm, text badly centered,
388                     font=\scriptsize}]
389\draw [<-] ([yshift=-1ex]cerco.north west) coordinate (t) --
390    node {C source}
391    (t-|left);
392\draw [->] (cerco) -- (cost);
393\draw [->] ([yshift=1ex]cerco.south west) coordinate (t) --
394    node {C source+\color{red}{cost annotations}}
395    (t-|left) coordinate (cerco out);
396\draw [->] ([yshift=1ex]cost.south west) coordinate (t) --
397    node {C source+\color{red}{cost annotations}\\+\color{blue}{synthesized assertions}}
398    (t-|left) coordinate (out);
399{[densely dashed]
400\draw [<-] ([yshift=-1ex]ded.north west) coordinate (t) --
401    node {C source+\color{red}{cost annotations}\\+\color{blue}{complexity assertions}}
402    (t-|left) coordinate (ded in) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
403\draw [->] ([yshift=1ex]ded.south west) coordinate (t) --
404    node {complexity obligations}
405    (t-|left) coordinate (out);
406\draw [<-] ([yshift=-1ex]check.north west) coordinate (t) --
407    node {complexity proof}
408    (t-|left) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
409\draw [dash phase=2.5pt] (cerco out) .. controls +(-1, 0) and +(-1, 0) ..
410    (ded in);
412%% user:
413% head
414\draw (current bounding box.west) ++(-.2,.5)
415    circle (.2) ++(0,-.2) -- ++(0,-.1) coordinate (t);
416% arms
417\draw (t) -- +(-.3,-.2);
418\draw (t) -- +(.3,-.2);
419% body
420\draw (t) -- ++(0,-.4) coordinate (t);
421% legs
422\draw (t) -- +(-.2,-.6);
423\draw (t) -- +(.2,-.6);
426\caption{On the left: code to count the array's elements
427 that are less than or equal to the threshold. On the right: CerCo's interaction
428 diagram. CerCo's components are drawn solid.}
431We illustrate the workflow we envisage (on the right of~\autoref{test1})
432on an example program (on the left of~\autoref{test1}).
433The user writes the program and feeds it to the CerCo compiler, which outputs
434an instrumented version of the same program that updates global variables
435that record the elapsed execution time and the stack space usage.
436The red lines in \autoref{itest1} are the instrumentation introduced by the
437compiler. The annotated program can then be enriched with complexity assertions
438in the style of Hoare logic, that are passed to a deductive platform (in our
439case Frama-C). We provide as a Frama-C cost plugin a simple automatic
440synthesiser for complexity assertions (the blue lines in \autoref{itest1}),
441which can be overridden by the user to increase or decrease accuracy. From the
442assertions, a general purpose deductive platform produces proof obligations
443which in turn can be closed by automatic or interactive provers, ending in a
444proof certificate.
446% NB: if you try this example with the live CD you should increase the timeout
448Twelve proof obligations are generated from~\autoref{itest1} (to prove
449that the loop invariant holds after one execution if it holds before,
450to prove that the whole program execution takes at most 1358 cycles,
451etc.).  Note that the synthesised time bound for \lstinline'count',
452$178+214*(1+\lstinline'len')$ cycles, is parametric in the length of
453the array. The CVC3 prover
454closes all obligations within half a minute on routine commodity
455hardware.  A simpler non-parametric version can be solved in a few
460        moredelim=[is][\color{blue}]{$}{$},
461        moredelim=[is][\color{red}]{|}{|}}
464|int __cost = 33, __stack = 5, __stack_max = 5;|
465|void __cost_incr(int incr) { __cost += incr; }|
466|void __stack_incr(int incr) {
467  __stack += incr;
468  __stack_max = __stack_max < __stack ? __stack : __stack_max;
471char a[4] = {3, 2, 7, 14};  char threshold = 4;
473/*@ behavior stack_cost:
474      ensures __stack_max <= __max(\old(__stack_max), 4+\old(__stack));
475      ensures __stack == \old(__stack);
476    behavior time_cost:
477      ensures __cost <= \old(__cost)+(178+214*__max(1+\at(len,Pre), 0));
479int count(unsigned char *p, int len) {
480  char j;  int found = 0;
481  |__stack_incr(4);  __cost_incr(111);|
482  $__l: /* internal */$
483  /*@ for time_cost: loop invariant
484        __cost <= \at(__cost,__l)+
485                  214*(__max(\at((len-j)+1,__l), 0)-__max(1+(len-j), 0));
486      for stack_cost: loop invariant
487        __stack_max == \at(__stack_max,__l);
488      for stack_cost: loop invariant
489        __stack == \at(__stack,__l);
490      loop variant len-j;
491  */
492  for (j = 0; j < len; j++) {
493    |__cost_incr(78);|
494    if (*p <= threshold) { |__cost_incr(136);| found ++; }
495    else { |__cost_incr(114);| }
496    p ++;
497  }
498  |__cost_incr(67);  __stack_incr(-4);|
499  return found;
502/*@ behavior stack_cost:
503      ensures __stack_max <= __max(\old(__stack_max), 6+\old(__stack));
504      ensures __stack == \old(__stack);
505    behavior time_cost:
506      ensures __cost <= \old(__cost)+1358;
508int main(void) {
509  int t;
510  |__stack_incr(2);  __cost_incr(110);|
511  t = count(a,4);
512  |__stack_incr(-2);|
513  return t;
516\caption{The instrumented version of the program in \autoref{test1},
517 with instrumentation added by the CerCo compiler in red and cost invariants
518 added by the CerCo Frama-C plugin in blue. The \texttt{\_\_cost},
519 \texttt{\_\_stack} and \texttt{\_\_stack\_max} variables hold the elapsed time
520in clock cycles and the current and maximum stack usage. Their initial values
521hold the clock cycles spent in initialising the global data before calling
522\texttt{main} and the space required by global data (and thus unavailable for
523the stack).
527\section{Main scientific and technical results}
528We will now review the main results that the CerCo project has achieved.
530% \emph{Dependent labeling~\cite{?}.} The basic labeling approach assumes a
531% bijective mapping between object code and source code O(1) blocks (called basic
532% blocks). This assumption is violated by many program optimizations (e.g. loop
533% peeling and loop unrolling). It also assumes the cost model computed on the
534% object code to be non parametric: every block must be assigned a cost that does
535% not depend on the state. This assumption is violated by stateful hardware like
536% pipelines, caches, branch prediction units. The dependent labeling approach is
537% an extension of the basic labeling approach that allows to handle parametric
538% cost models. We showed how the method allows to deal with loop optimizations and
539% pipelines, and we speculated about its applications to caches.
541% \emph{Techniques to exploit the induced cost model.} Every technique used for
542% the analysis of functional properties of programs can be adapted to analyse the
543% non-functional properties of the source code instrumented by compilers that
544% implement the labeling approach. In order to gain confidence in this claim, we
545% showed how to implement a cost invariant generator combining abstract
546% interpretation with separation logic ideas \cite{separation}. We integrated
547% everything in the Frama-C modular architecture, in order to compute proof
548% obligations from functional and cost invariants and to use automatic theorem
549% provers on them. This is an example of a new technique that is not currently
550% exploited in WCET analysis. It also shows how precise functional invariants
551% benefits the non-functional analysis too. Finally, we show how to fully
552% automatically analyse the reaction time of Lustre nodes that are first compiled
553% to C using a standard Lustre compiler and then processed by a C compiler that
554% implements the labeling approach.
556% \emph{The CerCo compiler.} This is a compiler from a large subset of the C
557% program to 8051/8052 object code,
558% integrating the labeling approach and a static analyser for 8051 executables.
559% The latter can be implemented easily and does not require dependent costs
560% because the 8051 microprocessor is a very simple one, with constant-cost
561% instruction. It was chosen to separate the issue of exact propagation of the
562% cost model from the orthogonal problem of the static analysis of object code
563% that may require approximations or dependent costs. The compiler comes in
564% several versions: some are prototypes implemented directly in OCaml, and they
565% implement both the basic and dependent labeling approaches; the final version
566% is extracted from a Matita certification and at the moment implements only the
567% basic approach.
569\subsection{The (basic) labeling approach}
570The labeling approach is the foundational insight that underlies all the developments in CerCo.
571It allows the tracking of the evolution of
572basic blocks during the compilation process in order to propagate the cost model from the
573object code to the source code without losing precision in the process.
574\paragraph{Problem statement.} Given a source program $P$, we want to obtain an
575instrumented source program $P'$,  written in the same programming language, and
576the object code $O$ such that: 1) $P'$ is obtained by inserting into $P$ some
577additional instructions to update global cost information like the amount of
578time spent during execution or the maximal stack space required; 2) $P$ and $P'$ 
579must have the same functional behaviour, i.e.\ they must produce that same output
580and intermediate observables; 3) $P$ and $O$ must have the same functional
581behaviour; 4) after execution and in interesting points during execution, the
582cost information computed by $P'$ must be an upper bound of the one spent by $O$ 
583to perform the corresponding operations (soundness property); 5) the difference
584between the costs computed by $P'$ and the execution costs of $O$ must be
585bounded by a program-dependent constant (precision property).
587\paragraph{The labeling software components.} We solve the problem in four
588stages \cite{labeling}, implemented by four software components that are used
589in sequence.
591The first component labels the source program $P$ by injecting label emission
592statements in appropriate positions to mark the beginning of basic blocks.
593% The
594% set of labels with their positions is called labeling.
595The syntax and semantics
596of the source programming language is augmented with label emission statements.
597The statement ``EMIT $\ell$'' behaves like a NOP instruction that does not affect the
598program state or control flow, but its execution is observable.
599% Therefore the observables of a run of a program becomes a stream
600% of label emissions: $\ell_1,\ldots,\ell_n$, called the program trace. We clarify the conditions
601% that the labeling must respect later.
603The second component is a labeling preserving compiler. It can be obtained from
604an existing compiler by adding label emission statements to every intermediate
605language and by propagating label emission statements during compilation. The
606compiler is correct if it preserves both the functional behaviour of the program
607and the traces of observables.
608% We may also ask that the function that erases the cost
609% emission statements commute with compilation. This optional property grants that
610% the labeling does not interfere with the original compiler behaviour. A further
611% set of requirements will be added later.
613The third component is a static object code analyser. It takes as input a labeled
614object code and it computes the scope of each of its label emission statements,
615i.e.\ the tree of instructions that may be executed after the statement and before
616a new label emission is encountered.
617It is a tree and not a sequence because the scope
618may contain a branching statement. In order to grant that such a finite tree
619exists, the object code must not contain any loop that is not broken by a label
620emission statement. This is the first requirement of a sound labeling. The
621analyser fails if the labeling is unsound. For each scope, the analyser
622computes an upper bound of the execution time required by the scope, using the
623maximum of the costs of the two branches in case of a conditional statement.
624Finally, the analyser computes the cost of a label by taking the maximum of the
625costs of the scopes of every statement that emits that label.
627The fourth and last component takes in input the cost model computed at step 3
628and the labelled code computed at step 1. It outputs a source program obtained
629by replacing each label emission statement with a statement that increments the
630global cost variable with the cost associated to the label by the cost model. 
631The obtained source code is the instrumented source code.
633\paragraph{Correctness.} Requirements 1, 2 and 3 of the problem statement
634obviously hold, with 2 and 3 being a consequence of the definition of a correct
635labeling preserving compiler. It is also obvious that the value of the global
636cost variable of an instrumented source code is at any time equal to the sum of
637the costs of the labels emitted by the corresponding labelled code. Moreover,
638because the compiler preserves all traces, the sum of the costs of the labels
639emitted in the source and target labelled code are the same. Therefore, to
640satisfy the fourth requirement, we need to grant that the time taken to execute
641the object code is equal to the sum of the costs of the labels emitted by the
642object code. We collect all the necessary conditions for this to happen in the
643definition of a sound labeling: a) all loops must be broken by a cost emission
644statement;  b) all program instructions must be in the scope of some cost
645emission statement. To satisfy also the fifth requirement, additional
646requirements must be imposed on the object code labeling to avoid all uses of
647the maximum in the cost computation: the labeling is precise if every label is
648emitted at most once and both branches of a conditional jump start with a label
649emission statement.
651The correctness and precision of the labeling approach only rely on the
652correctness and precision of the object code labeling. The simplest
653way to achieve them is to impose correctness and precision
654requirements also on the initial labeling produced by the first software
655component, and to ask the compiler to preserve these
656properties too. The latter requirement imposes serious limitations on the
657compilation strategy and optimizations: the compiler may not duplicate any code
658that contains label emission statements, like loop bodies. Therefore several
659loop optimisations like peeling or unrolling are prevented. Moreover, precision
660of the object code labeling is not sufficient \emph{per se} to obtain global
661precision: we implicitly assumed that a precise constant cost can be assigned
662to every instruction. This is not possible in
663the presence of stateful hardware whose state influences the cost of operations,
664like pipelines and caches. In the next subsection we will see an extension of the
665basic labeling approach to cover this situation.
667The labeling approach described in this section can be applied equally well and
668with minor modifications to imperative and functional languages
669\cite{labeling2}. We have tested it on a simple imperative language without
670functions (a `while' language), on a subset of C and on two compilation chains for
671a purely functional higher-order language. The two main changes required to deal
672with functional languages are: 1) because global variables and updates are not
673available, the instrumentation phase produces monadic code to `update' the
674global costs; 2) the requirements for a sound and precise labeling of the
675source code must be changed when the compilation is based on CPS translations.
676In particular, we need to introduce labels emitted before a statement is
677executed and also labels emitted after a statement is executed. The latter capture
678code that is inserted by the CPS translation and that would escape all label
681% Brian: one of the reviewers pointed out that standard Prolog implementations
682% do have some global state that apparently survives backtracking and might be
683% used.  As we haven't experimented with this, I think it's best to elide it
684% entirely.
686% Phases 1, 2 and 3 can be applied as well to logic languages (e.g. Prolog).
687% However, the instrumentation phase cannot: in standard Prolog there is no notion
688% of (global) variable whose state is not retracted during backtracking.
689% Therefore, the cost of executing computations that are later backtracked would
690% not be correctly counted in. Any extension of logic languages with
691% non-backtrackable state could support our labeling approach.
693\subsection{Dependent labeling}
694The core idea of the basic labeling approach is to establish a tight connection
695between basic blocks executed in the source and target languages. Once the
696connection is established, any cost model computed on the object code can be
697transferred to the source code, without affecting the code of the compiler or
698its proof. In particular, it is immediate that we can also transport cost models
699that associate to each label functions from hardware state to natural numbers.
700However, a problem arises during the instrumentation phase that replaces cost
701emission statements with increments of global cost variables. The global cost
702variable must be incremented with the result of applying the function associated
703to the label to the hardware state at the time of execution of the block.
704The hardware state comprises both the functional state that affects the
705computation (the value of the registers and memory) and the non-functional
706state that does not (the pipeline and cache contents for example). The former is
707in correspondence with the source code state, but reconstructing the
708correspondence may be hard and lifting the cost model to work on the source code
709state is likely to produce cost expressions that are too complex to understand and reason about.
710Luckily enough, in all modern architectures the cost of executing single
711instructions is either independent of the functional state or the jitter---the
712difference between the worst and best case execution times---is small enough
713to be bounded without losing too much precision. Therefore we can concentrate
714on dependencies over the `non-functional' parts of the state only.
716The non-functional state has no correspondence in the high level state and does
717not influence the functional properties. What can be done is to expose the
718non-functional state in the source code. We present here the basic
719intuition in a simplified form: the technical details that allow us to handle the
720general case are more complex and can be found in~\cite{paolo}. We add
721to the source code an additional global variable that represents the
722non-functional state and another one that remembers the last labels emitted. The
723state variable must be updated at every label emission statement, using an
724update function which is computed during static analysis too. The update
725function associates to each label a function from the recently emitted labels
726and old state to the new state. It is computed composing the semantics of every
727instruction in a basic block and restricting it to the non-functional part of
728the state.
730Not all the details of the non-functional state needs to be exposed, and the
731technique works better when the part of state that is required can be summarized
732in a simple data structure. For example, to handle simple but realistic
733pipelines it is sufficient to remember a short integer that encodes the position
734of bubbles (stuck instructions) in the pipeline. In any case, it is not necessary
735for the user to understand the meaning of the state to reason over the properties
736of the
737program. Moreover, at any moment the user, or the invariant generator tools that
738analyse the instrumented source code produced by the compiler, can decide to
739trade precision of the analysis for simplicity by approximating the parametric
740cost with safe non parametric bounds. Interestingly, the functional analysis of
741the code can determine which blocks are executed more frequently in order to
742approximate more aggressively the ones that are executed less.
744Dependent labeling can also be applied to allow the compiler to duplicate
745blocks that contain labels (e.g. in loop optimisations)~\cite{paolo}. The effect is to assign
746a different cost to the different occurrences of a duplicated label. For
747example, loop peeling turns a loop into the concatenation of a copy of the loop
748body (that executes the first iteration) with the conditional execution of the
749loop (for the successive iterations). Because of further optimisations, the two
750copies of the loop will be compiled differently, with the first body usually
751taking more time.
753By introducing a variable that keeps track of the iteration number, we can
754associate to the label a cost that is a function of the iteration number. The
755same technique works for loop unrolling without modifications: the function will
756assign a cost to the even iterations and another cost to the odd ones. The
757actual work to be done consists of introducing within the source code, and for each
758loop, a variable that counts the number of iterations. The loop optimisation code
759that duplicate the loop bodies must also modify the code to propagate correctly
760the update of the iteration numbers. The technical details are more complex and
761can be found in the CerCo reports and publications. The implementation, however,
762is quite simple and the changes to a loop optimising compiler are minimal.
764\subsection{Techniques to exploit the induced cost model}
765We review the cost synthesis techniques developed in the project.
766The starting hypothesis is that we have a certified methodology to annotate
767blocks in the source code with constants which provide a sound and sufficiently
768precise upper bound on the cost of executing the blocks after compilation to
769object code.
771The principle that we have followed in designing the cost synthesis tools is
772that the synthetic bounds should be expressed and proved within a general
773purpose tool built to reason on the source code. In particular, we rely on the
774Frama-C tool to reason on C code and on the Coq proof-assistant to reason on
775higher-order functional programs.
777This principle entails that: 1)
778The inferred synthetic bounds are indeed correct as long as the general purpose
779tool is; 2) there is no limitation on the class of programs that can be handled
780as long as the user is willing to carry on an interactive proof.
782Of course, automation is desirable whenever possible. Within this framework,
783automation means writing programs that give hints to the general purpose tool.
784These hints may take the form, say, of loop invariants/variants, of predicates
785describing the structure of the heap, or of types in a light logic. If these
786hints are correct and sufficiently precise the general purpose tool will produce
787a proof automatically, otherwise, user interaction is required.
789\paragraph{The Cost plugin and its application to the Lustre compiler.}
790Frama-C \cite{framac} is a set of analysers for C programs with a
791specification language called ACSL. New analyses can be dynamically added
792via a plugin system. For instance, the Jessie plugin allows deductive
793verification of C programs with respect to their specification in ACSL, with
794various provers as back-end tools.
795We developed the CerCo Cost plugin for the Frama-C platform as a proof of
796concept of an automatic environment exploiting the cost annotations produced by
797the CerCo compiler. It consists of an OCaml program which essentially
798takes the following actions: 1) it receives as input a C program, 2) it
799applies the CerCo compiler to produce a related C program with cost annotations,
8003) it applies some heuristics to produce a tentative bound on the cost of
801executing the C functions of the program as a function of the value of their
802parameters, 4) the user can then call the Jessie plugin to discharge the
803related proof obligations.
804In the following we elaborate on the soundness of the framework and the
805experiments we performed with the Cost tool on the C programs produced by a
806Lustre compiler.
808\paragraph{Soundness.} The soundness of the whole framework depends on the cost
809annotations added by the CerCo compiler, the synthetic costs produced by the
810cost plugin, the verification conditions (VCs) generated by Jessie, and the
811external provers discharging the VCs. The synthetic costs being in ACSL format,
812Jessie can be used to verify them. Thus, even if the added synthetic costs are
813incorrect (relatively to the cost annotations), the process as a whole is still
814correct: indeed, Jessie will not validate incorrect costs and no conclusion can
815be made about the WCET of the program in this case. In other terms, the
816soundness does not really depend on the action of the cost plugin, which can in
817principle produce any synthetic cost. However, in order to be able to actually
818prove a WCET of a C function, we need to add correct annotations in a way that
819Jessie and subsequent automatic provers have enough information to deduce their
820validity. In practice this is not straightforward even for very simple programs
821composed of branching and assignments (no loops and no recursion) because a fine
822analysis of the VCs associated with branching may lead to a complexity blow up.
823\paragraph{Experience with Lustre.} Lustre is a data-flow language for programming
824synchronous systems, with a compiler which targets C. We designed a
825wrapper for supporting Lustre files.
826The C function produced by the compiler implements the step function of the
827synchronous system and computing the WCET of the function amounts to obtain a
828bound on the reaction time of the system. We tested the Cost plugin and the
829Lustre wrapper on the C programs generated by the Lustre compiler. For programs
830consisting of a few hundred lines of code, the cost plugin computes a WCET and Alt-
831Ergo is able to discharge all VCs automatically.
833\paragraph{Handling C programs with simple loops.}
834The cost annotations added by the CerCo compiler take the form of C instructions
835that update by a constant a fresh global variable called the cost variable.
836Synthesizing a WCET of a C function thus consists in statically resolving an
837upper bound of the difference between the value of the cost variable before and
838after the execution of the function, i.e. find in the function the instructions
839that update the cost variable and establish the number of times they are passed
840through during the flow of execution. In order to do the analysis the plugin
841makes the following assumptions on the programs:
8421) there are no recursive functions;
8432) every loop must be annotated with a variant. The variants of `for' loops are
844automatically inferred.
846The plugin proceeds as follows.
847First the call graph of the program is computed.
848Then the computation of the cost of the function is performed by traversing its
849control flow graph. If the function $f$ calls the function $g$ 
850then the function $g$ 
851is processed before the function $f$. The cost at a node is the maximum of the
852costs of the successors.
853In the case of a loop with a body that has a constant cost for every step of the
854loop, the cost is the product of the cost of the body and of the variant taken
855at the start of the loop.
856In the case of a loop with a body whose cost depends on the values of some free
857variables, a fresh logic function $f$ is introduced to represent the cost of the
858loop in the logic assertions. This logic function takes the variant as a first
859parameter. The other parameters of $f$ are the free variables of the body of the
860loop. An axiom is added to account the fact that the cost is accumulated at each
861step of the loop.
862The cost of the function is directly added as post-condition of the function.
864The user can influence the annotation by two different means:
8651) by using more precise variants;
8662) by annotating functions with cost specifications. The plugin will use this cost
867for the function instead of computing it.
868\paragraph{C programs with pointers.}
869When it comes to verifying programs involving pointer-based data structures,
870such as linked lists, trees, or graphs, the use of traditional first-order logic
871to specify, and of SMT solvers to verify, shows some limitations. Separation
872logic~\cite{separation} is an elegant alternative. It is a program logic
873with a new notion of conjunction to express spatial heap separation. Bobot has
874recently introduced automatically generated separation
875predicates to simulate separation logic reasoning in the Jessie plugin where the specification language, the verification condition
876generator, and the theorem provers were not designed with separation logic in
877mind. CerCo's plugin can exploit the separation predicates to automatically
878reason on the cost of execution of simple heap manipulation programs such as an
879in-place list reversal.
881\subsection{The CerCo compiler}
882In CerCo we have developed a certain number of cost preserving compilers based
883on the labeling approach. Excluding an initial certified compiler for a `while'
884language, all remaining compilers target realistic source languages---a pure
885higher order functional language and a large subset of C with pointers, gotos
886and all data structures---and real world target processors---MIPS and the
887Intel 8051/8052 processor family. Moreover, they achieve a level of optimisation
888that ranges from moderate (comparable to GCC level 1) to intermediate (including
889loop peeling and unrolling, hoisting and late constant propagation). The so
890called \emph{Trusted CerCo Compiler} is the only one that was implemented in the
891interactive theorem prover Matita~\cite{matita} and its costs certified. The code distributed
892is extracted OCaml code from the Matita implementation. In the rest of
893this section we will only focus on the Trusted CerCo Compiler, that only targets
894the C language and the 8051/8052 family, and that does not implement any
895advanced optimisations yet. Its user interface, however, is the same as the one
896of the other versions, in order to trade safety with additional performances. In
897particular, the Frama-C CerCo plugin can work without recompilation with all
900The 8051/8052 microprocessor is a very simple one, with constant-cost
901instructions. It was chosen to separate the issue of exact propagation of the
902cost model from the orthogonal problem of the static analysis of object code
903that may require approximations or dependent costs.
905The (trusted) CerCo compiler implements the following optimisations: cast
906simplification, constant propagation in expressions, liveness analysis driven
907spilling of registers, dead code elimination, branch displacement, and tunneling.
908The two latter optimisations are performed by our optimising assembler
909\cite{correctness}. The back-end of the compiler works on three address
910instructions, preferred to static single assignment code for the simplicity of
911the formal certification.
913The CerCo compiler is loosely based on the CompCert compiler \cite{compcert}, a
914recently developed certified compiler from C to the PowerPC, ARM and x86
915microprocessors. Contrary to CompCert, both the CerCo code and its
916certification are open source. Some data structures and language definitions for
917the front-end are directly taken from CompCert, while the back-end is a redesign
918of a compiler from Pascal to MIPS used by Fran\c{c}ois Pottier for a course at the
919Ecole Polytechnique.
921The main peculiarities of the CerCo compiler are the following.
923\item All the intermediate languages include label emitting instructions to
924implement the labeling approach, and the compiler preserves execution traces.
925\item Traditionally compilers target an assembly language with additional
926macro-instructions to be expanded before assembly; for CerCo we need to go all
927the way down to object code in order to perform the static analysis. Therefore
928we integrated also an optimising assembler and a static analyser.
929\item In order to avoid implementing a linker and a loader, we do not support separate
930compilation and external calls. Adding them is a transparent
931process to the labeling approach and should create no unknown problem.
932\item We target an 8-bit processor, in contrast to CompCert's 32-bit targets. Targeting an 8-bit processor requires
933several changes and increased code size, but it is not fundamentally more
934complex. The proof of correctness, however, becomes much harder.
935\item We target a microprocessor that has a non uniform memory model, which is
936still often the case for microprocessors used in embedded systems and that is
937becoming common again in multi-core processors. Therefore the compiler has to
938keep track of data and it must move data between memory regions in the proper
939way. Moreover the size of pointers to different regions is not uniform. An
940additional difficulty was that the space available for the stack in internal
941memory in the 8051 is tiny, allowing only a minor number of nested calls. To
942support full recursion in order to test the CerCo tools also on recursive
943programs, the compiler implements a stack in external memory.
946\subsection{Formal certification of the CerCo compiler}
947We implemented the
948CerCo compiler in the interactive theorem prover Matita and have formally
949certified that the cost model induced on the source code correctly and precisely
950predicts the object code behaviour. We actually induce two cost models, one for
951time and one for stack space consumption. We show the correctness of the prediction
952only for those programs that do not exhaust the available machine stack space, a
953property that---thanks to the stack cost model---we can statically analyse on the
954source code using our Frama-C tool. The preservation of functional properties we
955take as an assumption, not itself formally proved in CerCo.  Other projects have
956already certified the preservation of functional semantics in similar compilers,
957and we have not attempted to directly repeat that work. In order to complete the
958proof for non-functional properties, we have introduced a new semantics for
959programming languages based on a new kind of structured observables with the
960relative notions of forward similarity and the study of the intentional
961consequences of forward similarity. We have also introduced a unified
962representation for back-end intermediate languages that was exploited to provide
963a uniform proof of forward similarity.
965The details on the proof techniques employed
967the proof sketch can be collected from the CerCo deliverables and papers.
968In this section we will only hint at the correctness statement, which turned
969out to be more complex than what we expected at the beginning.
971\paragraph{The statement.}
972Real time programs are often reactive programs that loop forever responding to
973events (inputs) by performing some computation followed by some action (output)
974and the return to the initial state. For looping programs the overall execution
975time does not make sense. The same happens for reactive programs that spend an
976unpredictable amount of time in I/O. What is interesting is the reaction time
977that measure the time spent between I/O events. Moreover, we are interested in
978predicting and ruling out programs that crash running out of space on a certain
980Therefore we need to look for a statement that talks about sub-runs of a
981program. The most natural statement is that the time predicted on the source
982code and spent on the object code by two corresponding sub-runs are the same.
983The problem to solve to make this statement formal is how to identify the
984corresponding sub-runs and how to single out those that are meaningful.
985The solution we found is based on the notion of measurability. We say that a run
986has a \emph{measurable sub-run} when both the prefix of the sub-run and the
987sub-run do not exhaust the stack space, the number of function calls and returns
988in the sub-run is the same, the sub-run does not perform any I/O and the sub-run
989starts with a label emission statement and ends with a return or another label
990emission statements. The stack usage is estimated using the stack usage model
991that is computed by the compiler.
993The statement that we formally proved is: for each C run with a measurable
994sub-run, there exists an object code run with a sub-run, such that the observables
995of the pairs of prefixes and sub-runs are the same and the time spent by the
996object code in the sub-run is the same as the one predicted on the source code
997using the time cost model generated by the compiler.
998We briefly discuss the constraints for measurability. Not exhausting the stack
999space is a clear requirement of meaningfulness of a run, because source programs
1000do not crash for lack of space while object code ones do. The balancing of
1001function calls and returns is a requirement for precision: the labeling approach
1002allows the scope of label emission statements to extend after function calls to
1003minimize the number of labels. Therefore a label pays for all the instructions
1004in a block, excluding those executed in nested function calls. If the number of
1005calls/returns is unbalanced, it means that there is a call we have not returned
1006to that could be followed by additional instructions whose cost has already been
1007taken in account. To make the statement true (but less precise) in this
1008situation, we could only say that the cost predicted on the source code is a
1009safe bound, not that it is exact. The last condition on the entry/exit points of
1010a run is used to identify sub-runs whose code correspond to a sequence of blocks
1011that we can measure precisely. Any other choice would start or end the run in the
1012middle of a block and we would be forced again to weaken the statement taking as
1013a bound the cost obtained counting in all the instructions that precede the
1014starting one in the block, or follow the final one in the block.
1015I/O operations can be performed in the prefix of the run, but not in the
1016measurable sub-run. Therefore we prove that we can predict reaction times, but
1017not I/O times, as it should be.
1019\section{Conclusions and future work}
1021All the CerCo software and deliverables can be found on the CerCo homepage at~\url{}.
1023The results obtained so far are encouraging and provide evidence that
1024it is possible to perform static time and space analysis at the source level
1025without losing accuracy, reducing the trusted code base and reconciling the
1026study of functional and non-functional properties of programs. The
1027techniques introduced seem to be scalable, cover both imperative and
1028functional languages and are compatible with every compiler optimisation
1029considered by us so far.
1031To prove that compilers can keep track of optimisations
1032and induce a precise cost model on the source code, we targeted a simple
1033architecture that admits a cost model that is execution history independent.
1034The most important future work is dealing with hardware architectures
1035characterized by history dependent stateful components, like caches and
1036pipelines. The main issue consists in assigning a parametric, dependent cost
1037to basic blocks that can be later transferred by the labeling approach to
1038the source code and represented in a meaningful way to the user. The dependent
1039labeling approach that we have studied seems a promising tool to achieve
1040this goal, but the cost model generated for a realistic processor could be too
1041large and complex to be exposed in the source code. Further study is required
1042to evaluate the technique on a realistic processor and to introduce early
1043approximations of the cost model to make the technique feasible.
1045Examples of further future work consist in improving the cost invariant
1046generator algorithms and the coverage of compiler optimizations, in combining
1047the labeling approach with the type and effect discipline of~\cite{typeffects}
1048to handle languages with implicit memory management, and in experimenting with
1049our tools in early development phases. Some larger case study is also necessary
1050to evaluate the CerCo's prototype on realistic, industrial-scale programs.
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