source: Papers/fopara2013/fopara13.tex @ 3419

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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, -4};
354char treshold = 4;
356int main() {
357  char j;
358  char *p = a;
359  int found = 0;
360  for (j=0; j < 4; j++) {
361    if (*p <= treshold)
362      { found++; }
363    p++;
364  }
365  return found;
369%  $\vcenter{\includegraphics[width=7.5cm]{interaction_diagram.pdf}}$
371    baseline={([yshift=-.5ex]current bounding box)},
372    element/.style={draw, text width=1.6cm, on chain, text badly centered},
373    >=stealth
374    ]
375{[start chain=going below, node distance=.5cm]
376\node [element] (cerco) {CerCo\\compiler};
377\node [element] (cost)  {CerCo\\cost plugin};
378{[densely dashed]
379\node [element] (ded)   {Deductive\\platform};
380\node [element] (check) {Proof\\checker};
383\coordinate [left=3.5cm of cerco] (left);
384{[every node/.style={above, text width=3.5cm, text badly centered,
385                     font=\scriptsize}]
386\draw [<-] ([yshift=-1ex]cerco.north west) coordinate (t) --
387    node {C source}
388    (t-|left);
389\draw [->] (cerco) -- (cost);
390\draw [->] ([yshift=1ex]cerco.south west) coordinate (t) --
391    node {C source+cost annotations}
392    (t-|left) coordinate (cerco out);
393\draw [->] ([yshift=1ex]cost.south west) coordinate (t) --
394    node {C source+cost annotations\\+synthesized assertions}
395    (t-|left) coordinate (out);
396{[densely dashed]
397\draw [<-] ([yshift=-1ex]ded.north west) coordinate (t) --
398    node {C source+cost annotations\\+complexity assertions}
399    (t-|left) coordinate (ded in) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
400\draw [->] ([yshift=1ex]ded.south west) coordinate (t) --
401    node {complexity obligations}
402    (t-|left) coordinate (out);
403\draw [<-] ([yshift=-1ex]check.north west) coordinate (t) --
404    node {complexity proof}
405    (t-|left) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
406\draw [dash phase=2.5pt] (cerco out) .. controls +(-1, 0) and +(-1, 0) ..
407    (ded in);
409%% user:
410% head
411\draw (current bounding box.west) ++(-.2,.5)
412    circle (.2) ++(0,-.2) -- ++(0,-.1) coordinate (t);
413% arms
414\draw (t) -- +(-.3,-.2);
415\draw (t) -- +(.3,-.2);
416% body
417\draw (t) -- ++(0,-.4) coordinate (t);
418% legs
419\draw (t) -- +(-.2,-.6);
420\draw (t) -- +(.2,-.6);
423\caption{On the left: code to count the array's elements
424 that are less than or equal to the treshold. On the right: CerCo's interaction
425 diagram. CerCo's components are drawn solid.}
428We illustrate the workflow we envisage (on the right of~\autoref{test1})
429on an example program (on the left of~\autoref{test1}).
430The user writes the program and feeds it to the CerCo compiler, which outputs
431an instrumented version of the same program that updates global variables
432that record the elapsed execution time and the stack space usage.
433The red lines in \autoref{itest1} are the instrumentation introduced by the
434compiler. The annotated program can then be enriched with complexity assertions
435in the style of Hoare logic, that are passed to a deductive platform (in our
436case Frama-C). We provide as a Frama-C cost plugin a simple automatic
437synthesiser for complexity assertions (the blue lines in \autoref{itest1}),
438which can be overridden by the user to increase or decrease accuracy. From the
439assertions, a general purpose deductive platform produces proof obligations
440which in turn can be closed by automatic or interactive provers, ending in a
441proof certificate. Nine proof obligations are generated
442from~\autoref{itest1} (to prove that the loop invariant holds after
443one execution if it holds before, to prove that the whole program execution
444takes at most 1228 cycles, etc.). The CVC3 prover closes all obligations
445in a few seconds on routine commodity hardware.
449        moredelim=[is][\color{blue}]{$}{$},
450        moredelim=[is][\color{red}]{|}{|}}
453|int __cost = 33; int __stack = 5, __stack_max = 5;|
454|void __cost_incr(int incr) { __cost = __cost + incr; }|
455|void __stack_incr(int incr) {
456  __stack = __stack + incr;
457  __stack_max = __stack_max < __stack ? __stack : __stack_max;
460char a[4] = { 3, 2, 7, 252, };
461char treshold = 4;
463/*@ behaviour stack_cost:
464      ensures __stack_max <=
465              __max(\old(__stack_max), \old(__stack));
466      ensures __stack == \old(__stack);
468    behaviour time_cost:
469      ensures __cost <= \old(__cost)+1228; */
470int main(void)
472  char j;
473  char *p;
474  int found;
475  |__stack_incr(0); __cost_incr(91);|
476  p = a;
477  found = 0;
478  $__l: /* internal */$
479  /*@ for time_cost: loop invariant
480        __cost <=
481        \at(__cost,__l)+220*(__max(\at(5-j,__l), 0)
482                       -__max (5-j, 0));
483      for stack_cost: loop invariant
484        __stack_max == \at(__stack_max,__l);
485      for stack_cost: loop invariant
486        __stack == \at(__stack,__l);
487      loop variant 4-j; */
488  for (j = 0; j < 4; j++) {
489    |__cost_incr(76);|
490    if (*p <= treshold) {
491      |__cost_incr(144);|
492      found++;
493    } else {
494      |__cost_incr(122);|
495    }
496    p++;
497  }
498  |__cost_incr(37); __stack_incr(-0);|
499  return found;
502\caption{The instrumented version of the program in \autoref{test1},
503 with instrumentation added by the CerCo compiler in red and cost invariants
504 added by the CerCo Frama-C plugin in blue. The \texttt{\_\_cost},
505 \texttt{\_\_stack} and \texttt{\_\_stack\_max} variables hold the elapsed time
506in clock cycles and the current and maximum stack usage. Their initial values
507hold the clock cycles spent in initialising the global data before calling
508\texttt{main} and the space required by global data (and thus unavailable for
509the stack).
513\section{Main scientific and technical results}
514We will now review the main results that the CerCo project has achieved.
516% \emph{Dependent labeling~\cite{?}.} The basic labeling approach assumes a
517% bijective mapping between object code and source code O(1) blocks (called basic
518% blocks). This assumption is violated by many program optimizations (e.g. loop
519% peeling and loop unrolling). It also assumes the cost model computed on the
520% object code to be non parametric: every block must be assigned a cost that does
521% not depend on the state. This assumption is violated by stateful hardware like
522% pipelines, caches, branch prediction units. The dependent labeling approach is
523% an extension of the basic labeling approach that allows to handle parametric
524% cost models. We showed how the method allows to deal with loop optimizations and
525% pipelines, and we speculated about its applications to caches.
527% \emph{Techniques to exploit the induced cost model.} Every technique used for
528% the analysis of functional properties of programs can be adapted to analyse the
529% non-functional properties of the source code instrumented by compilers that
530% implement the labeling approach. In order to gain confidence in this claim, we
531% showed how to implement a cost invariant generator combining abstract
532% interpretation with separation logic ideas \cite{separation}. We integrated
533% everything in the Frama-C modular architecture, in order to compute proof
534% obligations from functional and cost invariants and to use automatic theorem
535% provers on them. This is an example of a new technique that is not currently
536% exploited in WCET analysis. It also shows how precise functional invariants
537% benefits the non-functional analysis too. Finally, we show how to fully
538% automatically analyse the reaction time of Lustre nodes that are first compiled
539% to C using a standard Lustre compiler and then processed by a C compiler that
540% implements the labeling approach.
542% \emph{The CerCo compiler.} This is a compiler from a large subset of the C
543% program to 8051/8052 object code,
544% integrating the labeling approach and a static analyser for 8051 executables.
545% The latter can be implemented easily and does not require dependent costs
546% because the 8051 microprocessor is a very simple one, with constant-cost
547% instruction. It was chosen to separate the issue of exact propagation of the
548% cost model from the orthogonal problem of the static analysis of object code
549% that may require approximations or dependent costs. The compiler comes in
550% several versions: some are prototypes implemented directly in OCaml, and they
551% implement both the basic and dependent labeling approaches; the final version
552% is extracted from a Matita certification and at the moment implements only the
553% basic approach.
555\subsection{The (basic) labeling approach}
556The labeling approach is the foundational insight that underlies all the developments in CerCo.
557It allows the tracking of the evolution of
558basic blocks during the compilation process in order to propagate the cost model from the
559object code to the source code without losing precision in the process.
560\paragraph{Problem statement.} Given a source program $P$, we want to obtain an
561instrumented source program $P'$,  written in the same programming language, and
562the object code $O$ such that: 1) $P'$ is obtained by inserting into $P$ some
563additional instructions to update global cost information like the amount of
564time spent during execution or the maximal stack space required; 2) $P$ and $P'$ 
565must have the same functional behaviour, i.e.\ they must produce that same output
566and intermediate observables; 3) $P$ and $O$ must have the same functional
567behaviour; 4) after execution and in interesting points during execution, the
568cost information computed by $P'$ must be an upper bound of the one spent by $O$ 
569to perform the corresponding operations (soundness property); 5) the difference
570between the costs computed by $P'$ and the execution costs of $O$ must be
571bounded by a program-dependent constant (precision property).
573\paragraph{The labeling software components.} We solve the problem in four
574stages \cite{labeling}, implemented by four software components that are used
575in sequence.
577The first component labels the source program $P$ by injecting label emission
578statements in appropriate positions to mark the beginning of basic blocks.
579% The
580% set of labels with their positions is called labeling.
581The syntax and semantics
582of the source programming language is augmented with label emission statements.
583The statement ``EMIT $\ell$'' behaves like a NOP instruction that does not affect the
584program state or control flow, but its execution is observable.
585% Therefore the observables of a run of a program becomes a stream
586% of label emissions: $\ell_1,\ldots,\ell_n$, called the program trace. We clarify the conditions
587% that the labeling must respect later.
589The second component is a labeling preserving compiler. It can be obtained from
590an existing compiler by adding label emission statements to every intermediate
591language and by propagating label emission statements during compilation. The
592compiler is correct if it preserves both the functional behaviour of the program
593and the traces of observables.
594% We may also ask that the function that erases the cost
595% emission statements commute with compilation. This optional property grants that
596% the labeling does not interfere with the original compiler behaviour. A further
597% set of requirements will be added later.
599The third component is a static object code analyser. It takes as input a labeled
600object code and it computes the scope of each of its label emission statements,
601i.e.\ the tree of instructions that may be executed after the statement and before
602a new label emission is encountered.
603It is a tree and not a sequence because the scope
604may contain a branching statement. In order to grant that such a finite tree
605exists, the object code must not contain any loop that is not broken by a label
606emission statement. This is the first requirement of a sound labeling. The
607analyser fails if the labeling is unsound. For each scope, the analyser
608computes an upper bound of the execution time required by the scope, using the
609maximum of the costs of the two branches in case of a conditional statement.
610Finally, the analyser computes the cost of a label by taking the maximum of the
611costs of the scopes of every statement that emits that label.
613The fourth and last component takes in input the cost model computed at step 3
614and the labelled code computed at step 1. It outputs a source program obtained
615by replacing each label emission statement with a statement that increments the
616global cost variable with the cost associated to the label by the cost model. 
617The obtained source code is the instrumented source code.
619\paragraph{Correctness.} Requirements 1, 2 and 3 of the problem statement
620obviously hold, with 2 and 3 being a consequence of the definition of a correct
621labeling preserving compiler. It is also obvious that the value of the global
622cost variable of an instrumented source code is at any time equal to the sum of
623the costs of the labels emitted by the corresponding labelled code. Moreover,
624because the compiler preserves all traces, the sum of the costs of the labels
625emitted in the source and target labelled code are the same. Therefore, to
626satisfy the fourth requirement, we need to grant that the time taken to execute
627the object code is equal to the sum of the costs of the labels emitted by the
628object code. We collect all the necessary conditions for this to happen in the
629definition of a sound labeling: a) all loops must be broken by a cost emission
630statement;  b) all program instructions must be in the scope of some cost
631emission statement. To satisfy also the fifth requirement, additional
632requirements must be imposed on the object code labeling to avoid all uses of
633the maximum in the cost computation: the labeling is precise if every label is
634emitted at most once and both branches of a conditional jump start with a label
635emission statement.
637The correctness and precision of the labeling approach only rely on the
638correctness and precision of the object code labeling. The simplest
639way to achieve them is to impose correctness and precision
640requirements also on the initial labeling produced by the first software
641component, and to ask the compiler to preserve these
642properties too. The latter requirement imposes serious limitations on the
643compilation strategy and optimizations: the compiler may not duplicate any code
644that contains label emission statements, like loop bodies. Therefore several
645loop optimisations like peeling or unrolling are prevented. Moreover, precision
646of the object code labeling is not sufficient \emph{per se} to obtain global
647precision: we implicitly assumed that a precise constant cost can be assigned
648to every instruction. This is not possible in
649the presence of stateful hardware whose state influences the cost of operations,
650like pipelines and caches. In the next subsection we will see an extension of the
651basic labeling approach to cover this situation.
653The labeling approach described in this section can be applied equally well and
654with minor modifications to imperative and functional languages
655\cite{labeling2}. We have tested it on a simple imperative language without
656functions (a `while' language), on a subset of C and on two compilation chains for
657a purely functional higher-order language. The two main changes required to deal
658with functional languages are: 1) because global variables and updates are not
659available, the instrumentation phase produces monadic code to `update' the
660global costs; 2) the requirements for a sound and precise labeling of the
661source code must be changed when the compilation is based on CPS translations.
662In particular, we need to introduce labels emitted before a statement is
663executed and also labels emitted after a statement is executed. The latter capture
664code that is inserted by the CPS translation and that would escape all label
667% Brian: one of the reviewers pointed out that standard Prolog implementations
668% do have some global state that apparently survives backtracking and might be
669% used.  As we haven't experimented with this, I think it's best to elide it
670% entirely.
672% Phases 1, 2 and 3 can be applied as well to logic languages (e.g. Prolog).
673% However, the instrumentation phase cannot: in standard Prolog there is no notion
674% of (global) variable whose state is not retracted during backtracking.
675% Therefore, the cost of executing computations that are later backtracked would
676% not be correctly counted in. Any extension of logic languages with
677% non-backtrackable state could support our labeling approach.
679\subsection{Dependent labeling}
680The core idea of the basic labeling approach is to establish a tight connection
681between basic blocks executed in the source and target languages. Once the
682connection is established, any cost model computed on the object code can be
683transferred to the source code, without affecting the code of the compiler or
684its proof. In particular, it is immediate that we can also transport cost models
685that associate to each label functions from hardware state to natural numbers.
686However, a problem arises during the instrumentation phase that replaces cost
687emission statements with increments of global cost variables. The global cost
688variable must be incremented with the result of applying the function associated
689to the label to the hardware state at the time of execution of the block.
690The hardware state comprises both the functional state that affects the
691computation (the value of the registers and memory) and the non-functional
692state that does not (the pipeline and cache contents for example). The former is
693in correspondence with the source code state, but reconstructing the
694correspondence may be hard and lifting the cost model to work on the source code
695state is likely to produce cost expressions that are too complex to understand and reason about.
696Luckily enough, in all modern architectures the cost of executing single
697instructions is either independent of the functional state or the jitter---the
698difference between the worst and best case execution times---is small enough
699to be bounded without losing too much precision. Therefore we can concentrate
700on dependencies over the `non-functional' parts of the state only.
702The non-functional state has no correspondence in the high level state and does
703not influence the functional properties. What can be done is to expose the
704non-functional state in the source code. We present here the basic
705intuition in a simplified form: the technical details that allow us to handle the
706general case are more complex and can be found in~\cite{paolo}. We add
707to the source code an additional global variable that represents the
708non-functional state and another one that remembers the last labels emitted. The
709state variable must be updated at every label emission statement, using an
710update function which is computed during static analysis too. The update
711function associates to each label a function from the recently emitted labels
712and old state to the new state. It is computed composing the semantics of every
713instruction in a basic block and restricting it to the non-functional part of
714the state.
716Not all the details of the non-functional state needs to be exposed, and the
717technique works better when the part of state that is required can be summarized
718in a simple data structure. For example, to handle simple but realistic
719pipelines it is sufficient to remember a short integer that encodes the position
720of bubbles (stuck instructions) in the pipeline. In any case, it is not necessary
721for the user to understand the meaning of the state to reason over the properties
722of the
723program. Moreover, at any moment the user, or the invariant generator tools that
724analyse the instrumented source code produced by the compiler, can decide to
725trade precision of the analysis for simplicity by approximating the parametric
726cost with safe non parametric bounds. Interestingly, the functional analysis of
727the code can determine which blocks are executed more frequently in order to
728approximate more aggressively the ones that are executed less.
730Dependent labeling can also be applied to allow the compiler to duplicate
731blocks that contain labels (e.g. in loop optimisations)~\cite{paolo}. The effect is to assign
732a different cost to the different occurrences of a duplicated label. For
733example, loop peeling turns a loop into the concatenation of a copy of the loop
734body (that executes the first iteration) with the conditional execution of the
735loop (for the successive iterations). Because of further optimisations, the two
736copies of the loop will be compiled differently, with the first body usually
737taking more time.
739By introducing a variable that keeps track of the iteration number, we can
740associate to the label a cost that is a function of the iteration number. The
741same technique works for loop unrolling without modifications: the function will
742assign a cost to the even iterations and another cost to the odd ones. The
743actual work to be done consists of introducing within the source code, and for each
744loop, a variable that counts the number of iterations. The loop optimisation code
745that duplicate the loop bodies must also modify the code to propagate correctly
746the update of the iteration numbers. The technical details are more complex and
747can be found in the CerCo reports and publications. The implementation, however,
748is quite simple and the changes to a loop optimising compiler are minimal.
750\subsection{Techniques to exploit the induced cost model}
751We review the cost synthesis techniques developed in the project.
752The starting hypothesis is that we have a certified methodology to annotate
753blocks in the source code with constants which provide a sound and sufficiently
754precise upper bound on the cost of executing the blocks after compilation to
755object code.
757The principle that we have followed in designing the cost synthesis tools is
758that the synthetic bounds should be expressed and proved within a general
759purpose tool built to reason on the source code. In particular, we rely on the
760Frama-C tool to reason on C code and on the Coq proof-assistant to reason on
761higher-order functional programs.
763This principle entails that: 1)
764The inferred synthetic bounds are indeed correct as long as the general purpose
765tool is; 2) there is no limitation on the class of programs that can be handled
766as long as the user is willing to carry on an interactive proof.
768Of course, automation is desirable whenever possible. Within this framework,
769automation means writing programs that give hints to the general purpose tool.
770These hints may take the form, say, of loop invariants/variants, of predicates
771describing the structure of the heap, or of types in a light logic. If these
772hints are correct and sufficiently precise the general purpose tool will produce
773a proof automatically, otherwise, user interaction is required.
775\paragraph{The Cost plugin and its application to the Lustre compiler.}
776Frama-C \cite{framac} is a set of analysers for C programs with a
777specification language called ACSL. New analyses can be dynamically added
778via a plugin system. For instance, the Jessie plugin allows deductive
779verification of C programs with respect to their specification in ACSL, with
780various provers as back-end tools.
781We developed the CerCo Cost plugin for the Frama-C platform as a proof of
782concept of an automatic environment exploiting the cost annotations produced by
783the CerCo compiler. It consists of an OCaml program which essentially
784takes the following actions: 1) it receives as input a C program, 2) it
785applies the CerCo compiler to produce a related C program with cost annotations,
7863) it applies some heuristics to produce a tentative bound on the cost of
787executing the C functions of the program as a function of the value of their
788parameters, 4) the user can then call the Jessie plugin to discharge the
789related proof obligations.
790In the following we elaborate on the soundness of the framework and the
791experiments we performed with the Cost tool on the C programs produced by a
792Lustre compiler.
794\paragraph{Soundness.} The soundness of the whole framework depends on the cost
795annotations added by the CerCo compiler, the synthetic costs produced by the
796cost plugin, the verification conditions (VCs) generated by Jessie, and the
797external provers discharging the VCs. The synthetic costs being in ACSL format,
798Jessie can be used to verify them. Thus, even if the added synthetic costs are
799incorrect (relatively to the cost annotations), the process as a whole is still
800correct: indeed, Jessie will not validate incorrect costs and no conclusion can
801be made about the WCET of the program in this case. In other terms, the
802soundness does not really depend on the action of the cost plugin, which can in
803principle produce any synthetic cost. However, in order to be able to actually
804prove a WCET of a C function, we need to add correct annotations in a way that
805Jessie and subsequent automatic provers have enough information to deduce their
806validity. In practice this is not straightforward even for very simple programs
807composed of branching and assignments (no loops and no recursion) because a fine
808analysis of the VCs associated with branching may lead to a complexity blow up.
809\paragraph{Experience with Lustre.} Lustre is a data-flow language for programming
810synchronous systems, with a compiler which targets C. We designed a
811wrapper for supporting Lustre files.
812The C function produced by the compiler implements the step function of the
813synchronous system and computing the WCET of the function amounts to obtain a
814bound on the reaction time of the system. We tested the Cost plugin and the
815Lustre wrapper on the C programs generated by the Lustre compiler. For programs
816consisting of a few hundred lines of code, the cost plugin computes a WCET and Alt-
817Ergo is able to discharge all VCs automatically.
819\paragraph{Handling C programs with simple loops.}
820The cost annotations added by the CerCo compiler take the form of C instructions
821that update by a constant a fresh global variable called the cost variable.
822Synthesizing a WCET of a C function thus consists in statically resolving an
823upper bound of the difference between the value of the cost variable before and
824after the execution of the function, i.e. find in the function the instructions
825that update the cost variable and establish the number of times they are passed
826through during the flow of execution. In order to do the analysis the plugin
827makes the following assumptions on the programs:
8281) there are no recursive functions;
8292) every loop must be annotated with a variant. The variants of `for' loops are
830automatically inferred.
832The plugin proceeds as follows.
833First the call graph of the program is computed.
834Then the computation of the cost of the function is performed by traversing its
835control flow graph. If the function $f$ calls the function $g$ 
836then the function $g$ 
837is processed before the function $f$. The cost at a node is the maximum of the
838costs of the successors.
839In the case of a loop with a body that has a constant cost for every step of the
840loop, the cost is the product of the cost of the body and of the variant taken
841at the start of the loop.
842In the case of a loop with a body whose cost depends on the values of some free
843variables, a fresh logic function $f$ is introduced to represent the cost of the
844loop in the logic assertions. This logic function takes the variant as a first
845parameter. The other parameters of $f$ are the free variables of the body of the
846loop. An axiom is added to account the fact that the cost is accumulated at each
847step of the loop.
848The cost of the function is directly added as post-condition of the function.
850The user can influence the annotation by two different means:
8511) by using more precise variants;
8522) by annotating functions with cost specifications. The plugin will use this cost
853for the function instead of computing it.
854\paragraph{C programs with pointers.}
855When it comes to verifying programs involving pointer-based data structures,
856such as linked lists, trees, or graphs, the use of traditional first-order logic
857to specify, and of SMT solvers to verify, shows some limitations. Separation
858logic~\cite{separation} is an elegant alternative. It is a program logic
859with a new notion of conjunction to express spatial heap separation. Bobot has
860recently introduced automatically generated separation
861predicates to simulate separation logic reasoning in the Jessie plugin where the specification language, the verification condition
862generator, and the theorem provers were not designed with separation logic in
863mind. CerCo's plugin can exploit the separation predicates to automatically
864reason on the cost of execution of simple heap manipulation programs such as an
865in-place list reversal.
867\subsection{The CerCo compiler}
868In CerCo we have developed a certain number of cost preserving compilers based
869on the labeling approach. Excluding an initial certified compiler for a `while'
870language, all remaining compilers target realistic source languages---a pure
871higher order functional language and a large subset of C with pointers, gotos
872and all data structures---and real world target processors---MIPS and the
873Intel 8051/8052 processor family. Moreover, they achieve a level of optimisation
874that ranges from moderate (comparable to GCC level 1) to intermediate (including
875loop peeling and unrolling, hoisting and late constant propagation). The so
876called \emph{Trusted CerCo Compiler} is the only one that was implemented in the
877interactive theorem prover Matita~\cite{matita} and its costs certified. The code distributed
878is extracted OCaml code from the Matita implementation. In the rest of
879this section we will only focus on the Trusted CerCo Compiler, that only targets
880the C language and the 8051/8052 family, and that does not implement any
881advanced optimisations yet. Its user interface, however, is the same as the one
882of the other versions, in order to trade safety with additional performances. In
883particular, the Frama-C CerCo plugin can work without recompilation with all
886The 8051/8052 microprocessor is a very simple one, with constant-cost
887instructions. It was chosen to separate the issue of exact propagation of the
888cost model from the orthogonal problem of the static analysis of object code
889that may require approximations or dependent costs.
891The (trusted) CerCo compiler implements the following optimisations: cast
892simplification, constant propagation in expressions, liveness analysis driven
893spilling of registers, dead code elimination, branch displacement, and tunneling.
894The two latter optimisations are performed by our optimising assembler
895\cite{correctness}. The back-end of the compiler works on three address
896instructions, preferred to static single assignment code for the simplicity of
897the formal certification.
899The CerCo compiler is loosely based on the CompCert compiler \cite{compcert}, a
900recently developed certified compiler from C to the PowerPC, ARM and x86
901microprocessors. Contrary to CompCert, both the CerCo code and its
902certification are open source. Some data structures and language definitions for
903the front-end are directly taken from CompCert, while the back-end is a redesign
904of a compiler from Pascal to MIPS used by Fran\c{c}ois Pottier for a course at the
905Ecole Polytechnique.
907The main peculiarities of the CerCo compiler are the following.
909\item All the intermediate languages include label emitting instructions to
910implement the labeling approach, and the compiler preserves execution traces.
911\item Traditionally compilers target an assembly language with additional
912macro-instructions to be expanded before assembly; for CerCo we need to go all
913the way down to object code in order to perform the static analysis. Therefore
914we integrated also an optimising assembler and a static analyser.
915\item In order to avoid implementing a linker and a loader, we do not support separate
916compilation and external calls. Adding them is a transparent
917process to the labeling approach and should create no unknown problem.
918\item We target an 8-bit processor, in contrast to CompCert's 32-bit targets. Targeting an 8-bit processor requires
919several changes and increased code size, but it is not fundamentally more
920complex. The proof of correctness, however, becomes much harder.
921\item We target a microprocessor that has a non uniform memory model, which is
922still often the case for microprocessors used in embedded systems and that is
923becoming common again in multi-core processors. Therefore the compiler has to
924keep track of data and it must move data between memory regions in the proper
925way. Moreover the size of pointers to different regions is not uniform. An
926additional difficulty was that the space available for the stack in internal
927memory in the 8051 is tiny, allowing only a minor number of nested calls. To
928support full recursion in order to test the CerCo tools also on recursive
929programs, the compiler implements a stack in external memory.
932\subsection{Formal certification of the CerCo compiler}
933We implemented the
934CerCo compiler in the interactive theorem prover Matita and have formally
935certified that the cost model induced on the source code correctly and precisely
936predicts the object code behaviour. We actually induce two cost models, one for
937time and one for stack space consumption. We show the correctness of the prediction
938only for those programs that do not exhaust the available machine stack space, a
939property that---thanks to the stack cost model---we can statically analyse on the
940source code using our Frama-C tool. The preservation of functional properties we
941take as an assumption, not itself formally proved in CerCo.  Other projects have
942already certified the preservation of functional semantics in similar compilers,
943and we have not attempted to directly repeat that work. In order to complete the
944proof for non-functional properties, we have introduced a new semantics for
945programming languages based on a new kind of structured observables with the
946relative notions of forward similarity and the study of the intentional
947consequences of forward similarity. We have also introduced a unified
948representation for back-end intermediate languages that was exploited to provide
949a uniform proof of forward similarity.
951The details on the proof techniques employed
953the proof sketch can be collected from the CerCo deliverables and papers.
954In this section we will only hint at the correctness statement, which turned
955out to be more complex than what we expected at the beginning.
957\paragraph{The statement.}
958Real time programs are often reactive programs that loop forever responding to
959events (inputs) by performing some computation followed by some action (output)
960and the return to the initial state. For looping programs the overall execution
961time does not make sense. The same happens for reactive programs that spend an
962unpredictable amount of time in I/O. What is interesting is the reaction time
963that measure the time spent between I/O events. Moreover, we are interested in
964predicting and ruling out programs that crash running out of space on a certain
966Therefore we need to look for a statement that talks about sub-runs of a
967program. The most natural statement is that the time predicted on the source
968code and spent on the object code by two corresponding sub-runs are the same.
969The problem to solve to make this statement formal is how to identify the
970corresponding sub-runs and how to single out those that are meaningful.
971The solution we found is based on the notion of measurability. We say that a run
972has a \emph{measurable sub-run} when both the prefix of the sub-run and the
973sub-run do not exhaust the stack space, the number of function calls and returns
974in the sub-run is the same, the sub-run does not perform any I/O and the sub-run
975starts with a label emission statement and ends with a return or another label
976emission statements. The stack usage is estimated using the stack usage model
977that is computed by the compiler.
979The statement that we formally proved is: for each C run with a measurable
980sub-run, there exists an object code run with a sub-run, such that the observables
981of the pairs of prefixes and sub-runs are the same and the time spent by the
982object code in the sub-run is the same as the one predicted on the source code
983using the time cost model generated by the compiler.
984We briefly discuss the constraints for measurability. Not exhausting the stack
985space is a clear requirement of meaningfulness of a run, because source programs
986do not crash for lack of space while object code ones do. The balancing of
987function calls and returns is a requirement for precision: the labeling approach
988allows the scope of label emission statements to extend after function calls to
989minimize the number of labels. Therefore a label pays for all the instructions
990in a block, excluding those executed in nested function calls. If the number of
991calls/returns is unbalanced, it means that there is a call we have not returned
992to that could be followed by additional instructions whose cost has already been
993taken in account. To make the statement true (but less precise) in this
994situation, we could only say that the cost predicted on the source code is a
995safe bound, not that it is exact. The last condition on the entry/exit points of
996a run is used to identify sub-runs whose code correspond to a sequence of blocks
997that we can measure precisely. Any other choice would start or end the run in the
998middle of a block and we would be forced again to weaken the statement taking as
999a bound the cost obtained counting in all the instructions that precede the
1000starting one in the block, or follow the final one in the block.
1001I/O operations can be performed in the prefix of the run, but not in the
1002measurable sub-run. Therefore we prove that we can predict reaction times, but
1003not I/O times, as it should be.
1005\section{Conclusions and future work}
1007All the CerCo software and deliverables can be found on the CerCo homepage at~\url{}.
1009The results obtained so far are encouraging and provide evidence that
1010it is possible to perform static time and space analysis at the source level
1011without losing accuracy, reducing the trusted code base and reconciling the
1012study of functional and non-functional properties of programs. The
1013techniques introduced seem to be scalable, cover both imperative and
1014functional languages and are compatible with every compiler optimisation
1015considered by us so far.
1017To prove that compilers can keep track of optimisations
1018and induce a precise cost model on the source code, we targeted a simple
1019architecture that admits a cost model that is execution history independent.
1020The most important future work is dealing with hardware architectures
1021characterized by history dependent stateful components, like caches and
1022pipelines. The main issue consists in assigning a parametric, dependent cost
1023to basic blocks that can be later transferred by the labeling approach to
1024the source code and represented in a meaningful way to the user. The dependent
1025labeling approach that we have studied seems a promising tool to achieve
1026this goal, but the cost model generated for a realistic processor could be too
1027large and complex to be exposed in the source code. Further study is required
1028to evaluate the technique on a realistic processor and to introduce early
1029approximations of the cost model to make the technique feasible.
1031Examples of further future work consist in improving the cost invariant
1032generator algorithms and the coverage of compiler optimizations, in combining
1033the labeling approach with the type and effect discipline of~\cite{typeffects}
1034to handle languages with implicit memory management, and in experimenting with
1035our tools in early development phases. Some larger case study is also necessary
1036to evaluate the CerCo's prototype on realistic, industrial-scale programs.
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