source: Papers/fopara2013/fopara13.tex @ 3428

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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}}
47\author{
48%Roberto
49R.~M. Amadio$^{4}$ \and
50%Nicolas
51N.~Ayache$^{3,4}$ \and
52%François
53F.~Bobot$^{3,4}$ \and
54%Jacob
55J.~P. Boender$^1$ \and
56%Brian
57B.~Campbell$^2$ \and
58%Ilias
59I.~Garnier$^2$ \and
60%Antoine
61A.~Madet$^4$ \and
62%James
63J.~McKinna$^2$ \and
64%Dominic
65D.~P. Mulligan$^1$ \and
66%Mauro
67M.~Piccolo$^1$ \and
68%Randy
69R.~Pollack$^2$ \and
70%Yann
71Y.~R\'egis-Gianas$^{3,4}$ \and
72%Claudio
73C.~Sacerdoti Coen$^1$ \and
74%Ian
75I.~Stark$^2$ \and
76%Paolo
77P.~Tranquilli$^1$}
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 )
82\and
83Universit\`e Paris Diderot
84}
85
86\bibliographystyle{splncs03}
87
88\begin{document}
89
90\maketitle
91
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.
95\begin{abstract}
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
107code.
108\end{abstract}
109
110% ---------------------------------------------------------------------------- %
111% SECTION                                                                      %
112% ---------------------------------------------------------------------------- %
113\section{Introduction}
114
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.
124
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.
134
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.
138%
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.
144
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
158components.
159
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.
173
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.
185
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.
203
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}.
210
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.
220
221\subsection{Current object-code methods}
222
223Analysis currently takes place on object code for two main reasons.
224First, there cannot be a uniform, precise cost model for source
225code instructions (or even basic blocks). During compilation, high level
226instructions are torn apart and reassembled in context-specific ways so that
227identifying a fragment of object code and a single high level instruction is
228infeasible. Even the control flow of the object and source code can be very
229different as a result of optimisations, for example aggressive loop
230optimisations may completely transform source level loops. Despite the lack of a uniform, compilation- and
231program-independent cost model on the source language, the literature on the
232analysis of non-asymptotic execution time on high level languages that assumes
233such a model is growing and gaining momentum. However, unless we can provide a
234replacement for such cost models, this literature's future practical impact looks
235to be minimal. Some hope has been provided by the EmBounded project \cite{embounded}, which
236compositionally compiles high-level code to a byte code that is executed by an
237emulator with guarantees on the maximal execution time spent for each byte code
238instruction. That approach provides a uniform model at the price of the model's
239precision (each cost is a pessimistic upper bound) and performance of the
240executed code (because the byte code is emulated  compositionally instead of
241performing a fully non-compositional compilation).
242
243The second reason to perform the analysis on the object code is that bounding
244the worst case execution time of small code fragments in isolation (e.g. loop
245bodies) and then adding up the bounds yields very poor estimates because no
246knowledge on the hardware state can be assumed when executing the fragment. By
247analysing longer runs the bound obtained becomes more precise because the lack
248of knowledge on the initial state has less of an effect on longer computations.
249
250The cost of an execution is the sum of the cost of basic blocks multiplied by
251the number of times they are executed, which is a functional property of the
252program. Therefore, in order to perform (parametric) time analysis of programs,
253it is necessary to combine a cost model with control and data flow analysis.
254Current state of the art WCET technology such as AbsInt
255(see~\cite{stateart} for a survey) performs the analysis on the object code, where the logic of the
256program is harder to reconstruct and most information available at the source
257code level has been lost. Imprecision in the analysis leads to useless bounds. To
258augment precision, the tools ask the user to provide constraints on the object
259code control flow, usually in the form of bounds on the number of iterations of
260loops or linear inequalities on them. This requires the user to manually link
261the source and object code, translating his assumptions on the source code
262(which may be wrong) to object code constraints. The task is error prone and
263hard, especially in the presence of complex compiler optimisations.
264
265Traditional techniques for WCET that work on object code are also affected by
266another problem: they cannot be applied before the generation of the object
267code. Functional properties can be analysed in early development stages, while
268analysis of non-functional properties may come too late to avoid expensive
269changes to the program architecture.
270
271\subsection{CerCo}
272
273In CerCo we propose a radically new approach to the problem: we reject the idea
274of a uniform cost model and we propose that the compiler, which knows how the
275code is translated, must return the cost model for basic blocks of high level
276instructions. It must do so by keeping track of the control flow modifications
277to reverse them and by interfacing with static analysers.
278
279By embracing compilation, instead of avoiding it like EmBounded did, a CerCo
280compiler can at the same time produce efficient code and return costs that are
281as precise as the static analysis can be. Moreover, we allow our costs to be
282parametric: the cost of a block can depend on actual program data or on a
283summary of the execution history or on an approximated representation of the
284hardware state. For example, loop optimizations may assign to a loop body a cost
285that is a function of the number of iterations performed. As another example,
286the cost of a block may be a function of the vector of stalled pipeline states,
287which can be exposed in the source code and updated at each basic block exit. It
288is parametricity that allows one to analyse small code fragments without losing
289precision: in the analysis of the code fragment we do not have to ignore the
290initial hardware state. On the contrary, we can assume that we know exactly which
291state (or mode, as the WCET literature calls it) we are in.
292
293The CerCo approach has the potential to dramatically improve the state of the
294art. By performing control and data flow analyses on the source code, the error
295prone translation of invariants is completely avoided. It is in fact performed
296by the compiler itself when it induces the cost model on the source language.
297Moreover, any available technique for the verification of functional properties
298can be immediately reused and multiple techniques can collaborate together to
299infer and certify cost invariants for the program. Parametric cost analysis
300becomes the default one, with non parametric bounds used as last resorts when
301trading the complexity of the analysis with its precision. \emph{A priori}, no
302technique previously used in traditional WCET is lost: they can just be applied
303at the source code level.
304 Our approach can also work in the early
305stages of development by axiomatically attaching costs to unimplemented components.
306
307
308All software used to verify properties of programs must be as bug free as
309possible. The trusted code base for verification is made by the code that needs
310to be trusted to believe that the property holds. The trusted code base of
311state-of-the-art WCET tools is very large: one needs to trust the control flow
312analyser and the linear programming libraries it uses and also the formal models
313of the hardware. In CerCo we are moving the control flow analysis to the source
314code and we are introducing a non-standard compiler too. To reduce the trusted
315code base, we implemented a prototype and a static analyser in an interactive
316theorem prover, which was used to certify that the cost computed on the source
317code is indeed the one actually spent by the hardware. Formal models of the
318hardware and of the high level source languages were also implemented in the
319interactive theorem prover. Control flow analysis on the source code has been
320obtained using invariant generators, tools to produce proof obligations from
321generated invariants and automatic theorem provers to verify the obligations. If
322the automatic provers are able to generate proof traces that can be
323independently checked, the only remaining component that enters the trusted code
324base is an off-the-shelf invariant generator which, in turn, can be proved
325correct using an interactive theorem prover. Therefore we achieve the double
326objective of allowing the use of more off-the-shelf components (e.g. provers and
327invariant generators) whilst reducing the trusted code base at the same time.
328
329%\paragraph{Summary of the CerCo objectives.} To summarize, the goal of CerCo is
330% to reconcile functional with non-functional analysis by performing them together
331% on the source code, sharing common knowledge about execution invariants. We want
332% to achieve the goal implementing a new generation of compilers that induce a
333% parametric, precise cost model for basic blocks on the source code. The compiler
334% should be certified using an interactive theorem prover to minimize the trusted
335% code base of the analysis. Once the cost model is induced, off-the-shelf tools
336% and techniques can be combined together to infer and prove parametric cost
337% bounds.
338%The long term benefits of the CerCo vision are expected to be:
339%1. the possibility to perform static analysis during early development stages
340%2.  parametric bounds made easier
341%3.  the application of off-the-shelf techniques currently unused for the
342% analysis of non-functional properties, like automated proving and type systems
343%4. simpler and safer interaction with the user, that is still asked for
344% knowledge, but on the source code, with the additional possibility of actually
345% verifying the provided knowledge
346%5. a reduced trusted code base
347%6. the increased accuracy of the bounds themselves.
348%
349%The long term success of the project is hindered by the increased complexity of
350% the static prediction of the non-functional behaviour of modern hardware. In the
351% time frame of the European contribution we have focused on the general
352% methodology and on the difficulties related to the development and certification
353% of a cost model inducing compiler.
354
355\section{The typical CerCo workflow}
356\begin{figure}[!t]
357\begin{tabular}{l@{\hspace{0.2cm}}|@{\hspace{0.2cm}}l}
358\begin{lstlisting}
359char a[] = {3, 2, 7, 14};
360char threshold = 4;
361
362int count(char *p, int len) {
363  char j;
364  int found = 0;
365  for (j=0; j < len; j++) {
366    if (*p <= threshold)
367      found++;
368    p++;
369    }
370  return found;
371}
372
373int main() {
374  return count(a,4);
375}
376\end{lstlisting}
377&
378%  $\vcenter{\includegraphics[width=7.5cm]{interaction_diagram.pdf}}$
379\begin{tikzpicture}[
380    baseline={([yshift=-.5ex]current bounding box)},
381    element/.style={draw, text width=1.6cm, on chain, text badly centered},
382    >=stealth
383    ]
384{[start chain=going below, node distance=.5cm]
385\node [element] (cerco) {CerCo\\compiler};
386\node [element] (cost)  {CerCo\\cost plugin};
387{[densely dashed]
388\node [element] (ded)   {Deductive\\platform};
389\node [element] (check) {Proof\\checker};
390}
391}
392\coordinate [left=3.5cm of cerco] (left);
393{[every node/.style={above, text width=3.5cm, text badly centered,
394                     font=\scriptsize}]
395\draw [<-] ([yshift=-1ex]cerco.north west) coordinate (t) --
396    node {C source}
397    (t-|left);
398\draw [->] (cerco) -- (cost);
399\draw [->] ([yshift=1ex]cerco.south west) coordinate (t) --
400    node {C source+\color{red}{cost annotations}}
401    (t-|left) coordinate (cerco out);
402\draw [->] ([yshift=1ex]cost.south west) coordinate (t) --
403    node {C source+\color{red}{cost annotations}\\+\color{blue}{synthesized assertions}}
404    (t-|left) coordinate (out);
405{[densely dashed]
406\draw [<-] ([yshift=-1ex]ded.north west) coordinate (t) --
407    node {C source+\color{red}{cost annotations}\\+\color{blue}{complexity assertions}}
408    (t-|left) coordinate (ded in) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
409\draw [->] ([yshift=1ex]ded.south west) coordinate (t) --
410    node {complexity obligations}
411    (t-|left) coordinate (out);
412\draw [<-] ([yshift=-1ex]check.north west) coordinate (t) --
413    node {complexity proof}
414    (t-|left) .. controls +(-.5, 0) and +(-.5, 0) .. (out);
415\draw [dash phase=2.5pt] (cerco out) .. controls +(-1, 0) and +(-1, 0) ..
416    (ded in);
417}}
418%% user:
419% head
420\draw (current bounding box.west) ++(-.2,.5)
421    circle (.2) ++(0,-.2) -- ++(0,-.1) coordinate (t);
422% arms
423\draw (t) -- +(-.3,-.2);
424\draw (t) -- +(.3,-.2);
425% body
426\draw (t) -- ++(0,-.4) coordinate (t);
427% legs
428\draw (t) -- +(-.2,-.6);
429\draw (t) -- +(.2,-.6);
430\end{tikzpicture}
431\end{tabular}
432\caption{On the left: code to count the array's elements
433 that are less than or equal to the threshold. On the right: CerCo's interaction
434 diagram. CerCo's components are drawn solid.}
435\label{test1}
436\end{figure}
437We illustrate the workflow we envisage (on the right
438of~\autoref{test1}) on an example program (on the left
439of~\autoref{test1}).  The user writes the program and feeds it to the
440CerCo compiler, which outputs an instrumented version of the same
441program that updates global variables that record the elapsed
442execution time and the stack space usage.  The red lines in
443\autoref{itest1} introducing variables, functions and function calls
444starting \lstinline'__cost' and \lstinline'__stack' are the instrumentation introduced by the
445compiler.
446
447 The annotated program can then be enriched with complexity
448assertions in the style of Hoare logic, that are passed to a deductive
449platform (in our case Frama-C). We provide as a Frama-C cost plugin a
450simple automatic synthesiser for complexity assertions (the blue
451comments starting with \lstinline'/*@' in \autoref{itest1}), which can
452be overridden by the user to increase or decrease accuracy. From the
453assertions, a general purpose deductive platform produces proof
454obligations which in turn can be closed by automatic or interactive
455provers, ending in a proof certificate.
456
457% NB: if you try this example with the live CD you should increase the timeout
458
459Twelve proof obligations are generated from~\autoref{itest1} (to prove
460that the loop invariant holds after one execution if it holds before,
461to prove that the whole program execution takes at most 1358 cycles,
462etc.).  Note that the synthesised time bound for \lstinline'count',
463$178+214*(1+\lstinline'len')$ cycles, is parametric in the length of
464the array. The CVC3 prover
465closes all obligations within half a minute on routine commodity
466hardware.  A simpler non-parametric version can be solved in a few
467seconds.
468%
469\fvset{commandchars=\\\{\}}
470\lstset{morecomment=[s][\color{blue}]{/*@}{*/},
471        moredelim=[is][\color{blue}]{$}{$},
472        moredelim=[is][\color{red}]{|}{|}}
473\begin{figure}[!p]
474\begin{lstlisting}
475|int __cost = 33, __stack = 5, __stack_max = 5;|
476|void __cost_incr(int incr) { __cost += incr; }|
477|void __stack_incr(int incr) {
478  __stack += incr;
479  __stack_max = __stack_max < __stack ? __stack : __stack_max;
480}|
481
482char a[4] = {3, 2, 7, 14};  char threshold = 4;
483
484/*@ behavior stack_cost:
485      ensures __stack_max <= __max(\old(__stack_max), 4+\old(__stack));
486      ensures __stack == \old(__stack);
487    behavior time_cost:
488      ensures __cost <= \old(__cost)+(178+214*__max(1+\at(len,Pre), 0));
489*/
490int count(unsigned char *p, int len) {
491  char j;  int found = 0;
492  |__stack_incr(4);  __cost_incr(111);|
493  $__l: /* internal */$
494  /*@ for time_cost: loop invariant
495        __cost <= \at(__cost,__l)+
496                  214*(__max(\at((len-j)+1,__l), 0)-__max(1+(len-j), 0));
497      for stack_cost: loop invariant
498        __stack_max == \at(__stack_max,__l);
499      for stack_cost: loop invariant
500        __stack == \at(__stack,__l);
501      loop variant len-j;
502  */
503  for (j = 0; j < len; j++) {
504    |__cost_incr(78);|
505    if (*p <= threshold) { |__cost_incr(136);| found ++; }
506    else { |__cost_incr(114);| }
507    p ++;
508  }
509  |__cost_incr(67);  __stack_incr(-4);|
510  return found;
511}
512
513/*@ behavior stack_cost:
514      ensures __stack_max <= __max(\old(__stack_max), 6+\old(__stack));
515      ensures __stack == \old(__stack);
516    behavior time_cost:
517      ensures __cost <= \old(__cost)+1358;
518*/
519int main(void) {
520  int t;
521  |__stack_incr(2);  __cost_incr(110);|
522  t = count(a,4);
523  |__stack_incr(-2);|
524  return t;
525}
526\end{lstlisting}
527\caption{The instrumented version of the program in \autoref{test1},
528 with instrumentation added by the CerCo compiler in red and cost invariants
529 added by the CerCo Frama-C plugin in blue. The \texttt{\_\_cost},
530 \texttt{\_\_stack} and \texttt{\_\_stack\_max} variables hold the elapsed time
531in clock cycles and the current and maximum stack usage. Their initial values
532hold the clock cycles spent in initialising the global data before calling
533\texttt{main} and the space required by global data (and thus unavailable for
534the stack).
535}
536\label{itest1}
537\end{figure}
538\section{Main scientific and technical results}
539First we describe the basic labeling approach and our compiler
540implementations that use it.  This is suitable for basic architectures
541with simple cost models.  Then we will discuss the dependent labeling
542extension which is suitable for more advanced processor architectures
543and compiler optimisations.  At the end of this section we will
544demonstrate automated high level reasoning about the source level
545costs provided by the compilers.
546
547% \emph{Dependent labeling~\cite{?}.} The basic labeling approach assumes a
548% bijective mapping between object code and source code O(1) blocks (called basic
549% blocks). This assumption is violated by many program optimizations (e.g. loop
550% peeling and loop unrolling). It also assumes the cost model computed on the
551% object code to be non parametric: every block must be assigned a cost that does
552% not depend on the state. This assumption is violated by stateful hardware like
553% pipelines, caches, branch prediction units. The dependent labeling approach is
554% an extension of the basic labeling approach that allows to handle parametric
555% cost models. We showed how the method allows to deal with loop optimizations and
556% pipelines, and we speculated about its applications to caches.
557%
558% \emph{Techniques to exploit the induced cost model.} Every technique used for
559% the analysis of functional properties of programs can be adapted to analyse the
560% non-functional properties of the source code instrumented by compilers that
561% implement the labeling approach. In order to gain confidence in this claim, we
562% showed how to implement a cost invariant generator combining abstract
563% interpretation with separation logic ideas \cite{separation}. We integrated
564% everything in the Frama-C modular architecture, in order to compute proof
565% obligations from functional and cost invariants and to use automatic theorem
566% provers on them. This is an example of a new technique that is not currently
567% exploited in WCET analysis. It also shows how precise functional invariants
568% benefits the non-functional analysis too. Finally, we show how to fully
569% automatically analyse the reaction time of Lustre nodes that are first compiled
570% to C using a standard Lustre compiler and then processed by a C compiler that
571% implements the labeling approach.
572
573% \emph{The CerCo compiler.} This is a compiler from a large subset of the C
574% program to 8051/8052 object code,
575% integrating the labeling approach and a static analyser for 8051 executables.
576% The latter can be implemented easily and does not require dependent costs
577% because the 8051 microprocessor is a very simple one, with constant-cost
578% instruction. It was chosen to separate the issue of exact propagation of the
579% cost model from the orthogonal problem of the static analysis of object code
580% that may require approximations or dependent costs. The compiler comes in
581% several versions: some are prototypes implemented directly in OCaml, and they
582% implement both the basic and dependent labeling approaches; the final version
583% is extracted from a Matita certification and at the moment implements only the
584% basic approach.
585
586\subsection{The (basic) labeling approach}
587The labeling approach is the foundational insight that underlies all the developments in CerCo.
588It allows the tracking of the evolution of
589basic blocks during the compilation process in order to propagate the cost model from the
590object code to the source code without losing precision in the process.
591\paragraph{Problem statement.} Given a source program $P$, we want to obtain an
592instrumented source program $P'$,  written in the same programming language, and
593the object code $O$ such that: 1) $P'$ is obtained by inserting into $P$ some
594additional instructions to update global cost information like the amount of
595time spent during execution or the maximal stack space required; 2) $P$ and $P'$ 
596must have the same functional behaviour, i.e.\ they must produce that same output
597and intermediate observables; 3) $P$ and $O$ must have the same functional
598behaviour; 4) after execution and in interesting points during execution, the
599cost information computed by $P'$ must be an upper bound of the one spent by $O$ 
600to perform the corresponding operations (soundness property); 5) the difference
601between the costs computed by $P'$ and the execution costs of $O$ must be
602bounded by a program-dependent constant (precision property).
603
604\paragraph{The labeling software components.} We solve the problem in four
605stages \cite{labeling}, implemented by four software components that are used
606in sequence.
607
608The first component labels the source program $P$ by injecting label emission
609statements in appropriate positions to mark the beginning of basic blocks.
610% The
611% set of labels with their positions is called labeling.
612The syntax and semantics
613of the source programming language is augmented with label emission statements.
614The statement ``EMIT $\ell$'' behaves like a NOP instruction that does not affect the
615program state or control flow, but its execution is observable.
616% Therefore the observables of a run of a program becomes a stream
617% of label emissions: $\ell_1,\ldots,\ell_n$, called the program trace. We clarify the conditions
618% that the labeling must respect later.
619
620The second component is a labeling preserving compiler. It can be obtained from
621an existing compiler by adding label emission statements to every intermediate
622language and by propagating label emission statements during compilation. The
623compiler is correct if it preserves both the functional behaviour of the program
624and the traces of observables.
625% We may also ask that the function that erases the cost
626% emission statements commute with compilation. This optional property grants that
627% the labeling does not interfere with the original compiler behaviour. A further
628% set of requirements will be added later.
629
630The third component is a static object code analyser. It takes as input a labeled
631object code and it computes the scope of each of its label emission statements,
632i.e.\ the tree of instructions that may be executed after the statement and before
633a new label emission is encountered.
634It is a tree and not a sequence because the scope
635may contain a branching statement. In order to grant that such a finite tree
636exists, the object code must not contain any loop that is not broken by a label
637emission statement. This is the first requirement of a sound labeling. The
638analyser fails if the labeling is unsound. For each scope, the analyser
639computes an upper bound of the execution time required by the scope, using the
640maximum of the costs of the two branches in case of a conditional statement.
641Finally, the analyser computes the cost of a label by taking the maximum of the
642costs of the scopes of every statement that emits that label.
643
644The fourth and last component takes in input the cost model computed at step 3
645and the labelled code computed at step 1. It outputs a source program obtained
646by replacing each label emission statement with a statement that increments the
647global cost variable with the cost associated to the label by the cost model. 
648The obtained source code is the instrumented source code.
649
650\paragraph{Correctness.} Requirements 1, 2 and 3 of the problem statement
651obviously hold, with 2 and 3 being a consequence of the definition of a correct
652labeling preserving compiler. It is also obvious that the value of the global
653cost variable of an instrumented source code is at any time equal to the sum of
654the costs of the labels emitted by the corresponding labelled code. Moreover,
655because the compiler preserves all traces, the sum of the costs of the labels
656emitted in the source and target labelled code are the same. Therefore, to
657satisfy the fourth requirement, we need to grant that the time taken to execute
658the object code is equal to the sum of the costs of the labels emitted by the
659object code. We collect all the necessary conditions for this to happen in the
660definition of a sound labeling: a) all loops must be broken by a cost emission
661statement;  b) all program instructions must be in the scope of some cost
662emission statement. To satisfy also the fifth requirement, additional
663requirements must be imposed on the object code labeling to avoid all uses of
664the maximum in the cost computation: the labeling is precise if every label is
665emitted at most once and both branches of a conditional jump start with a label
666emission statement.
667
668The correctness and precision of the labeling approach only rely on the
669correctness and precision of the object code labeling. The simplest
670way to achieve them is to impose correctness and precision
671requirements also on the initial labeling produced by the first software
672component, and to ask the compiler to preserve these
673properties too. The latter requirement imposes serious limitations on the
674compilation strategy and optimizations: the compiler may not duplicate any code
675that contains label emission statements, like loop bodies. Therefore several
676loop optimisations like peeling or unrolling are prevented. Moreover, precision
677of the object code labeling is not sufficient \emph{per se} to obtain global
678precision: we implicitly assumed that a precise constant cost can be assigned
679to every instruction. This is not possible in
680the presence of stateful hardware whose state influences the cost of operations,
681like pipelines and caches. In Section~\ref{lab:deplabel} we will see an extension of the
682basic labeling approach to cover this situation.
683
684In CerCo we have developed several cost preserving compilers based on
685the labeling approach. Excluding an initial certified compiler for a
686`while' language, all remaining compilers target realistic source
687languages---a pure higher order functional language and a large subset
688of C with pointers, gotos and all data structures---and real world
689target processors---MIPS and the Intel 8051/8052 processor
690family. Moreover, they achieve a level of optimisation that ranges
691from moderate (comparable to GCC level 1) to intermediate (including
692loop peeling and unrolling, hoisting and late constant propagation).
693We describe the C compilers in detail in the following section.
694
695Two compilation chains were implemented for a purely functional
696higher-order language~\cite{labeling2}. The two main changes required
697to deal with functional languages are: 1) because global variables and
698updates are not available, the instrumentation phase produces monadic
699code to `update' the global costs; 2) the requirements for a sound and
700precise labeling of the source code must be changed when the
701compilation is based on CPS translations.  In particular, we need to
702introduce labels emitted before a statement is executed and also
703labels emitted after a statement is executed. The latter capture code
704that is inserted by the CPS translation and that would escape all
705label scopes.
706
707% Brian: one of the reviewers pointed out that standard Prolog implementations
708% do have some global state that apparently survives backtracking and might be
709% used.  As we haven't experimented with this, I think it's best to elide it
710% entirely.
711
712% Phases 1, 2 and 3 can be applied as well to logic languages (e.g. Prolog).
713% However, the instrumentation phase cannot: in standard Prolog there is no notion
714% of (global) variable whose state is not retracted during backtracking.
715% Therefore, the cost of executing computations that are later backtracked would
716% not be correctly counted in. Any extension of logic languages with
717% non-backtrackable state could support our labeling approach.
718
719\subsection{The CerCo C compilers}
720We implemented two C compilers, one implemented directly in OCaml and
721the other implemented in the interactive theorem prover
722Matita~\cite{matita}.  The first acted as a prototype for the second,
723but also supported MIPS and acted as a testbed for more advanced
724features such as the dependent labeling approach
725in Section~\ref{lab:deplabel}.
726
727The second C compiler is the \emph{Trusted CerCo Compiler}, whose cost
728predictions are formally verified. The code distributed
729is extracted OCaml code from the Matita implementation. In the rest of
730this section we will only focus on the Trusted CerCo Compiler, that only targets
731the C language and the 8051/8052 family, and that does not implement any
732advanced optimisations yet. Its user interface, however, is the same as the one
733of the other version, in order to trade safety with additional performances. In
734particular, the Frama-C CerCo plugin can work without recompilation
735with both of our C compilers.
736
737The 8051/8052 microprocessor is a very simple one, with constant-cost
738instructions. It was chosen to separate the issue of exact propagation of the
739cost model from the orthogonal problem of the static analysis of object code
740that may require approximations or dependent costs.
741
742The (trusted) CerCo compiler implements the following optimisations: cast
743simplification, constant propagation in expressions, liveness analysis driven
744spilling of registers, dead code elimination, branch displacement, and tunneling.
745The two latter optimisations are performed by our optimising assembler
746\cite{correctness}. The back-end of the compiler works on three address
747instructions, preferred to static single assignment code for the simplicity of
748the formal certification.
749
750The CerCo compiler is loosely based on the CompCert compiler \cite{compcert}, a
751recently developed certified compiler from C to the PowerPC, ARM and x86
752microprocessors. In contrast to CompCert, both the CerCo code and its
753certification are fully open source. Some data structures and language definitions for
754the front-end are directly taken from CompCert, while the back-end is a redesign
755of a compiler from Pascal to MIPS used by Fran\c{c}ois Pottier for a course at the
756Ecole Polytechnique.
757
758The main differences in the CerCo compiler are:
759\begin{enumerate}
760\item All the intermediate languages include label emitting instructions to
761implement the labeling approach, and the compiler preserves execution traces.
762\item Traditionally compilers target an assembly language with additional
763macro-instructions to be expanded before assembly; for CerCo we need to go all
764the way down to object code in order to perform the static analysis. Therefore
765we integrated also an optimising assembler and a static analyser.
766\item In order to avoid the additional work of implementing a linker
767 and a loader, we do not support separate
768compilation and external calls. Adding them is orthogonal
769to the labeling approach and should not introduce extra problems.
770\item We target an 8-bit processor, in contrast to CompCert's 32-bit targets. This requires
771many changes and more compiler code, but it is not fundamentally more
772complex. The proof of correctness, however, becomes much harder.
773\item We target a microprocessor that has a non uniform memory model, which is
774still often the case for microprocessors used in embedded systems and that is
775becoming common again in multi-core processors. Therefore the compiler has to
776keep track of data and it must move data between memory regions in the proper
777way. Moreover the size of pointers to different regions is not uniform. An
778additional difficulty was that the space available for the stack in internal
779memory in the 8051 is tiny, allowing only a minor number of nested calls. To
780support full recursion in order to test the CerCo tools also on recursive
781programs, the compiler implements a stack in external memory.
782\end{enumerate}
783
784\subsection{Formal certification of the CerCo compiler}
785We have formally
786certified in the Matita interactive proof assistant that the cost models induced on the source code by the
787Trusted CerCo Compiler correctly and precisely
788predict the object code behaviour. There are two cost models, one for execution
789time and one for stack space consumption. We show the correctness of the prediction
790only for those programs that do not exhaust the available machine stack space, a
791property that---thanks to the stack cost model---we can statically analyse on the
792source code using our Frama-C tool. The preservation of functional properties we
793take as an assumption, not itself formally proved in CerCo.  Other projects have
794already certified the preservation of functional semantics in similar compilers,
795and we have not attempted to directly repeat that work. In order to complete the
796proof for non-functional properties, we have introduced a new semantics for
797programming languages based on a new kind of structured observables with the
798relative notions of forward similarity and the study of the intensional
799consequences of forward similarity. We have also introduced a unified
800representation for back-end intermediate languages that was exploited to provide
801a uniform proof of forward similarity.
802
803The details on the proof techniques employed
804and
805the proof sketch can be collected from the CerCo deliverables and papers.
806In this section we will only hint at the correctness statement, which turned
807out to be more complex than what we expected at the beginning.
808
809\paragraph{The statement.}
810Real time programs are often reactive programs that loop forever responding to
811events (inputs) by performing some computation followed by some action (output)
812and the return to the initial state. For looping programs the overall execution
813time does not make sense. The same happens for reactive programs that spend an
814unpredictable amount of time in I/O. What is interesting is the reaction time
815that measure the time spent between I/O events. Moreover, we are interested in
816predicting and ruling out crashes due to running out of space on certain
817inputs.
818Therefore we need to look for a statement that talks about sub-runs of a
819program. The most natural statement is that the time predicted on the source
820code and spent on the object code by two corresponding sub-runs are the same.
821The problem to solve to make this statement formal is how to identify the
822corresponding sub-runs and how to single out those that are meaningful.
823The solution we found is based on the notion of measurability. We say that a run
824has a \emph{measurable sub-run} when both the prefix of the sub-run and the
825sub-run do not exhaust the stack space, the number of function calls and returns
826in the sub-run is the same, the sub-run does not perform any I/O and the sub-run
827starts with a label emission statement and ends with a return or another label
828emission statements. The stack usage is estimated using the stack usage model
829that is computed by the compiler.
830
831The statement that we formally proved is: for each C run with a measurable
832sub-run, there exists an object code run with a sub-run, such that the observables
833of the pairs of prefixes and sub-runs are the same and the time spent by the
834object code in the sub-run is the same as the one predicted on the source code
835using the time cost model generated by the compiler.
836We briefly discuss the constraints for measurability. Not exhausting the stack
837space is a clear requirement of meaningfulness of a run, because source programs
838do not crash for lack of space while object code ones do. The balancing of
839function calls and returns is a requirement for precision: the labeling approach
840allows the scope of label emission statements to extend after function calls to
841minimize the number of labels. Therefore a label pays for all the instructions
842in a block, excluding those executed in nested function calls. If the number of
843calls/returns is unbalanced, it means that there is a call we have not returned
844to that could be followed by additional instructions whose cost has already been
845taken in account. To make the statement true (but less precise) in this
846situation, we could only say that the cost predicted on the source code is a
847safe bound, not that it is exact. The last condition on the entry/exit points of
848a run is used to identify sub-runs whose code correspond to a sequence of blocks
849that we can measure precisely. Any other choice would start or end the run in the
850middle of a block and we would be forced again to weaken the statement taking as
851a bound the cost obtained counting in all the instructions that precede the
852starting one in the block, or follow the final one in the block.
853I/O operations can be performed in the prefix of the run, but not in the
854measurable sub-run. Therefore we prove that we can predict reaction times, but
855not I/O times, as desired.
856
857\subsection{Dependent labeling}
858\label{lab:deplabel}
859The core idea of the basic labeling approach is to establish a tight connection
860between basic blocks executed in the source and target languages. Once the
861connection is established, any cost model computed on the object code can be
862transferred to the source code, without affecting the code of the compiler or
863its proof. In particular, it is immediate that we can also transport cost models
864that associate to each label functions from hardware state to natural numbers.
865However, a problem arises during the instrumentation phase that replaces cost
866emission statements with increments of global cost variables. The global cost
867variable must be incremented with the result of applying the function associated
868to the label to the hardware state at the time of execution of the block.
869The hardware state comprises both the functional state that affects the
870computation (the value of the registers and memory) and the non-functional
871state that does not (the pipeline and cache contents, for example). The former is
872in correspondence with the source code state, but reconstructing the
873correspondence may be hard and lifting the cost model to work on the source code
874state is likely to produce cost expressions that are too complex to understand and reason about.
875Fortunately, in all modern architectures the cost of executing single
876instructions is either independent of the functional state or the jitter---the
877difference between the worst and best case execution times---is small enough
878to be bounded without losing too much precision. Therefore we only consider
879dependencies on the `non-functional' parts of the state.
880
881The non-functional state is not directly related to the high level
882state and does not influence the functional properties. What can be
883done is to expose key aspects of the non-functional state in the
884source code. We present here the basic intuition in a simplified form:
885the technical details that allow us to handle the general case are
886more complex and can be found in~\cite{paolo}. We add to the source
887code an additional global variable that represents the non-functional
888state and another one that remembers the last few labels emitted. The
889state variable must be updated at every label emission statement,
890using an update function which is computed during static analysis
891too. The update function associates to each label a function from the
892recently emitted labels and old state to the new state. It is computed
893by composing the semantics of every instruction in a basic block and
894restricting it to the non-functional part of the state.
895
896Not all the details of the non-functional state needs to be exposed, and the
897technique works better when the part of state that is required can be summarized
898in a simple data structure. For example, to handle simple but realistic
899pipelines it is sufficient to remember a short integer that encodes the position
900of bubbles (stuck instructions) in the pipeline. In any case, it is not necessary
901for the user to understand the meaning of the state to reason over the properties
902of the
903program. Moreover, at any moment the user, or the invariant generator tools that
904analyse the instrumented source code produced by the compiler, can decide to
905trade precision of the analysis for simplicity by approximating the
906cost with safe bounds that do not depend on the processor state. Interestingly, the functional analysis of
907the code could determine which blocks are executed more frequently in order to
908use more aggressive approximations for the ones that are executed least.
909
910Dependent labeling can also be applied to allow the compiler to duplicate
911blocks that contain labels (e.g. in loop optimisations)~\cite{paolo}. The effect is to assign
912a different cost to the different occurrences of a duplicated label. For
913example, loop peeling turns a loop into the concatenation of a copy of the loop
914body for the first iteration and the conditional execution of the
915loop for successive iterations. Further optimisations will compile the two
916copies of the loop differently, with the first body usually
917taking more time.
918
919By introducing a variable that keeps track of the iteration number, we can
920associate to the label a cost that is a function of the iteration number. The
921same technique works for loop unrolling without modifications: the function will
922assign a cost to the even iterations and another cost to the odd ones. The
923actual work to be done consists of introducing within the source code, and for each
924loop, a variable that counts the number of iterations. The loop optimisation code
925that duplicate the loop bodies must also modify the code to correctly propagate
926the update of the iteration numbers. The technical details are more complex and
927can be found in the CerCo reports and publications. The implementation, however,
928is quite simple (and forms part of our OCaml version of the compiler)
929and the changes to a loop optimising compiler are minimal.
930
931\subsection{Techniques to exploit the induced cost model}
932We now turn our attention to synthesising high-level costs, such as
933the execution time of a whole program.  We consider as our starting point source level costs
934provided by basic labeling, in other words annotations
935on the source code which are constants that provide a sound and sufficiently
936precise upper bound on the cost of executing the blocks after compilation to
937object code.
938
939The principle that we have followed in designing the cost synthesis tools is
940that the synthetic bounds should be expressed and proved within a general
941purpose tool built to reason on the source code. In particular, we rely on the
942Frama-C tool to reason on C code and on the Coq proof-assistant to reason on
943higher-order functional programs.
944
945This principle entails that: 1)
946The inferred synthetic bounds are indeed correct as long as the general purpose
947tool is; 2) there is no limitation on the class of programs that can be handled
948as long as the user is willing to carry on an interactive proof.
949
950Of course, automation is desirable whenever possible. Within this framework,
951automation means writing programs that give hints to the general purpose tool.
952These hints may take the form, say, of loop invariants/variants, of predicates
953describing the structure of the heap, or of types in a light logic. If these
954hints are correct and sufficiently precise the general purpose tool will produce
955a proof automatically, otherwise, user interaction is required.
956
957\paragraph{The Cost plugin and its application to the Lustre compiler.}
958Frama-C \cite{framac} is a set of analysers for C programs with a
959specification language called ACSL. New analyses can be dynamically added
960via a plugin system. For instance, the Jessie plugin allows deductive
961verification of C programs with respect to their specification in ACSL, with
962various provers as back-end tools.
963We developed the CerCo Cost plugin for the Frama-C platform as a proof of
964concept of an automatic environment exploiting the cost annotations produced by
965the CerCo compiler. It consists of an OCaml program which essentially
966takes the following actions: 1) it receives as input a C program, 2) it
967applies the CerCo compiler to produce a related C program with cost annotations,
9683) it applies some heuristics to produce a tentative bound on the cost of
969executing the C functions of the program as a function of the value of their
970parameters, 4) the user can then call the Jessie plugin to discharge the
971related proof obligations.
972In the following we elaborate on the soundness of the framework and the
973experiments we performed with the Cost tool on the C programs produced by a
974Lustre compiler.
975
976\paragraph{Soundness.} The soundness of the whole framework depends on the cost
977annotations added by the CerCo compiler, the synthetic costs produced by the
978cost plugin, the verification conditions (VCs) generated by Jessie, and the
979external provers discharging the VCs. The synthetic costs being in ACSL format,
980Jessie can be used to verify them. Thus, even if the added synthetic costs are
981incorrect (relatively to the cost annotations), the process as a whole is still
982correct: indeed, Jessie will not validate incorrect costs and no conclusion can
983be made about the WCET of the program in this case. In other terms, the
984soundness does not really depend on the action of the cost plugin, which can in
985principle produce any synthetic cost. However, in order to be able to actually
986prove a WCET of a C function, we need to add correct annotations in a way that
987Jessie and subsequent automatic provers have enough information to deduce their
988validity. In practice this is not straightforward even for very simple programs
989composed of branching and assignments (no loops and no recursion) because a fine
990analysis of the VCs associated with branching may lead to a complexity blow up.
991\paragraph{Experience with Lustre.} Lustre is a data-flow language for programming
992synchronous systems, with a compiler which targets C. We designed a
993wrapper for supporting Lustre files.
994The C function produced by the compiler implements the step function of the
995synchronous system and computing the WCET of the function amounts to obtain a
996bound on the reaction time of the system. We tested the Cost plugin and the
997Lustre wrapper on the C programs generated by the Lustre compiler. For programs
998consisting of a few hundred lines of code, the cost plugin computes a
999WCET and Alt-Ergo is able to discharge all VCs automatically.
1000
1001\paragraph{Handling C programs with simple loops.}
1002The cost annotations added by the CerCo compiler take the form of C instructions
1003that update a fresh global variable called the cost variable by a constant.
1004Synthesizing a WCET of a C function thus consists of statically resolving an
1005upper bound of the difference between the value of the cost variable before and
1006after the execution of the function, i.e. finding the instructions
1007that update the cost variable and establish the number of times they are passed
1008through during the flow of execution. To perform the analysis the plugin
1009makes the following assumptions on the programs:
10101) there are no recursive functions;
10112) every loop must be annotated with a variant. The variants of `for' loops are
1012automatically inferred.
1013
1014The plugin proceeds as follows.
1015First the call graph of the program is computed.
1016Then the computation of the cost of the function is performed by traversing its
1017control flow graph. If the function $f$ calls the function $g$ 
1018then the function $g$ 
1019is processed before the function $f$. The cost at a node is the maximum of the
1020costs of the successors.
1021In the case of a loop with a body that has a constant cost for every step of the
1022loop, the cost is the product of the cost of the body and of the variant taken
1023at the start of the loop.
1024In the case of a loop with a body whose cost depends on the values of some free
1025variables, a fresh logic function $f$ is introduced to represent the cost of the
1026loop in the logic assertions. This logic function takes the variant as a first
1027parameter. The other parameters of $f$ are the free variables of the body of the
1028loop. An axiom is added to account the fact that the cost is accumulated at each
1029step of the loop.
1030The cost of the function is directly added as post-condition of the function.
1031
1032The user can influence the annotation by two different means:
10331) by using more precise variants;
10342) by annotating functions with cost specifications. The plugin will use this cost
1035for the function instead of computing it.
1036\paragraph{C programs with pointers.}
1037When it comes to verifying programs involving pointer-based data structures,
1038such as linked lists, trees, or graphs, the use of traditional first-order logic
1039to specify, and of SMT solvers to verify, shows some limitations. Separation
1040logic~\cite{separation} is an elegant alternative. It is a program logic
1041with a new notion of conjunction to express spatial heap separation. Bobot has
1042recently introduced automatically generated separation
1043predicates to simulate separation logic reasoning in the Jessie plugin where the specification language, the verification condition
1044generator, and the theorem provers were not designed with separation logic in
1045mind. CerCo's plugin can exploit the separation predicates to automatically
1046reason on the cost of execution of simple heap manipulation programs such as an
1047in-place list reversal.
1048
1049\section{Conclusions and future work}
1050
1051All the CerCo software and deliverables can be found on the CerCo homepage at~\url{http://cerco.cs.unibo.it}.
1052
1053The results obtained so far are encouraging and provide evidence that
1054it is possible to perform static time and space analysis at the source level
1055without losing accuracy, reducing the trusted code base and reconciling the
1056study of functional and non-functional properties of programs. The
1057techniques introduced seem to be scalable, cover both imperative and
1058functional languages and are compatible with every compiler optimisation
1059considered by us so far.
1060
1061To prove that compilers can keep track of optimisations
1062and induce a precise cost model on the source code, we targeted a simple
1063architecture that admits a cost model that is execution history independent.
1064The most important future work is dealing with hardware architectures
1065characterized by history dependent stateful components, like caches and
1066pipelines. The main issue is to assign a parametric, dependent cost
1067to basic blocks that can be later transferred by the labeling approach to
1068the source code and represented in a meaningful way to the user. The dependent
1069labeling approach that we have studied seems a promising tool to achieve
1070this goal, but more work is required to provide good source level
1071approximations of the relevant processor state.
1072
1073Other examples of future work are to improve the cost invariant
1074generator algorithms and the coverage of compiler optimizations, to combining
1075the labeling approach with the type and effect discipline of~\cite{typeffects}
1076to handle languages with implicit memory management, and to experiment with
1077our tools in the early phases of development. Larger case studies are also necessary
1078to evaluate the CerCo's prototype on realistic, industrial-scale programs.
1079
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1164
1165\end{document}
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