# source:Deliverables/D1.2/CompilerProofOutline/outline.tex@1776

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[1716]1\documentclass[a4paper, 10pt]{article}
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3\usepackage{a4wide}
4\usepackage{amsfonts}
5\usepackage{amsmath}
6\usepackage{amssymb}
7\usepackage[english]{babel}
[1723]8\usepackage{color}
[1717]9\usepackage{diagrams}
[1723]10\usepackage{graphicx}
[1723]12\usepackage[utf8x]{inputenc}
13\usepackage{listings}
[1716]14\usepackage{microtype}
[1718]15\usepackage{skull}
16\usepackage{stmaryrd}
[1750]17\usepackage{array}
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[1741]26\usepackage{wasysym}
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[1750]28
[1723]29\lstdefinelanguage{matita-ocaml} {
30  mathescape=true
31}
32\lstset{
33  language=matita-ocaml,basicstyle=\tt,columns=flexible,breaklines=false,
34  showspaces=false, showstringspaces=false, extendedchars=false,
35  inputencoding=utf8x, tabsize=2
36}
37
[1716]38\title{Proof outline for the correctness of the CerCo compiler}
39\date{\today}
40\author{The CerCo team}
41
42\begin{document}
43
44\maketitle
45
46\section{Introduction}
47\label{sect.introduction}
48
[1749]49In the last project review of the CerCo project, the project reviewers
50recommended us to quickly outline a paper-and-pencil correctness proof
51for each of the stages of the CerCo compiler in order to allow for an
52estimation of the complexity and time required to complete the formalization
53of the proof. This has been possible starting from month 18 when we have
54completed the formalization in Matita of the datastructures and code of
55the compiler.
[1728]56
[1749]57In this document we provide a very high-level, pen-and-paper
58sketch of what we view as the best path to completing the correctness proof
[1758]59for the compiler. In particular, for every translation between two intermediate languages, in both the front- and back-ends, we identify the key translation steps, and identify some invariants that we view as being important for the correctness proof.  We sketch the overall correctness results, and also briefly describe the parts of the proof that have already
[1749]60been completed at the end of the First Period.
61
62In the last section we finally present an estimation of the effort required
[1752]63for the certification in Matita of the compiler and we draw conclusions.
[1749]64
[1731]65\section{Front-end: Clight to RTLabs}
66
67The front-end of the CerCo compiler consists of several stages:
68
69\begin{center}
70\begin{minipage}{.8\linewidth}
71\begin{tabbing}
72\quad \= $\downarrow$ \quad \= \kill
73\textsf{Clight}\\
74\> $\downarrow$ \> cast removal\\
75\> $\downarrow$ \> add runtime functions\footnote{Following the last project
76meeting we intend to move this transformation to the back-end}\\
77\> $\downarrow$ \> cost labelling\\
[1741]78\> $\downarrow$ \> loop optimizations\footnote{\label{lab:opt2}To be ported from the untrusted compiler and certified only in case of early completion of the certification of the other passes.} (an endo-transformation)\\
79\> $\downarrow$ \> partial redundancy elimination$^{\mbox{\scriptsize \ref{lab:opt2}}}$ (an endo-transformation)\\
[1731]80\> $\downarrow$ \> stack variable allocation and control structure
81 simplification\\
82\textsf{Cminor}\\
83\> $\downarrow$ \> generate global variable initialisation code\\
84\> $\downarrow$ \> transform to RTL graph\\
85\textsf{RTLabs}\\
[1741]86\> $\downarrow$ \> \\
87\>\,\vdots
[1731]88\end{tabbing}
89\end{minipage}
90\end{center}
91
[1740]92Here, by endo-transformation', we mean a mapping from language back to itself:
93the loop optimization step maps the Clight language to itself.
94
[1731]95%Our overall statements of correctness with respect to costs will
96%require a correctly labelled program
97There are three layers in most of the proofs proposed:
98\begin{enumerate}
99\item invariants closely tied to the syntax and transformations using
[1758]100  dependent types (such as the presence of variable names in environments),
101\item a forward simulation proof relating each small-step of the
102  source to zero or more steps of the target, and
103\item proofs about syntactic properties of the cost labelling.
[1731]104\end{enumerate}
[1758]105The first will support both functional correctness and allow us to
106show the totality of some of the compiler stages (that is, those
107stages of the compiler cannot fail).  The second provides the main
108functional correctness result, including the preservation of cost
109labels in the traces, and the last will be crucial for applying
110correctness results about the costings from the back-end by showing
111that they appear in enough places so that we can assign all of the
112execution costs to them.
[1731]113
114We will also prove that a suitably labelled RTLabs trace can be turned
115into a \emph{structured trace} which splits the execution trace into
[1746]116cost-label to cost-label chunks with nested function calls.  This
[1731]117structure was identified during work on the correctness of the
118back-end cost analysis as retaining important information about the
119structure of the execution that is difficult to reconstruct later in
120the compiler.
121
[1746]122\subsection{Clight cast removal}
[1731]123
[1746]124This transformation removes some casts inserted by the parser to make
125arithmetic promotion explicit but which are superfluous (such as
126\lstinline[language=C]'c = (short)((int)a + (int)b);' where
127\lstinline'a' and \lstinline'b' are \lstinline[language=C]'short').
128This is necessary for producing good code for our target architecture.
[1731]129
[1746]130It only affects Clight expressions, recursively detecting casts that
131can be safely eliminated.  The semantics provides a big-step
132definition for expression, so we should be able to show a lock-step
133forward simulation between otherwise identical states using a lemma
134showing that cast elimination does not change the evaluation of
135expressions.  This lemma will follow from a structural induction on
136the source expression.  We have already proved a few of the underlying
137arithmetic results necessary to validate the approach.
[1731]138
139\subsection{Clight cost labelling}
140
141This adds cost labels before and after selected statements and
142expressions, and the execution traces ought to be equivalent modulo
[1746]143the new cost labels.  Hence it requires a simple forward simulation
144with a limited amount of stuttering whereever a new cost label is
145introduced.  A bound can be given for the amount of stuttering allowed
[1731]146based on the statement or continuation to be evaluated next.
147
148We also intend to show three syntactic properties about the cost
149labelling:
[1746]150\begin{enumerate}
[1731]151\item every function starts with a cost label,
[1746]152\item every branching instruction is followed by a cost label (note that
[1731]153  exiting a loop is treated as a branch), and
154\item the head of every loop (and any \lstinline'goto' destination) is
155  a cost label.
[1746]156\end{enumerate}
[1731]157These can be shown by structural induction on the source term.
158
159\subsection{Clight to Cminor translation}
160
[1732]161This translation is the first to introduce some invariants, with the
162proofs closely tied to the implementation by dependent typing.  These
163are largely complete and show that the generated code enjoys:
164\begin{itemize}
165\item some minimal type safety shown by explicit checks on the
166  Cminor types during the transformation (a little more work remains
167  to be done here, but follows the same form);
168\item that variables named in the parameter and local variable
169  environments are distinct from one another, again by an explicit
170  check;
171\item that variables used in the generated code are present in the
172  resulting environment (either by checking their presence in the
[1746]173  source environment, or from a list of freshly generated temporary variables);
[1732]174  and
175\item that all \lstinline[language=C]'goto' labels are present (by
176  checking them against a list of source labels and proving that all
177  source labels are preserved).
178\end{itemize}
179
180The simulation will be similar to the relevant stages of CompCert
181(Clight to Csharpminor and Csharpminor to Cminor --- in the event that
[1746]182the direct proof is unwieldy we could introduce an intermediate
183language corresponding to Csharpminor).  During early experimentation
184with porting CompCert definitions to the Matita proof assistant we
185found little difficulty reproving the results for the memory model, so
186we plan to port the memory injection properties and use them to relate
187Clight in-memory variables with either the value of the local variable or a
188stack slot, depending on how it was classified.
[1732]189
190This should be sufficient to show the equivalence of (big-step)
191expression evaluation.  The simulation can then be shown by relating
192corresponding blocks of statement and continuations with their Cminor
193counterparts and proving that a few steps reaches the next matching
194state.
195
196The syntactic properties required for cost labels remain similar and a
197structural induction on the function bodies should be sufficient to
198show that they are preserved.
199
[1731]200\subsection{Cminor global initialisation code}
201
[1732]202This short phase replaces the global variable initialisation data with
203code that executes when the program starts.  Each piece of
204initialisation data in the source is matched by a new statement
205storing that data.  As each global variable is allocated a distinct
206memory block, the program state after the initialisation statements
207will be the same as the original program's state at the start of
208execution, and will proceed in the same manner afterwards.
209
210% Actually, the above is wrong...
211% ... this ought to be in a fresh main function with a fresh cost label
212
[1731]213\subsection{Cminor to RTLabs translation}
214
[1733]215In this part of the compiler we transform the program's functions into
216control flow graphs.  It is closely related to CompCert's Cminorsel to
217RTL transformation, albeit with target-independent operations.
218
219We already enforce several invariants with dependent types: some type
[1746]220safety, mostly shown using the type information from Cminor; and
[1733]221that the graph is closed (by showing that each successor was recently
222added, or corresponds to a \lstinline[language=C]'goto' label which
223are all added before the end).  Note that this relies on a
224monotonicity property; CompCert maintains a similar property in a
225similar way while building RTL graphs.  We will also add a result
226showing that all of the pseudo-register names are distinct for use by
227later stages using the same method as Cminor.
228
[1746]229The simulation will relate Cminor states to RTLabs states which are about to
230execute the code corresponding to the Cminor statement or continuation.
[1733]231Each Cminor statement becomes zero or more RTLabs statements, with a
232decreasing measure based on the statement and continuations similar to
233CompCert's.  We may also follow CompCert in using a relational
234specification of this stage so as to abstract away from the functional
235(and highly dependently typed) definition.
236
237The first two labelling properties remain as before; we will show that
238cost labels are preserved, so the function entry point will be a cost
[1746]239label, and successors to any statement that are cost labels map still
240map to cost labels, preserving the condition on branches.  We replace
241the property for loops with the notion that we will always reach a
242cost label or the end of the function after following a bounded number of
243successors.  This can be easily seen in Cminor using the requirement
244for cost labels at the head of loops and after gotos.  It remains to
245show that this is preserved by the translation to RTLabs.  % how?
[1733]246
[1731]247\subsection{RTLabs structured trace generation}
248
[1733]249This proof-only step incorporates the function call structure and cost
250labelling properties into the execution trace.  As the function calls
251are nested within the trace, we need to distinguish between
[1746]252terminating and non-terminating function calls.  Thus we use the
253excluded middle (specialised to a function termination property) to do
254this.
[1733]255
[1746]256Structured traces for terminating functions are built by following the
257flat trace, breaking it into chunks between cost labels and
258recursively processing function calls.  The main difficulties here are
259the non-structurally recursive nature of the function (instead we use
260the size of the termination proof as a measure) and using the RTLabs
261cost labelling properties to show that the constraints of the
262structured traces are observed.  We also show that the lower stack
263frames are preserved during function calls in order to prove that
264after returning from a function call we resume execution of the
265correct code.  This part of the work has already been constructed, but
266still requires a simple proof to show that flattening the structured
[1733]267trace recreates the original flat trace.
268
[1746]269The non-terminating case follows the trace like the terminating
270version to build up chunks of trace from cost-label to cost-label
271(which, by the finite distance to a cost label property shown before,
272can be represented by an inductive type).  These chunks are chained
273together in a coinductive data structure that can represent
274non-terminating traces.  The excluded middle is used to decide whether
275function calls terminate, in which case the function described above
276constructs an inductive terminating structured trace which is nested
277in the caller's trace.  Otherwise, another coinductive constructor is
278used to embed the non-terminating trace of the callee, generated by
279corecursion.  This part of the trace transformation is currently under
280construction, and will also need a flattening result to show that it
281is correct.
[1733]282
[1746]283
[1734]284\section{Backend: RTLabs to machine code}
285\label{sect.backend.rtlabs.machine.code}
[1716]286
[1734]287The compiler backend consists of the following intermediate languages, and stages of translation:
288
289\begin{center}
290\begin{minipage}{.8\linewidth}
291\begin{tabbing}
293\> $\downarrow$ \>\\
294\> $\downarrow$ \quad \= \kill
[1734]295\textsf{RTLabs}\\
[1739]296\> $\downarrow$ \> copy propagation\footnote{\label{lab:opt}To be ported from the untrusted compiler and certified only in case of early completion of the certification of the other passes.} (an endo-transformation) \\
[1734]297\> $\downarrow$ \> instruction selection\\
[1748]298\> $\downarrow$ \> change of memory models in compiler\\
[1734]299\textsf{RTL}\\
[1739]300\> $\downarrow$ \> constant propagation$^{\mbox{\scriptsize \ref{lab:opt}}}$ (an endo-transformation) \\
[1734]301\> $\downarrow$ \> calling convention made explicit \\
[1741]302\> $\downarrow$ \> layout of activation records \\
[1734]303\textsf{ERTL}\\
304\> $\downarrow$ \> register allocation and spilling\\
305\> $\downarrow$ \> dead code elimination\\
306\textsf{LTL}\\
307\> $\downarrow$ \> function linearisation\\
308\> $\downarrow$ \> branch compression (an endo-transformation) \\
309\textsf{LIN}\\
310\> $\downarrow$ \> relabeling\\
311\textsf{ASM}\\
312\> $\downarrow$ \> pseudoinstruction expansion\\
[1741]313\textsf{MCS-51 machine code}\\
[1734]314\end{tabbing}
315\end{minipage}
316\end{center}
317
[1769]318\subsection{Graph translations}
[1773]319RTLabs and most intermediate languages in the back-end have a graph
320representation:
321the code of each function is represented by a graph of instructions.
322The graph maps a set of labels (the names of the nodes) to the instruction
323stored at that label (the nodes of the graph).
324Instructions reference zero or more additional labels that are the immediate
325successors of the instruction: zero for return from functions; more than one
326for conditional jumps and calls; one in all other cases. The references
327from one instruction to its immediates are the arcs of the graph.
[1769]328
[1773]329Status of graph languages always have a program counter that holds a
330representation of a reference to the current instruction.
331
332A translation between two consecutive graph languages maps each instruction
333stored at location $l$ in the first graph and with immediate successors
334$\{l_1,\ldots,l_n\}$ to a subgraph of the output graph that has a single
335entry point at location $l$ and exit arcs to $\{l_1,\ldots,l_n\}$. Moreover,
336the labels of all non entry nodes in the subgraph are distinct from all the
337labels in the source graph.
338
339In order to simplify the translations and the relative proofs of forward
340simulation, after the release of D4.2 and D4.3, we have provided:
341\begin{itemize}
342 \item a new data type (called \texttt{blist}) that represents a
343   sequence of instructions to be added to the output graph.
344   The b'' in the name stands for binder, since a \texttt{blist} is
345   either empty, an extension of a \texttt{blist} with an instruction
346   at the front, or the generation of a fresh quantity followed by a
347   \texttt{blist}. The latter feature is used, for instance, to generate
348   fresh register names. The instructions in the list are unlabelled and
349   all of them but the last one are also sequential, like in a linear program.
350 \item a new iterator (called \texttt{b\_graph\_translate}) of type
351\begin{displaymath}
352\mathtt{b\_graph\_translate}: (\mathtt{label} \rightarrow \mathtt{blist})
353\rightarrow \mathtt{graph} \rightarrow \mathtt{graph}
354\end{displaymath}
355   The iterator transform the input graph in the output graph by replacing
356   each node with the graph that corresponds to the linear \texttt{blist}
357   obtained by applying the function in input to the node label.
358\end{itemize}
359
360Using the iterator above, the code can be written in such a way that
361the programmer does not see any distinction between writing a transformation
362on linear or graph languages.
363
364In order to prove simulations for translations obtained using the iterator,
365we will prove the following theorem:
366
367\begin{align*}
368\mathtt{theorem} &\ \mathtt{b\_graph\_translate\_ok}: \\
369& \forall  f.\forall G_{i}.\mathtt{let}\ G_{\sigma} := \mathtt{b\_graph\_translate}\ f\ G_{i}\ \mathtt{in} \\
370&       \forall l \in G_{i}.\mathtt{subgraph}\ (f\ l)\ l\ (next \ l \ G_i)\ G_{\sigma}
371\end{align*}
372
373Here \texttt{subgraph} is a computational predicate that given a \texttt{blist}
374$[i_1, \ldots, i_n]$, an entry label $l$, an exit label $l'$ and a graph $G$
375expands to the fact that fetching from $G$ at address $l$ one retrieves a node
376$i_1$ with a successor $l_1$ that, when fetched, yields a node $i_2$ with a
377successor $l_2$ such that \ldots. The successor of $i_n$ is $l'$.
378
379Proving a forward simulation diagram of the following kind using the theorem
380above is now as simple as doing the same using standard small step operational
381semantics over linear languages.
382
383\begin{align*}
384\mathtt{lemma} &\ \mathtt{execute\_1\_step\_ok}: \\
385&       \forall s.  \mathtt{let}\ s' := s\ \sigma\ \mathtt{in} \\
386&       \mathtt{let}\ l := pc\ s\ \mathtt{in} \\
387&       s \stackrel{1}{\rightarrow} s^{*} \Rightarrow \exists n. s' \stackrel{n}{\rightarrow} s'^{*} \wedge s'^{*} = s'\ \sigma
388\end{align*}
389
390Because of the fact that graph translation preserves entry and exit labels of
391translated statements, the state translation function $\sigma$ will simply
392preserve the value of the program counter. The program code, which is
393part of the state, is translated using the iterator.
394
395The proof is then roughly the following. Let $l$ be the program counter of the
396input state $s$. We proceed by cases on the current instruction of $s$.
397Let $[i_1, \ldots, i_n]$ be the \texttt{blist} associated to $l$ and $s$
398by the translation function. The witness required for the existential
399statement is simply $n$. By applying the theorem above we know that the
400next $n$ instructions that will be fetched from $s\ \sigma$ will be
401$[i_1, \ldots, i_n]$ and it is now sufficient to prove that they simulate
402the original instruction.
403
[1734]404\subsection{The RTLabs to RTL translation}
405\label{subsect.rtlabs.rtl.translation}
406
[1748]407The RTLabs to RTL translation pass marks the frontier between the two memory models used in the CerCo project.
408As a result, we require some method of translating between the values that the two memory models permit.
409Suppose we have such a translation, $\sigma$.
410Then the translation between values of the two memory models may be pictured with:
[1716]411
412\begin{displaymath}
[1752]413\mathtt{Value} ::= \bot \mid \mathtt{int(size)} \mid \mathtt{float} \mid \mathtt{null} \mid \mathtt{ptr} \quad\stackrel{\sigma}{\longrightarrow}\quad \mathtt{BEValue} ::= \bot \mid \mathtt{byte} \mid \mathtt{null}_i \mid \mathtt{ptr}_i
[1716]414\end{displaymath}
415
[1752]416In the front-end, we have both integer and float values, where integer values are sized', along with null values and pointers. Some frontenv values are
417representables in a byte, but some others require more bits.
[1748]418
[1752]419In the back-end model all values are meant to be represented in a single byte.
420Values can thefore be undefined, be one byte long integers or be indexed
421fragments of a pointer, null or not. Floats values are no longer present, as floating point arithmetic is not supported by the CerCo compiler.
[1716]422
[1752]423The $\sigma$ map implements a one-to-many relation: a single front-end value
424is mapped to a sequence of back-end values when its size is more then one byte.
425
426We further require a map, $\sigma$, which maps the front-end \texttt{Memory} and the back-end's notion of \texttt{BEMemory}. Both kinds of memory can be
427thought as an instance of a generic \texttt{Mem} data type parameterized over
428the kind of values stored in memory.
429
[1716]430\begin{displaymath}
431\mathtt{Mem}\ \alpha = \mathtt{Block} \rightarrow (\mathbb{Z} \rightarrow \alpha)
432\end{displaymath}
433
[1748]434Here, \texttt{Block} consists of a \texttt{Region} paired with an identifier.
[1716]435
436\begin{displaymath}
[1752]437\mathtt{Block} ::= \mathtt{Region} \times \mathtt{ID}
[1716]438\end{displaymath}
439
[1748]440We now have what we need for defining what is meant by the memory' in the backend memory model.
441Namely, we instantiate the previously defined \texttt{Mem} type with the type of back-end memory values.
442
[1716]443\begin{displaymath}
[1768]444\mathtt{BEMem} = \mathtt{Mem}~\mathtt{BEValue}
[1716]445\end{displaymath}
446
[1748]447Memory addresses consist of a pair of back-end memory values:
448
[1716]449\begin{displaymath}
450\mathtt{Address} = \mathtt{BEValue} \times  \mathtt{BEValue} \\
451\end{displaymath}
452
[1751]453The back- and front-end memory models differ in how they represent sized integeer values in memory.
454In particular, the front-end stores integer values as a header, with size information, followed by a string of continuation' blocks, marking out the full representation of the value in memory.
455In contrast, the layout of sized integer values in the back-end memory model consists of a series of byte-sized chunks':
[1752]456
[1716]457\begin{center}
[1752]458\begin{picture}(0, 25)
459\put(-125,0){\framebox(25,25)[c]{\texttt{v,4}}}
460\put(-100,0){\framebox(25,25)[c]{\texttt{cont}}}
461\put(-75,0){\framebox(25,25)[c]{\texttt{cont}}}
462\put(-50,0){\framebox(25,25)[c]{\texttt{cont}}}
463\put(-15,10){\vector(1, 0){30}}
464\put(25,0){\framebox(25,25)[c]{\texttt{\texttt{v$_1$}}}}
465\put(50,0){\framebox(25,25)[c]{\texttt{\texttt{v$_2$}}}}
466\put(75,0){\framebox(25,25)[c]{\texttt{\texttt{v$_3$}}}}
467\put(100,0){\framebox(25,25)[c]{\texttt{\texttt{v$_4$}}}}
[1716]468\end{picture}
469\end{center}
[1752]470
[1760]471Chunks for pointers are pairs made of the original pointer and the index of the chunk.
472Therefore, when assembling the chunks together, we can always recognize if all chunks refer to the same value or if the operation is meaningless.
[1752]473
[1751]474The differing memory representations of values in the two memory models imply the need for a series of lemmas on the actions of \texttt{load} and \texttt{store} to ensure correctness.
475The first lemma required has the following statement:
[1716]476\begin{displaymath}
477\mathtt{load}\ s\ a\ M = \mathtt{Some}\ v \rightarrow \forall i \leq s.\ \mathtt{load}\ s\ (a + i)\ \sigma(M) = \mathtt{Some}\ v_i
478\end{displaymath}
[1768]479That is, if we are successful in reading a value of size $s$ from memory at address $a$ in front-end memory, then we should successfully be able to read all of its chunks from memory in the back-end memory at appropriate address (from address $a$ up to and including address $a + i$, where $i \leq s$).
[1716]480
[1760]481Next, we must show that \texttt{store} properly commutes with the $\sigma$-map between memory spaces:
[1716]482\begin{displaymath}
[1760]483\sigma(\mathtt{store}\ a\ v\ M) = \mathtt{store}\ \sigma(v)\ \sigma(a)\ \sigma(M)
[1716]484\end{displaymath}
[1760]485That is, if we store a value \texttt{v} in the front-end memory \texttt{M} at address \texttt{a} and transform the resulting memory with $\sigma$, then this is equivalent to storing a transformed value $\mathtt{\sigma(v)}$ at address $\mathtt{\sigma(a)}$ into the back-end memory $\mathtt{\sigma(M)}$.
[1716]486
[1768]487Finally, the commutation properties between \texttt{load} and \texttt{store} are weakened in the $\sigma$-image of the memory.
488Writing \texttt{load}$^*$ for the multiple consecutive iterations of \texttt{load} used to fetch all chunks of a value, we must prove that, when $a \neq a'$:
[1716]489\begin{displaymath}
[1743]491\end{displaymath}
[1768]492That is, suppose we store a transformed value $\mathtt{\sigma(v)}$ into a back-end memory $\mathtt{\sigma(M)}$ at address $\mathtt{\sigma(a')}$, using \texttt{store}, and then load from the address $\sigma(a)$. Even if $a$ and $a'$ are
493distinct by hypothesis, there is a priori no guarantee that the consecutive
494bytes for the value stored at $\sigma(a)$ are disjoint from those for the
495values stored at $\sigma(a')$. The fact that this holds is a non-trivial
496property of $\sigma$ to be proved.
[1743]497
[1763]498RTLabs states come in three flavours:
[1743]499\begin{displaymath}
[1716]500\begin{array}{rll}
501\mathtt{State} & ::=  & (\mathtt{State} : \mathtt{Frame}^* \times \mathtt{Frame} \\
502               & \mid & \mathtt{Call} : \mathtt{Frame}^* \times \mathtt{Args} \times \mathtt{Return} \times \mathtt{Fun} \\
503               & \mid & \mathtt{Return} : \mathtt{Frame}^* \times \mathtt{Value} \times \mathtt{Return}) \times \mathtt{Mem}
504\end{array}
505\end{displaymath}
[1763]506\texttt{State} is the default state in which RTLabs programs are almost always in.
507The \texttt{Call} state is only entered when a call instruction is being executed, and then we immediately return to being in \texttt{State}.
508Similarly, \texttt{Return} is only entered when a return instruction is being executed, before returning immediately to \texttt{State}.
509All RTLabs states are accompanied by a memory, \texttt{Mem}, with \texttt{Call} and \texttt{Return} keeping track of arguments, return addresses and the results of functions.
510\texttt{State} keeps track of a list of stack frames.
[1716]511
[1763]512RTL states differ from their RTLabs counterparts, in including a program counter \texttt{PC}, stack-pointer \texttt{SP}, internal stack pointer \texttt{ISP}, a carry flag \texttt{CARRY} and a set of registers \texttt{REGS}:
[1716]513\begin{displaymath}
514\mathtt{State} ::= \mathtt{Frame}^* \times \mathtt{PC} \times \mathtt{SP} \times \mathtt{ISP} \times \mathtt{CARRY} \times \mathtt{REGS}
515\end{displaymath}
[1763]516The internal stack pointer \texttt{ISP}, and its relationship with the stack pointer \texttt{SP}, needs some comment.
517Due to the design of the MCS-51, and its minuscule stack, it was decided that the compiler would implement an emulated stack in external memory.
518As a result, we have two stack pointers in our state: \texttt{ISP}, which is the real, hardware stack, and \texttt{SP}, which is the stack pointer of the emulated stack in memory.
519The emulated stack is used for pushing and popping stack frames when calling or returning from function calls, however this is done using the hardware stack, indexed by \texttt{ISP} as an intermediary.
[1768]520Instructions like \texttt{LCALL} and \texttt{ACALL} are hardwired by the processor's design to push the return address on to the hardware stack. Therefore after a call has been made, and before a call returns, the compiler emits code to move the return address back and forth the two stacks. Parameters, return values
521and local variables are only present in the external stack.
[1763]522As a result, for most of the execution of the processor, the hardware stack is empty, or contains a single item ready to be moved into external memory.
[1716]523
[1768]524Once more, we require a relation $\sigma$ between RTLabs states and RTL states.
525Because $\sigma$ is one-to-many and, morally, a multi-function,
526we use in the following the functional notation for $\sigma$, using $\star$
527in the output of $\sigma$ to mean that any value is accepted.
[1716]528\begin{displaymath}
529\mathtt{State} \stackrel{\sigma}{\longrightarrow} \mathtt{State}
530\end{displaymath}
531
[1763]532Translating an RTLabs state to an RTL state proceeds by cases on the particular type of state we are trying to translate, either a \texttt{State}, \texttt{Call} or a \texttt{Return}.
533For \texttt{State} we perform a further case analysis of the top stack frame, which decomposes into a tuple holding the current program counter value, the current stack pointer and the value of the registers:
[1717]534\begin{displaymath}
[1768]535\sigma(\mathtt{State} (\mathtt{Frame}^* \times \mathtt{\langle PC, REGS, SP \rangle})) \longrightarrow ((\sigma(\mathtt{Frame}^*), \sigma(\mathtt{PC}), \sigma(\mathtt{SP}), \star, \star, \sigma(\mathtt{REGS})), \sigma(\mathtt{Mem}))
[1717]536\end{displaymath}
[1768]537Translation then proceeds by translating the remaining stack frames, as well as the contents of the top stack frame. Any value for the internal stack pointer
538and the carry bit is admitted.
[1717]539
[1763]540Translating \texttt{Call} and \texttt{Return} states is more involved, as a commutation between a single step of execution and the translation process must hold:
[1717]541\begin{displaymath}
542\sigma(\mathtt{Return}(-)) \longrightarrow \sigma \circ \text{return one step}
543\end{displaymath}
544
545\begin{displaymath}
[1718]546\sigma(\mathtt{Call}(-)) \longrightarrow \sigma \circ \text{call one step}
[1717]547\end{displaymath}
548
[1763]549Here \emph{return one step} and \emph{call one step} refer to a pair of commuting diagrams relating the one-step execution of a call and return state and translation of both.
[1768]550We provide the one step commuting diagrams in Figure~\ref{fig.commuting.diagrams}. The fact that one execution step in the source language is not performed
551in the target language is not problematic for preservation of divergence
552because it is easy to show that every step from a \texttt{Call} or
553\texttt{Return} state is always preceeded/followed by one step that is always
554simulated.
[1718]555
[1763]556\begin{figure}
[1718]557\begin{displaymath}
[1717]558\begin{diagram}
[1718]559s & \rTo^{\text{one step of execution}} & s'   \\
560  & \rdTo                             & \dTo \\
561  &                                   & \llbracket s'' \rrbracket
[1717]562\end{diagram}
[1718]563\end{displaymath}
[1717]564
[1718]565\begin{displaymath}
566\begin{diagram}
567s & \rTo^{\text{one step of execution}} & s'   \\
568  & \rdTo                             & \dTo \\
569  &                                   & \llbracket s'' \rrbracket
570\end{diagram}
571\end{displaymath}
[1763]572\caption{The one-step commuting diagrams for \texttt{Call} and \texttt{Return} state translations}
573\label{fig.commuting.diagrams}
574\end{figure}
[1718]575
[1771]576The forward simulation proof for all steps that do not involve function calls are lengthy, but routine.
577They consist of simulating a front-end operation on front-end pseudo-registers and the front-end memory with sequences of back-end operations on the back-end pseudo-registers and back-end memory.
578The properties of $\sigma$ presented before that relate values and memories will need to be heavily exploited.
[1768]579
[1771]580The simulation of invocation of functions and returns from functions is less obvious.
581We sketch here what happens on the source code and on its translation.
[1769]582
[1718]583\begin{displaymath}
[1743]584\begin{array}{rcl}
[1763]585\mathtt{Call(id,\ args,\ dst,\ pc),\ State(Frame^*, Frame)} & \longrightarrow & \mathtt{Call(M(args), dst)}, \\
586                                                           &                 & \mathtt{PUSH(Frame[PC := after\_return])}
[1743]587\end{array}
[1718]588\end{displaymath}
[1763]589Suppose we are given a \texttt{State} with a list of stack frames, with the top frame being \texttt{Frame}.
590Suppose also that the program counter in \texttt{Frame} points to a \texttt{Call} instruction, complete with arguments and destination address.
[1771]591Then this is executed by entering into a \texttt{Call} state where the arguments are loaded from memory, and the address pointing to the instruction immediately following the \texttt{Call} instruction is filled in, with the current stack frame being pushed on top of the stack with the return address substituted for the program counter.
[1718]592
[1763]593Now, what happens next depends on whether we are executing an internal or an external function.
[1743]594In the case where the call is to an external function, we have:
[1718]595\begin{displaymath}
596\begin{array}{rcl}
[1743]597\mathtt{Call(M(args), dst)},                       & \stackrel{\mathtt{ret\_val = f(M(args))}}{\longrightarrow} & \mathtt{Return(ret\_val,\ dst,\ PUSH(...))} \\
598\mathtt{PUSH(current\_frame[PC := after\_return])} &                                                            &
599\end{array}
600\end{displaymath}
[1766]601That is, the call to the external function enters a return state after first computing the return value by executing the external function on the arguments.
[1767]602Then the return state restores the program counter by popping the stack, and execution proceeds in a new \texttt{State}:
[1743]603\begin{displaymath}
604\begin{array}{rcl}
[1767]605\mathtt{Return(ret\_val,\ dst,\ PUSH(...))} & \longrightarrow & \mathtt{pc = POP\_STACK(regs[dst := M(ret\_val)],\ pc)} \\
606                                            &                 & \mathtt{State(regs[dst := M(ret\_val),\ pc)}
[1743]607\end{array}
608\end{displaymath}
609
[1767]610Suppose we are executing an internal function, however:
[1743]611\begin{displaymath}
612\begin{array}{rcl}
[1772]613\mathtt{Call(M(args), dst)}                        & \longrightarrow & \mathtt{SP = alloc,\ regs = \emptyset[- := params]} \\
614\mathtt{PUSH(current\_frame[PC := after\_return])} &                 & \mathtt{State(regs,\ sp,\ pc_\emptyset,\ dst)}
[1743]615\end{array}
616\end{displaymath}
[1767]617Here, execution of the \texttt{Call} state first pushes the current frame with the program counter set to the address following the function call.
618The stack pointer allocates more space, the register map is initialized first to the empty map, assigning an undefined value to all register, before the value of the parameters is inserted into the map into the argument registers, and a new \texttt{State} follows.
[1770]619After this, the stack pointer is freed and a \texttt{Return} state is entered:
[1743]620\begin{displaymath}
621\begin{array}{rcl}
[1772]622\mathtt{sp = alloc,\ regs = \emptyset[- := PARAMS]} & \longrightarrow & \mathtt{free(sp)} \\
623\mathtt{State(regs,\ sp,\ pc_\emptyset,\ dst)}     &                 & \mathtt{Return(M(ret\_val), dst, Frames)}
[1743]624\end{array}
625\end{displaymath}
[1770]626Then the return state restores the program counter by popping the stack, and execution proceeds in a new \texttt{State}, like the case for external functions:
[1743]627\begin{displaymath}
628\begin{array}{rcl}
629\mathtt{free(sp)}                         & \longrightarrow & \mathtt{pc = POP\_STACK(regs[dst := M(ret\_val)],\ pc)} \\
[1770]630\mathtt{Return(M(ret\_val), dst, frames)} &                 & \mathtt{State(regs[dst := M(ret\_val),\ pc)}
[1743]631\end{array}
632\end{displaymath}
633
[1718]634\begin{displaymath}
635\begin{diagram}
[1724]636& & \llbracket \mathtt{CALL\_ID}(\mathtt{id}, \mathtt{args}, \mathtt{dst}, \mathtt{pc})\rrbracket & & \\
637& \ldTo^{\text{external}} & & \rdTo^{\text{internal}} & \\
638\skull & & & & \mathtt{regs} = [\mathtt{params}/-] \\
639& & & & \mathtt{sp} = \mathtt{ALLOC} \\
640& & & & \mathtt{PUSH}(\mathtt{carry}, \mathtt{regs}, \mathtt{dst}, \mathtt{return\_addr}), \mathtt{pc}_{0}, \mathtt{regs}, \mathtt{sp} \\
[1718]641\end{diagram}
642\end{displaymath}
643
[1724]644\begin{align*}
645\llbracket \mathtt{RETURN} \rrbracket \\
647v*                    & := m(\mathtt{rv\_regs}) \\
648\mathtt{dst}, \mathtt{sp}, \mathtt{carry}, \mathtt{regs} & := \mathtt{pop} \\
649\mathtt{regs}[v* / \mathtt{dst}] \\
650\end{align*}
651
652\begin{displaymath}
653\begin{diagram}
654s    & \rTo^1 & s' & \rTo^1 & s'' \\
655\dTo &        &    & \rdTo  & \dTo \\
656\llbracket s \rrbracket & \rTo(1,3)^1 & & & \llbracket s'' \rrbracket \\
657\mathtt{CALL} \\
658\end{diagram}
659\end{displaymath}
660
661\begin{displaymath}
662\begin{diagram}
663s    & \rTo^1 & s' & \rTo^1 & s'' \\
664\dTo &        &    & \rdTo  & \dTo \\
665\  & \rTo(1,3) & & & \ \\
666\mathtt{RETURN} \\
667\end{diagram}
668\end{displaymath}
669
670\begin{align*}
[1727]671\mathrm{RTL\ status} & \ \ \mathrm{ERTL\ status} \\
672\mathtt{sp} & = \mathtt{spl} / \mathtt{sph} \\
[1750]673\mathtt{graph} &  \mathtt{graph} + \mathtt{prologue}(s) + \mathtt{epilogue}(s) \\
[1727]674& \mathrm{where}\ s = \mathrm{callee\ saved} + \nu \mathrm{RA} \\
675\end{align*}
676
677\begin{displaymath}
678\begin{diagram}
679\mathtt{CALL} & \rTo^1 & \mathtt{inside\ function} \\
680\dTo & & \dTo \\
681\underbrace{\ldots}_{\llbracket \mathtt{CALL} \rrbracket} & \rTo &
682\underbrace{\ldots}_{\mathtt{prologue}} \\
683\end{diagram}
684\end{displaymath}
685
686\begin{displaymath}
687\begin{diagram}
688\mathtt{RETURN} & \rTo^1 & \mathtt{.} \\
689\dTo & & \dTo \\
690\underbrace{\ldots}_{\mathtt{epilogue}} & \rTo &
691\underbrace{\ldots} \\
692\end{diagram}
693\end{displaymath}
694
695\begin{align*}
696\mathtt{prologue}(s) = & \mathtt{create\_new\_frame}; \\
697                       & \mathtt{pop\ ra}; \\
698                       & \mathtt{save\ callee\_saved}; \\
699                                                                                         & \mathtt{get\_params} \\
700                                                                                         & \ \ \mathtt{reg\_params}: \mathtt{move} \\
701                                                                                         & \ \ \mathtt{stack\_params}: \mathtt{push}/\mathtt{pop}/\mathtt{move} \\
702\end{align*}
703
704\begin{align*}
705\mathtt{epilogue}(s) = & \mathtt{save\ return\ to\ tmp\ real\ regs}; \\
706                                                                                         & \mathtt{restore\_registers}; \\
707                       & \mathtt{push\ ra}; \\
708                       & \mathtt{delete\_frame}; \\
709                       & \mathtt{save return} \\
710\end{align*}
711
712\begin{displaymath}
[1738]713\mathtt{CALL}\ id \mapsto \mathtt{set\_params};\ \mathtt{CALL}\ id;\ \mathtt{fetch\_result}
[1727]714\end{displaymath}
715
[1734]716\subsection{The ERTL to LTL translation}
717\label{subsect.ertl.ltl.translation}
719\declsf{Livebefore}
720\declsf{Liveafter}
721\declsf{Defined}
722\declsf{Used}
723\declsf{Eliminable}
724\declsf{StatementSem}
725For the liveness analysis, we aim at a map
726$\ell \in \mathtt{label} \mapsto$ live registers at $\ell$.
727We define the following operators on ERTL statements.
728$$729\begin{array}{lL>{(ex. }L<{)}} [1753]730\Defined(s) & registers defined at s & r_1\leftarrow r_2+r_3 \mapsto \{r_1,C\}, \mathtt{CALL}~id\mapsto \text{caller-save} [1750]731\\ [1753]732\Used(s) & registers used at s & r_1\leftarrow r_2+r_3 \mapsto \{r_2,r_3\}, \mathtt{CALL}~id\mapsto \text{parameters} [1750]733\end{array} 734$$
735Given $LA:\mathtt{label}\to\mathtt{lattice}$ (where $\mathtt{lattice}$
736is the type of sets of registers\footnote{More precisely, it is thethe lattice
737of pairs of sets of pseudo-registers and sets of hardware registers,
738with pointwise operations.}, we also have have the following
739predicates:
740$$741\begin{array}{lL} 742\Eliminable_{LA}(\ell) & iff s(\ell) has side-effects only on r\notin LA(\ell) 743\\& 744(ex.\ \ell : r_1\leftarrow r_2+r_3 \mapsto (\{r_1,C\}\cap LA(\ell)\neq\emptyset, 745 \mathtt{CALL}id\mapsto \text{never}) 746\\ 747\Livebefore_{LA}(\ell) &:= 748 \begin{cases} 749 LA(\ell) &\text{if \Eliminable_{LA}(\ell),}\\ 750 (LA(\ell)\setminus \Defined(s(\ell)))\cup \Used(s(\ell) &\text{otherwise}. 751 \end{cases} 752\end{array} 753$$
754In particular, $\Livebefore$ has type $(\mathtt{label}\to\mathtt{lattice})\to 755\mathtt{label}\to\mathtt{lattice}$.
[1716]756
[1750]757The equation on which we build the fixpoint is then
758$$\Liveafter(\ell) \doteq \bigcup_{\ell' >_1 \ell} \Livebefore_{\Liveafter}(\ell')$$
759where $\ell' >_1 \ell$ denotes that $\ell'$ is an immediate successor of $\ell$
760in the graph. We do not require the fixpoint to be the least one, so the hypothesis
761on $\Liveafter$ that we require is
762$$\Liveafter(\ell) \supseteq \bigcup_{\ell' >_1 \ell} \Livebefore(\ell')$$
763(for shortness we drop the subscript from $\Livebefore$).
[1734]764\subsection{The LTL to LIN translation}
765\label{subsect.ltl.lin.translation}
[1762]766Ad detailed elsewhere in the reports, due to the parameterized representation of
767the back-end languages, the pass described here is actually much more generic
768than the translation from LTL to LIN. It consists in a linearization pass
769that maps any graph-based back-end language to its corresponding linear form,
770preserving its semantics. In the rest of the section, however, we will keep
771the names LTL and LIN for the two partial instantiations of the parameterized
772language.
[1716]773
[1721]774We require a map, $\sigma$, from LTL statuses, where program counters are represented as labels in a graph data structure, to LIN statuses, where program counters are natural numbers:
775\begin{displaymath}
776\mathtt{pc : label} \stackrel{\sigma}{\longrightarrow} \mathbb{N}
777\end{displaymath}
778
[1723]779The LTL to LIN translation pass also linearises the graph data structure into a list of instructions.
780Pseudocode for the linearisation process is as follows:
[1721]781
[1723]782\begin{lstlisting}
783let rec linearise graph visited required generated todo :=
784  match todo with
785  | l::todo ->
786    if l $\in$ visited then
787      let generated := generated $\cup\ \{$ Goto l $\}$ in
788      let required := required $\cup$ l in
789        linearise graph visited required generated todo
790    else
[1725]791      -- Get the instruction at label l' in the graph
[1723]792      let lookup := graph(l) in
793      let generated := generated $\cup\ \{$ lookup $\}$ in
[1725]794      -- Find the successor of the instruction at label l' in the graph
[1723]795      let successor := succ(l, graph) in
796      let todo := successor::todo in
797        linearise graph visited required generated todo
798  | []      -> (required, generated)
799\end{lstlisting}
[1721]800
[1725]801It is easy to see that this linearisation process eventually terminates.
802In particular, the size of the visited label set is monotonically increasing, and is bounded above by the size of the graph that we are linearising.
[1721]803
[1725]804The initial call to \texttt{linearise} sees the \texttt{visited}, \texttt{required} and \texttt{generated} sets set to the empty set, and \texttt{todo} initialized with the singleton list consisting of the entry point of the graph.
805We envisage needing to prove the following invariants on the linearisation function above:
806
807\begin{enumerate}
808\item
809$\mathtt{visited} \approx \mathtt{generated}$, where $\approx$ is \emph{multiset} equality, as \texttt{generated} is a set of instructions where instructions may mention labels multiple times, and \texttt{visited} is a set of labels,
810\item
811$\forall \mathtt{l} \in \mathtt{generated}.\ \mathtt{succ(l,\ graph)} \subseteq \mathtt{required} \cup \mathtt{todo}$,
812\item
813$\mathtt{required} \subseteq \mathtt{visited}$,
814\item
815$\mathtt{visited} \cap \mathtt{todo} = \emptyset$.
816\end{enumerate}
817
818The invariants collectively imply the following properties, crucial to correctness, about the linearisation process:
819
820\begin{enumerate}
821\item
822Every graph node is visited at most once,
823\item
824Every instruction that is generated is generated due to some graph node being visited,
825\item
826The successor instruction of every instruction that has been visited already will eventually be visited too.
827\end{enumerate}
828
[1762]829Note, because the LTL to LIN transformation is the first time the code of
830a function is linearised in the back-end, we must discover a notion of well formed function code' suitable for linearised forms.
[1725]831In particular, we see the notion of well formedness (yet to be formally defined) resting on the following conditions:
832
833\begin{enumerate}
834\item
[1762]835For every jump to a label in a linearised function code, the target label exists at some point in the function code,
[1725]836\item
[1762]837Each label is unique, appearing only once in the function code,
[1725]838\item
[1762]839The final instruction of a function code must be a return or an unconditional
840jump.
[1725]841\end{enumerate}
842
[1726]843We assume that these properties will be easy consequences of the invariants on the linearisation function defined above.
844
[1725]845The final condition above is potentially a little opaque, so we explain further.
[1762]846The only instructions that can reasonably appear in final position at the end of a function code are returns or backward jumps, as any other instruction would cause execution to fall out' of the end of the program (for example, when a function invoked with \texttt{CALL} returns, it returns to the next instruction past the \texttt{CALL} that invoked it).
[1725]847
[1734]848\subsection{The LIN to ASM and ASM to MCS-51 machine code translations}
849\label{subsect.lin.asm.translation}
[1716]850
[1721]851The LIN to ASM translation step is trivial, being almost the identity function.
852The only non-trivial feature of the LIN to ASM translation is that all labels are named apart' so that there is no chance of freshly generated labels from different namespaces clashing with labels from another namespace.
[1716]853
[1721]854The ASM to MCS-51 machine code translation step, and the required statements of correctness, are found in an unpublished manuscript attached to this document.
[1757]855This is the most complex translation because of the huge number of cases
856to be addressed and because of the complexity of the two semantics.
857Moreover, in the assembly code we have conditional and unconditional jumps
858to arbitrary locations in the code, which are not supported by the MCS-51
859instruction set. The latter has several kind of jumps characterized by a
860different instruction size and execution time, but limited in range. For
861instance, conditional jumps to locations whose destination is more than
862$2^7$ bytes away from the jump instruction location are not supported at
863all and need to be emulated with a code transformation. The problem, which
864is known in the litterature as branch displacement and that applies also
865to modern architectures, is known to be hard and is often NP. As far as we
866know, we will provide the first formally verified proof of correctness for
867an assembler that implements branch displacement. We are also providing
868the first verified proof of correctness of a mildly optimizing branch
869displacement algorithm that, at the moment, is almost finished, but not
870described in the companion paper. This proof by itself took about 6 men
871months.
[1721]872
[1757]873\section{Correctness of cost prediction}
874Roughly speaking,
875the proof of correctness of cost prediction shows that the cost of executing
876a labelled object code program is the same as the sum over all labels in the
[1761]877program execution trace of the cost statically associated to the label and
[1757]878computed on the object code itself.
879
[1761]880In presence of object level function calls, the previous statement is, however,
881incorrect. The reason is twofold. First of all, a function call may diverge.
882To the last labels that comes before the call, however, we also associate
883the cost of the instructions that follow the call. Therefore, in the
884sum over all labels, when we meet a label we pre-pay for the instructions
885after function calls, assuming all calls to be terminating. This choice is
886driven by considerations on the source code. Functions can be called also
887inside expressions and it would be too disruptive to put labels inside
888expressions to capture the cost of instructions that follow a call. Moreover,
889adding a label after each call would produce a much higher number of proof
890obligations in the certification of source programs using Frama-C. The
891proof obligations, moreover, would be guarded by termination of all functions
892involved, that also generates lots of additional complex proof obligations
893that have little to do with execution costs. With our approach, instead, we
894put less burden on the user, at the price of proving a weaker statement:
895the estimated and actual costs will be the same if and only if the high level
896program is converging. For prefixes of diverging programs we can provide
897a similar result where the equality is replaced by an inequality (loss of
898precision).
[1757]899
[1761]900Assuming totality of functions is however not sufficient yet at the object
901level. Even if a function returns, there is no guarantee that it will transfer
902control back to the calling point. For instance, the function could have
903manipulated the return address from its stack frame. Moreover, an object level
904program can forge any address and transfer control to it, with no guarantee
905on the execution behaviour and labelling properties of the called program.
[1757]906
[1761]907To solve the problem, we introduced the notion of \emph{structured trace}
908that come in two flavours: structured traces for total programs (an inductive
909type) and structured traces for diverging programs (a co-inductive type based
910on the previous one). Roughly speaking, a structured trace represents the
911execution of a well behaved program that is subject to several constraints
912like
913\begin{enumerate}
914 \item All function calls return control just after the calling point
915 \item The execution of all function bodies start with a label and end with
916   a RET (even the ones reached by invoking a function pointer)
917 \item All instructions are covered by a label (required by correctness of
918   the labelling approach)
919 \item The target of all conditional jumps must be labelled (a sufficient
920   but not necessary condition for precision of the labelling approach)
921 \item \label{prop5} Two structured traces with the same structure yield the same
922   cost traces.
923\end{enumerate}
924
925Correctness of cost predictions is proved only for structured execution traces,
926i.e. well behaved programs. The forward simulation proof for all back-end
927passes will actually be a proof of preservation of the structure of
928the structured traces that, because of property \ref{prop5}, will imply
929correctness of the cost prediction for the back-end. The Clight to RTLabs
930will also include a proof that associates to each converging execution its
931converging structured trace and to each diverging execution its diverging
932structured trace.
933
934There are also other two issues that invalidate the naive statement of
935correctness of cost prediciton given above. The algorithm that statically
936computes the cost of blocks is correct only when the object code is \emph{well
937formed} and the program counter is \emph{reachable}.
938A well formed object code is such that
939the program counter will never overflow after the execution step of
940the processor. An overflow that occurs during fetching but is overwritten
941during execution is, however, correct and necessary to accept correct
942programs that are as large as the program memory. Temporary overflows add
943complications to the proof. A reachable address is an address that can be
944obtained by fetching (not executing!) a finite number of times from the
945beginning of the code memory without ever overflowing. The complication is that
946the static prediction traverses the code memory assuming that the memory will
947be read sequentially from the beginning and that all jumps jump only to
948reachable addresses. When this property is violated, the way the code memory
949is interpreted is uncorrect and the cost computed is totally meaningless.
950The reachability relation is closed by fetching for well formed programs.
951The property that calls to function pointers only target reachable and
952well labelled locations, however, is not statically predictable and it is
953enforced in the structured trace.
954
955The proof of correctness of cost predictions has been quite complex. Setting
956up the good invariants (structured traces, well formed programs, reachability)
957and completing the proof has required more than 3 men months while the initally
958estimated effort was much lower. In the paper-and-pencil proof for IMP, the
959corresponding proof was obvious and only took two lines.
960
961The proof itself is quite involved. We
962basically need to show as an important lemma that the sum of the execution
963costs over a structured trace, where the costs are summed in execution order,
964is equivalent to the sum of the execution costs in the order of pre-payment.
965The two orders are quite different and the proof is by mutual recursion over
966the definition of the converging structured traces, which is a family of three
967mutual inductive types. The fact that this property only holds for converging
968function calls in hidden in the definition of the structured traces.
969Then we need to show that the order of pre-payment
970corresponds to the order induced by the cost traces extracted from the
971structured trace. Finally, we need to show that the statically computed cost
972for one block corresponds to the cost dinamically computed in pre-payment
973order.
974
[1758]975\section{Overall results}
976
977Functional correctness of the compiled code can be shown by composing
978the simulations to show that the target behaviour matches the
979behaviour of the source program, if the source program does not go
980wrong'.  More precisely, we show that there is a forward simulation
981between the source trace and a (flattened structured) trace of the
982output, and conclude equivalence because the target's semantics are
983in the form of an executable function, and hence
984deterministic.
985
986Combining this with the correctness of the assignment of costs to cost
987labels at the ASM level for a structured trace, we can show that the
988cost of executing any compiled function (including the main function)
989is equal to the sum of all the values for cost labels encountered in
990the \emph{source code's} trace of the function.
991
[1741]992\section{Estimated effort}
[1742]993Based on the rough analysis performed so far we can estimate the total
994effort for the certification of the compiler. We obtain this estimation by
[1741]995combining, for each pass: 1) the number of lines of code to be certified;
[1742]9962) the ratio of number of lines of proof to number of lines of code from
997the CompCert project~\cite{compcert} for the CompCert pass that is closest to
[1741]998ours; 3) an estimation of the complexity of the pass according to the
[1775]999analysis above. The result is shown in Table~\ref{table}.
[1741]1000
[1775]1001\begin{table}{h}
[1756]1002\begin{tabular}{lrlrr}
[1747]1003Pass origin & Code lines & CompCert ratio & Estimated effort & Estimated effort \\
1004            &            &                & (based on CompCert) & \\
[1744]1005\hline
[1756]1006Common &  4864 & 4.25 \permil & 20.67 & 17.0 \\
1007Cminor &  1057 & 5.23 \permil & 5.53  &  6.0 \\
1008Clight &  1856 & 5.23 \permil & 9.71  & 10.0 \\
1009RTLabs &  1252 & 1.17 \permil & 1.48  &  5.0 \\
1010RTL    &   469 & 4.17 \permil & 1.95  &  2.0 \\
[1754]1011ERTL   &   789 & 3.01 \permil & 2.38  & 2.5 \\
1012LTL    &    92 & 5.94 \permil & 0.55  & 0.5 \\
[1756]1013LIN    &   354 & 6.54 \permil & 2.31  &   1.0 \\
1014ASM    &   984 & 4.80 \permil & 4.72  &  10.0 \\
[1744]1015\hline
[1756]1016Total common    &  4864 & 4.25 \permil & 20.67 & 17.0 \\
1017Total front-end &  2913 & 5.23 \permil & 15.24 & 16.0 \\
1018Total back-end  &  6853 & 4.17 \permil & 13.39 & 21.0 \\
[1747]1019\hline
[1756]1020Total           & 14630 & 3.75 \permil & 49.30 & 54.0 \\
[1741]1021\end{tabular}
[1775]1022\caption{\label{table}Estimated effort}
1023\end{table}
[1741]1024
[1745]1025We provide now some additional informations on the methodology used in the
1026computation. The passes in Cerco and CompCert front-end closely match each
1027other. However, there is no clear correspondence between the two back-ends.
1028For instance, we enforce the calling convention immediately after instruction
1029selection, whereas in CompCert this is performed in a later phase. Or we
1030linearize the code at the very end, whereas CompCert performs linearization
1031as soon as possible. Therefore, the first part of the exercise has consisted
1032in shuffling and partitioning the CompCert code in order to assign to each
1033CerCo pass the CompCert code that performs the same transformation.
1034
1035After this preliminary step, using the data given in~\cite{compcert} (which
1036are relative to an early version of CompCert) we computed the ratio between
1037men months and lines of code in CompCert for each CerCo pass. This is shown
1038in the third column of Table~\ref{wildguess}. For those CerCo passes that
1039have no correspondence in CompCert (like the optimizing assembler) or where
1040we have insufficient data, we have used the average of the ratios computed
1041above.
1042
1043The first column of the table shows the number of lines of code for each
1044pass in CerCo. The third column is obtained multiplying the first with the
1045CompCert ratio. It provides an estimate of the effort required (in men months)
1046if the complexity of the proofs for CerCo and Compcert would be the same.
1047
1048The two proof styles, however, are on purpose completely different. Where
1049CompCert uses non executable semantics, describing the various semantics with
1050inductive types, we have preferred executable semantics. Therefore, CompCert
1051proofs by induction and inversion become proof by functional inversion,
1052performed using the Russel methodology (now called Program in Coq, but whose
1053behaviour differs from Matita's one). Moreover, CompCert code is written using
1054only types that belong to the Hindley-Milner fragment, whereas we have
1055heavily exploited dependent types all over the code. The dependent type
1056discipline offers many advantages from the point of view of clarity of the
1057invariants involved and early detection of errors and it naturally combines
1058well with the Russel approach which is based on dependent types. However, it
1059is also well known to introduce technical problems all over the code, like
1060the need to explicitly prove type equalities to be able to manipulate
1061expressions in certain ways. In many situations, the difficulties encountered
1062with manipulating dependent types are better addressed by improving the Matita
1063system, according to the formalization driven system development. For this
1064reason, and assuming a pessimistic point of view on our performance, the
1065fourth columns presents the final estimation of the effort required, that also
1066takes in account the complexity of the proof suggested by the informal proofs
1067sketched in the previous section.
1068
[1774]1069\subsection{Contingency plan}
1070On the basis of the proof strategy sketched in this document and the
1071estimated effort, we can refine our contingency plan. In case we will end
1072the certification of the basic compiler in advance we will have the choice
1073of either proving loop optimizations and/or partial redundancy elimination
1074correct (both tasks that seem difficult to achieve in a short time) or
1075considering the MCS-51 specific extensions introduced during the first period
1076and under-used in the formalized prototype. Yet another possibility would be
1077to better study retargeting of the code and the commutation property between
1078different compiler passes. The latter study is easily enabled by our
1079approach where all back-end languages are instances of the same parameterized
1080language.
1081
1082In the case of a consistent delay in the certification of some
1083components, we will address first the passes that are more likely to have
1084undetected bugs and we will follow a top-down approach, axiomatizing
1085the properties of the data structured used in the compiler to focus more
1086on the algorithms. The rational is that data structures are easier then
1087algorithms to test using well known methodologies.
1088The effort table clearly shows that commond definitions
1089and data structures are 1/4th of the size of the code and require slightly
1090less than 1/3rd of the total effort. At least half of this effort really goes
1091into simple data structures (vectors, bounded and unbounded integers, tries
[1775]1092and maps) whose certification is not interesting and whose code could be
1093taken without re-proving it from the library of any other theorem prover.
[1774]1094
[1762]1095\section{Conclusions}
1096The overall exercise, whose details have been only minimally reported here,
1097has been very useful. It has allowed to spot in an early moment some criticities
1098of the proof that have required major changes in the proof plan. It has also
1099shown that the last passes of the compilation (e.g. assembly) and cost
1100prediction on the object code are much more involved than more high level
1101passes.
1102
1103The final estimation for the effort is surely affected by a low degree of
1104confidence. It is however sufficient to conclude that the effort required
1105is in line with the man power that was scheduled for the second half of the
1106second period and for the third period. Compared to the number of men months
1107declared in Annex I of the contract, we will need more men months. However,
1108both at UNIBO and UEDIN there have been major differences in hiring with
1109respect to the Annex. Therefore both sites have now an higher number of men
1110months available, with the trade-off of a lower level of maturity of the
1111people employed.
1112
1113The reviewers suggested that we use this estimation to compare two possible
1114scenarios: a) proceed as planned, porting all the CompCert proofs to Matita
1115or b) port D3.1 and D4.1 to Coq and re-use the CompCert proofs.
1116We remark here again that the back-end of the two compilers, from the
1117memory model on, are sensibly different: we are not re-proving correctness
1118of the same piece of code. Moreover, the proof techniques are different for
1119the front-end too. Switching to the CompCert formalization would imply
1120the abandon of the untrusted compiler, the abandon of the experiment with
1121a different proof technique, the abandon of the quest for an open source
1122proof, and the abandon of the co-development of the formalization and the
1123Matita proof assistant. In the Commitment Letter~\cite{letter} delivered
1124to the Officer in May we clarified our personal perspective on the project
1125goals and objectives. We do not re-describe here the point of view presented
1126in the letter that we can condense in we value diversity''.
1127
1128Clearly, if the execise would have suggested the infeasability in terms of
1129effort of concluding the formalization or getting close to that, we would have
1130abandoned our path and embraced the reviewer's suggestion. However, we
1131have been comforted in the analysis we did in autumn and further progress done
1132during the winter does not show yet any major delay with respect to the
1133proof schedule. We are thus planning to continue the certification according
1134to the more detailed proof plan that came out from the exercise reported in
1135this manuscript.
1136
[1776]1137\begin{thebibliography}{2}
1138\bibitem{compcert} X. Leroy, A Formally Verified Compiler back-end'',
1139Journal of Automated Reasoning 43(4)):363-446, 2009.
1140
1141\bibitem{letter}The CerCo team, `Commitment to the Consideration of Reviewer's Reccomendation'', 16/05/2011.
1142\end{thebibliography}
1143
1144
[1716]1145\end{document}
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