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32\vspace*{1cm}Project FP7-ICT-2009-C-243881 {\cerco}}
[1778]34\date{ }
54Proof outline for the correctness of the CerCo compiler
61Version 1.0
68Main Authors:\\
69B. Campbell, D. Mulligan, P. Tranquilli, C. Sacerdoti Coen
75Project Acronym: {\cerco}\\
76Project full title: Certified Complexity\\
77Proposal/Contract no.: FP7-ICT-2009-C-243881 {\cerco}\\
79\clearpage \pagestyle{myheadings} \markright{{\cerco}, FP7-ICT-2009-C-243881}
[1749]86In the last project review of the CerCo project, the project reviewers
87recommended us to quickly outline a paper-and-pencil correctness proof
88for each of the stages of the CerCo compiler in order to allow for an
89estimation of the complexity and time required to complete the formalization
90of the proof. This has been possible starting from month 18 when we have
91completed the formalization in Matita of the datastructures and code of
92the compiler.
[1749]94In this document we provide a very high-level, pen-and-paper
95sketch of what we view as the best path to completing the correctness proof
[1758]96for 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]97been completed at the end of the First Period.
99In the last section we finally present an estimation of the effort required
[1752]100for the certification in Matita of the compiler and we draw conclusions.
[1731]102\section{Front-end: Clight to RTLabs}
104The front-end of the CerCo compiler consists of several stages:
109\quad \= $\downarrow$ \quad \= \kill
111\> $\downarrow$ \> cast removal\\
112\> $\downarrow$ \> add runtime functions\footnote{Following the last project
113meeting we intend to move this transformation to the back-end}\\
114\> $\downarrow$ \> cost labelling\\
[1741]115\> $\downarrow$ \> 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)\\
116\> $\downarrow$ \> partial redundancy elimination$^{\mbox{\scriptsize \ref{lab:opt2}}}$ (an endo-transformation)\\
[1731]117\> $\downarrow$ \> stack variable allocation and control structure
118 simplification\\
120\> $\downarrow$ \> generate global variable initialisation code\\
121\> $\downarrow$ \> transform to RTL graph\\
[1741]123\> $\downarrow$ \> \\
[1740]129Here, by `endo-transformation', we mean a mapping from language back to itself:
130the loop optimization step maps the Clight language to itself.
[1731]132%Our overall statements of correctness with respect to costs will
133%require a correctly labelled program
134There are three layers in most of the proofs proposed:
136\item invariants closely tied to the syntax and transformations using
[1758]137  dependent types (such as the presence of variable names in environments),
138\item a forward simulation proof relating each small-step of the
139  source to zero or more steps of the target, and
140\item proofs about syntactic properties of the cost labelling.
[1758]142The first will support both functional correctness and allow us to
143show the totality of some of the compiler stages (that is, those
144stages of the compiler cannot fail).  The second provides the main
145functional correctness result, including the preservation of cost
146labels in the traces, and the last will be crucial for applying
147correctness results about the costings from the back-end by showing
148that they appear in enough places so that we can assign all of the
149execution costs to them.
151We will also prove that a suitably labelled RTLabs trace can be turned
152into a \emph{structured trace} which splits the execution trace into
[1746]153cost-label to cost-label chunks with nested function calls.  This
[1731]154structure was identified during work on the correctness of the
155back-end cost analysis as retaining important information about the
156structure of the execution that is difficult to reconstruct later in
157the compiler.
[1746]159\subsection{Clight cast removal}
[1746]161This transformation removes some casts inserted by the parser to make
162arithmetic promotion explicit but which are superfluous (such as
163\lstinline[language=C]'c = (short)((int)a + (int)b);' where
164\lstinline'a' and \lstinline'b' are \lstinline[language=C]'short').
165This is necessary for producing good code for our target architecture.
[1746]167It only affects Clight expressions, recursively detecting casts that
168can be safely eliminated.  The semantics provides a big-step
169definition for expression, so we should be able to show a lock-step
170forward simulation between otherwise identical states using a lemma
171showing that cast elimination does not change the evaluation of
172expressions.  This lemma will follow from a structural induction on
173the source expression.  We have already proved a few of the underlying
174arithmetic results necessary to validate the approach.
176\subsection{Clight cost labelling}
178This adds cost labels before and after selected statements and
179expressions, and the execution traces ought to be equivalent modulo
[1746]180the new cost labels.  Hence it requires a simple forward simulation
181with a limited amount of stuttering whereever a new cost label is
182introduced.  A bound can be given for the amount of stuttering allowed
[1731]183based on the statement or continuation to be evaluated next.
185We also intend to show three syntactic properties about the cost
[1731]188\item every function starts with a cost label,
[1746]189\item every branching instruction is followed by a cost label (note that
[1731]190  exiting a loop is treated as a branch), and
191\item the head of every loop (and any \lstinline'goto' destination) is
192  a cost label.
[1731]194These can be shown by structural induction on the source term.
196\subsection{Clight to Cminor translation}
[1732]198This translation is the first to introduce some invariants, with the
199proofs closely tied to the implementation by dependent typing.  These
200are largely complete and show that the generated code enjoys:
202\item some minimal type safety shown by explicit checks on the
203  Cminor types during the transformation (a little more work remains
204  to be done here, but follows the same form);
205\item that variables named in the parameter and local variable
206  environments are distinct from one another, again by an explicit
207  check;
208\item that variables used in the generated code are present in the
209  resulting environment (either by checking their presence in the
[1746]210  source environment, or from a list of freshly generated temporary variables);
[1732]211  and
212\item that all \lstinline[language=C]'goto' labels are present (by
213  checking them against a list of source labels and proving that all
214  source labels are preserved).
217The simulation will be similar to the relevant stages of CompCert
218(Clight to Csharpminor and Csharpminor to Cminor --- in the event that
[1746]219the direct proof is unwieldy we could introduce an intermediate
220language corresponding to Csharpminor).  During early experimentation
221with porting CompCert definitions to the Matita proof assistant we
222found little difficulty reproving the results for the memory model, so
223we plan to port the memory injection properties and use them to relate
224Clight in-memory variables with either the value of the local variable or a
225stack slot, depending on how it was classified.
227This should be sufficient to show the equivalence of (big-step)
228expression evaluation.  The simulation can then be shown by relating
229corresponding blocks of statement and continuations with their Cminor
230counterparts and proving that a few steps reaches the next matching
233The syntactic properties required for cost labels remain similar and a
234structural induction on the function bodies should be sufficient to
235show that they are preserved.
[1731]237\subsection{Cminor global initialisation code}
[1732]239This short phase replaces the global variable initialisation data with
240code that executes when the program starts.  Each piece of
241initialisation data in the source is matched by a new statement
242storing that data.  As each global variable is allocated a distinct
243memory block, the program state after the initialisation statements
244will be the same as the original program's state at the start of
245execution, and will proceed in the same manner afterwards.
247% Actually, the above is wrong...
248% ... this ought to be in a fresh main function with a fresh cost label
[1731]250\subsection{Cminor to RTLabs translation}
[1733]252In this part of the compiler we transform the program's functions into
253control flow graphs.  It is closely related to CompCert's Cminorsel to
254RTL transformation, albeit with target-independent operations.
256We already enforce several invariants with dependent types: some type
[1746]257safety, mostly shown using the type information from Cminor; and
[1733]258that the graph is closed (by showing that each successor was recently
259added, or corresponds to a \lstinline[language=C]'goto' label which
260are all added before the end).  Note that this relies on a
261monotonicity property; CompCert maintains a similar property in a
262similar way while building RTL graphs.  We will also add a result
263showing that all of the pseudo-register names are distinct for use by
264later stages using the same method as Cminor.
[1746]266The simulation will relate Cminor states to RTLabs states which are about to
267execute the code corresponding to the Cminor statement or continuation.
[1733]268Each Cminor statement becomes zero or more RTLabs statements, with a
269decreasing measure based on the statement and continuations similar to
270CompCert's.  We may also follow CompCert in using a relational
271specification of this stage so as to abstract away from the functional
272(and highly dependently typed) definition.
274The first two labelling properties remain as before; we will show that
275cost labels are preserved, so the function entry point will be a cost
[1746]276label, and successors to any statement that are cost labels map still
277map to cost labels, preserving the condition on branches.  We replace
278the property for loops with the notion that we will always reach a
279cost label or the end of the function after following a bounded number of
280successors.  This can be easily seen in Cminor using the requirement
281for cost labels at the head of loops and after gotos.  It remains to
282show that this is preserved by the translation to RTLabs.  % how?
[1731]284\subsection{RTLabs structured trace generation}
[1733]286This proof-only step incorporates the function call structure and cost
287labelling properties into the execution trace.  As the function calls
288are nested within the trace, we need to distinguish between
[1746]289terminating and non-terminating function calls.  Thus we use the
290excluded middle (specialised to a function termination property) to do
[1746]293Structured traces for terminating functions are built by following the
294flat trace, breaking it into chunks between cost labels and
295recursively processing function calls.  The main difficulties here are
296the non-structurally recursive nature of the function (instead we use
297the size of the termination proof as a measure) and using the RTLabs
298cost labelling properties to show that the constraints of the
299structured traces are observed.  We also show that the lower stack
300frames are preserved during function calls in order to prove that
301after returning from a function call we resume execution of the
302correct code.  This part of the work has already been constructed, but
303still requires a simple proof to show that flattening the structured
[1733]304trace recreates the original flat trace.
[1746]306The non-terminating case follows the trace like the terminating
307version to build up chunks of trace from cost-label to cost-label
308(which, by the finite distance to a cost label property shown before,
309can be represented by an inductive type).  These chunks are chained
310together in a coinductive data structure that can represent
311non-terminating traces.  The excluded middle is used to decide whether
312function calls terminate, in which case the function described above
313constructs an inductive terminating structured trace which is nested
314in the caller's trace.  Otherwise, another coinductive constructor is
315used to embed the non-terminating trace of the callee, generated by
316corecursion.  This part of the trace transformation is currently under
317construction, and will also need a flattening result to show that it
318is correct.
[1734]321\section{Backend: RTLabs to machine code}
[1734]324The compiler backend consists of the following intermediate languages, and stages of translation:
[1741]329\quad \=\,\vdots\= \\
330\> $\downarrow$ \>\\
331\> $\downarrow$ \quad \= \kill
[1739]333\> $\downarrow$ \> 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]334\> $\downarrow$ \> instruction selection\\
[1748]335\> $\downarrow$ \> change of memory models in compiler\\
[1739]337\> $\downarrow$ \> constant propagation$^{\mbox{\scriptsize \ref{lab:opt}}}$ (an endo-transformation) \\
[1734]338\> $\downarrow$ \> calling convention made explicit \\
[1741]339\> $\downarrow$ \> layout of activation records \\
341\> $\downarrow$ \> register allocation and spilling\\
342\> $\downarrow$ \> dead code elimination\\
344\> $\downarrow$ \> function linearisation\\
345\> $\downarrow$ \> branch compression (an endo-transformation) \\
347\> $\downarrow$ \> relabeling\\
349\> $\downarrow$ \> pseudoinstruction expansion\\
[1741]350\textsf{MCS-51 machine code}\\
[1769]355\subsection{Graph translations}
[1773]356RTLabs and most intermediate languages in the back-end have a graph
358the code of each function is represented by a graph of instructions.
359The graph maps a set of labels (the names of the nodes) to the instruction
360stored at that label (the nodes of the graph).
361Instructions reference zero or more additional labels that are the immediate
362successors of the instruction: zero for return from functions; more than one
363for conditional jumps and calls; one in all other cases. The references
364from one instruction to its immediates are the arcs of the graph.
[1773]366Status of graph languages always have a program counter that holds a
367representation of a reference to the current instruction.
369A translation between two consecutive graph languages maps each instruction
370stored at location $l$ in the first graph and with immediate successors
371$\{l_1,\ldots,l_n\}$ to a subgraph of the output graph that has a single
372entry point at location $l$ and exit arcs to $\{l_1,\ldots,l_n\}$. Moreover,
373the labels of all non entry nodes in the subgraph are distinct from all the
374labels in the source graph.
376In order to simplify the translations and the relative proofs of forward
377simulation, after the release of D4.2 and D4.3, we have provided:
379 \item a new data type (called \texttt{blist}) that represents a
380   sequence of instructions to be added to the output graph.
381   The ``b'' in the name stands for binder, since a \texttt{blist} is
382   either empty, an extension of a \texttt{blist} with an instruction
383   at the front, or the generation of a fresh quantity followed by a
384   \texttt{blist}. The latter feature is used, for instance, to generate
385   fresh register names. The instructions in the list are unlabelled and
386   all of them but the last one are also sequential, like in a linear program.
387 \item a new iterator (called \texttt{b\_graph\_translate}) of type
389\mathtt{b\_graph\_translate}: (\mathtt{label} \rightarrow \mathtt{blist})
390\rightarrow \mathtt{graph} \rightarrow \mathtt{graph}
392   The iterator transform the input graph in the output graph by replacing
393   each node with the graph that corresponds to the linear \texttt{blist}
394   obtained by applying the function in input to the node label.
397Using the iterator above, the code can be written in such a way that
398the programmer does not see any distinction between writing a transformation
399on linear or graph languages.
401In order to prove simulations for translations obtained using the iterator,
402we will prove the following theorem:
405\mathtt{theorem} &\ \mathtt{b\_graph\_translate\_ok}: \\
406& \forall  f.\forall G_{i}.\mathtt{let}\ G_{\sigma} := \mathtt{b\_graph\_translate}\ f\ G_{i}\ \mathtt{in} \\
407&       \forall l \in G_{i}.\mathtt{subgraph}\ (f\ l)\ l\ (next \ l \ G_i)\ G_{\sigma}
410Here \texttt{subgraph} is a computational predicate that given a \texttt{blist}
411$[i_1, \ldots, i_n]$, an entry label $l$, an exit label $l'$ and a graph $G$
412expands to the fact that fetching from $G$ at address $l$ one retrieves a node
413$i_1$ with a successor $l_1$ that, when fetched, yields a node $i_2$ with a
414successor $l_2$ such that \ldots. The successor of $i_n$ is $l'$.
416Proving a forward simulation diagram of the following kind using the theorem
417above is now as simple as doing the same using standard small step operational
418semantics over linear languages.
421\mathtt{lemma} &\ \mathtt{execute\_1\_step\_ok}: \\
422&       \forall s.  \mathtt{let}\ s' := s\ \sigma\ \mathtt{in} \\
423&       \mathtt{let}\ l := pc\ s\ \mathtt{in} \\
424&       s \stackrel{1}{\rightarrow} s^{*} \Rightarrow \exists n. s' \stackrel{n}{\rightarrow} s'^{*} \wedge s'^{*} = s'\ \sigma
427Because of the fact that graph translation preserves entry and exit labels of
428translated statements, the state translation function $\sigma$ will simply
429preserve the value of the program counter. The program code, which is
430part of the state, is translated using the iterator.
432The proof is then roughly the following. Let $l$ be the program counter of the
433input state $s$. We proceed by cases on the current instruction of $s$.
434Let $[i_1, \ldots, i_n]$ be the \texttt{blist} associated to $l$ and $s$
435by the translation function. The witness required for the existential
436statement is simply $n$. By applying the theorem above we know that the
437next $n$ instructions that will be fetched from $s\ \sigma$ will be
438$[i_1, \ldots, i_n]$ and it is now sufficient to prove that they simulate
439the original instruction.
[1734]441\subsection{The RTLabs to RTL translation}
[1748]444The RTLabs to RTL translation pass marks the frontier between the two memory models used in the CerCo project.
445As a result, we require some method of translating between the values that the two memory models permit.
446Suppose we have such a translation, $\sigma$.
447Then the translation between values of the two memory models may be pictured with:
[1752]450\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
[1752]453In 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
454representables in a byte, but some others require more bits.
[1752]456In the back-end model all values are meant to be represented in a single byte.
457Values can thefore be undefined, be one byte long integers or be indexed
458fragments of a pointer, null or not. Floats values are no longer present, as floating point arithmetic is not supported by the CerCo compiler.
[1752]460The $\sigma$ map implements a one-to-many relation: a single front-end value
461is mapped to a sequence of back-end values when its size is more then one byte.
463We 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
464thought as an instance of a generic \texttt{Mem} data type parameterized over
465the kind of values stored in memory.
468\mathtt{Mem}\ \alpha = \mathtt{Block} \rightarrow (\mathbb{Z} \rightarrow \alpha)
[1748]471Here, \texttt{Block} consists of a \texttt{Region} paired with an identifier.
[1752]474\mathtt{Block} ::= \mathtt{Region} \times \mathtt{ID}
[1748]477We now have what we need for defining what is meant by the `memory' in the backend memory model.
478Namely, we instantiate the previously defined \texttt{Mem} type with the type of back-end memory values.
[1768]481\mathtt{BEMem} = \mathtt{Mem}~\mathtt{BEValue}
[1748]484Memory addresses consist of a pair of back-end memory values:
487\mathtt{Address} = \mathtt{BEValue} \times  \mathtt{BEValue} \\
[1751]490The back- and front-end memory models differ in how they represent sized integeer values in memory.
491In 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.
492In contrast, the layout of sized integer values in the back-end memory model consists of a series of byte-sized `chunks':
[1752]495\begin{picture}(0, 25)
500\put(-15,10){\vector(1, 0){30}}
[1760]508Chunks for pointers are pairs made of the original pointer and the index of the chunk.
509Therefore, when assembling the chunks together, we can always recognize if all chunks refer to the same value or if the operation is meaningless.
[1751]511The 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.
512The first lemma required has the following statement:
514\mathtt{load}\ s\ a\ M = \mathtt{Some}\ v \rightarrow \forall i \leq s.\ \mathtt{load}\ s\ (a + i)\ \sigma(M) = \mathtt{Some}\ v_i
[1768]516That 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 $i \leq s$).
[1760]518Next, we must show that \texttt{store} properly commutes with the $\sigma$-map between memory spaces:
[1760]520\sigma(\mathtt{store}\ a\ v\ M) = \mathtt{store}\ \sigma(v)\ \sigma(a)\ \sigma(M)
[1760]522That 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 $\mathtt{\sigma(M)}$.
[1768]524Finally, the commutation properties between \texttt{load} and \texttt{store} are weakened in the $\sigma$-image of the memory.
525Writing \texttt{load}$^*$ for the multiple consecutive iterations of \texttt{load} used to fetch all chunks of a value, we must prove that, when $a \neq a'$:
[1768]527\texttt{load}^* \sigma(a)\ (\mathtt{store}\ \sigma(a')\ \sigma(v)\ \sigma(M)) = \mathtt{load}^*\ \sigma(s)\ \sigma(a)\ \sigma(M)
[1768]529That 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
530distinct by hypothesis, there is a priori no guarantee that the consecutive
531bytes for the value stored at $\sigma(a)$ are disjoint from those for the
532values stored at $\sigma(a')$. The fact that this holds is a non-trivial
533property of $\sigma$ to be proved.
[1763]535RTLabs states come in three flavours:
538\mathtt{State} & ::=  & (\mathtt{State} : \mathtt{Frame}^* \times \mathtt{Frame} \\
539               & \mid & \mathtt{Call} : \mathtt{Frame}^* \times \mathtt{Args} \times \mathtt{Return} \times \mathtt{Fun} \\
540               & \mid & \mathtt{Return} : \mathtt{Frame}^* \times \mathtt{Value} \times \mathtt{Return}) \times \mathtt{Mem}
[1763]543\texttt{State} is the default state in which RTLabs programs are almost always in.
544The \texttt{Call} state is only entered when a call instruction is being executed, and then we immediately return to being in \texttt{State}.
545Similarly, \texttt{Return} is only entered when a return instruction is being executed, before returning immediately to \texttt{State}.
546All 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.
547\texttt{State} keeps track of a list of stack frames.
[1763]549RTL 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}:
551\mathtt{State} ::= \mathtt{Frame}^* \times \mathtt{PC} \times \mathtt{SP} \times \mathtt{ISP} \times \mathtt{CARRY} \times \mathtt{REGS}
[1763]553The internal stack pointer \texttt{ISP}, and its relationship with the stack pointer \texttt{SP}, needs some comment.
554Due 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.
555As 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.
556The 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]557Instructions 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
558and local variables are only present in the external stack.
[1763]559As 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.
[1768]561Once more, we require a relation $\sigma$ between RTLabs states and RTL states.
562Because $\sigma$ is one-to-many and, morally, a multi-function,
563we use in the following the functional notation for $\sigma$, using $\star$
564in the output of $\sigma$ to mean that any value is accepted.
566\mathtt{State} \stackrel{\sigma}{\longrightarrow} \mathtt{State}
[1763]569Translating 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}.
570For \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:
[1768]572\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}))
[1768]574Translation 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
575and the carry bit is admitted.
[1763]577Translating \texttt{Call} and \texttt{Return} states is more involved, as a commutation between a single step of execution and the translation process must hold:
579\sigma(\mathtt{Return}(-)) \longrightarrow \sigma \circ \text{return one step}
[1718]583\sigma(\mathtt{Call}(-)) \longrightarrow \sigma \circ \text{call one step}
[1763]586Here \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]587We 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
588in the target language is not problematic for preservation of divergence
589because it is easy to show that every step from a \texttt{Call} or
590\texttt{Return} state is always preceeded/followed by one step that is always
[1718]596s & \rTo^{\text{one step of execution}} & s'   \\
597  & \rdTo                             & \dTo \\
598  &                                   & \llbracket s'' \rrbracket
604s & \rTo^{\text{one step of execution}} & s'   \\
605  & \rdTo                             & \dTo \\
606  &                                   & \llbracket s'' \rrbracket
[1763]609\caption{The one-step commuting diagrams for \texttt{Call} and \texttt{Return} state translations}
[1771]613The forward simulation proof for all steps that do not involve function calls are lengthy, but routine.
614They 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.
615The properties of $\sigma$ presented before that relate values and memories will need to be heavily exploited.
[1771]617The simulation of invocation of functions and returns from functions is less obvious.
618We sketch here what happens on the source code and on its translation.
[1763]622\mathtt{Call(id,\ args,\ dst,\ pc),\ State(Frame^*, Frame)} & \longrightarrow & \mathtt{Call(M(args), dst)}, \\
623                                                           &                 & \mathtt{PUSH(Frame[PC := after\_return])}
[1763]626Suppose we are given a \texttt{State} with a list of stack frames, with the top frame being \texttt{Frame}.
627Suppose also that the program counter in \texttt{Frame} points to a \texttt{Call} instruction, complete with arguments and destination address.
[1771]628Then 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.
[1763]630Now, what happens next depends on whether we are executing an internal or an external function.
[1743]631In the case where the call is to an external function, we have:
[1743]634\mathtt{Call(M(args), dst)},                       & \stackrel{\mathtt{ret\_val = f(M(args))}}{\longrightarrow} & \mathtt{Return(ret\_val,\ dst,\ PUSH(...))} \\
635\mathtt{PUSH(current\_frame[PC := after\_return])} &                                                            & 
[1766]638That 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]639Then the return state restores the program counter by popping the stack, and execution proceeds in a new \texttt{State}:
[1767]642\mathtt{Return(ret\_val,\ dst,\ PUSH(...))} & \longrightarrow & \mathtt{pc = POP\_STACK(regs[dst := M(ret\_val)],\ pc)} \\
643                                            &                 & \mathtt{State(regs[dst := M(ret\_val),\ pc)}
[1767]647Suppose we are executing an internal function, however:
[1772]650\mathtt{Call(M(args), dst)}                        & \longrightarrow & \mathtt{SP = alloc,\ regs = \emptyset[- := params]} \\
651\mathtt{PUSH(current\_frame[PC := after\_return])} &                 & \mathtt{State(regs,\ sp,\ pc_\emptyset,\ dst)}
[1767]654Here, execution of the \texttt{Call} state first pushes the current frame with the program counter set to the address following the function call.
655The 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]656After this, the stack pointer is freed and a \texttt{Return} state is entered:
[1772]659\mathtt{sp = alloc,\ regs = \emptyset[- := PARAMS]} & \longrightarrow & \mathtt{free(sp)} \\
660\mathtt{State(regs,\ sp,\ pc_\emptyset,\ dst)}     &                 & \mathtt{Return(M(ret\_val), dst, Frames)}
[1770]663Then 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:
666\mathtt{free(sp)}                         & \longrightarrow & \mathtt{pc = POP\_STACK(regs[dst := M(ret\_val)],\ pc)} \\
[1770]667\mathtt{Return(M(ret\_val), dst, frames)} &                 & \mathtt{State(regs[dst := M(ret\_val),\ pc)}
[1777]671Translation from RTLabs to RTL states proceeds as follows.
672Return states are translated as is:
[1777]674\mathtt{Return} \longrightarrow \mathtt{Return}
677\texttt{Call} states are translated to \texttt{Call\_ID} states:
679\mathtt{Call(id,\ args,\ dst,\ pc)} \longrightarrow \mathtt{Call\_ID(id,\ \sigma'(args),\ \sigma(dst),\ pc)}
681Here, $\sigma$ and $\sigma'$ are two maps to be defined between pseudo-registers and lists of pseudo-registers, of the type:
684\sigma: \mathtt{register} \rightarrow \mathtt{list\ register}
690\sigma': \mathtt{list\ register} \rightarrow \mathtt{list\ register}
693where $\sigma'$ is implemented as:
696\sigma' = \mathtt{flatten} \circ \sigma
699In the case of RTL, execution proceeds as follows.
700Suppose we are executing a \texttt{CALL\_ID} instruction.
701Then a case split occurs depending on whether we are executing an internal or an external function, as in the RTLabs case:
[1724]704& & \llbracket \mathtt{CALL\_ID}(\mathtt{id}, \mathtt{args}, \mathtt{dst}, \mathtt{pc})\rrbracket & & \\
705& \ldTo^{\text{external}} & & \rdTo^{\text{internal}} & \\
706\skull & & & & \mathtt{regs} = [\mathtt{params}/-] \\
707& & & & \mathtt{sp} = \mathtt{ALLOC} \\
708& & & & \mathtt{PUSH}(\mathtt{carry}, \mathtt{regs}, \mathtt{dst}, \mathtt{return\_addr}), \mathtt{pc}_{0}, \mathtt{regs}, \mathtt{sp} \\
[1777]711Here, however, we differ from RTLabs when we attempt to execute an external function, in that we use a daemon (i.e. an axiom that can close any goal) to artificially close the case, as we have not yet implemented external functions in the backend.
712The reason for this lack of implementation is as follows.
713Though we have implemented an optimising assembler as the target of the compiler's backend, we have not yet implemented a linker for that assembler, so external functions can not yet be called.
714Whilst external functions are carried forth throughout the entirety of the compiler's frontend, we choose not to do the same for the backend, instead eliminating them in RTL.
715However, it is plausible that we could have carried external functions forth, in order to eliminate them at a later stage (i.e. when translating from LIN to assembly).
[1777]717In the case of an internal function being executed, we proceed as follows.
718The register map is initialized to the empty map, where all registers are assigned the undefined value, and then the registers corresponding to the function parameters are assigned the value of the parameters.
719Further, the stack pointer is reallocated to make room for an extra stack frame, then a frame is pushed onto the stack with the correct address to jump back to in place of the program counter.
721Note, in particular, that this final act of pushing a frame on the stack leaves us in an identical state to the RTLabs case, where the instruction
723\mathtt{PUSH(current\_frame[PC := after\_return])}
726was executed.
728The execution of \texttt{Return} in RTL is similarly straightforward, with the return address, stack pointer, and so on, being computed by popping off the top of the stack, and the return value computed by the function being retrieved from memory:
730\mathtt{return\_addr} & := \mathtt{top}(\mathtt{stack}) \\
[1777]731v*                    & := M(\mathtt{rv\_regs}) \\
[1724]732\mathtt{dst}, \mathtt{sp}, \mathtt{carry}, \mathtt{regs} & := \mathtt{pop} \\
733\mathtt{regs}[v* / \mathtt{dst}] \\
[1779]736Translation and execution must satisfy a pair of commutation properties for the \texttt{Return} and \texttt{Call} cases.
737Starting from any \texttt{Return} or \texttt{Call} state, translating and then executing a single step must be the same as executing exactly two steps and then translating, with the intermediate state obtained by executing once also being translatable to the final state.
738This is exemplified by the following diagram:
741s    & \rTo^1 & s' & \rTo^1 & s'' \\
742\dTo &        &    & \rdTo  & \dTo \\
743\llbracket s \rrbracket & \rTo(1,3)^1 & & & \llbracket s'' \rrbracket \\ 
[1780]747\subsection{The RTL to ERTL translation}
750We map RTL statuses to ERTL statuses as follows:
[1780]752\mathtt{sp} & = \mathtt{RegisterSPH} / \mathtt{RegisterSPL} \\
[1750]753\mathtt{graph} &  \mathtt{graph} + \mathtt{prologue}(s) + \mathtt{epilogue}(s) \\
[1727]754& \mathrm{where}\ s = \mathrm{callee\ saved} + \nu \mathrm{RA} \\
[1780]756The 16-bit RTL stack pointer \texttt{SP} is mapped to a pair of 8-bit hardware registers \texttt{RegisterSPH} and \texttt{RegisterSPL}.
757The internal function graphs of RTL are augmented with an epilogue and a prologue, indexed by a set of registers, consisting of a fresh pair of registers \texttt{RA} and the set of registers that must be saved by the callee of a function.
[1780]759The prologue and epilogue that are added to the function graph do the following:
761\mathtt{prologue}(s) = & \mathtt{create\_new\_frame}; \\
762                       & \mathtt{pop\ ra}; \\
763                       & \mathtt{save\ callee\_saved}; \\
764                                                                                         & \mathtt{get\_params} \\
765                                                                                         & \ \ \mathtt{reg\_params}: \mathtt{move} \\
766                                                                                         & \ \ \mathtt{stack\_params}: \mathtt{push}/\mathtt{pop}/\mathtt{move} \\
[1780]768That is, the prologue first creates a new stack frame, pops the return address from the stack, saves all the callee saved registers (i.e. the set \texttt{s}), fetches the parameters that are passed via registers and the stack and moves them into the correct registers.
769In other words, the prologue of a function correctly sets up the calling convention used in the compiler when calling a function.
770On the other hand, the epilogue undoes the action of the prologue:
772\mathtt{epilogue}(s) = & \mathtt{save\ return\ to\ tmp\ real\ regs}; \\
773                                                                                         & \mathtt{restore\_registers}; \\
774                       & \mathtt{push\ ra}; \\
775                       & \mathtt{delete\_frame}; \\
776                       & \mathtt{save return} \\
[1780]778That is, the epilogue first saves the return value to a temporary register, restores all the registers, pushes the return address on to the stack, deletes the stack frame that the prologue created, and saves the return value.
[1780]780The \texttt{CALL} instruction is translated as follows:
[1738]782\mathtt{CALL}\ id \mapsto \mathtt{set\_params};\ \mathtt{CALL}\ id;\ \mathtt{fetch\_result}
[1780]784Here, \texttt{set\_params} and \texttt{fetch\_result} are functions that implement what the caller of the function needs to do when calling a function, as opposed to the epilogue and prologue which implement what the callee must do.
[1780]786The translation from RTL to ERTL and execution functions must satisfy the following properties for \texttt{CALL} and \texttt{RETURN} instructions appropriately:
789\mathtt{CALL} & \rTo^1 & \mathtt{inside\ function} \\
790\dTo & & \dTo \\
791\underbrace{\ldots}_{\llbracket \mathtt{CALL} \rrbracket} & \rTo &
792\underbrace{\ldots}_{\mathtt{prologue}} \\
795That is, if we start in a RTL \texttt{CALL} instruction, and translate this to an ERTL \texttt{CALL} instruction, then executing the RTL \texttt{CALL} instruction for one step and translating should land us in the prologue of the translated function.
796A similar property for \texttt{RETURN} should also hold, substituting the prologue for the epilogue of the function being translated:
799\mathtt{RETURN} & \rTo^1 & \mathtt{.} \\
800\dTo & & \dTo \\
801\underbrace{\ldots}_{\mathtt{epilogue}} & \rTo &
802\underbrace{\ldots} \\
[1734]806\subsection{The ERTL to LTL translation}
[1750]808\newcommand{\declsf}[1]{\expandafter\newcommand\expandafter{\csname #1\endcsname}{\mathop{\mathsf{#1}}\nolimits}}
815For the liveness analysis, we aim at a map
816$\ell \in \mathtt{label} \mapsto $ live registers at $\ell$.
817We define the following operators on ERTL statements.
819\begin{array}{lL>{(ex. $}L<{)$}}
[1753]820\Defined(s) & registers defined at $s$ & r_1\leftarrow r_2+r_3 \mapsto \{r_1,C\}, \mathtt{CALL}~id\mapsto \text{caller-save}
[1753]822\Used(s) & registers used at $s$ & r_1\leftarrow r_2+r_3 \mapsto \{r_2,r_3\}, \mathtt{CALL}~id\mapsto \text{parameters}
825Given $LA:\mathtt{label}\to\mathtt{lattice}$ (where $\mathtt{lattice}$
826is the type of sets of registers\footnote{More precisely, it is thethe lattice
827of pairs of sets of pseudo-registers and sets of hardware registers,
828with pointwise operations.}, we also have have the following
832\Eliminable_{LA}(\ell) & iff $s(\ell)$ has side-effects only on $r\notin LA(\ell)$
834(ex.\ $\ell : r_1\leftarrow r_2+r_3 \mapsto (\{r_1,C\}\cap LA(\ell)\neq\emptyset,
835  \mathtt{CALL}id\mapsto \text{never}$)
837\Livebefore_{LA}(\ell) &$:=
838  \begin{cases}
839    LA(\ell) &\text{if $\Eliminable_{LA}(\ell)$,}\\
840    (LA(\ell)\setminus \Defined(s(\ell)))\cup \Used(s(\ell) &\text{otherwise}.
841  \end{cases}$
844In particular, $\Livebefore$ has type $(\mathtt{label}\to\mathtt{lattice})\to
[1750]847The equation on which we build the fixpoint is then
848$$\Liveafter(\ell) \doteq \bigcup_{\ell' >_1 \ell} \Livebefore_{\Liveafter}(\ell')$$
849where $\ell' >_1 \ell$ denotes that $\ell'$ is an immediate successor of $\ell$
850in the graph. We do not require the fixpoint to be the least one, so the hypothesis
851on $\Liveafter$ that we require is
852$$\Liveafter(\ell) \supseteq \bigcup_{\ell' >_1 \ell} \Livebefore(\ell')$$
853(for shortness we drop the subscript from $\Livebefore$).
[1734]854\subsection{The LTL to LIN translation}
[1762]856Ad detailed elsewhere in the reports, due to the parameterized representation of
857the back-end languages, the pass described here is actually much more generic
858than the translation from LTL to LIN. It consists in a linearization pass
859that maps any graph-based back-end language to its corresponding linear form,
860preserving its semantics. In the rest of the section, however, we will keep
861the names LTL and LIN for the two partial instantiations of the parameterized
[1721]864We 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:
866\mathtt{pc : label} \stackrel{\sigma}{\longrightarrow} \mathbb{N}
[1723]869The LTL to LIN translation pass also linearises the graph data structure into a list of instructions.
870Pseudocode for the linearisation process is as follows:
873let rec linearise graph visited required generated todo :=
874  match todo with
875  | l::todo ->
876    if l $\in$ visited then
877      let generated := generated $\cup\ \{$ Goto l $\}$ in
878      let required := required $\cup$ l in
879        linearise graph visited required generated todo
880    else
[1725]881      -- Get the instruction at label `l' in the graph
[1723]882      let lookup := graph(l) in
883      let generated := generated $\cup\ \{$ lookup $\}$ in
[1725]884      -- Find the successor of the instruction at label `l' in the graph
[1723]885      let successor := succ(l, graph) in
886      let todo := successor::todo in
887        linearise graph visited required generated todo
888  | []      -> (required, generated)
[1725]891It is easy to see that this linearisation process eventually terminates.
892In 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.
[1725]894The 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.
895We envisage needing to prove the following invariants on the linearisation function above:
899$\mathtt{visited} \approx \mathtt{generated}$, 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,
901$\forall \mathtt{l} \in \mathtt{generated}.\ \mathtt{succ(l,\ graph)} \subseteq \mathtt{required} \cup \mathtt{todo}$,
903$\mathtt{required} \subseteq \mathtt{visited}$,
905$\mathtt{visited} \cap \mathtt{todo} = \emptyset$.
908The invariants collectively imply the following properties, crucial to correctness, about the linearisation process:
912Every graph node is visited at most once,
914Every instruction that is generated is generated due to some graph node being visited,
916The successor instruction of every instruction that has been visited already will eventually be visited too.
[1762]919Note, because the LTL to LIN transformation is the first time the code of
920a function is linearised in the back-end, we must discover a notion of `well formed function code' suitable for linearised forms.
[1725]921In particular, we see the notion of well formedness (yet to be formally defined) resting on the following conditions:
[1762]925For every jump to a label in a linearised function code, the target label exists at some point in the function code,
[1762]927Each label is unique, appearing only once in the function code,
[1762]929The final instruction of a function code must be a return or an unconditional
[1726]933We assume that these properties will be easy consequences of the invariants on the linearisation function defined above.
[1725]935The final condition above is potentially a little opaque, so we explain further.
[1762]936The 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).
[1734]938\subsection{The LIN to ASM and ASM to MCS-51 machine code translations}
[1721]941The LIN to ASM translation step is trivial, being almost the identity function.
942The 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.
[1721]944The 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]945This is the most complex translation because of the huge number of cases
946to be addressed and because of the complexity of the two semantics.
947Moreover, in the assembly code we have conditional and unconditional jumps
948to arbitrary locations in the code, which are not supported by the MCS-51
949instruction set. The latter has several kind of jumps characterized by a
950different instruction size and execution time, but limited in range. For
951instance, conditional jumps to locations whose destination is more than
952$2^7$ bytes away from the jump instruction location are not supported at
953all and need to be emulated with a code transformation. The problem, which
954is known in the litterature as branch displacement and that applies also
955to modern architectures, is known to be hard and is often NP. As far as we
956know, we will provide the first formally verified proof of correctness for
957an assembler that implements branch displacement. We are also providing
958the first verified proof of correctness of a mildly optimizing branch
959displacement algorithm that, at the moment, is almost finished, but not
960described in the companion paper. This proof by itself took about 6 men
[1757]963\section{Correctness of cost prediction}
964Roughly speaking,
965the proof of correctness of cost prediction shows that the cost of executing
966a labelled object code program is the same as the sum over all labels in the
[1761]967program execution trace of the cost statically associated to the label and
[1757]968computed on the object code itself.
[1761]970In presence of object level function calls, the previous statement is, however,
971incorrect. The reason is twofold. First of all, a function call may diverge.
972To the last labels that comes before the call, however, we also associate
973the cost of the instructions that follow the call. Therefore, in the
974sum over all labels, when we meet a label we pre-pay for the instructions
975after function calls, assuming all calls to be terminating. This choice is
976driven by considerations on the source code. Functions can be called also
977inside expressions and it would be too disruptive to put labels inside
978expressions to capture the cost of instructions that follow a call. Moreover,
979adding a label after each call would produce a much higher number of proof
980obligations in the certification of source programs using Frama-C. The
981proof obligations, moreover, would be guarded by termination of all functions
982involved, that also generates lots of additional complex proof obligations
983that have little to do with execution costs. With our approach, instead, we
984put less burden on the user, at the price of proving a weaker statement:
985the estimated and actual costs will be the same if and only if the high level
986program is converging. For prefixes of diverging programs we can provide
987a similar result where the equality is replaced by an inequality (loss of
[1761]990Assuming totality of functions is however not sufficient yet at the object
991level. Even if a function returns, there is no guarantee that it will transfer
992control back to the calling point. For instance, the function could have
993manipulated the return address from its stack frame. Moreover, an object level
994program can forge any address and transfer control to it, with no guarantee
995on the execution behaviour and labelling properties of the called program.
[1761]997To solve the problem, we introduced the notion of \emph{structured trace}
998that come in two flavours: structured traces for total programs (an inductive
999type) and structured traces for diverging programs (a co-inductive type based
1000on the previous one). Roughly speaking, a structured trace represents the
1001execution of a well behaved program that is subject to several constraints
1004 \item All function calls return control just after the calling point
1005 \item The execution of all function bodies start with a label and end with
1006   a RET (even the ones reached by invoking a function pointer)
1007 \item All instructions are covered by a label (required by correctness of
1008   the labelling approach)
1009 \item The target of all conditional jumps must be labelled (a sufficient
1010   but not necessary condition for precision of the labelling approach)
1011 \item \label{prop5} Two structured traces with the same structure yield the same
1012   cost traces.
1015Correctness of cost predictions is proved only for structured execution traces,
1016i.e. well behaved programs. The forward simulation proof for all back-end
1017passes will actually be a proof of preservation of the structure of
1018the structured traces that, because of property \ref{prop5}, will imply
1019correctness of the cost prediction for the back-end. The Clight to RTLabs
1020will also include a proof that associates to each converging execution its
1021converging structured trace and to each diverging execution its diverging
1022structured trace.
1024There are also other two issues that invalidate the naive statement of
1025correctness of cost prediciton given above. The algorithm that statically
1026computes the cost of blocks is correct only when the object code is \emph{well
1027formed} and the program counter is \emph{reachable}.
1028A well formed object code is such that
1029the program counter will never overflow after the execution step of
1030the processor. An overflow that occurs during fetching but is overwritten
1031during execution is, however, correct and necessary to accept correct
1032programs that are as large as the program memory. Temporary overflows add
1033complications to the proof. A reachable address is an address that can be
1034obtained by fetching (not executing!) a finite number of times from the
1035beginning of the code memory without ever overflowing. The complication is that
1036the static prediction traverses the code memory assuming that the memory will
1037be read sequentially from the beginning and that all jumps jump only to
1038reachable addresses. When this property is violated, the way the code memory
1039is interpreted is uncorrect and the cost computed is totally meaningless.
1040The reachability relation is closed by fetching for well formed programs.
1041The property that calls to function pointers only target reachable and
1042well labelled locations, however, is not statically predictable and it is
1043enforced in the structured trace.
1045The proof of correctness of cost predictions has been quite complex. Setting
1046up the good invariants (structured traces, well formed programs, reachability)
1047and completing the proof has required more than 3 men months while the initally
1048estimated effort was much lower. In the paper-and-pencil proof for IMP, the
1049corresponding proof was obvious and only took two lines.
1051The proof itself is quite involved. We
1052basically need to show as an important lemma that the sum of the execution
1053costs over a structured trace, where the costs are summed in execution order,
1054is equivalent to the sum of the execution costs in the order of pre-payment.
1055The two orders are quite different and the proof is by mutual recursion over
1056the definition of the converging structured traces, which is a family of three
1057mutual inductive types. The fact that this property only holds for converging
1058function calls in hidden in the definition of the structured traces.
1059Then we need to show that the order of pre-payment
1060corresponds to the order induced by the cost traces extracted from the
1061structured trace. Finally, we need to show that the statically computed cost
1062for one block corresponds to the cost dinamically computed in pre-payment
[1758]1065\section{Overall results}
1067Functional correctness of the compiled code can be shown by composing
1068the simulations to show that the target behaviour matches the
1069behaviour of the source program, if the source program does not `go
1070wrong'.  More precisely, we show that there is a forward simulation
1071between the source trace and a (flattened structured) trace of the
1072output, and conclude equivalence because the target's semantics are
1073in the form of an executable function, and hence
1076Combining this with the correctness of the assignment of costs to cost
1077labels at the ASM level for a structured trace, we can show that the
1078cost of executing any compiled function (including the main function)
1079is equal to the sum of all the values for cost labels encountered in
1080the \emph{source code's} trace of the function.
[1741]1082\section{Estimated effort}
[1742]1083Based on the rough analysis performed so far we can estimate the total
1084effort for the certification of the compiler. We obtain this estimation by
[1741]1085combining, for each pass: 1) the number of lines of code to be certified;
[1742]10862) the ratio of number of lines of proof to number of lines of code from
1087the CompCert project~\cite{compcert} for the CompCert pass that is closest to
[1741]1088ours; 3) an estimation of the complexity of the pass according to the
[1775]1089analysis above. The result is shown in Table~\ref{table}.
[1747]1093Pass origin & Code lines & CompCert ratio & Estimated effort & Estimated effort \\
1094            &            &                & (based on CompCert) & \\
[1756]1096Common &  4864 & 4.25 \permil & 20.67 & 17.0 \\
1097Cminor &  1057 & 5.23 \permil & 5.53  &  6.0 \\
1098Clight &  1856 & 5.23 \permil & 9.71  & 10.0 \\ 
1099RTLabs &  1252 & 1.17 \permil & 1.48  &  5.0 \\
1100RTL    &   469 & 4.17 \permil & 1.95  &  2.0 \\
[1754]1101ERTL   &   789 & 3.01 \permil & 2.38  & 2.5 \\
1102LTL    &    92 & 5.94 \permil & 0.55  & 0.5 \\
[1756]1103LIN    &   354 & 6.54 \permil & 2.31  &   1.0 \\
1104ASM    &   984 & 4.80 \permil & 4.72  &  10.0 \\
[1756]1106Total common    &  4864 & 4.25 \permil & 20.67 & 17.0 \\
1107Total front-end &  2913 & 5.23 \permil & 15.24 & 16.0 \\
1108Total back-end  &  6853 & 4.17 \permil & 13.39 & 21.0 \\
[1756]1110Total           & 14630 & 3.75 \permil & 49.30 & 54.0 \\
[1775]1112\caption{\label{table}Estimated effort}
[1745]1115We provide now some additional informations on the methodology used in the
1116computation. The passes in Cerco and CompCert front-end closely match each
1117other. However, there is no clear correspondence between the two back-ends.
1118For instance, we enforce the calling convention immediately after instruction
1119selection, whereas in CompCert this is performed in a later phase. Or we
1120linearize the code at the very end, whereas CompCert performs linearization
1121as soon as possible. Therefore, the first part of the exercise has consisted
1122in shuffling and partitioning the CompCert code in order to assign to each
1123CerCo pass the CompCert code that performs the same transformation.
1125After this preliminary step, using the data given in~\cite{compcert} (which
1126are relative to an early version of CompCert) we computed the ratio between
1127men months and lines of code in CompCert for each CerCo pass. This is shown
1128in the third column of Table~\ref{wildguess}. For those CerCo passes that
1129have no correspondence in CompCert (like the optimizing assembler) or where
1130we have insufficient data, we have used the average of the ratios computed
1133The first column of the table shows the number of lines of code for each
1134pass in CerCo. The third column is obtained multiplying the first with the
1135CompCert ratio. It provides an estimate of the effort required (in men months)
1136if the complexity of the proofs for CerCo and Compcert would be the same.
1138The two proof styles, however, are on purpose completely different. Where
1139CompCert uses non executable semantics, describing the various semantics with
1140inductive types, we have preferred executable semantics. Therefore, CompCert
1141proofs by induction and inversion become proof by functional inversion,
1142performed using the Russel methodology (now called Program in Coq, but whose
1143behaviour differs from Matita's one). Moreover, CompCert code is written using
1144only types that belong to the Hindley-Milner fragment, whereas we have
1145heavily exploited dependent types all over the code. The dependent type
1146discipline offers many advantages from the point of view of clarity of the
1147invariants involved and early detection of errors and it naturally combines
1148well with the Russel approach which is based on dependent types. However, it
1149is also well known to introduce technical problems all over the code, like
1150the need to explicitly prove type equalities to be able to manipulate
1151expressions in certain ways. In many situations, the difficulties encountered
1152with manipulating dependent types are better addressed by improving the Matita
1153system, according to the formalization driven system development. For this
1154reason, and assuming a pessimistic point of view on our performance, the
1155fourth columns presents the final estimation of the effort required, that also
1156takes in account the complexity of the proof suggested by the informal proofs
1157sketched in the previous section.
[1774]1159\subsection{Contingency plan}
1160On the basis of the proof strategy sketched in this document and the
1161estimated effort, we can refine our contingency plan. In case we will end
1162the certification of the basic compiler in advance we will have the choice
1163of either proving loop optimizations and/or partial redundancy elimination
1164correct (both tasks that seem difficult to achieve in a short time) or
1165considering the MCS-51 specific extensions introduced during the first period
1166and under-used in the formalized prototype. Yet another possibility would be
1167to better study retargeting of the code and the commutation property between
1168different compiler passes. The latter study is easily enabled by our
1169approach where all back-end languages are instances of the same parameterized
1172In the case of a consistent delay in the certification of some
1173components, we will address first the passes that are more likely to have
1174undetected bugs and we will follow a top-down approach, axiomatizing
1175the properties of the data structured used in the compiler to focus more
1176on the algorithms. The rational is that data structures are easier then
1177algorithms to test using well known methodologies.
1178The effort table clearly shows that commond definitions
1179and data structures are 1/4th of the size of the code and require slightly
1180less than 1/3rd of the total effort. At least half of this effort really goes
1181into simple data structures (vectors, bounded and unbounded integers, tries
[1775]1182and maps) whose certification is not interesting and whose code could be
1183taken without re-proving it from the library of any other theorem prover.
1186The overall exercise, whose details have been only minimally reported here,
1187has been very useful. It has allowed to spot in an early moment some criticities
1188of the proof that have required major changes in the proof plan. It has also
1189shown that the last passes of the compilation (e.g. assembly) and cost
1190prediction on the object code are much more involved than more high level
1193The final estimation for the effort is surely affected by a low degree of
1194confidence. It is however sufficient to conclude that the effort required
1195is in line with the man power that was scheduled for the second half of the
1196second period and for the third period. Compared to the number of men months
1197declared in Annex I of the contract, we will need more men months. However,
1198both at UNIBO and UEDIN there have been major differences in hiring with
1199respect to the Annex. Therefore both sites have now an higher number of men
1200months available, with the trade-off of a lower level of maturity of the
1201people employed.
1203The reviewers suggested that we use this estimation to compare two possible
1204scenarios: a) proceed as planned, porting all the CompCert proofs to Matita
1205or b) port D3.1 and D4.1 to Coq and re-use the CompCert proofs.
1206We remark here again that the back-end of the two compilers, from the
1207memory model on, are sensibly different: we are not re-proving correctness
1208of the same piece of code. Moreover, the proof techniques are different for
1209the front-end too. Switching to the CompCert formalization would imply
1210the abandon of the untrusted compiler, the abandon of the experiment with
1211a different proof technique, the abandon of the quest for an open source
1212proof, and the abandon of the co-development of the formalization and the
1213Matita proof assistant. In the Commitment Letter~\cite{letter} delivered
1214to the Officer in May we clarified our personal perspective on the project
1215goals and objectives. We do not re-describe here the point of view presented
1216in the letter that we can condense in ``we value diversity''.
1218Clearly, if the execise would have suggested the infeasability in terms of
1219effort of concluding the formalization or getting close to that, we would have
1220abandoned our path and embraced the reviewer's suggestion. However, we
1221have been comforted in the analysis we did in autumn and further progress done
1222during the winter does not show yet any major delay with respect to the
1223proof schedule. We are thus planning to continue the certification according
1224to the more detailed proof plan that came out from the exercise reported in
1225this manuscript.
1228\bibitem{compcert} X. Leroy, ``A Formally Verified Compiler back-end'',
1229Journal of Automated Reasoning 43(4)):363-446, 2009.
1231\bibitem{letter}The CerCo team, ``Commitment to the Consideration of Reviewer's Reccomendation'', 16/05/2011.
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