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60\vspace*{1cm}Project FP7-ICT-2009-C-243881 {\cerco}}
[1778]62\date{ }
[1790]82Proof outline for the correctness of\\the CerCo compiler
89Version 1.0
96Main Authors:\\
[1790]97J. Boender, B. Campbell, D. Mulligan, P. Tranquilli, C. Sacerdoti Coen
103Project Acronym: {\cerco}\\
104Project full title: Certified Complexity\\
105Proposal/Contract no.: FP7-ICT-2009-C-243881 {\cerco}\\
107\clearpage \pagestyle{myheadings} \markright{{\cerco}, FP7-ICT-2009-C-243881}
[1790]114In the last project review of the CerCo project, the project reviewers recommended that we briefly outline a pencil-and-paper correctness proof for each of the stages of the CerCo compiler in order to facilitate an estimation of the complexity and time required to complete the formalisation of the proof.
115This has been possible starting from month eighteen, as we have now completed the formalisation in Matita of the data structures and code of the compiler.
[1790]117In this document we provide a very high-level, pencil-and-paper sketch of what we view as the best path to completing the correctness proof for the compiler.
118In 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.
119We sketch the overall correctness results, and also briefly describe the parts of the proof that have already been completed at the end of the First Period.
[1790]121Finally, in the last section we present an estimation of the effort required for the certification in Matita of the compiler and draw conclusions.
[1731]123\section{Front-end: Clight to RTLabs}
125The front-end of the CerCo compiler consists of several stages:
130\quad \= $\downarrow$ \quad \= \kill
132\> $\downarrow$ \> cast removal\\
133\> $\downarrow$ \> add runtime functions\footnote{Following the last project
134meeting we intend to move this transformation to the back-end}\\
135\> $\downarrow$ \> cost labelling\\
[1790]136\> $\downarrow$ \> loop optimisations\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)\\
[1741]137\> $\downarrow$ \> partial redundancy elimination$^{\mbox{\scriptsize \ref{lab:opt2}}}$ (an endo-transformation)\\
[1731]138\> $\downarrow$ \> stack variable allocation and control structure
139 simplification\\
141\> $\downarrow$ \> generate global variable initialisation code\\
142\> $\downarrow$ \> transform to RTL graph\\
[1741]144\> $\downarrow$ \> \\
[1740]150Here, by `endo-transformation', we mean a mapping from language back to itself:
[1790]151the loop optimisation step maps the Clight language to itself.
[1731]153%Our overall statements of correctness with respect to costs will
154%require a correctly labelled program
155There are three layers in most of the proofs proposed:
157\item invariants closely tied to the syntax and transformations using
[1758]158  dependent types (such as the presence of variable names in environments),
159\item a forward simulation proof relating each small-step of the
160  source to zero or more steps of the target, and
161\item proofs about syntactic properties of the cost labelling.
[1758]163The first will support both functional correctness and allow us to
164show the totality of some of the compiler stages (that is, those
165stages of the compiler cannot fail).  The second provides the main
166functional correctness result, including the preservation of cost
167labels in the traces, and the last will be crucial for applying
168correctness results about the costings from the back-end by showing
169that they appear in enough places so that we can assign all of the
170execution costs to them.
172We will also prove that a suitably labelled RTLabs trace can be turned
173into a \emph{structured trace} which splits the execution trace into
[1746]174cost-label to cost-label chunks with nested function calls.  This
[1731]175structure was identified during work on the correctness of the
176back-end cost analysis as retaining important information about the
177structure of the execution that is difficult to reconstruct later in
178the compiler.
[1746]180\subsection{Clight cast removal}
[1746]182This transformation removes some casts inserted by the parser to make
183arithmetic promotion explicit but which are superfluous (such as
184\lstinline[language=C]'c = (short)((int)a + (int)b);' where
185\lstinline'a' and \lstinline'b' are \lstinline[language=C]'short').
186This is necessary for producing good code for our target architecture.
[1746]188It only affects Clight expressions, recursively detecting casts that
189can be safely eliminated.  The semantics provides a big-step
190definition for expression, so we should be able to show a lock-step
191forward simulation between otherwise identical states using a lemma
192showing that cast elimination does not change the evaluation of
193expressions.  This lemma will follow from a structural induction on
194the source expression.  We have already proved a few of the underlying
195arithmetic results necessary to validate the approach.
197\subsection{Clight cost labelling}
199This adds cost labels before and after selected statements and
200expressions, and the execution traces ought to be equivalent modulo
[1746]201the new cost labels.  Hence it requires a simple forward simulation
202with a limited amount of stuttering whereever a new cost label is
203introduced.  A bound can be given for the amount of stuttering allowed
[1731]204based on the statement or continuation to be evaluated next.
206We also intend to show three syntactic properties about the cost
[1731]209\item every function starts with a cost label,
[1746]210\item every branching instruction is followed by a cost label (note that
[1731]211  exiting a loop is treated as a branch), and
212\item the head of every loop (and any \lstinline'goto' destination) is
213  a cost label.
[1731]215These can be shown by structural induction on the source term.
217\subsection{Clight to Cminor translation}
[1732]219This translation is the first to introduce some invariants, with the
220proofs closely tied to the implementation by dependent typing.  These
221are largely complete and show that the generated code enjoys:
223\item some minimal type safety shown by explicit checks on the
224  Cminor types during the transformation (a little more work remains
225  to be done here, but follows the same form);
226\item that variables named in the parameter and local variable
227  environments are distinct from one another, again by an explicit
228  check;
229\item that variables used in the generated code are present in the
230  resulting environment (either by checking their presence in the
[1746]231  source environment, or from a list of freshly generated temporary variables);
[1732]232  and
233\item that all \lstinline[language=C]'goto' labels are present (by
234  checking them against a list of source labels and proving that all
235  source labels are preserved).
238The simulation will be similar to the relevant stages of CompCert
239(Clight to Csharpminor and Csharpminor to Cminor --- in the event that
[1746]240the direct proof is unwieldy we could introduce an intermediate
241language corresponding to Csharpminor).  During early experimentation
242with porting CompCert definitions to the Matita proof assistant we
243found little difficulty reproving the results for the memory model, so
244we plan to port the memory injection properties and use them to relate
245Clight in-memory variables with either the value of the local variable or a
246stack slot, depending on how it was classified.
248This should be sufficient to show the equivalence of (big-step)
249expression evaluation.  The simulation can then be shown by relating
250corresponding blocks of statement and continuations with their Cminor
251counterparts and proving that a few steps reaches the next matching
254The syntactic properties required for cost labels remain similar and a
255structural induction on the function bodies should be sufficient to
256show that they are preserved.
[1731]258\subsection{Cminor global initialisation code}
[1732]260This short phase replaces the global variable initialisation data with
261code that executes when the program starts.  Each piece of
262initialisation data in the source is matched by a new statement
263storing that data.  As each global variable is allocated a distinct
264memory block, the program state after the initialisation statements
265will be the same as the original program's state at the start of
266execution, and will proceed in the same manner afterwards.
268% Actually, the above is wrong...
269% ... this ought to be in a fresh main function with a fresh cost label
[1731]271\subsection{Cminor to RTLabs translation}
[1733]273In this part of the compiler we transform the program's functions into
274control flow graphs.  It is closely related to CompCert's Cminorsel to
275RTL transformation, albeit with target-independent operations.
277We already enforce several invariants with dependent types: some type
[1746]278safety, mostly shown using the type information from Cminor; and
[1733]279that the graph is closed (by showing that each successor was recently
280added, or corresponds to a \lstinline[language=C]'goto' label which
281are all added before the end).  Note that this relies on a
282monotonicity property; CompCert maintains a similar property in a
283similar way while building RTL graphs.  We will also add a result
284showing that all of the pseudo-register names are distinct for use by
285later stages using the same method as Cminor.
[1746]287The simulation will relate Cminor states to RTLabs states which are about to
288execute the code corresponding to the Cminor statement or continuation.
[1733]289Each Cminor statement becomes zero or more RTLabs statements, with a
290decreasing measure based on the statement and continuations similar to
291CompCert's.  We may also follow CompCert in using a relational
292specification of this stage so as to abstract away from the functional
293(and highly dependently typed) definition.
295The first two labelling properties remain as before; we will show that
296cost labels are preserved, so the function entry point will be a cost
[1746]297label, and successors to any statement that are cost labels map still
298map to cost labels, preserving the condition on branches.  We replace
299the property for loops with the notion that we will always reach a
300cost label or the end of the function after following a bounded number of
301successors.  This can be easily seen in Cminor using the requirement
302for cost labels at the head of loops and after gotos.  It remains to
303show that this is preserved by the translation to RTLabs.  % how?
[1731]305\subsection{RTLabs structured trace generation}
[1733]307This proof-only step incorporates the function call structure and cost
308labelling properties into the execution trace.  As the function calls
309are nested within the trace, we need to distinguish between
[1746]310terminating and non-terminating function calls.  Thus we use the
311excluded middle (specialised to a function termination property) to do
[1746]314Structured traces for terminating functions are built by following the
315flat trace, breaking it into chunks between cost labels and
316recursively processing function calls.  The main difficulties here are
317the non-structurally recursive nature of the function (instead we use
318the size of the termination proof as a measure) and using the RTLabs
319cost labelling properties to show that the constraints of the
320structured traces are observed.  We also show that the lower stack
321frames are preserved during function calls in order to prove that
322after returning from a function call we resume execution of the
323correct code.  This part of the work has already been constructed, but
324still requires a simple proof to show that flattening the structured
[1733]325trace recreates the original flat trace.
[1746]327The non-terminating case follows the trace like the terminating
328version to build up chunks of trace from cost-label to cost-label
329(which, by the finite distance to a cost label property shown before,
330can be represented by an inductive type).  These chunks are chained
331together in a coinductive data structure that can represent
332non-terminating traces.  The excluded middle is used to decide whether
333function calls terminate, in which case the function described above
334constructs an inductive terminating structured trace which is nested
335in the caller's trace.  Otherwise, another coinductive constructor is
336used to embed the non-terminating trace of the callee, generated by
337corecursion.  This part of the trace transformation is currently under
338construction, and will also need a flattening result to show that it
339is correct.
[1734]342\section{Backend: RTLabs to machine code}
[1734]345The compiler backend consists of the following intermediate languages, and stages of translation:
[1741]350\quad \=\,\vdots\= \\
351\> $\downarrow$ \>\\
352\> $\downarrow$ \quad \= \kill
[1790]354\> $\downarrow$ \> copy propagation\footnote{\label{lab:opt}To be ported from the untrusted compiler and certified only in the case of an early completion of the certification of the other passes.} (an endo-transformation) \\
[1734]355\> $\downarrow$ \> instruction selection\\
[1748]356\> $\downarrow$ \> change of memory models in compiler\\
[1739]358\> $\downarrow$ \> constant propagation$^{\mbox{\scriptsize \ref{lab:opt}}}$ (an endo-transformation) \\
[1734]359\> $\downarrow$ \> calling convention made explicit \\
[1741]360\> $\downarrow$ \> layout of activation records \\
362\> $\downarrow$ \> register allocation and spilling\\
363\> $\downarrow$ \> dead code elimination\\
365\> $\downarrow$ \> function linearisation\\
366\> $\downarrow$ \> branch compression (an endo-transformation) \\
368\> $\downarrow$ \> relabeling\\
370\> $\downarrow$ \> pseudoinstruction expansion\\
[1741]371\textsf{MCS-51 machine code}\\
[1769]376\subsection{Graph translations}
[1790]377RTLabs and most intermediate languages in the back-end have a graph representation: the code of each function is represented by a graph of instructions.
378The graph maps a set of labels (the names of the nodes) to the instruction stored at that label (the nodes of the graph).
379Instructions reference zero or more additional labels that are the immediate successors of the instruction: zero for return from functions, more than one for conditional jumps and calls, one in all other cases.
380The references from one instruction to its immediate successors are the arcs of the graph.
[1790]382The statuses of graph languages always contain a program counter that holds a representation of a reference to the current instruction.
[1790]384A translation between two consecutive graph languages maps each instruction stored at location $l$ in the first graph and with immediate successors $\{l_1,\ldots,l_n\}$ to a subgraph of the output graph that has a single entry point at location $l$ and exit arcs to $\{l_1,\ldots,l_n\}$.
385Moreover, the labels of all non-entry nodes in the subgraph are distinct from all the labels in the source graph.
[1790]387In order to simplify the translations and the relative proofs of forward simulation, after the release of D4.2 and D4.3, we have provided:
390A new data type (called \texttt{blist}) that represents a sequence of instructions to be added to the output graph.
391The ``b'' in the name stands for binder, since a \texttt{blist} is either empty, an extension of a \texttt{blist} with an instruction at the front, or the generation of a fresh quantity followed by a \texttt{blist}.
392The latter feature is used, for instance, to generate fresh register names.
393The instructions in the list are unlabelled and all of them but the last one are also sequential, like in a linear program.
395A new iterator (called \texttt{b\_graph\_translate}) of type
397\mathtt{b\_graph\_translate}: (\mathtt{label} \rightarrow \mathtt{blist})
398\rightarrow \mathtt{graph} \rightarrow \mathtt{graph}
[1790]400The iterator transform the input graph in the output graph by replacing each node with the graph that corresponds to the linear \texttt{blist} obtained by applying the function in input to the node label.
[1790]403Using the iterator above, the code can be written in such a way that the programmer does not see any distinction between writing a transformatio on linear or graph languages.
[1790]405In order to prove simulations for translations obtained using the iterator, we will prove the following theorem:
408\mathtt{theorem} &\ \mathtt{b\_graph\_translate\_ok}: \\
409& \forall  f.\forall G_{i}.\mathtt{let}\ G_{\sigma} := \mathtt{b\_graph\_translate}\ f\ G_{i}\ \mathtt{in} \\
410&       \forall l \in G_{i}.\mathtt{subgraph}\ (f\ l)\ l\ (next \ l \ G_i)\ G_{\sigma}
[1790]413Here \texttt{subgraph} is a computational predicate that given a \texttt{blist} $[i_1, \ldots, i_n]$, an entry label $l$, an exit label $l'$ and a graph $G$ expands to the fact that fetching from $G$ at address $l$ one retrieves a node $i_1$ with a successor $l_1$ that, when fetched, yields a node $i_2$ with a successor $l_2$ such that \ldots. The successor of $i_n$ is $l'$.
[1790]415Proving a forward simulation diagram of the following kind using the aforementioned theorem is now as straightforward as doing the same using standard small-step operational semantics over linear languages.
418\mathtt{lemma} &\ \mathtt{execute\_1\_step\_ok}: \\
419&       \forall s.  \mathtt{let}\ s' := s\ \sigma\ \mathtt{in} \\
420&       \mathtt{let}\ l := pc\ s\ \mathtt{in} \\
421&       s \stackrel{1}{\rightarrow} s^{*} \Rightarrow \exists n. s' \stackrel{n}{\rightarrow} s'^{*} \wedge s'^{*} = s'\ \sigma
[1790]424Because graph translations preserve entry and exit labels of translated statements, the state translation function $\sigma$ will simply preserve the value of the program counter.
425The program code, which is part of the state, is translated using the iterator.
[1790]427The proof is then roughly as follows.
428Let $l$ be the program counter of the input state $s$.
429We proceed by cases on the current instruction of $s$.
430Let $[i_1, \ldots, i_n]$ be the \texttt{blist} associated to $l$ and $s$ by the translation function.
431The witness required for the existential statement is simply $n$.
432By applying the theorem above we know that the next $n$ instructions that will be fetched from $s\ \sigma$ will be $[i_1, \ldots, i_n]$ and it is now sufficient to prove that they simulate the original instruction.
[1734]434\subsection{The RTLabs to RTL translation}
[1786]437\subsubsection*{Translation of values and memory}
[1748]439The RTLabs to RTL translation pass marks the frontier between the two memory models used in the CerCo project.
440As a result, we require some method of translating between the values that the two memory models permit.
441Suppose we have such a translation, $\sigma$.
442Then the translation between values of the two memory models may be pictured with:
[1752]445\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
[1790]448In the front-end, we have both integer and float values, where integer values are `sized', along with null values and pointers.
449Some front-end values are representables in a byte, but some others require more bits.
[1752]451In the back-end model all values are meant to be represented in a single byte.
[1790]452Values can thefore be undefined, be one byte long integers or be indexed fragments of a pointer, null or not.
453Floats values are no longer present, as floating point arithmetic is not supported by the CerCo compiler.
[1790]455The $\sigma$ map implements a one-to-many relation: a single front-end value is mapped to a sequence of back-end values when its size is more then one byte.
[1790]457We further require a map, $\sigma$, which maps the front-end \texttt{Memory} and the back-end's notion of \texttt{BEMemory}.
458Both kinds of memory can be thought as an instance of a generic \texttt{Mem} data type parameterized over the kind of values stored in memory:
461\mathtt{Mem}\ \alpha = \mathtt{Block} \rightarrow (\mathbb{Z} \rightarrow \alpha)
[1748]463Here, \texttt{Block} consists of a \texttt{Region} paired with an identifier.
[1752]466\mathtt{Block} ::= \mathtt{Region} \times \mathtt{ID}
[1748]468We now have what we need for defining what is meant by the `memory' in the backend memory model.
469Namely, we instantiate the previously defined \texttt{Mem} type with the type of back-end memory values.
[1768]472\mathtt{BEMem} = \mathtt{Mem}~\mathtt{BEValue}
[1748]474Memory addresses consist of a pair of back-end memory values:
477\mathtt{Address} = \mathtt{BEValue} \times  \mathtt{BEValue} \\
[1751]479The back- and front-end memory models differ in how they represent sized integeer values in memory.
480In 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.
481In contrast, the layout of sized integer values in the back-end memory model consists of a series of byte-sized `chunks':
[1752]484\begin{picture}(0, 25)
489\put(-15,10){\vector(1, 0){30}}
[1760]497Chunks for pointers are pairs made of the original pointer and the index of the chunk.
498Therefore, when assembling the chunks together, we can always recognize if all chunks refer to the same value or if the operation is meaningless.
[1751]500The 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.
501The first lemma required has the following statement:
[1791]503\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]505That 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]507Next, we must show that \texttt{store} properly commutes with the $\sigma$-map between memory spaces:
[1791]509\sigma(\mathtt{store\ a\ v\ M}) = \mathtt{store\ \sigma(v)\ \sigma(a)\ \sigma(M)}
[1760]511That 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]513Finally, the commutation properties between \texttt{load} and \texttt{store} are weakened in the $\sigma$-image of the memory.
514Writing \texttt{load}$^*$ for the multiple consecutive iterations of \texttt{load} used to fetch all chunks of a value, we must prove that, when $a \neq a'$:
[1791]516\mathtt{load^* \sigma(a)\ (\mathtt{store}\ \sigma(a')\ \sigma(v)\ \sigma(M))} = \mathtt{load^*\ \sigma(s)\ \sigma(a)\ \sigma(M)}
[1790]518That 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 distinct by hypothesis, there is \emph{a priori} no guarantee that the consecutive bytes for the value stored at $\sigma(a)$ are disjoint from those for the values stored at $\sigma(a')$.
519The fact that this holds is a non-trivial property of $\sigma$ that must be explicitly proved.
[1786]521\subsubsection*{Translation of RTLabs states}
[1763]523RTLabs states come in three flavours:
[1792]526\mathtt{StateRTLabs} & ::=  & (\mathtt{State} : \mathtt{Frame}^* \times \mathtt{Frame} \\
[1716]527               & \mid & \mathtt{Call} : \mathtt{Frame}^* \times \mathtt{Args} \times \mathtt{Return} \times \mathtt{Fun} \\
528               & \mid & \mathtt{Return} : \mathtt{Frame}^* \times \mathtt{Value} \times \mathtt{Return}) \times \mathtt{Mem}
[1790]531\texttt{State} is the default state in which RTLabs programs are almost always in for the duration of their execution.
[1763]532The \texttt{Call} state is only entered when a call instruction is being executed, and then we immediately return to being in \texttt{State}.
533Similarly, \texttt{Return} is only entered when a return instruction is being executed, before returning immediately to \texttt{State}.
534All 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.
535\texttt{State} keeps track of a list of stack frames.
[1763]537RTL 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}:
[1792]540\mathtt{StateRTL} ::= (\mathtt{Frame}^* \times \mathtt{PC} \times \mathtt{SP} \times \mathtt{ISP} \times \mathtt{CARRY} \times \mathtt{REGS}) \times \mathtt{Mem}
[1763]542The internal stack pointer \texttt{ISP}, and its relationship with the stack pointer \texttt{SP}, needs some comment.
543Due 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.
544As 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.
545The 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.
[1790]546Instructions like \texttt{LCALL} and \texttt{ACALL} are hardwired by the processor's design to push the return address on to the hardware stack.
547Therefore 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.
548Parameters, return values and local variables are only present in the external stack.
[1763]549As 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]551Once more, we require a relation $\sigma$ between RTLabs states and RTL states.
[1790]552Because $\sigma$ is one-to-many and, morally, a multivalued function, we use in the sequel the functional notation for $\sigma$, using $\star$ as a distinct marker in the range of $\sigma$ to mean that any value is accepted.
[1792]554\mathtt{StateRTLabs} \stackrel{\sigma}{\longrightarrow} \mathtt{StateRTL}s
[1763]557Translating 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}.
558For \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:
[1791]560\sigma(\mathtt{State} (\mathtt{Frame}^* \times \mathtt{\langle PC, REGS, SP, \ldots \rangle})) \longrightarrow ((\sigma(\mathtt{Frame}^*), \sigma(\mathtt{PC}), \sigma(\mathtt{SP}), \star, \star, \sigma(\mathtt{REGS})), \sigma(\mathtt{Mem}))
[1790]562Translation then proceeds by translating the remaining stack frames, as well as the contents of the top stack frame.
563An arbitrary value for the internal stack pointer and the carry bit is admitted.
[1763]565Translating \texttt{Call} and \texttt{Return} states is more involved, as a commutation between a single step of execution and the translation process must hold:
567\sigma(\mathtt{Return}(-)) \longrightarrow \sigma \circ \text{return one step}
[1718]571\sigma(\mathtt{Call}(-)) \longrightarrow \sigma \circ \text{call one step}
[1763]574Here \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.
[1790]575We provide the one step commuting diagrams in Figure~\ref{fig.commuting.diagrams}.
576The fact that one execution step in the source language is not performed in the target language is not problematic for preservation of divergence because it is easy to show that every step from a \texttt{Call} or \texttt{Return} state is always preceeded or followed by one step that is always simulated.
[1793]581s & \rTo^1 & s'   \\
[1718]582  & \rdTo                             & \dTo \\
583  &                                   & \llbracket s'' \rrbracket
[1793]587\caption{The one-step commuting diagrams for \texttt{Call} and \texttt{Return} state translations.}
[1786]591\subsubsection*{The forward simulation proof}
593The forward simulation proofs for all steps that do not involve function calls are lengthy, but routine.
[1771]594They 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.
595The properties of $\sigma$ presented before that relate values and memories will need to be heavily exploited.
[1771]597The simulation of invocation of functions and returns from functions is less obvious.
598We sketch here what happens on the source code and on its translation.
[1786]600\subparagraph{Function call/return in RTLabs}
[1792]604\mathtt{CALL(id,\ args,\ dst,\ pc) \in State(Frame^*, Frame)} & \longrightarrow & \mathtt{PUSH(Frame[pc := after\_return])}, \\
605                                                           &                 &  \mathtt{Call(M(args), dst)}
[1763]608Suppose we are given a \texttt{State} with a list of stack frames, with the top frame being \texttt{Frame}.
[1791]609Suppose also that the program counter in \texttt{Frame} points to a \texttt{CALL} instruction, complete with arguments and destination address.
[1771]610Then 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]612Now, what happens next depends on whether we are executing an internal or an external function.
[1743]613In the case where the call is to an external function, we have:
[1792]616\mathtt{PUSH(Frame[pc := after\_return])}& \stackrel{\mathtt{ret\_val = f(M(args))}}{\longrightarrow} & \mathtt{Return(ret\_val,\ dst,\ PUSH(...))} \\
617\mathtt{Call(M(args), dst)}                 &                                                            & 
[1766]620That 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]621Then the return state restores the program counter by popping the stack, and execution proceeds in a new \texttt{State}:
[1767]624\mathtt{Return(ret\_val,\ dst,\ PUSH(...))} & \longrightarrow & \mathtt{pc = POP\_STACK(regs[dst := M(ret\_val)],\ pc)} \\
625                                            &                 & \mathtt{State(regs[dst := M(ret\_val),\ pc)}
[1767]629Suppose we are executing an internal function, however:
[1786]632\mathtt{Call(M(args), dst)}                        & \longrightarrow & \mathtt{sp = alloc,\ regs = \emptyset[- := params]} \\
[1791]633\mathtt{PUSH(Frame[pc := after\_return])} &                 & \mathtt{State(regs,\ sp,\ pc_\emptyset,\ dst)}
[1786]636A new stack frame is allocated and its address is stored in the stack pointer.
637The 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.
[1790]640Eventually, a \texttt{RET} instruction is faced, the return value is fetched from the registers map, the stack frame is deallocated and a \texttt{Return} state is entered:
[1790]643\mathtt{RET(id,\ args,\ dst,\ pc) \in State(Frame^*, Frame)} & \longrightarrow & \mathtt{free(sp)} \\
[1786]644   &                 & \mathtt{Return(M(ret\_val), dst, Frames)}
[1790]648Then the \texttt{Return} state restores the program counter by popping the stack, and execution proceeds in a new \texttt{State}, like the case for external functions:
651\mathtt{free(sp)}                         & \longrightarrow & \mathtt{pc = POP\_STACK(regs[dst := M(ret\_val)],\ pc)} \\
[1791]652\mathtt{Return(M(ret\_val), dst, Frame^*)} &                 & \mathtt{State(regs[dst := M(ret\_val),\ pc)}
[1786]656\subparagraph{The RTLabs to RTL translation for function calls}
[1786]658Return instructions are translated to return instructions:
[1790]660\mathtt{RET} \longrightarrow \mathtt{RET}
[1790]663\texttt{CALL} instructions are translated to \texttt{CALL\_ID} instructions:
[1790]665\mathtt{CALL(id,\ args,\ dst,\ pc)} \longrightarrow \mathtt{CALL\_ID(id,\ \Sigma'(args),\ \Sigma(dst),\ pc)}
[1790]667Here $\Sigma$ is the map, computed by the compiler, that translate pseudo-registers holding front-end values to list of pseudo-registers holding the chunks for the front-end values.
[1786]668The specification for $\Sigma$ is that for every state $s$,
670\sigma(s(r)) = (\sigma(s))(\Sigma(r))
[1786]673\subparagraph{Function call/return in RTL}
675In the case of RTL, execution proceeds as follows.
676Suppose we are executing a \texttt{CALL\_ID} instruction.
677Then a case split occurs depending on whether we are executing an internal or an external function, as in the RTLabs case:
[1724]681& & \llbracket \mathtt{CALL\_ID}(\mathtt{id}, \mathtt{args}, \mathtt{dst}, \mathtt{pc})\rrbracket & & \\
682& \ldTo^{\text{external}} & & \rdTo^{\text{internal}} & \\
[1786]683\skull & & & &
685\mathtt{sp = alloc,\ regs = \emptyset[- := params]} \\
686\mathtt{PUSH}(\mathtt{carry}, \mathtt{regs}, \mathtt{dst}, \mathtt{return\_addr}), \mathtt{pc}_{0}, \mathtt{regs}, \mathtt{sp}
[1777]690Here, 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.
691The reason for this lack of implementation is as follows.
692Though 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.
693Whilst 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.
694However, 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]696In the case of an internal function being executed, we proceed as follows.
697The 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.
698Further, 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.
700Note, 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
[1792]702\mathtt{PUSH(Frame[pc := after\_return])}
[1790]704was executed.
705To summarize, up to the different numer of transitions required to do the job, the RTL code for internal function calls closely simulates the RTLabs code.
707The 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:
709\mathtt{return\_addr} & := \mathtt{top}(\mathtt{stack}) \\
[1786]710v^*                    & := M(\mathtt{rv\_regs}) \\
[1724]711\mathtt{dst}, \mathtt{sp}, \mathtt{carry}, \mathtt{regs} & := \mathtt{pop} \\
[1786]712\mathtt{regs}[v^* / \mathtt{dst}] \\
[1786]715To summarize, the forward simulation diagrams for function call/return
[1793]716have the following form where the triangle is the one given in
717Figure~\ref{fig.commuting.diagrams}, the next instruction to be
718executed in state $s$ is either a function call or return and the intermediate
719state $s'$ is either a \texttt{Call} or a \texttt{Return} state.
722s    & \rTo^1 & s' & \rTo^1 & s'' \\
723\dTo &        &    & \rdTo  & \dTo \\
724\llbracket s \rrbracket & \rTo(1,3)^1 & & & \llbracket s'' \rrbracket \\ 
[1793]727Two steps of execution are simulated by a single step.
[1780]729\subsection{The RTL to ERTL translation}
732We map RTL statuses to ERTL statuses as follows:
[1780]734\mathtt{sp} & = \mathtt{RegisterSPH} / \mathtt{RegisterSPL} \\
[1791]735\mathtt{graph} & = \mathtt{graph} + \mathtt{prologue}(s) + \mathtt{epilogue}(s) \\
[1727]736& \mathrm{where}\ s = \mathrm{callee\ saved} + \nu \mathrm{RA} \\
[1780]738The 16-bit RTL stack pointer \texttt{SP} is mapped to a pair of 8-bit hardware registers \texttt{RegisterSPH} and \texttt{RegisterSPL}.
739The 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]741The prologue and epilogue that are added to the function graph do the following:
743\mathtt{prologue}(s) = & \mathtt{create\_new\_frame}; \\
744                       & \mathtt{pop\ ra}; \\
745                       & \mathtt{save\ callee\_saved}; \\
746                                                                                         & \mathtt{get\_params} \\
747                                                                                         & \ \ \mathtt{reg\_params}: \mathtt{move} \\
748                                                                                         & \ \ \mathtt{stack\_params}: \mathtt{push}/\mathtt{pop}/\mathtt{move} \\
[1780]750That 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.
751In other words, the prologue of a function correctly sets up the calling convention used in the compiler when calling a function.
752On the other hand, the epilogue undoes the action of the prologue:
754\mathtt{epilogue}(s) = & \mathtt{save\ return\ to\ tmp\ real\ regs}; \\
755                                                                                         & \mathtt{restore\_registers}; \\
756                       & \mathtt{push\ ra}; \\
757                       & \mathtt{delete\_frame}; \\
758                       & \mathtt{save return} \\
[1780]760That 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]762The \texttt{CALL} instruction is translated as follows:
[1738]764\mathtt{CALL}\ id \mapsto \mathtt{set\_params};\ \mathtt{CALL}\ id;\ \mathtt{fetch\_result}
[1780]766Here, \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]768The translation from RTL to ERTL and execution functions must satisfy the following properties for \texttt{CALL} and \texttt{RETURN} instructions appropriately:
771\mathtt{CALL} & \rTo^1 & \mathtt{inside\ function} \\
772\dTo & & \dTo \\
773\underbrace{\ldots}_{\llbracket \mathtt{CALL} \rrbracket} & \rTo &
774\underbrace{\ldots}_{\mathtt{prologue}} \\
777That 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.
778A similar property for \texttt{RETURN} should also hold, substituting the prologue for the epilogue of the function being translated:
781\mathtt{RETURN} & \rTo^1 & \mathtt{.} \\
782\dTo & & \dTo \\
783\underbrace{\ldots}_{\mathtt{epilogue}} & \rTo &
784\underbrace{\ldots} \\
[1734]788\subsection{The ERTL to LTL translation}
[1790]790During the ERTL to LTL translation pseudo-registers are stored in hardware registers or spilled onto the stack frame.
791The decision is based on a liveness analysis performed on the ERTL code to determine what pair of pseudo-registers are live at the same time for a given location.
792A colouring algorithm is then used to choose where to store the pseudo-registers, permitting pseudo-registers that are deemed never to be live at the same time to share the same location.
[1790]794We will not certify any colouring algorithm or control flow analysis.
795Instead, we axiomatically assume the existence of `oracles' that implement the colouring and liveness analyses.
796In a later phase we plan to validate the solutions by writing and certifying the code of a validator.
[1790]798We describe the liveness analysis and colouring analysis first and then the ERTL to LTL translation after.
[1790]800Throughout this section, we denote pseudoregisters with the type $\mathtt{register}$ and hardware registers with $\mathtt{hdwregister}$.
[1786]801\subsubsection{Liveness analysis}
[1750]802\newcommand{\declsf}[1]{\expandafter\newcommand\expandafter{\csname #1\endcsname}{\mathop{\mathsf{#1}}\nolimits}}
[1790]810For the liveness analysis, we aim to construct a map
[1750]811$\ell \in \mathtt{label} \mapsto $ live registers at $\ell$.
[1790]812We define the following operators on ERTL statements:
[1750]815\begin{array}{lL>{(ex. $}L<{)$}}
[1785]816\Defined(\ell) & registers defined at $\ell$ & \ell:r_1\leftarrow r_2+r_3 \mapsto \{r_1,C\}, \ell:\mathtt{CALL}~id\mapsto \text{caller-save}
[1785]818\Used(\ell) & registers used at $\ell$ & \ell:r_1\leftarrow r_2+r_3 \mapsto \{r_2,r_3\}, \ell:\mathtt{CALL}~id\mapsto \text{parameters}
821Given $LA:\mathtt{label}\to\mathtt{lattice}$ (where $\mathtt{lattice}$ is the type of sets of registers\footnote{More precisely, it is the lattice $\mathtt{set}(\mathtt{register}) \times \mathtt{set}(\mathtt{hdwregister})$, with pointwise operations.}), we also have have the following predicates:
[1785]824\Eliminable_{LA}(\ell) & iff executing $\ell$ has side-effects only on $r\notin LA(\ell)$
[1785]826(ex.\ $\ell : r_1\leftarrow r_2+r_3 \mapsto (\{r_1,C\}\cap LA(\ell)\neq\emptyset),
[1750]827  \mathtt{CALL}id\mapsto \text{never}$)
829\Livebefore_{LA}(\ell) &$:=
830  \begin{cases}
831    LA(\ell) &\text{if $\Eliminable_{LA}(\ell)$,}\\
[1785]832    (LA(\ell)\setminus \Defined(\ell))\cup \Used(\ell) &\text{otherwise}.
[1750]833  \end{cases}$
836In particular, $\Livebefore$ has type $(\mathtt{label}\to\mathtt{lattice})\to \mathtt{label}\to\mathtt{lattice}$.
[1790]838The equation upon which we build the fixpoint is then
840\Liveafter(\ell) \doteq \bigcup_{\ell <_1 \ell'} \Livebefore_{\Liveafter}(\ell')
842where $\ell <_1 \ell'$ denotes that $\ell'$ is an immediate successor of $\ell$ in the graph.
843We do not require the fixpoint to be the least one, so the hypothesis on $\Liveafter$ that we require is
846\Liveafter(\ell) \supseteq \bigcup_{\ell <_1 \ell'} \Livebefore(\ell')
[1790]848(for brevity we drop the subscript from $\Livebefore$).
855The aim of liveness analysis is to define what properties we need of the colouring function, which is a map (computed separately for each internal function)
859which identifies pseudoregisters with hardware ones if it is able to, otherwise it spills them to the stack.
860We will just state what property we require from such a map.
861First, we extend the definition to all types of registers by:
[1785]864   \Colour^+:\mathtt{hdwregister}+\mathtt{register} &\to \mathtt{hdwregister}+\mathtt{nat}\\
865   r & \mapsto
867  \Colour(r) &\text{if $r\in\mathtt{register}$,}\\
868  r &\text{if $r\in\mathtt{hdwregister}$,}.
872The other piece of information we compute for each function is a \emph{similarity} relation, which is an equivalence relation on all kinds of registers which depends on the point of the program.
873We write
875x\sim y \at \ell
877to state that registers $x$ and $y$ are similar at $\ell$.
878The formal definition of this relation's property will be given next, but intuitively it means that those two registers \emph{must} have the same value after $\ell$.
879The analysis that produces this information can be coarse: in our case, we just set two different registers to be similar at $\ell$ if at $\ell$ itself there is a move instruction between the two.
881The property required of colouring is the following:
884\forall \ell.\forall x,y. x,y\in \Liveafter(\ell)\Rightarrow
885  \Colour^+(x)=\Colour^+(y) \Rightarrow x\sim y \at\ell.
888\subsubsection{The translation}
[1785]890For example:
892\ell : r_1\leftarrow r_2+r_3 \mapsto \begin{cases}
[1785]893                                 \varepsilon & \text{if $\Eliminable(\ell)$},\\
894                                 \Colour(r_1) \leftarrow \Colour(r_2) + \Colour(r_3) & \text{otherwise}.
[1790]895                                \end{cases}
897where $\varepsilon$ is the empty block of instructions (i.e.\ a \texttt{GOTO}), and $\Colour(r_1) \leftarrow \Colour(r_2) + \Colour(r_3)$ is a notation for a block of instructions that take into account:
900Load and store ops on the stack if any colouring is in fact a spilling;
902Using the accumulator to store intermediate values.
[1790]904The overall effect is that if $T$ is an LTL state with $\ell(T)=\ell$ then we will have $T\to^+T'$ where $T'(\Colour(r_1))=T(\Colour(r_2))+T(\Colour(r_2))$, while $T'(y)=T(y)$ for any $y$ \emph{in the image of $\Colour$} different from $\Colour(r_1)$.
905Some hardware registers that are used for book-keeping and which are explicitly excluded from colouring may have different values.
[1788]907We skip the details of correctly dealing with the stack and its size.
[1785]909\subsubsection{The relation between ERTL and LTL states}
911Given a state $S$ in ERTL, we abuse notation by using $S$ as the underlying map
913S : \mathtt{hdwregister}+\mathtt{register} \to \mathtt{Value}
915We write $\ell(S)$ for the program counter in $S$.
916At this point we can state the property asked from similarity:
919\forall S,S'.S\to S' \Rightarrow \forall x,y.x\sim y \at \ell(S) \Rightarrow S'(x) = S'(y).
[1790]922Next, we relate ERTL states with LTL ones.
923For a state $T$ in LTL we again abuse notation using $T$ as a map
925T: \mathtt{hdwregister}+\mathtt{nat} \to \mathtt{Value}
927which maps hardware registers and \emph{local stack offsets} to values (in particular, $T$ as a map depends on the saved frames for computing the correct absolute stack values).
[1790]929The relation existing between the states at the two sides of this translation step, which depends on liveness and colouring, is then defined as
931S\mathrel\sigma T \iff \ldots \wedge \forall x. x\in \Livebefore(\ell(S))\Rightarrow T(\Colour^+(x)) = S(x)
933The ellipsis stands for other straightforward preservation, among which the properties $\ell(T) = \ell(S)$ and, inductively, the preservation of frames.
935\subsubsection{Proof of preservation}
[1785]937We will prove the following proposition:
939\forall S, T. S \mathrel\sigma T \Rightarrow S \to S' \Rightarrow \exists T'.T\to^+ T' \wedge S'\mathrel\sigma T'
941(with appropriate cost-labelled trace preservation which we omit).
942We will call $S\mathrel \sigma T$ the inductive hypothsis, as it will be such in the complete proof by induction on the trace of the program.
943As usual, this step is done by cases on the statement at $\ell(S)$ and how it is translated.
944We carry out the case of a binary operation on registers in some detail.
946Suppose that $\ell(S):r_1 \leftarrow r_2+r_3$, so that
948S'(x)=\begin{cases}S(r_1)+S(r_2) &\text{if $x=r_1$,}\\S(x) &\text{otherwise.}\end{cases}
[1785]951\paragraph*{Case $\Eliminable(\ell(S))$.}
[1790]953By definition we have $r_1\notin \Liveafter(\ell(S))$, and the translation of the operation yields a \texttt{GOTO}.
954We take $T'$ the immediate successor of $T$.
[1785]956Now in order to prove $S'\mathrel\sigma T'$, take any
958x\in\Livebefore(\ell(S'))\subseteq \Liveafter(\ell(S)) = \Livebefore(\ell(S))
960where we have used property~\eqref{eq:livefixpoint} and the definition of $\Livebefore$ when $\Eliminable(\ell(S))$.
961We get the following chain of equalities:
964T'(\Colour^+(x))\stackrel 1=T(\Colour^+(x))\stackrel 2=S(x) \stackrel 3= S'(x)
969follows as $T'$ has the same store as $T$,
971follows from the inductive hypothesis as $x\in\Livebefore(\ell(S))$,
973follows as $x\neq r_1$, as $r_1\notin \Liveafter(\ell(S))\ni x$.
[1785]976\paragraph*{Case $\neg\Eliminable(\ell(S))$.}
[1785]978We then have $r_1\in\Liveafter(\ell(S))$, and
983Moreover the statement is translated to $\Colour(r_1)\leftarrow\Colour(r_2)+\Colour(r_3)$, and we take the $T'\leftarrow^+T$ such that $T'(\Colour(r_1))=T(\Colour(r_2))+T(\Colour(r_3))$ and $T'(\Colour^+(x))=T(\Colour^+(x))$ for all $x$ with $\Colour^+(x)\neq\Colour(r_1)$.
[1785]985Take any $x\in\Livebefore(\ell(S'))\subseteq \Liveafter(\ell(S))$ (by property~\eqref{eq:livefixpoint}).
[1790]987If $\Colour^+(x)=\Colour(r_1)$, we have by property~\eqref{eq:colourprop} that $x\sim r_1\at \ell(S)$ (as both $r_1,x\in\Liveafter(\ell(S))$, so that
989T'(\Colour^+(x))=T(\Colour(r_2))+T(\Colour(r_3))\stackrel 1=S(r_2)+S(r_3)=S'(r_1)\stackrel 2=S(x)
994follows from two uses of the inductive hypothesis, as $r_2,r_3\in\Livebefore(\ell(S))$,
996follows from property~\eqref{eq:similprop}\footnote{Notice that in our particular implementation for this case of binary op $x\sim r_1\at\ell(S)$ implies $x=r_1$.
997However, nothing prevents us from employing more finegrained heuristics for similarity.}.
[1790]1000If $\Colour^+(x)\neq\Colour(r_1)$ (so in particular $x\neq r_1$), then $x\in\Livebefore(\ell(S))$, so by inductive hypothesis we have
[1734]1005\subsection{The LTL to LIN translation}
[1790]1007As detailed elsewhere in the reports, due to the parameterised representation of the back-end languages, the pass described here is actually much more generic than the translation from LTL to LIN.
1008It consists in a linearisation pass that maps any graph-based back-end language to its corresponding linear form, preserving its semantics.
1009In the rest of the section, however, we will keep the names LTL and LIN for the two partial instantiations of the parameterized language for convenience.
[1721]1011We 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:
1013\mathtt{pc : label} \stackrel{\sigma}{\longrightarrow} \mathbb{N}
[1723]1016The LTL to LIN translation pass also linearises the graph data structure into a list of instructions.
1017Pseudocode for the linearisation process is as follows:
1020let rec linearise graph visited required generated todo :=
1021  match todo with
1022  | l::todo ->
1023    if l $\in$ visited then
1024      let generated := generated $\cup\ \{$ Goto l $\}$ in
1025      let required := required $\cup$ l in
1026        linearise graph visited required generated todo
1027    else
[1725]1028      -- Get the instruction at label `l' in the graph
[1723]1029      let lookup := graph(l) in
1030      let generated := generated $\cup\ \{$ lookup $\}$ in
[1725]1031      -- Find the successor of the instruction at label `l' in the graph
[1723]1032      let successor := succ(l, graph) in
1033      let todo := successor::todo in
1034        linearise graph visited required generated todo
1035  | []      -> (required, generated)
[1725]1038It is easy to see that this linearisation process eventually terminates.
1039In 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]1041The 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.
1042We envisage needing to prove the following invariants on the linearisation function above:
1046$\mathtt{visited} \approx \mathtt{generated}$, 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,
1048$\forall \mathtt{l} \in \mathtt{generated}.\ \mathtt{succ(l,\ graph)} \subseteq \mathtt{required} \cup \mathtt{todo}$,
1050$\mathtt{required} \subseteq \mathtt{visited}$,
1052$\mathtt{visited} \cap \mathtt{todo} = \emptyset$.
1055The invariants collectively imply the following properties, crucial to correctness, about the linearisation process:
1059Every graph node is visited at most once,
1061Every instruction that is generated is generated due to some graph node being visited,
1063The successor instruction of every instruction that has been visited already will eventually be visited too.
[1790]1066Note, because the LTL to LIN transformation is the first time the code of a function is linearised in the back-end, we must discover a notion of `well-formed function code' suitable for linearised forms.
1067In particular, we see the notion of well-formedness (yet to be formally defined) resting on the following conditions:
[1762]1071For every jump to a label in a linearised function code, the target label exists at some point in the function code,
[1762]1073Each label is unique, appearing only once in the function code,
[1790]1075The final instruction of a function code must be a return or an unconditional jump.
[1726]1078We assume that these properties will be easy consequences of the invariants on the linearisation function defined above.
[1725]1080The final condition above is potentially a little opaque, so we explain further.
[1762]1081The 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]1083\subsection{The LIN to ASM and ASM to MCS-51 machine code translations}
[1721]1086The LIN to ASM translation step is trivial, being almost the identity function.
1087The 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]1089The ASM to MCS-51 machine code translation step, and the required statements of correctness, are found in an unpublished manuscript attached to this document.
[1790]1090This is the most complex translation because of the huge number of cases to be addressed and because of the complexity of the two semantics.
1091Moreover, in the assembly code we have conditional and unconditional jumps to arbitrary locations in the code, which are not supported by the MCS-51 instruction set.
1092The latter has several kind of jumps characterized by a different instruction size and execution time, but limited in range.
1093For instance, conditional jumps to locations whose destination is more than $2^7$ bytes away from the jump instruction location are not supported at all and need to be emulated with a code transformation.
1094This problem, which is known in the literature as branch displacement and is a universal problem for all architectures of microcontroller, is known to be computationally hard, often lying inside NP, depending on the exact characteristics of the target architecture.
1095As far as we know, we will provide the first formally verified proof of correctness for an assembler that implements branch displacement.
1096We are also providing the first verified proof of correctness of a mildly optimising branch displacement algorithm that, at the moment, is almost finished, but not described in the companion paper.
1097This proof, in isolation, took around 6 man months.
[1757]1099\section{Correctness of cost prediction}
[1790]1101Roughly speaking, the proof of correctness of cost prediction shows that the cost of executing a labelled object-code program is the same as the summation over all labels in the program execution trace of the cost statically associated to the label and computed on the object code itself.
[1790]1103In the presence of object-level function calls, the previous statement is, however, incorrect.
1104The reason is twofold.
1105First of all, a function call may diverge.
1106However, to the labels that appears just before a call, we also associate the cost of the instructions that follow the call.
1107Therefore, in the summation over all labels, when we meet a label we pre-pay for the instructions that appear after function calls, assuming all calls to be terminating.
[1790]1109This choice is driven by several considerations on the source code.
1110Namely, functions can be called inside expressions, and it would be too disruptive to put labels inside expressions to capture the cost of instructions that follow a call.
1111Moreover, adding a label after each call would produce a much higher number of proof obligations in the certification of source programs using Frama-C.
1112The proof obligations, further, would be guarded by a requirement to demonstrate the termination of all functions involved, that also generates lots of additional complex proof obligations that have little to do with execution costs.
1114With our approach, instead, we put less burden on the user, at the price of proving a weaker statement: the estimated and actual costs will be the same if and only if the high-level program is converging.
1115For prefixes of diverging programs we can provide a similar result where the equality is replaced by an inequality (loss of precision).
1117Assuming totality of functions is however not a sufficiently strong condition at the object-level.
1118Even if a function returns, there is no guarantee that it will transfer control back to the calling point.
1119For instance, the function could have manipulated the return address from its stack frame.
1120Moreover, an object-level program can forge any address and transfer control to it, with no guarantees about the execution behaviour and labelling properties of the called program.
1122To solve the problem, we introduced the notion of a \emph{structured trace} that come in two flavours: structured traces for total programs (an inductive type) and structured traces for diverging programs (a co-inductive type based on the previous one).
1123Roughly speaking, a structured trace represents the execution of a well behaved program that is subject to several constraints, such as:
1126All function calls return control just after the calling point,
1128The execution of all function bodies start with a label and end with a \texttt{RET} instruction (even those reached by invoking a function pointer),
1130All instructions are covered by a label (required by the correctness of the labelling approach),
1132The target of all conditional jumps must be labelled (a sufficient but not necessary condition for precision of the labelling approach)
1135Two structured traces with the same structure yield the same cost traces.
[1790]1138Correctness of cost predictions is only proved for structured execution traces, i.e. well behaved programs.
1139The forward simulation proof for all back-end passes will actually be a proof of preservation of the structure of the structured traces that, because of property \ref{prop5}, will imply correctness of the cost prediction for the back-end.
1140The Clight to RTLabs correctness proof will also include a proof that associates to each converging execution its converging structured trace and to each diverging execution its diverging structured trace.
[1790]1142There are also two more issues that invalidate the na\"ive statement of correctness for cost prediciton given above.
1143The algorithm that statically computes the costs of blocks is correct only when the object code is \emph{well-formed} and the program counter is \emph{reachable}.
1144A well-formed object-code is such that the program counter will never overflow after any execution step of the processor.
1145An overflow that occurs during fetching but is overwritten during execution is, however, correct and necessary to accept correct programs that are as large as the processor's code memory.
1146Temporary overflows add complications to the proof.
1147A reachable address is an address that can be obtained by fetching (\emph{not executing!}) a finite number of times from the beginning of the code memory without ever overflowing.
1148The complication is that the static prediction traverses the code memory assuming that the memory will be read sequentially from the beginning and that all jumps jump only to reachable addresses.
1149When this property is violated, the way the code memory is interpreted is incorrect and the cost computed is meaningless.
1150The reachability relation is closed by fetching for well-formed programs.
1151The property that states that function pointers only target reachable and well-labelled locations, however, is not statically predictable and it is therefore enforced in the structured trace.
[1790]1153The proof of correctness for cost predictions, and even discovering the correct statement, has been quite complex.
1154Setting up good invariants (i.e. structured traces, well formed programs, reachability) and completing the proof has required more than 3 man months, while the initally estimated effort was much lower.
1155In the paper-and-pencil proof for IMP, the corresponding proof was `obvious' and only took two lines.
[1790]1157The proof itself is quite involved.
1158We must show, as an important lemma, that the sum of the execution costs over a structured trace, where the costs are summed in execution order, is equivalent to the sum of the execution costs in the order of pre-payment.
1159The two orders are quite different and the proof is by mutual recursion over the definition of the converging structured traces, which is a family of three mutual inductive types.
1160The fact that this property only holds for converging function calls is hidden in the definition of the structured traces.
1161Then we need to show that the order of pre-payment corresponds to the order induced by the cost traces extracted from the structured trace.
1162Finally, we need to show that the statically computed cost for one block corresponds to the cost dynamically computed in pre-payment order.
[1758]1164\section{Overall results}
[1790]1166Functional correctness of the compiled code can be shown by composing the simulations to show that the target behaviour matches the behaviour of the source program, if the source program does not `go wrong'.
1167More precisely, we show that there is a forward simulation between the source trace and a (flattened structured) trace of the output, and conclude equivalence because the target's semantics are in the form of an executable function, and hence deterministic.
[1790]1169Combining this with the correctness of the assignment of costs to cost labels at the ASM level for a structured trace, we can show that the cost of executing any compiled function (including the top-level, main function of a C program) is equal to the sum of all the values for cost labels encountered in the \emph{source code's} trace of the function.
[1741]1171\section{Estimated effort}
[1790]1173Based on a rough analysis performed so far we can estimate the total effort required for the certification of the whole compiler.
1174We obtain this estimation by combining, for each pass:
1177The number of lines of code to be certified,
1179The ratio of number of lines of proof to number of lines of code from the CompCert project~\cite{compcert} for the CompCert pass that is closest in spirit to our own,
1181An estimation of the complexity of the pass according to the analysis above.
1183The result is shown in Table~\ref{table}.
[1747]1188Pass origin & Code lines & CompCert ratio & Estimated effort & Estimated effort \\
1189            &            &                & (based on CompCert) & \\
[1756]1191Common &  4864 & 4.25 \permil & 20.67 & 17.0 \\
1192Cminor &  1057 & 5.23 \permil & 5.53  &  6.0 \\
1193Clight &  1856 & 5.23 \permil & 9.71  & 10.0 \\ 
1194RTLabs &  1252 & 1.17 \permil & 1.48  &  5.0 \\
1195RTL    &   469 & 4.17 \permil & 1.95  &  2.0 \\
[1754]1196ERTL   &   789 & 3.01 \permil & 2.38  & 2.5 \\
1197LTL    &    92 & 5.94 \permil & 0.55  & 0.5 \\
[1756]1198LIN    &   354 & 6.54 \permil & 2.31  &   1.0 \\
1199ASM    &   984 & 4.80 \permil & 4.72  &  10.0 \\
[1756]1201Total common    &  4864 & 4.25 \permil & 20.67 & 17.0 \\
1202Total front-end &  2913 & 5.23 \permil & 15.24 & 16.0 \\
1203Total back-end  &  6853 & 4.17 \permil & 13.39 & 21.0 \\
[1756]1205Total           & 14630 & 3.75 \permil & 49.30 & 54.0 \\
[1786]1208\caption{\label{table} Estimated effort}
[1790]1211We provide now some additional informations on the methodology used in the computation.
1212The passes in Cerco and CompCert front-end closely match each other.
1213However, there is no clear correspondence between the two back-ends.
1214For instance, we enforce the calling convention immediately after instruction selection, whereas in CompCert this is performed in a later phase.
1215Further, we linearise the code at the very end, whereas CompCert performs linearisation as soon as possible.
1216Therefore, the first part of the estimation exercise has consisted of shuffling and partitioning the CompCert code in order to assign to each CerCo pass the CompCert code that performs a similar transformation.
[1790]1218After this preliminary step, using the data given in~\cite{compcert} (which refers to an early version of CompCert) we computed the ratio between man months and lines of code in CompCert for each CerCo pass.
1219This is shown in the third column of Table~\ref{table}.
1220For those CerCo passes that have no correspondence in CompCert (like the optimising assembler) or where we have insufficient data, we have used the average of the ratios computed above.
[1790]1222The first column of the table shows the number of lines of code for each pass in CerCo.
1223The third column is obtained multiplying the first with the CompCert ratio.
1224It provides an estimate of the effort required (in man months) if the complexity of the proofs for CerCo and CompCert would be the same.
[1790]1226The two proof styles, however, are purposefully completely different.
1227Where CompCert uses non-executable semantics, describing the various semantics of languages with inductive types, we have preferred executable semantics.
1228Therefore, CompCert proofs by induction and inversion become proofs by functional inversion, performed using the Russell methodology (now called Program in Coq, but whose behaviour differs from Matita's one).
1229Moreover, CompCert code is written using only types that belong to the Hindley-Milner fragment, whereas we have heavily exploited dependent types throughout the codebase.
1230The dependent type discipline offers many advantages, especially from the point of view of clarity of the invariants involved and early detection of errors.
1231It also naturally combines well with the Russell approach which is based on dependent types.
1232However, it is also known to introduce several technical problems, like the need to explicitly prove type equalities to be able to manipulate expressions in certain ways.
1233In many situations, the difficulties encountered with manipulating dependent types are better addressed by improving the Matita system, according to the formalisation driven system development.
1234For this reason, and assuming a pessimistic estimation of our performance, the fourth columns presents the final estimation of the effort required, that also takes into account the complexity of the proof suggested by the informal proofs sketched in the previous section.
[1774]1236\subsection{Contingency plan}
[1790]1238On the basis of the proof strategy sketched in this document and the estimated effort, we can refine our contingency plan.
1239In case we will end the certification of the basic compiler in advance we will have the choice of either proving loop optimisations and/or partial redundancy elimination correct (both tasks seem difficult to achieve in a short time) or considering the MCS-51 specific extensions introduced during the first period and under-used in the formalised prototype.
1240Yet another possibility would be to better study retargeting of the code and the commutation property between different compiler passes.
1241The latter study is easily enabled by our approach where all back-end languages are instances of the same parameterized language.
[1790]1243In the case of a consistent delay in the certification of some components, we will first address the passes that are more likely to have undetected bugs and we will follow a top-down approach, axiomatizing the properties of the data structures used in the compiler to focus more on the algorithms.
1244The rationale is that data structures are easier than algorithms to test using well-known methodologies.
1245The effort table clearly shows that common definitions and data structures are one quarter of the size of the current codebase code and require slightly less than one third of the total effort.
1246At least half of this effort really goes into simple data structures (vectors, bounded and unbounded integers, tries and maps) whose certification is not interesting and whose code could be taken without re-proving it from the library of any other theorem prover.
[1790]1250The overall exercise, whose details have only been sketched here, has been very useful.
1251It has brought to light some errors in our proof that have required major changes in the proof plan.
1252It has also shown that the last passes of the compilation (e.g. assembly) and cost prediction on the object-code are much more involved than more high-level passes.
[1790]1254The final estimation for the effort required to complete the proof suffers from a low degree of confidence engendered in the numbers, due to the difficulty in relating our work, and compiler design, with that of CompCert.
1255It is however sufficient to conclude that the effort required is in line with the man power that was scheduled for the second half of the second period and for the third period.
1256Compared to the number of man months declared in Annex I of the contract, we will need more man months.
1257However, at both UNIBO and UEDIN there have been major differences in hiring with respect to the Annex.
1258Therefore both sites now have more manpower available, though with the associated trade-off of a lower level of maturity for the people employed.
[1790]1260Our reviewers suggested that we use this estimation to compare two possible scenarios: a) proceed as planned, porting all the CompCert proofs to Matita or b) port D3.1 and D4.1 to Coq and re-use the CompCert proofs.
1261We remark here again that the back-end of the two compilers, from the memory model on, are quite different: we are not re-proving correctness of the same piece of code.
1262Moreover, the proof techniques are different for the front-end too.
1263Switching to the CompCert formalisation would imply the abandoning of the untrusted compiler, the abandoning of the experiment with a different proof technique, the abandoning of the quest for an open-source proof, and the abandoning of the co-development of the formalisation and the Matita proof assistant.
1264In the Commitment Letter~\cite{letter}, delivered to the Officer in May, we clarified our personal perspectives on the project goals and objectives.
1265We do not re-describe here the point of view presented in the letter, other than the condensed soundbite that ``we value diversity''.
[1790]1267Clearly, if the execise would have suggested the infeasability in terms of effort of concluding the formalisation, or getting close to that, we would have abandoned our path and embraced the reviewer's suggestion.
1268However, we have been comforted in the analysis that we did in Autumn and further progress completed during the winter does not yet show any major delay with respect to the proof schedule.
1269We are thus planning to continue the certification according to the more detailed proof plan that came out from the exercise reported in this manuscript.
1272\bibitem{compcert} X. Leroy, ``A Formally Verified Compiler back-end'',
1273Journal of Automated Reasoning 43(4)):363-446, 2009.
1275\bibitem{letter}The CerCo team, ``Commitment to the Consideration of Reviewer's Reccomendation'', 16/05/2011.
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