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37\title{On the correctness of an assembler for the Intel MCS-51 microprocessor}
38\author{Jaap Boender \and Dominic P. Mulligan \and Claudio Sacerdoti Coen}
39\institute{Dipartimento di Scienze dell'Informazione, Universit\'a di Bologna}
40
41\bibliographystyle{splncs03}
42
43\begin{document}
44
45\maketitle
46
47\begin{abstract}
48We consider the formalisation of an assembler for Intel MCS-51 assembly language in the Matita proof assistant.
49This formalisation forms a major component of the EU-funded CerCo project, concering the construction and formalisation of a concrete complexity preserving compiler for a large subset of the C programming language.
50
51The efficient expansion of pseudoinstructions---particularly jumps---into MCS-51 machine instructions is complex.
52We employ a strategy, involving the use of `policies', that separates the decision making over how jumps should be expanded from the expansion process itself.
53This makes the proof of correctness for the assembler significantly more straightforward.
54
55We prove, under the assumption of the existence of a correct policy, that the assembly process preserves the semantics of assembly programs.
56Correct policies fail to exist only in a limited number of pathological circumstances.
57\end{abstract}
58
59% ---------------------------------------------------------------------------- %
60% SECTION                                                                      %
61% ---------------------------------------------------------------------------- %
62\section{Introduction}
63\label{sect.introduction}
64
65We consider the formalisation of an assembler for the Intel MCS-51 8-bit microprocessor in the Matita proof assistant~\cite{asperti:user:2007}.
66This formalisation forms a major component of the EU-funded CerCo project~\cite{cerco:2011}, concering the construction and formalisation of a concrete complexity preserving compiler for a large subset of the C programming language.
67
68The MCS-51 dates from the early 1980s and commonly called the 8051/8052.
69Despite the microprocessor's age, derivatives are still widely manufactured by a number of semiconductor foundries.
70As a result the processor is widely used, especially in embedded systems development, where well-tested, cheap, predictable microprocessors find their niche.
71
72The MCS-51 has a relative paucity of features compared to its more modern brethren.
73In particular, the MCS-51 does not possess a cache or any instruction pipelining that would make predicting the concrete cost of executing a single instruction an involved process.
74Instead, each semiconductor foundry that produces an MCS-51 derivative is able to provide accurate timing information in clock cycles for each instruction in their derivative's instruction set.
75It is important to stress that this timing information, unlike in more sophisticated processors, is not an estimate, it is a definition.
76For the MCS-51, if a manufacturer states that a particular opcode takes three clock cycles to execute, then that opcode \emph{always} takes three clock cycles to execute.
77
78This predicability of timing information is especially attractive to the CerCo consortium.
79We are in the process of constructing a certified, concrete complexity compiler for a realistic processor, and not for building and formalising the worst case execution time (WCET) tools that would be necessary to achieve the same result with, for example, a modern ARM or PowerPC microprocessor.
80
81However, the MCS-51's paucity of features is a double edged sword.
82In particular, the MCS-51 features relatively miniscule memory spaces (including read-only code memory, stack and internal/external random access memory) by modern standards.
83As a result our compiler, to have any sort of hope of successfully compiling realistic C programs, ought to produce `tight' machine code.
84This is not simple.
85
86We here focus on a single issue in the MCS-51's instruction set: unconditional jumps.
87The MCS-51 features three conditional jump instructions: \texttt{LJMP} and \texttt{SJMP}---`long jump' and `short jump' respectively---and an 11-bit oddity of the MCS-51, \texttt{AJMP}, that the prototype CerCo compiler~\cite{cerco-report-code:2011} ignores for simplicity's sake.\footnote{Ignoring \texttt{AJMP} and its analogue \texttt{ACALL} is not idiosyncratic.  The Small Device C Compiler (SDCC)~\cite{sdcc:2011}, the leading open source C compiler for the MCS-51, also seemingly does not produce \texttt{AJMP} and \texttt{ACALL} instructions.  Their utility in a modern context remains unclear.}
88Each of these three instructions expects arguments in different sizes and behaves in different ways.
89\texttt{SJMP} may only perform a `local jump' whereas \texttt{LJMP} may jump to any address in the MCS-51's memory space.
90Consequently, the size of each opcode is different, and to squeeze as much code as possible into the MCS-51's limited code memory, the smallest possible opcode should be selected.
91
92The prototype CerCo C compiler does not attempt to select the smallest jump opcode in this manner, as this was thought to unneccessarily complicate the compilation chain.
93Instead, the compiler targets an assembly language, complete with pseudoinstructions including bespoke \texttt{Jmp} and \texttt{Call} instructions.
94Labels, conditional jumps to labels, a program preamble containing global data and a \texttt{MOV} instruction for moving this global data into the MCS-51's one 16-bit register also feature.
95This latter feature will ease any later consideration of separate compilation in the CerCo compiler.
96An assembler is used to expand pseudoinstructions into MCS-51 machine code.
97
98However, this assembly process is not trivial, for numerous reasons.
99For example, our conditional jumps to labels behave differently from their machine code counterparts.
100At the machine code level, conditional jumps may only jump to a relative offset, expressed in a byte, of the current program counter, limiting their range.
101However, at the assembly level, conditional jumps may jump to a label that appears anywhere in the program, significantly liberalising the use of conditional jumps and further simplifying the design of the CerCo compiler.
102
103Further, trying to na\"ively relate assembly programs with their machine code counterparts simply does not work.
104Machine code programs that fetch from code memory and programs that combine the program counter with constant shifts do not make sense at the assembly level.
105More generally, memory addresses can only be compared with other memory addresses.
106However, checking that memory addresses are only compared against each other at the assembly level is in fact undecidable.
107In short, the full preservation of the semantics of the two languages is impossible.
108
109Yet more complications are added by the peculiarities of the CerCo project itself.
110As mentioned, the CerCo consortium is in the business of constructing a verified compiler for the C programming language.
111However, unlike CompCert~\cite{compcert:2011,leroy:formal:2009,leroy:formally:2009}---which currently represents the state of the art for `industrial grade' verified compilers---CerCo considers not just the \emph{intensional correctness} of the compiler, but also its \emph{extensional correctness}.
112That is, CompCert focusses solely on the preservation of the \emph{meaning} of a program during the compilation process, guaranteeing that the program's meaning does not change as it is gradually transformed into assembly code.
113However in any realistic compiler (even the CompCert compiler!) there is no guarantee that the program's time properties are preserved during the compilation process; a compiler's `optimisations' could, in theory, even conspire to degrade the concrete complexity of certain classes of programs.
114CerCo aims to expand the current state of the art by producing a compiler where this temporal degradation is guaranteed not to happen.
115
116In order to achieve this CerCo imposes a cost model on programs, or more specifically, on simple blocks of instructions.
117This cost model is induced by the compilation process itself, and its non-compositional nature allows us to assign different costs to identical blocks of instructions depending on how they are compiled.
118In short, we aim to obtain a very precise costing for a program by embracing the compilation process, not ignoring it.
119This, however, complicates the proof of correctness for the compiler proper: for every translation pass from intermediate language to intermediate language, we must prove that not only has the meaning of a program been preserved, but also its complexity characteristics.
120This also applies for the translation from assembly language to machine code.
121
122How do we assign a cost to a pseudoinstruction?
123As mentioned, conditional jumps at the assembly level can jump to a label appearing anywhere in the program.
124However, at the machine code level, conditional jumps are limited to jumping `locally', using a measly byte offset.
125To translate a jump to a label, a single conditional jump pseudoinstruction may be translated into a block of three real instructions, as follows (here, \texttt{JZ} is `jump if accumulator is zero'):
126\begin{displaymath}
127\begin{array}{r@{\quad}l@{\;\;}l@{\qquad}c@{\qquad}l@{\;\;}l}
128       & \mathtt{JZ}  & label                      &                 & \mathtt{JZ}   & \text{size of \texttt{SJMP} instruction} \\
129       & \ldots       &                            & \text{translates to}   & \mathtt{SJMP} & \text{size of \texttt{LJMP} instruction} \\
130label: & \mathtt{MOV} & \mathtt{A}\;\;\mathtt{B}   & \Longrightarrow & \mathtt{LJMP} & \text{address of \textit{label}} \\
131       &              &                            &                 & \ldots        & \\
132       &              &                            &                 & \mathtt{MOV}  & \mathtt{A}\;\;\mathtt{B}
133\end{array}
134\end{displaymath}
135In the translation, if \texttt{JZ} fails, we fall through to the \texttt{SJMP} which jumps over the \texttt{LJMP}.
136Naturally, if \textit{label} is close enough, a conditional jump pseudoinstruction is mapped directly to a conditional jump machine instruction; the above translation only applies if \textit{label} is not sufficiently local.
137This leaves the problem, addressed below, of calculating whether a label is indeed `close enough' for the simpler translation to be used.
138
139Crucially, the above translation demonstrates the difficulty in predicting how many clock cycles a pseudoinstruction will take to execute.
140A conditional jump may be mapped to a single machine instruction or a block of three.
141Perhaps more insidious, the number of cycles needed to execute the instructions in the two branches of a translated conditional jump may be different.
142Depending on the particular MCS-51 derivative at hand, an \texttt{SJMP} could in theory take a different number of clock cycles to execute than an \texttt{LJMP}.
143These issues must also be dealt with in order to prove that the translation pass preserves the concrete complexity of the code.
144
145The question remains: how do we decide whether to expand a jump into an \texttt{SJMP} or an \texttt{LJMP}?
146To understand why this problem is not trivial, consider the following snippet of assembly code:
147\begin{displaymath}
148\text{dpm: finish me}
149\end{displaymath}
150
151As our example shows, given an occurence $l$ of an \texttt{LJMP} instruction, it may be possible to shrink $l$ to an occurence of an \texttt{SJMP} providing we can shrink any \texttt{LJMP}s that exist between $l$ and its target location.
152However, shrinking these \texttt{LJMP}s may in turn depend on shrinking $l$ to an \texttt{SJMP}, as it is perfectly possible to jump backwards.
153In short, unless we can somehow break this loop of circularity, and similar knotty configurations of jumps, we are stuck with a suboptimal solution to the expanding jumps problem.
154
155How we went about resolving this problem affected the shape of our proof of correctness for the whole assembler in a rather profound way.
156We first attempted to synthesize a solution bottom up: starting with no solution, we gradually refine a solution using the same functions that implement the jump expansion.
157Using this technique, solutions can fail to exist, and the proof quickly descends into a diabolical quagmire.
158
159Abandoning this attempt, we instead split the `policy'---the decision over how any particular jump should be expanded---from the implementation that actually expands assembly programs into machine code.
160Assuming the existence of a correct policy, we proved the implementation of the assembler correct.
161Further, we proved that the assembler fails to assemble an assembly program if and only if a correct policy does not exist.
162Policies do not exist in only a limited number of circumstances: namely, if a pseudoinstruction attempts to jump to a label that does not exist, or the program is too large to fit in code memory.
163The first case would constitute a serious compiler error, and hopefully certifying the rest of the compiler would rule this possibility out.
164The second case is unavoidable---certified compiler or not, trying to load a huge program into a small code memory will break \emph{something}.
165
166The rest of this paper is a detailed description of this proof.
167
168% ---------------------------------------------------------------------------- %
169% SECTION                                                                      %
170% ---------------------------------------------------------------------------- %
171\subsection{Overview of the paper}
172\label{subsect.overview.of.the.paper}
173In Section~\ref{sect.matita} we provide a brief overview of the Matita proof assistant for the unfamiliar reader.
174In Section~\ref{sect.the.proof} we discuss the design and implementation of the proof proper.
175In Section~\ref{sect.conclusions} we conclude.
176
177% ---------------------------------------------------------------------------- %
178% SECTION                                                                      %
179% ---------------------------------------------------------------------------- %
180\section{Matita}
181\label{sect.matita}
182
183Matita is a proof assistant based on the (Co)inductive Calculus of Constructions~\cite{asperti:user:2007}.
184For those familiar with Coq, Matita's syntax and mode of operation should be entirely familiar.
185We take time here to explain one of Matita's syntactic idiosyncracies, however.
186The use of `$\mathtt{?}$' or `$\mathtt{\ldots}$' in an argument position denotes a type or types to be inferred automatically by unification respectively.
187The use of `$\mathtt{?}$' in the body of a definition, lemma or axiom denotes an incomplete term that is to be closed, by hand, using tactics.
188
189% ---------------------------------------------------------------------------- %
190% SECTION                                                                      %
191% ---------------------------------------------------------------------------- %
192\section{The proof}
193\label{sect.the.proof}
194
195\subsection{The assembler and semantics of machine code}
196\label{subsect.the.assembler.and.semantics.of.machine.code}
197
198The formalisation in Matita of the semantics of MCS-51 machine code is described in~\cite{mulligan:executable:2011}.
199We merely describe enough here to understand the rest of the proof.
200
201At heart, the MCS-51 emulator centres around a \texttt{Status} record, describing the current state of the microprocessor.
202This record contains fields corresponding to the microprocessor's program counter, special function registers, and so on.
203At the machine code level, code memory is implemented as a trie of bytes, addressed by the program counter.
204Machine code programs are loaded into \texttt{Status} using the \texttt{load\_code\_memory} function.
205
206We may execut a single step of a machine code program using the \texttt{execute\_1} function, which returns an updated \texttt{Status}:
207\begin{lstlisting}
208definition execute_1: Status $\rightarrow$ Status := $\ldots$
209\end{lstlisting}
210The function \texttt{execute} allows one to execute an arbitrary, but fixed (due to Matita's normalisation requirement!) number of steps of a program.
211
212Naturally, assembly programs have analogues.
213The counterpart of the \texttt{Status} record is \texttt{PseudoStatus}.
214Instead of code memory being implemented as tries of bytes, code memory is here implemented as lists of pseudoinstructions, and program counters are merely indices into this list.
215Our analogue of \texttt{execute\_1} is \texttt{execute\_1\_pseudo\_instruction}:
216\begin{lstlisting}
217definition execute_1_pseudo_instruction: (Word $\rightarrow$ nat $\times$ nat) $\rightarrow$
218                                         PseudoStatus $\rightarrow$ PseudoStatus := $\ldots$
219\end{lstlisting}
220Notice, here, that the emulation function for pseudoprograms takes an additional argument.
221This is a function that maps program counters (for the pseudoprogram) to pairs of natural numbers representing the number of clock ticks that the pseudoinstruction needs to execute, post expansion.
222We call this function a \emph{costing}, and note that the costing is induced by the particular strategy we use to expand pseudoinstructions.
223If we change how we expand conditional jumps to labels, for instance, then the costing needs to change, hence \texttt{execute\_1\_pseudo\_instruction}'s parametricity in the costing.
224
225The costing returns \emph{pairs} of natural numbers because, in the case of expanding conditional jumps to labels, the expansion of the `true branch' and `false branch' may differ in the number of clock ticks needed for execution.
226This timing information is used inside \texttt{execute\_1\_pseudo\_instruction} to update the clock of the \texttt{PseudoStatus}.
227During the proof of correctness of the assembler we relate the clocks of \texttt{Status}es and \texttt{PseudoStatus}es.
228
229The assembler, mapping programs consisting of lists of pseudoinstructions to lists of bytes, operates in a mostly straightforward manner.
230To a degree of approximation, the assembler on an assembly program, consisting of $n$ pseudoinstructions $\mathtt{P_i}$ for $1 \leq i \leq n$, works as in the following diagram:
231\begin{displaymath}
232[\mathtt{P_1}, \ldots \mathtt{P_n}] \xrightarrow{\mathtt{flatten}\left(\mathtt{P_i} \xrightarrow{\mbox{\fontsize{7}{9}\selectfont$\mathtt{expand}$}} \mathtt{[I_1^i, \ldots I^q_i]} \xrightarrow{\mbox{\fontsize{7}{9}\selectfont$\mathtt{assembly1}^*$}} \mathtt{[0110]}\right)^{*}} \mathtt{[010101]}
233\end{displaymath}
234Here $\mathtt{I^i_j}$ for $1 \leq j \leq q$ are the $q$ machine code instructions obtained by expanding, with \texttt{expand\_pseudo\_instruction}, a single pseudoinstruction.
235Each machine code instruction $\mathtt{I^i_j}$ is then assembled, using the \texttt{assembly1} function, into a list of bytes.
236This process is iterated for each pseudoinstruction, before the lists are flattened into a single bit list representation of the original assembly program.
237
238By inspecting the above diagram, it would appear that the best way to proceed with a proof that the assembly process does not change the semantics of an assembly program is via a decomposition of the problem into two subproblems.
239Namely, we first expand any and all pseudoinstructions into lists of machine instructions, and provide a proof that this process does not change our program's semantics.
240Finally, we assemble all machine code instructions into machine code---lists of bytes---and prove once more that this process does not have an adverse effect on a program's semantics.
241By composition, we then have that the whole assembly process is semantics preserving.
242
243This is a tempting approach to the proof, but ultimately the wrong approach.
244In particular, it is important that we track how the program counter indexing into the assembly program, and the machine's program counter evolve, so that we can relate them.
245Expanding pseudoinstructions requires that the machine's program counter be incremented by $n$ steps, for $1 \leq n$, for every increment of the assembly program's program counter.
246Keeping track of the correspondence between the two program counters quickly becomes unfeasible using a compositional approach, and hence the proof must be monolithic.
247
248% ---------------------------------------------------------------------------- %
249% SECTION                                                                      %
250% ---------------------------------------------------------------------------- %
251\subsection{Policies}
252\label{subsect.policies}
253
254Policies exist to dictate how conditional and unconditional jumps at the assembly level should be expanded into machine code instructions.
255Using policies, we are able to completely decouple the decision over how jumps are expanded with the act of expansion, simplifying our proofs.
256As mentioned, the MCS-51 instruction set includes three different jump instructions: \texttt{SJMP}, \texttt{AJMP} and \texttt{LJMP}; call these `short', `medium' and `long' jumps, respectively:
257\begin{lstlisting}
258inductive jump_length: Type[0] :=
259  | short_jump: jump_length
260  | medium_jump: jump_length
261  | long_jump: jump_length.
262\end{lstlisting}
263A \texttt{jump\_expansion\_policy} is a map from \texttt{Word}s to \texttt{jump\_length}s, implemented as a trie.
264Intuitively, a policy maps positions in a program (indexed using program counters implemented as \texttt{Word}s) to a particular variety of jump.
265\begin{lstlisting}
266definition jump_expansion_policy := BitVectorTrie jump_length 16.
267\end{lstlisting}
268Next, we require a series of `sigma' functions.
269These functions map assembly program counters to their machine code counterparts, establishing the correspondence between `positions' in an assembly program and `positions' in a machine code program.
270At the heart of this process is \texttt{sigma0} which traverses an assembly program building maps from program counter to program counter.
271This function fails if and only if an internal call to \texttt{assembly\_1\_pseudoinstruction} fails:
272\begin{lstlisting}
273definition sigma0: pseudo_assembly_program
274  $\rightarrow$ option (nat $\times$ (nat $\times$ (BitVectorTrie Word 16))) := $\ldots$
275\end{lstlisting}
276We eventually lift this to functions from program counters to program counters:
277\begin{lstlisting}
278definition sigma_safe:
279  pseudo_assembly_program $\rightarrow$ option (Word $\rightarrow$ Word) := $\ldots$
280\end{lstlisting}
281Now, it's possible to define what a `good policy' is i.e. one that does not cause \texttt{sigma\_safe} to fail.
282As mentioned, \texttt{sigma\_safe} can only fail if an assembly program fails to be assembled:
283\begin{lstlisting}
284definition policy_ok := $\lambda$p. sigma_safe p $\neq$ None $\ldots$.
285\end{lstlisting}
286Finally, we obtain \texttt{sigma}, a map from program counters to program counters, which is guranteed not to fail as we internally provide a that
287\begin{lstlisting}
288definition sigma: pseudo_assembly_program $\rightarrow$ Word $\rightarrow$ Word := $\ldots$
289\end{lstlisting}
290
291% ---------------------------------------------------------------------------- %
292% SECTION                                                                      %
293% ---------------------------------------------------------------------------- %
294\subsection{Total correctness of the assembler}
295\label{subsect.total.correctness.of.the.assembler}
296
297Using our policies, we now work toward proving the total correctness of the assembler.
298By total correctness, we mean that the assembly process does not change the semantics of an assembly program.
299Naturally, this necessitates keeping some sort of correspondence between program counters at the assembly level, and program counters at the machine code level.
300For this, we use the \texttt{sigma} machinery defined at the end of Subsection~\ref{subsect.policies}.
301
302We expand pseudoinstructions using the function \texttt{expand\_pseudo\_instruction}.
303This function accepts a `policy decision'---an element of type \texttt{jump\_length}---that is used when expanding a \texttt{Call}, \texttt{Jmp} or conditional jump to a label into the correct machine instruction.
304This \texttt{policy\_decision} is asssumed to originate from a policy as defined in Subsection~\ref{subsect.policies}.
305\begin{lstlisting}
306definition expand_pseudo_instruction:
307  ∀lookup_labels, lookup_datalabels, pc, policy_decision.
308    pseudo_instruction $\rightarrow$ option list instruction := $\ldots$
309\end{lstlisting}
310Under the assumption that a correct policy exists, \texttt{expand\_pseudo\_instruction} should never fail, and therefore the option type may be dispensed with.
311This is because the only failure conditions for \texttt{expand\_pseudo\_instruction} result from trying to expand a pseudoinstruction into an `impossible' combination of machine code instructions.
312For instance, if the policy decision dictates that we should expand a \texttt{Call} pseudoinstruction into a `short jump', then we fail, as the MCS-51's instruction set only features instructions \texttt{ACALL} and \texttt{LCALL}.
313
314% dpm todo
315\begin{lstlisting}
316axiom assembly_ok: ∀program,assembled,costs,labels.
317  Some $\ldots$ $\langle$labels, costs$\rangle$ = build_maps program $\rightarrow$
318  Some $\ldots$ $\langle$assembled, costs$\rangle$ = assembly program $\rightarrow$
319  let code_memory := load_code_memory assembled in
320  let preamble := $\pi_1$ program in
321  let datalabels := construct_datalabels preamble in
322  let lk_labels :=
323    $\lambda$x. sigma program (address_of_word_labels_code_mem ($\pi_2$ program) x) in
324  let lk_dlabels := $\lambda$x. lookup ? ? x datalabels (zero ?) in
325   ∀ppc,len,assembledi.
326    let $\langle$pi, newppc$\rangle$ := fetch_pseudo_instruction ($\pi_2$ program) ppc in
327    let assembly' := assembly_1_pseudoinstruction program ppc
328      (sigma program ppc) lk_labels lk_dlabels pi in
329    let newpc := (sigma program ppc) + len in
330    let echeck :=
331      encoding_check code_memory (sigma program ppc) slen assembledi in
332     Some $\ldots$ $\langle$len, assembledi$\rangle$ = assembly' $\rightarrow$
333      echeck $\wedge$ sigma program newppc = newpc.
334\end{lstlisting}
335
336% dpm todo
337\begin{lstlisting}
338theorem fetch_assembly: $\forall$pc, i, cmem, assembled.
339  assembled = assembly1 i $\rightarrow$
340  let len := length $\ldots$ assembled in
341    encoding_check cmem pc (pc + len) assembled $\rightarrow$
342    let fetched := fetch code_memory (bitvector_of_nat $\ldots$ pc) in
343    let $\langle$instr_pc, ticks$\rangle$ := fetched in
344    let $\langle$instr, pc'$\rangle$ := instr_pc in
345      (eq_instruction instr i $\wedge$
346       eqb ticks (ticks_of_instruction instr) $\wedge$
347       eq_bv $\ldots$ pc' (pc + len)) = true.
348\end{lstlisting}
349
350Lemma \texttt{fetch\_assembly\_pseudo} establishes a basic property between \texttt{expand\_pseudo\_instruction} and \texttt{assembly\_1\_pseudoinstruction}:
351\begin{lstlisting}
352lemma fetch_assembly_pseudo: $\forall$program, ppc, lk_labels, lk_dlabels.
353  $\forall$pi, code_memory, len, assembled, instructions, pc.
354  let jexp := jump_expansion ppc program in
355  let exp :=
356    expand_pseudo_instruction lk_labels lk_dlabels pc jexp pi
357  let ass :=
358    assembly_1_pseudoinstruction program ppc pc lk_labels lk_dlabels pi in
359  Some ? instructions = exp $\rightarrow$
360    Some $\ldots$ $\langle$len, assembled$\rangle$ = ass $\rightarrow$
361      encoding_check code_memory pc (pc + len) assembled $\rightarrow$
362        fetch_many code_memory (pc + len) pc instructions.
363\end{lstlisting}
364Here, \texttt{len} is the number of machine code instructions the pseudoinstruction at hand has been expanded into, \texttt{encoding\_check} is a recursive function that checks for any possible corruption of the code memory, resulting from expanding the pseudoinstruction.
365We assemble a single pseudoinstruction with \texttt{assembly\_1\_pseudoinstruction}, which internally calls \texttt{jump\_expansion} and \texttt{expand\_pseudo\_instruction}.
366The function \texttt{fetch\_many} fetches multiple machine code instructions from code memory and performs some routine checks.
367
368Intuitively, Lemma \texttt{fetch\_assembly\_pseudo} can be read as follows.
369Suppose our policy \texttt{jump\_expansion} dictates that the pseudoinstruction indexed by the pseudo program counter \texttt{ppc} in assembly program \texttt{program} gives us the policy decision \texttt{jexp}.
370Further, suppose we expand the pseudoinstruction at \texttt{ppc} with the policy decision \texttt{jexp}, obtaining an (optional) list of machine code instructions \texttt{exp}.
371Suppose we also assemble the pseudoinstruction at \texttt{ppc} to obtain \texttt{ass}, a list of bytes.
372Then, under the assumption that neither the expansion of the pseudoinstruction to obtain \texttt{exp}, nor the assembly of the pseudoinstruction to obtain \texttt{ass}, failed, we check with \texttt{fetch\_many} that the number of machine instructions that were fetched matches the number of instruction that \texttt{expand\_pseudo\_instruction} expanded.
373
374At first sight, Lemma \texttt{fetch\_assembly\_pseudo2} appears to nearly establish the correctness of the assembler:
375\begin{lstlisting}
376lemma fetch_assembly_pseudo2: $\forall$program, assembled, costs, labels.
377  Some $\ldots$ $\langle$labels, costs$\rangle$ = build_maps program $\rightarrow$
378  Some $\ldots$ $\langle$assembled, costs$\rangle$ = assembly program $\rightarrow$ $\forall$ppc.
379  let code_memory := load_code_memory assembled in
380  let preamble := $\pi_1$ program in
381  let data_labels := construct_datalabels preamble in
382  let lk_labels :=
383    λx. sigma program (address_of_word_labels_code_mem ($\pi_2$ program) x) in
384  let lk_dlabels := λx. lookup ? ? x data_labels (zero ?) in
385  let expansion := jump_expansion ppc program in
386  let $\langle$pi, newppc$\rangle$ := fetch_pseudo_instruction ($\pi_2$ program) ppc in
387  let ppc' := sigma program ppc in
388  let newppc' := sigma program newppc in
389  let instructions' :=
390    expand_pseudo_instruction lk_labels lk_dlabels ppc' expansion pi in
391  let fetched := $\lambda$instr. fetch_many code_memory newppc' ppc' instr in
392    $\exists$instrs. Some ? instrs = instructions' $\wedge$ fetched instrs.
393\end{lstlisting}
394Intuitively, we may read \texttt{fetch\_assembly\_pseudo2} as follows.
395Suppose we are able to successfully assemble an assembly program using \texttt{assembly} and produce a code memory, \texttt{code\_memory}.
396Then there exists some list of machine instructions equal to the expansion of a pseudoinstruction and the number of machine instructions that need to be fetched is equal to the number of machine instructions that the pseudoinstruction was expanded into.
397
398However, this property is \emph{not} strong enough to establish that the semantics of an assembly program has been preserved by the assembly process.
399In particular, \texttt{fetch\_assembly\_pseudo2} says nothing about how
400
401An \texttt{internal\_pseudo\_address\_map} positions in the memory of a \texttt{PseudoStatus} with a physical memory address.
402\begin{lstlisting}
403definition internal_pseudo_address_map := list (BitVector 8).
404\end{lstlisting}
405We use \texttt{internal\_pseudo\_address\_map}s to convert the lower internal RAM of a \texttt{PseudoStatus} into the lower internal RAM of a \texttt{Status}.
406Notice, the MCS-51's internal RAM is addressed with a 7-bit `byte'.
407% dpm: ugly English, fix
408The whole of the internal RAM space is addressed with bytes: the first bit is used to distinguish between the programmer addressing low and high internal memory.
409\begin{lstlisting}
410axiom low_internal_ram_of_pseudo_low_internal_ram:
411  internal_pseudo_address_map $\rightarrow$ BitVectorTrie Byte 7 $\rightarrow$ BitVectorTrie Byte 7.
412\end{lstlisting}
413A similar axiom exists for high internal RAM.
414
415Next, we are able to translate \texttt{PseudoStatus} records into \texttt{Status} records using \texttt{status\_of\_pseudo\_status}.
416Translating a \texttt{PseudoStatus}'s code memory requires we expand pseudoinstructions and then assemble to obtain a trie of bytes.
417This can fail, as mentioned, in a limited number of situations, related to improper use of labels in an assembly program.
418However, it is possible to `tighten' the type of \texttt{status\_of\_pseudo\_status}, removing the option type, by using the fact that if any `good policy' exists, assembly will never fail.
419\begin{lstlisting}
420definition status_of_pseudo_status:
421 internal_pseudo_address_map → PseudoStatus → option Status
422\end{lstlisting}
423After fetching an assembly instruction we must update any \texttt{internal\_pseudo\hyp{}\_address\_map}s that may be laying around.
424This is done with the following function:
425\begin{lstlisting}
426definition next_internal_pseudo_address_map: internal_pseudo_address_map
427  $\rightarrow$ PseudoStatus $\rightarrow$ option internal_pseudo_address_map
428\end{lstlisting}
429Finally, we are able to state and prove our main theorem.
430This relates the execution of a single assembly instruction and the execution of (possibly) many machine code instructions.
431That is, the assembly process preserves the semantics of an assembly program, as it is translated into machine code:
432\begin{lstlisting}
433theorem main_thm:
434  ∀M,M',ps,s,s''.
435    next_internal_pseudo_address_map M ps = Some $\ldots$ M' $\rightarrow$
436      status_of_pseudo_status M ps = Some $\ldots$ s $\rightarrow$
437        status_of_pseudo_status M'
438          (execute_1_pseudo_instruction
439            (ticks_of (code_memory $\ldots$ ps)) ps) = Some $\ldots$ s'' $\rightarrow$
440              $\exists$n. execute n s = s''.
441\end{lstlisting}
442The statement can be given an intuitive reading as follows.
443Suppose our \texttt{PseudoStatus}, \texttt{ps}, can be successfully converted into a \texttt{Status}, \texttt{s}.
444Suppose further that, after executing a single assembly instruction and converting the resulting \texttt{PseudoStatus} into a \texttt{Status}, we obtain \texttt{s''}, being careful to track the number of ticks executed.
445Then, there exists some number \texttt{n}, so that executing \texttt{n} machine code instructions in \texttt{Status} \texttt{s} gives us \texttt{Status} \texttt{s''}.
446Theorem \texttt{main\_thm} establishes the correctness of the assembly process.
447
448% ---------------------------------------------------------------------------- %
449% SECTION                                                                      %
450% ---------------------------------------------------------------------------- %
451\section{Conclusions}
452\label{sect.conclusions}
453
454We have proved the total correctness of an assembler for MCS-51 assembly language.
455In particular, our assembly language featured labels, arbitrary conditional and unconditional jumps to labels, global data and instructions for moving this data into the MCS-51's single 16-bit register.
456Expanding these pseudoinstructions into machine code instructions is not trivial, and the proof that the assembly process is `correct', in that the semantics of an assembly program are not changed is complex.
457
458The formalisation is a key component of the CerCo project, which aims to produce a verified concrete complexity preserving compiler for a large subset of the C programming language.
459The verified assembler, complete with the underlying formalisation of the semantics of MCS-51 machine code (described fully in~\cite{mulligan:executable:2011}), will form the bedrock layer upon which the rest of the CerCo project will build its verified compiler platform.
460
461Aside from their application in verified compiler projects such as CerCo, verified assemblers such as ours could also be applied to the verification of operating system kernels.
462Of particular note is the verified seL4 kernel~\cite{klein:sel4:2009,klein:sel4:2010}.
463This verification explicitly assumes the existence of, amongst other things, a trustworthy assembler and compiler.
464
465\paragraph{Use of dependent types and Russell}
466Our formalisation makes sparing use of dependent types.
467In certain datastructures, such as tries and vectors, they are used to guarantee invariants.
468However, we have currently shyed away from making extensive use of dependent types and inductive predicates in the proof of correctness for the assembler itself.
469This is because complex dependent types and inductive predicates tend not to co\"operate particularly well with tactics such as inversion.
470
471However, there are certain cases where the use of dependent types is unavoidable.
472For instance, when proving that the \texttt{build\_maps} function is correct, a function that collates the cost and data labels of an assembly program into map datastructures.
473In cases such as these we make use of Matita's implementation of Russell~\cite{sozeau:subset:2006}.
474In Matita, Russell may be implemented with two coercions and some notational sugaring.
475
476\subsection{Related work}
477\label{subsect.related.work}
478
479% piton
480We are not the first to consider the total correctness of an assembler for a non-trivial assembly language.
481Perhaps the most impressive piece of work in this domain is the Piton stack~\cite{moore:piton:1996,moore:grand:2005}.
482This was a stack of verified components, written and verified in ACL2, ranging from a proprietary FM9001 microprocessor verified at the gate level, to assemblers and compilers for two high-level languages---a dialect of Lisp and $\mu$Gypsy~\cite{moore:grand:2005}.
483%dpm more: weirich paper?
484
485% jinja
486Klein and Nipkow consider a Java-like programming language, Jinja~\cite{klein:machine:2006,klein:machine:2010}.
487They provide a compiler, virtual machine and operational semantics for the programming language and virtual machine, and prove that their compiler is semantics and type preserving.
488
489Finally, mention should be made of CompCert~\cite{compcert:2011,blazy:formal:2006,leroy:formal:2009,leroy:formally:2009}, another verified compiler project related to CerCo.
490As previously mentioned, CompCert considers only extensional correctness of the compiler, and not intensional correctness, which CerCo focusses on.
491However, CerCo also extends CompCert in other ways.
492Namely, the CompCert verified compilation chain terminates at the PowerPC or ARM assembly level, and takes for granted the existence of a trustworthy assembler.
493CerCo chooses to go further, by considering a verified compilation chain all the way down to the machine code level.
494In essence, the work presented in this publication is one part of CerCo's extension over CompCert.
495
496\bibliography{cpp-2011.bib}
497
498\end{document}\renewcommand{\verb}{\lstinline}
499\def\lstlanguagefiles{lst-grafite.tex}
500\lstset{language=Grafite}
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