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1\documentclass[twocolumn,draft]{svjour3}
2\usepackage{algpseudocode}
3%\usepackage{algorithmicx}
4\usepackage{alltt}
5\usepackage{amsfonts}
6\usepackage{amsmath}
7\usepackage[british]{babel}
8\usepackage{caption}
9\usepackage{hyperref}
10\usepackage[utf8]{inputenc}
11\usepackage{listings}
12\usepackage{subcaption}
13
14\renewcommand{\verb}{\lstinline}
15\def\lstlanguagefiles{lst-grafite.tex}
16\lstset{language=Grafite}
17
18\begin{document}
19
20\title{On the proof of correctness of a verified optimising assembler
21\thanks{Research supported by the CerCo project, within the Future and Emerging Technologies (FET) programme of the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 243881}}
22\author{Jaap Boender \and Dominic P. Mulligan \and Claudio Sacerdoti Coen}
23\institute{Department of Computer Science, University of Middlesex \and Computer Laboratory, University of Cambridge \and Dipartimento di Informatica, University of Bologna}
24
25\maketitle
26
27\begin{abstract}
28Optimising assemblers face the `branch displacement' or `jump encoding' problem, i.e. how best to choose between concrete machine code jump instructions --- typically of differing instruction and offset sizes --- when expanding pseudo-instructions.
29Ideally, an optimising assembler would choose the set of jump expansions that minimises the size of the resulting machine code program, a task that is provably \textsc{np}-hard.
30
31As part of CerCo (`Certified Complexity') --- an \textsc{eu}-funded project to develop a verified concrete complexity preserving compiler for a large subset of the C programming language --- we have implemented and proved correct an assembler within an interactive theorem prover.
32Our assembler targets the instruction set of a typical micro-controller, the Intel \textsc{mcs}-51 series.
33As is common in embedded systems development, this micro-controller has a paucity of available code memory and therefore we face an additional pressure in reducing the size of any assembled machine code program.
34Out of necessity, then, our assembler implements an algorithm for solving the branch displacement problem, and we must prove that this algorithm is correct.
35
36We discuss wider problems associated with proving an optimising assembler correct, discuss possible solutions to those problems, and detail our chosen solutions and their proofs of correctness.
37
38\keywords{formal verification, interactive theorem proving, assembler, branch displacement optimisation}
39\end{abstract}
40
41\section{Introduction}
42
43The problem of branch displacement optimisation, also known as jump encoding, is
44a well-known problem in assembler design~\cite{Hyde2006}. Its origin lies in the
45fact that in many architecture sets, the encoding (and therefore size) of some
46instructions depends on the distance to their operand (the instruction 'span').
47The branch displacement optimisation problem consists of encoding these
48span-dependent instructions in such a way that the resulting program is as
49small as possible.
50
51This problem is the subject of the present paper. After introducing the problem
52in more detail, we will discuss the solutions used by other compilers, present
53the algorithm we use in the CerCo assembler, and discuss its verification,
54that is the proofs of termination and correctness using the Matita proof
55assistant~\cite{Asperti2007}.
56
57Formulating the final statement of correctness and finding the loop invariants
58have been non-trivial tasks and are, indeed, the main contribution of this
59paper. It has required considerable care and fine-tuning to formulate not
60only the minimal statement required for the ulterior proof of correctness of
61the assembler, but also the minimal set of invariants needed for the proof
62of correctness of the algorithm.
63
64The research presented in this paper has been executed within the CerCo project
65which aims at formally verifying a C compiler with cost annotations. The
66target architecture for this project is the MCS-51, whose instruction set
67contains span-dependent instructions. Furthermore, its maximum addressable
68memory size is very small (64 Kb), which makes it important to generate
69programs that are as small as possible. With this optimisation, however, comes increased complexity and hence increased possibility for error. We must make sure that the branch instructions are encoded correctly, otherwise the assembled program will behave unpredictably.
70
71All Matita files related to this development can be found on the CerCo
72website, \url{http://cerco.cs.unibo.it}. The specific part that contains the
73branch displacement algorithm is in the {\tt ASM} subdirectory, in the files
74{\tt PolicyFront.ma}, {\tt PolicyStep.ma} and {\tt Policy.ma}.
75
76\section{The branch displacement optimisation problem}
77
78In most modern instruction sets that have them, the only span-dependent
79instructions are branch instructions. Taking the ubiquitous x86-64 instruction
80set as an example, we find that it contains eleven different forms of the
81unconditional branch instruction, all with different ranges, instruction sizes
82and semantics (only six are valid in 64-bit mode, for example). Some examples
83are shown in Figure~\ref{f:x86jumps} (see also~\cite{IntelDev}).
84
85\begin{figure}[h]
86\begin{center}
87\begin{tabular}{|l|l|l|}
88\hline
89Instruction & Size (bytes) & Displacement range \\
90\hline
91Short jump & 2 & -128 to 127 bytes \\
92Relative near jump & 5 & $-2^{32}$ to $2^{32}-1$ bytes \\
93Absolute near jump & 6 & one segment (64-bit address) \\
94Far jump & 8 & entire memory (indirect jump) \\
95\hline
96\end{tabular}
97\end{center}
98\caption{List of x86 branch instructions}
99\label{f:x86jumps}
100\end{figure}
101
102The chosen target architecture of the CerCo project is the Intel MCS-51, which
103features three types of branch instructions (or jump instructions; the two terms
104are used interchangeably), as shown in Figure~\ref{f:mcs51jumps}.
105
106\begin{figure}[h]
107\begin{center}
108\begin{tabular}{|l|l|l|l|}
109\hline
110Instruction & Size    & Execution time & Displacement range \\
111            & (bytes) & (cycles) & \\
112\hline
113SJMP (`short jump') & 2 & 2 & -128 to 127 bytes \\
114AJMP (`absolute jump') & 2 & 2 & one segment (11-bit address) \\
115LJMP (`long jump') & 3 & 3 & entire memory \\
116\hline
117\end{tabular}
118\end{center}
119\caption{List of MCS-51 branch instructions}
120\label{f:mcs51jumps}
121\end{figure}
122
123Conditional branch instructions are only available in short form, which
124means that a conditional branch outside the short address range has to be
125encoded using three branch instructions (for instructions whose logical
126negation is available, it can be done with two branch instructions, but for
127some instructions this is not the case). The call instruction is
128only available in absolute and long forms.
129
130Note that even though the MCS-51 architecture is much less advanced and much
131simpler than the x86-64 architecture, the basic types of branch instruction
132remain the same: a short jump with a limited range, an intra-segment jump and a
133jump that can reach the entire available memory.
134 
135Generally, in code fed to the assembler as input, the only
136difference between branch instructions is semantics, not span. This
137means that a distinction is made between an unconditional branch and the
138several kinds of conditional branch, but not between their short, absolute or
139long variants.
140
141The algorithm used by the assembler to encode these branch instructions into
142the different machine instructions is known as the {\em branch displacement
143algorithm}. The optimisation problem consists of finding as small an encoding as
144possible, thus minimising program length and execution time.
145
146Similar problems, e.g. the branch displacement optimisation problem for other
147architectures, are known to be NP-complete~\cite{Robertson1979,Szymanski1978},
148which could make finding an optimal solution very time-consuming.
149
150The canonical solution, as shown by Szymanski~\cite{Szymanski1978} or more
151recently by Dickson~\cite{Dickson2008} for the x86 instruction set, is to use a
152fixed point algorithm that starts with the shortest possible encoding (all
153branch instruction encoded as short jumps, which is likely not a correct
154solution) and then iterates over the source to re-encode those branch
155instructions whose target is outside their range.
156
157\subsection*{Adding absolute jumps}
158
159In both papers mentioned above, the encoding of a jump is only dependent on the
160distance between the jump and its target: below a certain value a short jump
161can be used; above this value the jump must be encoded as a long jump.
162
163Here, termination of the smallest fixed point algorithm is easy to prove. All
164branch instructions start out encoded as short jumps, which means that the
165distance between any branch instruction and its target is as short as possible
166(all the intervening jumps are short).
167If, in this situation, there is a branch instruction $b$ whose span is not
168within the range for a short jump, we can be sure that we can never reach a
169situation where the span of $j$ is so small that it can be encoded as a short
170jump. This argument continues to hold throughout the subsequent iterations of
171the algorithm: short jumps can change into long jumps, but not \emph{vice versa},
172as spans only increase. Hence, the algorithm either terminates early when a fixed
173point is reached or when all short jumps have been changed into long jumps.
174
175Also, we can be certain that we have reached an optimal solution: a short jump
176is only changed into a long jump if it is absolutely necessary.
177
178However, neither of these claims (termination nor optimality) hold when we add
179the absolute jump. With absolute jumps, the encoding of a branch
180instruction no longer depends only on the distance between the branch
181instruction and its target. An absolute jump is possible when instruction and
182target are in the same segment (for the MCS-51, this means that the first 5
183bytes of their addresses have to be equal). It is therefore entirely possible
184for two branch instructions with the same span to be encoded in different ways
185(absolute if the branch instruction and its target are in the same segment,
186long if this is not the case).
187
188\begin{figure}[t]
189\begin{subfigure}[b]{.45\linewidth}
190\small
191\begin{alltt}
192    jmp X
193    \ldots
194L\(\sb{0}\): \ldots
195% Start of new segment if
196% jmp X is encoded as short
197    \ldots
198    jmp L\(\sb{0}\)
199\end{alltt}
200\caption{Example of a program where a long jump becomes absolute}
201\label{f:term_example}
202\end{subfigure}
203\hfill
204\begin{subfigure}[b]{.45\linewidth}
205\small
206\begin{alltt}
207L\(\sb{0}\): jmp X
208X:  \ldots
209    \ldots
210L\(\sb{1}\): \ldots
211% Start of new segment if
212% jmp X is encoded as short
213    \ldots
214    jmp L\(\sb{1}\)
215    \ldots
216    jmp L\(\sb{1}\)
217    \ldots
218    jmp L\(\sb{1}\) 
219    \ldots
220\end{alltt}
221\caption{Example of a program where the fixed-point algorithm is not optimal}
222\label{f:opt_example}
223\end{subfigure}
224\end{figure}
225
226This invalidates our earlier termination argument: a branch instruction, once encoded
227as a long jump, can be re-encoded during a later iteration as an absolute jump.
228Consider the program shown in Figure~\ref{f:term_example}. At the start of the
229first iteration, both the branch to {\tt X} and the branch to $\mathtt{L}_{0}$
230are encoded as small jumps. Let us assume that in this case, the placement of
231$\mathtt{L}_{0}$ and the branch to it are such that $\mathtt{L}_{0}$ is just
232outside the segment that contains this branch. Let us also assume that the
233distance between $\mathtt{L}_{0}$ and the branch to it is too large for the
234branch instruction to be encoded as a short jump.
235
236All this means that in the second iteration, the branch to $\mathtt{L}_{0}$ will
237be encoded as a long jump. If we assume that the branch to {\tt X} is encoded as
238a long jump as well, the size of the branch instruction will increase and
239$\mathtt{L}_{0}$ will be `propelled' into the same segment as its branch
240instruction, because every subsequent instruction will move one byte forward.
241Hence, in the third iteration, the branch to $\mathtt{L}_{0}$ can be encoded as
242an absolute jump. At first glance, there is nothing that prevents us from
243constructing a configuration where two branch instructions interact in such a
244way as to iterate indefinitely between long and absolute encodings.
245
246This situation mirrors the explanation by Szymanski~\cite{Szymanski1978} of why
247the branch displacement optimisation problem is NP-complete. In this explanation,
248a condition for NP-completeness is the fact that programs be allowed to contain
249{\em pathological} jumps. These are branch instructions that can normally not be
250encoded as a short(er) jump, but gain this property when some other branch
251instructions are encoded as a long(er) jump. This is exactly what happens in
252Figure~\ref{f:term_example}. By encoding the first branch instruction as a long
253jump, another branch instruction switches from long to absolute (which is
254shorter).
255
256In addition, our previous optimality argument no longer holds. Consider the
257program shown in Figure~\ref{f:opt_example}. Suppose that the distance between
258$\mathtt{L}_{0}$ and $\mathtt{L}_{1}$ is such that if {\tt jmp X} is encoded
259as a short jump, there is a segment border just after $\mathtt{L}_{1}$. Let
260us also assume that all three branches to $\mathtt{L}_{1}$ are in the same
261segment, but far enough away from $\mathtt{L}_{1}$ that they cannot be encoded
262as short jumps.
263
264Then, if {\tt jmp X} were to be encoded as a short jump, which is clearly
265possible, all of the branches to $\mathtt{L}_{1}$ would have to be encoded as
266long jumps. However, if {\tt jmp X} were to be encoded as a long jump, and
267therefore increase in size, $\mathtt{L}_{1}$ would be `propelled' across the
268segment border, so that the three branches to $\mathtt{L}_{1}$ could be encoded
269as absolute jumps. Depending on the relative sizes of long and absolute jumps,
270this solution might actually be smaller than the one reached by the smallest
271fixed point algorithm.
272
273\section{Our algorithm}
274
275\subsection{Design decisions}
276
277Given the NP-completeness of the problem, finding optimal solutions
278(using, for example, a constraint solver) can potentially be very costly.
279
280The SDCC compiler~\cite{SDCC2011}, which has a backend targeting the MCS-51
281instruction set, simply encodes every branch instruction as a long jump
282without taking the distance into account. While certainly correct (the long
283jump can reach any destination in memory) and a very fast solution to compute,
284it results in a less than optimal solution in terms of output size and
285execution time.
286
287On the other hand, the {\tt gcc} compiler suite, while compiling
288C on the x86 architecture, uses a greatest fix point algorithm. In other words,
289it starts with all branch instructions encoded as the largest jumps
290available, and then tries to reduce the size of branch instructions as much as
291possible.
292
293Such an algorithm has the advantage that any intermediate result it returns
294is correct: the solution where every branch instruction is encoded as a large
295jump is always possible, and the algorithm only reduces those branch
296instructions whose destination address is in range for a shorter jump.
297The algorithm can thus be stopped after a determined number of steps without
298sacrificing correctness.
299
300The result, however, is not necessarily optimal. Even if the algorithm is run
301until it terminates naturally, the fixed point reached is the {\em greatest}
302fixed point, not the least fixed point. Furthermore, {\tt gcc} (at least for
303the x86 architecture) only uses short and long jumps. This makes the algorithm
304more efficient, as shown in the previous section, but also results in a less
305optimal solution.
306
307In the CerCo assembler, we opted at first for a least fixed point algorithm,
308taking absolute jumps into account.
309
310Here, we ran into a problem with proving termination, as explained in the
311previous section: if we only take short and long jumps into account, the jump
312encoding can only switch from short to long, but never in the other direction.
313When we add absolute jumps, however, it is theoretically possible for a branch
314instruction to switch from absolute to long and back, as previously explained.
315Proving termination then becomes difficult, because there is nothing that
316precludes a branch instruction from oscillating back and forth between absolute
317and long jumps indefinitely.
318
319To keep the algorithm in the same complexity class and more easily
320prove termination, we decided to explicitly enforce the `branch instructions
321must always grow longer' requirement: if a branch instruction is encoded as a
322long jump in one iteration, it will also be encoded as a long jump in all the
323following iterations. Therefore the encoding of any branch instruction
324can change at most two times: once from short to absolute (or long), and once
325from absolute to long.
326
327There is one complicating factor. Suppose that a branch instruction is encoded
328in step $n$ as an absolute jump, but in step $n+1$ it is determined that
329(because of changes elsewhere) it can now be encoded as a short jump. Due to
330the requirement that the branch instructions must always grow longer,
331the branch encoding will be encoded as an absolute jump in step
332$n+1$ as well.
333
334This is not necessarily correct. A branch instruction that can be
335encoded as a short jump cannot always also be encoded as an absolute jump, as a
336short jump can bridge segments, whereas an absolute jump cannot. Therefore,
337in this situation we have decided to encode the branch instruction as a long
338jump, which is always correct.
339
340The resulting algorithm, therefore, will not return the least fixed point, as
341it might have too many long jumps. However, it is still better than the
342algorithms from SDCC and {\tt gcc}, since even in the worst case, it will still
343return a smaller or equal solution.
344
345Experimenting with our algorithm on the test suite of C programs included with
346gcc 2.3.3 has shown that on average, about 25 percent of jumps are encoded as short or absolute jumps by the algorithm. As not all instructions are jumps, this does not make for a large reduction in size, but it can make for a reduction in execution time: if jumps
347are executed multiple times, for example in loops, the fact that short jumps take less cycles to execute than long jumps can have great effect.
348
349As for complexity, there are at most $2n$ iterations, where $n$ is the number of
350branch instructions. Practical tests within the CerCo project on small to
351medium pieces of code have shown that in almost all cases, a fixed point is
352reached in 3 passes. Only in one case did the algorithm need 4. This is not surprising: after all, the difference between short/absolute and
353long jumps is only one byte (three for conditional jumps). For a change from
354short/absolute to long to have an effect on other jumps is therefore relatively
355uncommon, which explains why a fixed point is reached so quickly.
356
357\subsection{The algorithm in detail}
358
359The branch displacement algorithm forms part of the translation from
360pseudocode to assembler. More specifically, it is used by the function that
361translates pseudo-addresses (natural numbers indicating the position of the
362instruction in the program) to actual addresses in memory. Note that in pseudocode, all instructions are of size 1.
363
364Our original intention was to have two different functions, one function
365$\mathtt{policy}: \mathbb{N} \rightarrow \{\mathtt{short\_jump},
366\mathtt{absolute\_jump}, \mathtt{long\_jump}\}$ to associate jumps to their
367intended encoding, and a function $\sigma: \mathbb{N} \rightarrow
368\mathtt{Word}$ to associate pseudo-addresses to machine addresses. $\sigma$
369would use $\mathtt{policy}$ to determine the size of jump instructions. This turned out to be suboptimal from the algorithmic point of view and
370impossible to prove correct.
371
372From the algorithmic point of view, in order to create the $\mathtt{policy}$
373function, we must necessarily have a translation from pseudo-addresses
374to machine addresses (i.e. a $\sigma$ function): in order to judge the distance
375between a jump and its destination, we must know their memory locations.
376Conversely, in order to create the $\sigma$ function, we need to have the
377$\mathtt{policy}$ function, otherwise we do not know the sizes of the jump
378instructions in the program.
379
380Much the same problem appears when we try to prove the algorithm correct: the
381correctness of $\mathtt{policy}$ depends on the correctness of $\sigma$, and
382the correctness of $\sigma$ depends on the correctness of $\mathtt{policy}$.
383
384We solved this problem by integrating the $\mathtt{policy}$ and $\sigma$
385algorithms. We now have a function
386$\sigma: \mathbb{N} \rightarrow \mathtt{Word} \times \mathtt{bool}$ which
387associates a pseudo-address to a machine address. The boolean denotes a forced
388long jump; as noted in the previous section, if during the fixed point
389computation an absolute jump changes to be potentially re-encoded as a short
390jump, the result is actually a long jump. It might therefore be the case that
391jumps are encoded as long jumps without this actually being necessary, and this
392information needs to be passed to the code generating function.
393
394The assembler function encodes the jumps by checking the distance between
395source and destination according to $\sigma$, so it could select an absolute
396jump in a situation where there should be a long jump. The boolean is there
397to prevent this from happening by indicating the locations where a long jump
398should be encoded, even if a shorter jump is possible. This has no effect on
399correctness, since a long jump is applicable in any situation.
400
401\begin{figure}[t]
402\small
403\begin{algorithmic}
404\Function{f}{$labels$,$old\_sigma$,$instr$,$ppc$,$acc$}
405  \State $\langle added, pc, sigma \rangle \gets acc$
406  \If {$instr$ is a backward jump to $j$}
407    \State $length \gets \mathrm{jump\_size}(pc,sigma_1(labels(j)))$
408    \Comment compute jump distance
409  \ElsIf {$instr$ is a forward jump to $j$}
410    \State $length \gets \mathrm{jump\_size}(pc,old\_sigma_1(labels(j))+added)$
411  \EndIf
412  \State $old\_length \gets \mathrm{old\_sigma_1}(ppc)$
413  \State $new\_length \gets \mathrm{max}(old\_length, length)$
414  \Comment length must never decrease
415  \State $old\_size \gets \mathrm{old\_sigma_2}(ppc)$
416  \State $new\_size \gets \mathrm{instruction\_size}(instr,new\_length)$
417  \Comment compute size in bytes
418  \State $new\_added \gets added+(new\_size-old\_size)$
419  \Comment keep track of total added bytes
420  \State $new\_sigma \gets old\_sigma$
421  \State $new\_sigma_1(ppc+1) \gets pc+new\_size$
422  \State $new\_sigma_2(ppc) \gets new\_length$
423  \Comment update $\sigma$ \\
424  \Return $\langle new\_added, pc+new\_size, new\_sigma \rangle$
425\EndFunction
426\end{algorithmic}
427\caption{The heart of the algorithm}
428\label{f:jump_expansion_step}
429\end{figure}
430
431The algorithm, shown in Figure~\ref{f:jump_expansion_step}, works by folding the
432function {\sc f} over the entire program, thus gradually constructing $sigma$.
433This constitutes one step in the fixed point calculation; successive steps
434repeat the fold until a fixed point is reached. We have abstracted away the case where an instruction is not a jump, since the size of these instructions is constant.
435
436Parameters of the function {\sc f} are:
437\begin{itemize}
438  \item a function $labels$ that associates a label to its pseudo-address;
439  \item $old\_sigma$, the $\sigma$ function returned by the previous
440    iteration of the fixed point calculation;
441  \item $instr$, the instruction currently under consideration;
442  \item $ppc$, the pseudo-address of $instr$;
443  \item $acc$, the fold accumulator, which contains $added$ (the number of
444  bytes added to the program size with respect to the previous iteration), $pc$
445  (the highest memory address reached so far), and of course $sigma$, the
446    $\sigma$ function under construction.
447\end{itemize}
448The first two are parameters that remain the same through one iteration, the
449final three are standard parameters for a fold function (including $ppc$,
450which is simply the number of instructions of the program already processed).
451
452The $\sigma$ functions used by {\sc f} are not of the same type as the final
453$\sigma$ function: they are of type
454$\sigma: \mathbb{N} \rightarrow \mathbb{N} \times \{\mathtt{short\_jump},
455\mathtt{absolute\_jump},\mathtt{long\_jump}\}$; a function that associates a
456pseudo-address with a memory address and a jump length. We do this to
457ease the comparison of jump lengths between iterations. In the algorithm,
458we use the notation $sigma_1(x)$ to denote the memory address corresponding to
459$x$, and $sigma_2(x)$ for the jump length corresponding to $x$.
460
461Note that the $\sigma$ function used for label lookup varies depending on
462whether the label is behind our current position or ahead of it. For
463backward branches, where the label is behind our current position, we can use
464$sigma$ for lookup, since its memory address is already known. However, for
465forward branches, the memory address of the address of the label is not yet
466known, so we must use $old\_sigma$.
467
468We cannot use $old\_sigma$ without change: it might be the case that we have
469already increased the size of some branch instructions before, making the
470program longer and moving every instruction forward. We must compensate for this
471by adding the size increase of the program to the label's memory address
472according to $old\_sigma$, so that branch instruction spans do not get
473compromised.
474
475%Note also that we add the pc to $sigma$ at location $ppc+1$, whereas we add the
476%jump length at location $ppc$. We do this so that $sigma(ppc)$ will always
477%return a pair with the start address of the instruction at $ppc$ and the
478%length of its branch instruction (if any); the end address of the program can
479%be found at $sigma(n+1)$, where $n$ is the number of instructions in the
480%program.
481
482\section{The proof}
483
484In this section, we present the correctness proof for the algorithm in more
485detail. The main correctness statement is shown, slightly simplified, in~Figure~\ref{statement}.
486%
487\begin{figure}[t]
488\small
489\begin{alignat*}{6}
490\mathtt{sigma}&\omit\rlap{$\mathtt{\_policy\_specification} \equiv
491\lambda program.\lambda sigma.$} \notag\\
492  & \omit\rlap{$sigma\ 0 = 0\ \wedge$} \notag\\
493  & \mathbf{let}\ & & \omit\rlap{$instr\_list \equiv code\ program\ \mathbf{in}$} \notag\\
494  &&& \omit\rlap{$\forall ppc.ppc < |instr\_list| \rightarrow$} \notag\\
495  &&& \mathbf{let}\ && pc \equiv sigma\ ppc\ \mathbf{in} \notag\\
496  &&& \mathbf{let}\ && instruction \equiv \mathtt{fetch\_pseudo\_instruction}\ instr\_list\ ppc\ \mathbf{in} \notag\\
497  &&& \mathbf{let}\ && next\_pc \equiv sigma\ (ppc+1)\ \mathbf{in}\notag\\
498  &&&&& next\_pc = pc + \mathtt{instruction\_size}\ sigma\ ppc\ instruction\ \wedge\notag\\
499  &&&&& (pc + \mathtt{instruction\_size}\ sigma\ ppc\ instruction < 2^{16}\ \vee\notag\\
500  &&&&& (\forall ppc'.ppc' < |instr\_list| \rightarrow ppc < ppc' \rightarrow \notag\\
501  &&&&& \mathbf{let}\ instruction' \equiv \mathtt{fetch\_pseudo\_instruction}\ instr\_list\ ppc'\ \mathbf{in} \notag\\
502  &&&&&\ \mathtt{instruction\_size}\ sigma\ ppc'\ instruction' = 0)\ \wedge \notag\\
503  &&&&& pc + \mathtt{instruction\_size}\ sigma\ ppc\ instruction = 2^{16})
504\end{alignat*}
505\caption{Main correctness statement\label{statement}}
506\label{sigmapolspec}
507\end{figure}
508%
509Informally, this means that when fetching a pseudo-instruction at $ppc$, the
510translation by $\sigma$ of $ppc+1$ is the same as $\sigma(ppc)$ plus the size
511of the instruction at $ppc$.  That is, an instruction is placed consecutively
512after the previous one, and there are no overlaps. The rest of the statement deals with memory size: either the next instruction fits within memory ($next\_pc < 2^{16}$) or it ends exactly at the limit memory,
513in which case it must be the last translated instruction in the program (enforced by specfiying that the size of all subsequent instructions is 0: there may be comments or cost annotations that are not translated).
514
515Finally, we enforce that the program starts at address 0, i.e. $\sigma(0) = 0$. It may seem strange that we do not explicitly include a safety property stating that every jump instruction is of the right type with respect to its target (akin to the lemma from Figure~\ref{sigmasafe}), but this is not necessary. The distance is recalculated according to the instruction addresses from $\sigma$, which implicitly expresses safety.
516
517Since our computation is a least fixed point computation, we must prove
518termination in order to prove correctness: if the algorithm is halted after
519a number of steps without reaching a fixed point, the solution is not
520guaranteed to be correct. More specifically, branch instructions might be
521encoded which do not coincide with the span between their location and their
522destination.
523
524Proof of termination rests on the fact that the encoding of branch
525instructions can only grow larger, which means that we must reach a fixed point
526after at most $2n$ iterations, with $n$ the number of branch instructions in
527the program. This worst case is reached if at every iteration, we change the
528encoding of exactly one branch instruction; since the encoding of any branch
529instruction can change first from short to absolute, and then to long, there
530can be at most $2n$ changes.
531
532%The proof has been carried out using the ``Russell'' style from~\cite{Sozeau2006}.
533%We have proven some invariants of the {\sc f} function from the previous
534%section; these invariants are then used to prove properties that hold for every
535%iteration of the fixed point computation; and finally, we can prove some
536%properties of the fixed point.
537
538\subsection{Fold invariants}
539
540In this section, we present the invariants that hold during the fold of {\sc f}
541over the program. These will be used later on to prove the properties of the
542iteration. During the fixed point computation, the $\sigma$ function is
543implemented as a trie for ease of access; computing $\sigma(x)$ is achieved by
544looking up the value of $x$ in the trie. Actually, during the fold, the value
545we pass along is a pair $\mathbb{N} \times \mathtt{ppc\_pc\_map}$. The first
546component is the number of bytes added to the program so far with respect to
547the previous iteration, and the second component, {\tt ppc\_pc\_map}, is the
548actual $\sigma$ trie (which we'll call $strie$ to avoid confusion).
549%
550{\small
551\begin{alignat*}{2}
552\mathtt{out} & \mathtt{\_of\_program\_none} \equiv \lambda prefix.\lambda strie. \notag\\
553& \forall i.i < 2^{16} \rightarrow (i > |prefix| \leftrightarrow
554 \mathtt{lookup\_opt}\ i\ (\mathtt{snd}\ strie) = \mathtt{None})
555\end{alignat*}}
556%
557The first invariant states that any pseudo-address not yet examined is not
558present in the lookup trie.
559%
560{\small
561\begin{alignat*}{2}
562\mathtt{not} & \mathtt{\_jump\_default} \equiv \lambda prefix.\lambda strie.\forall i.i < |prefix| \rightarrow\notag\\
563& \neg\mathtt{is\_jump}\ (\mathtt{nth}\ i\ prefix) \rightarrow \mathtt{lookup}\ i\ (\mathtt{snd}\ strie) = \mathtt{short\_jump}
564\end{alignat*}}
565%
566This invariant states that when we try to look up the jump length of a
567pseudo-address where there is no branch instruction, we will get the default
568value, a short jump.
569%
570{\small
571\begin{alignat*}{4}
572\mathtt{jump} & \mathtt{\_increase} \equiv \lambda pc.\lambda op.\lambda p.\forall i.i < |prefix| \rightarrow \notag\\ 
573& \mathbf{let}\  oj \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ op)\ \mathbf{in} \notag\\
574& \mathbf{let}\ j \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ p)\ \mathbf{in}\ \mathtt{jmpleq}\ oj\ j
575\end{alignat*}}
576%
577This invariant states that between iterations (with $op$ being the previous
578iteration, and $p$ the current one), jump lengths either remain equal or
579increase. It is needed for proving termination.
580%
581\begin{figure}[h]
582\small
583\begin{alignat*}{6}
584\mathtt{sigma} & \omit\rlap{$\mathtt{\_compact\_unsafe} \equiv \lambda prefix.\lambda strie.\forall n.n < |prefix| \rightarrow$}\notag\\
585& \mathbf{match}\ && \omit\rlap{$\mathtt{lookup\_opt}\ n\ (\mathtt{snd}\ strie)\ \mathbf{with}$}\notag\\
586&&& \omit\rlap{$\mathtt{None} \Rightarrow \mathrm{False}$} \notag\\
587&&& \omit\rlap{$\mathtt{Some}\ \langle pc, j \rangle \Rightarrow$} \notag\\
588&&& \mathbf{match}\ && \mathtt{lookup\_opt}\ (n+1)\ (\mathtt{snd}\ strie)\ \mathbf{with}\notag\\
589&&&&& \mathtt{None} \Rightarrow \mathrm{False} \notag\\
590&&&&& \mathtt{Some}\ \langle pc_1, j_1 \rangle \Rightarrow
591    pc_1 = pc + \notag\\
592&&&&& \ \ \mathtt{instruction\_size\_jmplen}\ j\ (\mathtt{nth}\ n\ prefix)
593\end{alignat*}
594\caption{Temporary safety property}
595\label{sigmacompactunsafe}
596\end{figure}
597%
598We now proceed with the safety lemmas. The lemma in
599Figure~\ref{sigmacompactunsafe} is a temporary formulation of the main
600property {\tt sigma\_policy\_specification}. Its main difference from the
601final version is that it uses {\tt instruction\_size\_jmplen} to compute the
602instruction size. This function uses $j$ to compute the span of branch
603instructions  (i.e. it uses the $\sigma$ under construction), instead
604of looking at the distance between source and destination. This is because
605$\sigma$ is still under construction; we will prove below that after the
606final iteration, {\tt sigma\_compact\_unsafe} is equivalent to the main
607property in Figure~\ref{sigmasafe} which holds at the end of the computation.
608%
609\begin{figure}[h]
610\small
611\begin{alignat*}{6}
612\mathtt{sigma} & \omit\rlap{$\mathtt{\_safe} \equiv \lambda prefix.\lambda labels.\lambda old\_strie.\lambda strie.\forall i.i < |prefix| \rightarrow$} \notag\\
613& \omit\rlap{$\forall dest\_label.\mathtt{is\_jump\_to\ (\mathtt{nth}\ i\ prefix})\ dest\_label \rightarrow$} \notag\\
614& \mathbf{let} && \omit\rlap{$\ paddr \equiv \mathtt{lookup}\ labels\ dest\_label\ \mathbf{in}$} \notag\\
615& \mathbf{let} && \omit\rlap{$\ \langle j, src, dest \rangle \equiv \mathbf{if} \ paddr\ \leq\ i\ \mathbf{then}$}\notag\\
616&&&&& \mathbf{let}\ \langle \_, j \rangle \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ strie)\ \mathbf{in} \notag\\
617&&&&& \mathbf{let}\ \langle pc\_plus\_jl, \_ \rangle \equiv \mathtt{lookup}\ (i+1)\ (\mathtt{snd}\ strie)\ \mathbf{in}\notag\\
618&&&&& \mathbf{let}\ \langle addr, \_ \rangle \equiv \mathtt{lookup}\ paddr\ (\mathtt{snd}\ strie)\ \mathbf{in}\notag\\
619&&&&& \langle j, pc\_plus\_jl, addr \rangle\notag\\
620&&&\mathbf{else} \notag\\
621&&&&&\mathbf{let}\ \langle \_, j \rangle \equiv \mathtt{lookup}\ i\ (\mathtt{snd}\ strie)\ \mathbf{in} \notag\\
622&&&&&\mathbf{let}\ \langle pc\_plus\_jl, \_ \rangle \equiv \mathtt{lookup}\ (i+1)\ (\mathtt{snd}\ old\_strie)\ \mathbf{in}\notag\\
623&&&&&\mathbf{let}\ \langle addr, \_ \rangle \equiv \mathtt{lookup}\ paddr\ (\mathtt{snd}\ old\_strie)\ \mathbf{in}\notag\\
624&&&&&\langle j, pc\_plus\_jl, addr \rangle \mathbf{in}\ \notag\\
625&&&\mathbf{match} && \ j\ \mathbf{with} \notag\\
626&&&&&\mathrm{short\_jump} \Rightarrow \mathtt{short\_jump\_valid}\ src\ dest\notag\\
627&&&&&\mathrm{absolute\_jump} \Rightarrow \mathtt{absolute\_jump\_valid}\ src\ dest\notag\\
628&&&&&\mathrm{long\_jump} \Rightarrow \mathrm{True}
629\end{alignat*}
630\caption{Safety property}
631\label{sigmasafe}
632\end{figure}
633%
634We compute the distance using the memory address of the instruction
635plus its size. This follows the behaviour of the MCS-51 microprocessor, which
636increases the program counter directly after fetching, and only then executes
637the branch instruction (by changing the program counter again).
638
639There are also some simple, properties to make sure that our policy
640remains consistent, and to keep track of whether the fixed point has been
641reached. We do not include them here in detail. Two of these properties give the values of $\sigma$ for the start and end of the program; $\sigma(0) = 0$ and $\sigma(n)$, where $n$ is the number of instructions up until now, is equal to the maximum memory address so far. There are also two properties that deal with what happens when the previous
642iteration does not change with respect to the current one. $added$ is a
643variable that keeps track of the number of bytes we have added to the program
644size by changing the encoding of branch instructions. If $added$ is 0, the program
645has not changed and vice versa.
646
647%{\small
648%\begin{align*}
649%& \mathtt{lookup}\ 0\ (\mathtt{snd}\ strie) = 0 \notag\\
650%& \mathtt{lookup}\ |prefix|\ (\mathtt{snd}\ strie) = \mathtt{fst}\ strie
651%\end{align*}}
652
653
654%{\small
655%\begin{align*}
656%& added = 0\ \rightarrow\ \mathtt{policy\_pc\_equal}\ prefix\ old\_strie\ strie \notag\\
657%& \mathtt{policy\_jump\_equal}\ prefix\ old\_strie\ strie\ \rightarrow\ added = 0
658%\end{align*}}
659
660We need to use two different formulations, because the fact that $added$ is 0
661does not guarantee that no branch instructions have changed.  For instance,
662it is possible that we have replaced a short jump with an absolute jump, which
663does not change the size of the branch instruction. Therefore {\tt policy\_pc\_equal} states that $old\_sigma_1(x) = sigma_1(x)$, whereas {\tt policy\_jump\_equal} states that $old\_sigma_2(x) = sigma_2(x)$. This formulation is sufficient to prove termination and compactness. 
664
665Proving these invariants is simple, usually by induction on the prefix length.
666
667\subsection{Iteration invariants}
668
669These are invariants that hold after the completion of an iteration. The main
670difference between these invariants and the fold invariants is that after the
671completion of the fold, we check whether the program size does not supersede
67264 Kb, the maximum memory size the MCS-51 can address. The type of an iteration therefore becomes an option type: {\tt None} in case
673the program becomes larger than 64 Kb, or $\mathtt{Some}\ \sigma$
674otherwise. We also no longer pass along the number of bytes added to the
675program size, but a boolean that indicates whether we have changed something
676during the iteration or not.
677
678If the iteration returns {\tt None}, which means that it has become too large for memory, there is an invariant that states that the previous iteration cannot
679have every branch instruction encoded as a long jump. This is needed later in the proof of termination. If the iteration returns $\mathtt{Some}\ \sigma$, the fold invariants are retained without change.
680
681Instead of using {\tt sigma\_compact\_unsafe}, we can now use the proper
682invariant:
683%
684{\small
685\begin{alignat*}{6}
686\mathtt{sigma} & \omit\rlap{$\mathtt{\_compact} \equiv \lambda program.\lambda sigma.$} \notag\\
687& \omit\rlap{$\forall n.n < |program|\ \rightarrow$} \notag\\
688& \mathbf{match}\ && \omit\rlap{$\mathtt{lookup\_opt}\ n\ (\mathtt{snd}\ sigma)\ \mathbf{with}$}\notag\\
689&&& \omit\rlap{$\mathrm{None}\ \Rightarrow\ \mathrm{False}$}\notag\\
690&&& \omit\rlap{$\mathrm{Some}\ \langle pc, j \rangle \Rightarrow$}\notag\\
691&&& \mathbf{match}\ && \mathtt{lookup\_opt}\ (n+1)\ (\mathtt{snd}\ sigma)\ \mathbf{with}\notag\\
692&&&&& \mathrm{None}\ \Rightarrow\ \mathrm{False}\notag\\
693&&&&& \mathrm{Some} \langle pc1, j1 \rangle \Rightarrow\notag\\
694&&&&& \ \ pc1 = pc + \mathtt{instruction\_size}\ n\ (\mathtt{nth}\ n\ program)
695\end{alignat*}}
696%
697This is almost the same invariant as ${\tt sigma\_compact\_unsafe}$, but differs in that it
698computes the sizes of branch instructions by looking at the distance between
699position and destination using $\sigma$. In actual use, the invariant is qualified: $\sigma$ is compact if there have
700been no changes (i.e. the boolean passed along is {\tt true}). This is to
701reflect the fact that we are doing a least fixed point computation: the result
702is only correct when we have reached the fixed point.
703
704There is another, trivial, invariant in case the iteration returns
705$\mathtt{Some}\ \sigma$: it must hold that $\mathtt{fst}\ sigma < 2^{16}$.
706We need this invariant to make sure that addresses do not overflow.
707
708The proof of {\tt nec\_plus\_ultra} goes as follows: if we return {\tt None},
709then the program size must be greater than 64 Kb. However, since the
710previous iteration did not return {\tt None} (because otherwise we would
711terminate immediately), the program size in the previous iteration must have
712been smaller than 64 Kb.
713
714Suppose that all the branch instructions in the previous iteration are
715encoded as long jumps. This means that all branch instructions in this
716iteration are long jumps as well, and therefore that both iterations are equal
717in the encoding of their branch instructions. Per the invariant, this means that
718$added = 0$, and therefore that all addresses in both iterations are equal.
719But if all addresses are equal, the program sizes must be equal too, which
720means that the program size in the current iteration must be smaller than
72164 Kb. This contradicts the earlier hypothesis, hence not all branch
722instructions in the previous iteration are encoded as long jumps.
723
724The proof of {\tt sigma\_compact} follows from {\tt sigma\_compact\_unsafe} and
725the fact that we have reached a fixed point, i.e. the previous iteration and
726the current iteration are the same. This means that the results of
727{\tt instruction\_size\_jmplen} and {\tt instruction\_size} are the same.
728
729\subsection{Final properties}
730
731These are the invariants that hold after $2n$ iterations, where $n$ is the
732program size (we use the program size for convenience; we could also use the
733number of branch instructions, but this is more complex). Here, we only
734need {\tt out\_of\_program\_none}, {\tt sigma\_compact} and the fact that
735$\sigma(0) = 0$.
736
737Termination can now be proved using the fact that there is a $k \leq 2n$, with
738$n$ the length of the program, such that iteration $k$ is equal to iteration
739$k+1$. There are two possibilities: either there is a $k < 2n$ such that this
740property holds, or every iteration up to $2n$ is different. In the latter case,
741since the only changes between the iterations can be from shorter jumps to
742longer jumps, in iteration $2n$ every branch instruction must be encoded as
743a long jump. In this case, iteration $2n$ is equal to iteration $2n+1$ and the
744fixed point is reached.
745
746\section{Conclusion}
747
748In the previous sections we have discussed the branch displacement optimisation
749problem, presented an optimised solution, and discussed the proof of
750termination and correctness for this algorithm, as formalised in Matita.
751
752The algorithm we have presented is fast and correct, but not optimal; a true
753optimal solution would need techniques like constraint solvers. While outside
754the scope of the present research, it would be interesting to see if enough
755heuristics could be found to make such a solution practical for implementing
756in an existing compiler; this would be especially useful for embedded systems,
757where it is important to have as small a solution as possible.
758
759In itself the algorithm is already useful, as it results in a smaller solution
760than the simple `every branch instruction is long' used up until now---and with
761only 64 Kb of memory, every byte counts. It also results in a smaller solution
762than the greatest fixed point algorithm that {\tt gcc} uses. It does this
763without sacrificing speed or correctness.
764
765The certification of an assembler that relies on the branch displacement
766algorithm described in this paper was presented in~\cite{lastyear}.
767The assembler computes the $\sigma$ map as described in this paper and
768then works in two passes. In the first pass it builds a map
769from instruction labels to addresses in the assembly code. In the
770second pass it iterates over the code, translating every pseudo jump
771at address $src$ to a label l associated to the assembly instruction at
772address $dst$ to a jump of the size dictated by $(\sigma\ src)$ to
773$(\sigma\ dst)$. In case of conditional jumps, the translated jump may be
774implemented with a series of instructions.
775
776The proof of correctness abstracts over the algorithm used and only relies on
777{\tt sigma\_policy\_specification} (page~\ref{sigmapolspec}). It is a variation
778of a standard 1-to-many forward simulation proof~\cite{Leroy2009}. The
779relation R between states just maps every code address $ppc$ stored in
780registers or memory to $(\sigma\ ppc)$. To identify the code addresses,
781an additional data structure is always kept together with the source
782state and is updated by the semantics. The semantics is preserved
783only for those programs whose source code operations
784$(f\ ppc_1\ \ldots\ ppc_n)$ applied to code addresses $ppc_1 \ldots ppc_n$ are
785such that $(f\ (\sigma\ ppc_1)\ ...\ (\sigma\ ppc_n) = f\ ppc_1\ ppc_n))$. For
786example, an injective $\sigma$ preserves a binary equality test f for code
787addresses, but not pointer subtraction.
788
789The main lemma (fetching simulation), which relies on\\
790{\tt sigma\_policy\_specification} and is established by structural induction
791over the source code, says that fetching an assembly instruction at
792position ppc is equal to fetching the translation of the instruction at
793position $(\sigma\ ppc)$, and that the new incremented program counter is at
794the beginning of the next instruction (compactness). The only exception is
795when the instruction fetched is placed at the end of code memory and is
796followed only by dead code. Execution simulation is trivial because of the
797restriction over well behaved programs w.r.t. sigma. The condition
798$\sigma\ 0 = 0$ is necessary because the hardware model prescribes that the
799first instruction to be executed will be at address 0. For the details
800see~\cite{lastyear}.
801
802Instead of verifying the algorithm directly, another solution to the problem
803would be to run an optimisation algorithm, and then verify the safety of the
804result using a verified validator. Such a validator would be easier to verify
805than the algorithm itself and it would also be efficient, requiring only a
806linear pass over the source code to test the specification. However, it is
807surely also interesting to formally prove that the assembler never rejects
808programs that should be accepted, i.e. that the algorithm itself is correct.
809This is the topic of the current paper.
810
811\subsection{Related work}
812
813As far as we are aware, this is the first formal discussion of the branch
814displacement optimisation algorithm.
815
816The CompCert project is another verified compiler project.
817Their backend~\cite{Leroy2009} generates assembly code for (amongst others) subsets of the
818PowerPC and x86 (32-bit) architectures. At the assembly code stage, there is
819no distinction between the span-dependent jump instructions, so a branch
820displacement optimisation algorithm is not needed.
821
822%An offshoot of the CompCert project is the CompCertTSO project, which adds
823%thread concurrency and synchronisation to the CompCert compiler~\cite{TSO2011}.
824%This compiler also generates assembly code and therefore does not include a
825%branch displacement algorithm.
826 
827%Finally, there is also the Piton stack~\cite{Moore1996}, which not only includes the
828%formal verification of a compiler, but also of the machine architecture
829%targeted by that compiler, a microprocessor called the FM9001.
830%However, this architecture does not have different
831%jump sizes (branching is simulated by assigning values to the program counter),
832%so the branch displacement problem is irrelevant.
833 
834
835
836\bibliography{biblio}
837\bibliographystyle{spbasic}
838
839\end{document}
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