source: src/ASM/TACAS2013-policy/algorithm.tex @ 3393

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1\section{Our algorithm}
2
3\subsection{Design decisions}
4
5Given the NP-completeness of the problem, finding optimal solutions
6(using, for example, a constraint solver) can potentially be very costly.
7
8The SDCC compiler~\cite{SDCC2011}, which has a backend targeting the MCS-51
9instruction set, simply encodes every branch instruction as a long jump
10without taking the distance into account. While certainly correct (the long
11jump can reach any destination in memory) and a very fast solution to compute,
12it results in a less than optimal solution in terms of output size and
13execution time.
14
15On the other hand, the {\tt gcc} compiler suite, while compiling
16C on the x86 architecture, uses a greatest fix point algorithm. In other words,
17it starts with all branch instructions encoded as the largest jumps
18available, and then tries to reduce the size of branch instructions as much as
19possible.
20
21Such an algorithm has the advantage that any intermediate result it returns
22is correct: the solution where every branch instruction is encoded as a large
23jump is always possible, and the algorithm only reduces those branch
24instructions whose destination address is in range for a shorter jump.
25The algorithm can thus be stopped after a determined number of steps without
26sacrificing correctness.
27
28The result, however, is not necessarily optimal. Even if the algorithm is run
29until it terminates naturally, the fixed point reached is the {\em greatest}
30fixed point, not the least fixed point. Furthermore, {\tt gcc} (at least for
31the x86 architecture) only uses short and long jumps. This makes the algorithm
32more efficient, as shown in the previous section, but also results in a less
33optimal solution.
34
35In the CerCo assembler, we opted at first for a least fixed point algorithm,
36taking absolute jumps into account.
37
38Here, we ran into a problem with proving termination, as explained in the
39previous section: if we only take short and long jumps into account, the jump
40encoding can only switch from short to long, but never in the other direction.
41When we add absolute jumps, however, it is theoretically possible for a branch
42instruction to switch from absolute to long and back, as previously explained.
43Proving termination then becomes difficult, because there is nothing that
44precludes a branch instruction from oscillating back and forth between absolute
45and long jumps indefinitely.
46
47To keep the algorithm in the same complexity class and more easily
48prove termination, we decided to explicitly enforce the `branch instructions
49must always grow longer' requirement: if a branch instruction is encoded as a
50long jump in one iteration, it will also be encoded as a long jump in all the
51following iterations. Therefore the encoding of any branch instruction
52can change at most two times: once from short to absolute (or long), and once
53from absolute to long.
54
55There is one complicating factor. Suppose that a branch instruction is encoded
56in step $n$ as an absolute jump, but in step $n+1$ it is determined that
57(because of changes elsewhere) it can now be encoded as a short jump. Due to
58the requirement that the branch instructions must always grow longer,
59the branch encoding will be encoded as an absolute jump in step
60$n+1$ as well.
61
62This is not necessarily correct. A branch instruction that can be
63encoded as a short jump cannot always also be encoded as an absolute jump, as a
64short jump can bridge segments, whereas an absolute jump cannot. Therefore,
65in this situation we have decided to encode the branch instruction as a long
66jump, which is always correct.
67
68The resulting algorithm, therefore, will not return the least fixed point, as
69it might have too many long jumps. However, it is still better than the
70algorithms from SDCC and {\tt gcc}, since even in the worst case, it will still
71return a smaller or equal solution.
72
73Experiments on the gcc 2.3.3 test suite of C programs have shown that on
74average, 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
75are executed multiple times, for example in loops, the fact that short jumps take less cycles to execute than long jumps can have great effect.
76
77As for complexity, there are at most $2n$ iterations, with $n$ the number of
78branch instructions. Practical tests within the CerCo project on small to
79medium pieces of code have shown that in almost all cases, a fixed point is
80reached in 3 passes. Only in one case did the algorithm need 4. This is not surprising: after all, the difference between short/absolute and
81long jumps is only one byte (three for conditional jumps). For a change from
82short/absolute to long to have an effect on other jumps is therefore relatively
83uncommon, which explains why a fixed point is reached so quickly.
84
85\subsection{The algorithm in detail}
86
87The branch displacement algorithm forms part of the translation from
88pseudocode to assembler. More specifically, it is used by the function that
89translates pseudo-addresses (natural numbers indicating the position of the
90instruction in the program) to actual addresses in memory. Note that in pseudocode, all instructions are of size 1.
91
92Our original intention was to have two different functions, one function
93$\mathtt{policy}: \mathbb{N} \rightarrow \{\mathtt{short\_jump},
94\mathtt{absolute\_jump}, \mathtt{long\_jump}\}$ to associate jumps to their
95intended encoding, and a function $\sigma: \mathbb{N} \rightarrow
96\mathtt{Word}$ to associate pseudo-addresses to machine addresses. $\sigma$
97would use $\mathtt{policy}$ to determine the size of jump instructions. This turned out to be suboptimal from the algorithmic point of view and
98impossible to prove correct.
99
100From the algorithmic point of view, in order to create the $\mathtt{policy}$
101function, we must necessarily have a translation from pseudo-addresses
102to machine addresses (i.e. a $\sigma$ function): in order to judge the distance
103between a jump and its destination, we must know their memory locations.
104Conversely, in order to create the $\sigma$ function, we need to have the
105$\mathtt{policy}$ function, otherwise we do not know the sizes of the jump
106instructions in the program.
107
108Much the same problem appears when we try to prove the algorithm correct: the
109correctness of $\mathtt{policy}$ depends on the correctness of $\sigma$, and
110the correctness of $\sigma$ depends on the correctness of $\mathtt{policy}$.
111
112We solved this problem by integrating the $\mathtt{policy}$ and $\sigma$
113algorithms. We now have a function
114$\sigma: \mathbb{N} \rightarrow \mathtt{Word} \times \mathtt{bool}$ which
115associates a pseudo-address to a machine address. The boolean denotes a forced
116long jump; as noted in the previous section, if during the fixed point
117computation an absolute jump changes to be potentially re-encoded as a short
118jump, the result is actually a long jump. It might therefore be the case that
119jumps are encoded as long jumps without this actually being necessary, and this
120information needs to be passed to the code generating function.
121
122The assembler function encodes the jumps by checking the distance between
123source and destination according to $\sigma$, so it could select an absolute
124jump in a situation where there should be a long jump. The boolean is there
125to prevent this from happening by indicating the locations where a long jump
126should be encoded, even if a shorter jump is possible. This has no effect on
127correctness, since a long jump is applicable in any situation.
128
129\begin{figure}[t]
130\small
131\begin{algorithmic}
132\Function{f}{$labels$,$old\_sigma$,$instr$,$ppc$,$acc$}
133        \State $\langle added, pc, sigma \rangle \gets acc$
134        \If {$instr$ is a backward jump to $j$}
135                \State $length \gets \mathrm{jump\_size}(pc,sigma_1(labels(j)))$
136        \ElsIf {$instr$ is a forward jump to $j$}
137                \State $length \gets \mathrm{jump\_size}(pc,old\_sigma_1(labels(j))+added)$
138        \EndIf
139        \State $old\_length \gets \mathrm{old\_sigma_1}(ppc)$
140        \State $new\_length \gets \mathrm{max}(old\_length, length)$
141        \State $old\_size \gets \mathrm{old\_sigma_2}(ppc)$
142        \State $new\_size \gets \mathrm{instruction\_size}(instr,new\_length)$
143        \State $new\_added \gets added+(new\_size-old\_size)$
144        \State $new\_sigma \gets old\_sigma$
145        \State $new\_sigma_1(ppc+1) \gets pc+new\_size$
146        \State $new\_sigma_2(ppc) \gets new\_length$ \\
147        \Return $\langle new\_added, pc+new\_size, new\_sigma \rangle$
148\EndFunction
149\end{algorithmic}
150\caption{The heart of the algorithm}
151\label{f:jump_expansion_step}
152\end{figure}
153
154The algorithm, shown in Figure~\ref{f:jump_expansion_step}, works by folding the
155function {\sc f} over the entire program, thus gradually constructing $sigma$.
156This constitutes one step in the fixed point calculation; successive steps
157repeat 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.
158
159Parameters of the function {\sc f} are:
160\begin{itemize}
161        \item a function $labels$ that associates a label to its pseudo-address;
162        \item $old\_sigma$, the $\sigma$ function returned by the previous
163                iteration of the fixed point calculation;
164        \item $instr$, the instruction currently under consideration;
165        \item $ppc$, the pseudo-address of $instr$;
166        \item $acc$, the fold accumulator, which contains $pc$ (the highest memory
167                address reached so far), $added$ (the number of bytes added to the program
168                size with respect to the previous iteration), and of course $sigma$, the
169                $\sigma$ function under construction.
170\end{itemize}
171The first two are parameters that remain the same through one iteration, the
172final three are standard parameters for a fold function (including $ppc$,
173which is simply the number of instructions of the program already processed).
174
175The $\sigma$ functions used by {\sc f} are not of the same type as the final
176$\sigma$ function: they are of type
177$\sigma: \mathbb{N} \rightarrow \mathbb{N} \times \{\mathtt{short\_jump},
178\mathtt{absolute\_jump},\mathtt{long\_jump}\}$; a function that associates a
179pseudo-address with a memory address and a jump length. We do this to
180ease the comparison of jump lengths between iterations. In the algorithm,
181we use the notation $sigma_1(x)$ to denote the memory address corresponding to
182$x$, and $sigma_2(x)$ for the jump length corresponding to $x$.
183
184Note that the $\sigma$ function used for label lookup varies depending on
185whether the label is behind our current position or ahead of it. For
186backward branches, where the label is behind our current position, we can use
187$sigma$ for lookup, since its memory address is already known. However, for
188forward branches, the memory address of the address of the label is not yet
189known, so we must use $old\_sigma$.
190
191We cannot use $old\_sigma$ without change: it might be the case that we have
192already increased the size of some branch instructions before, making the
193program longer and moving every instruction forward. We must compensate for this
194by adding the size increase of the program to the label's memory address
195according to $old\_sigma$, so that branch instruction spans do not get
196compromised.
197
198%Note also that we add the pc to $sigma$ at location $ppc+1$, whereas we add the
199%jump length at location $ppc$. We do this so that $sigma(ppc)$ will always
200%return a pair with the start address of the instruction at $ppc$ and the
201%length of its branch instruction (if any); the end address of the program can
202%be found at $sigma(n+1)$, where $n$ is the number of instructions in the
203%program.
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