1 | \section{Our algorithm} |
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2 | |
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3 | \subsection{Design decisions} |
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4 | |
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5 | Given the NP-completeness of the problem, to arrive at an optimal solution |
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6 | within a short space of time (using, for example, a constraint solver) will |
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7 | potentially take a great amount of time. |
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8 | |
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9 | The {\tt gcc} compiler suite, for the x86 architecture, uses a greatest fix |
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10 | point algorithm. In other words, it starts off with all jumps encoded as the |
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11 | largest jumps possible, and then tries to reduce jumps as much as possible. |
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12 | |
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13 | Such an algorithm has the advantage that any intermediate results it returns |
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14 | are correct: the solution where every jump is encoded as a large jump is |
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15 | always possible, and the algorithm only reduces those jumps where the |
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16 | destination address is in range for a shorter jump instruction. The algorithm |
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17 | can thus be stopped after a determined amount of steps without losing |
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18 | correctness. |
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19 | |
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20 | The result, however, is not necessarily optimal, even if the algorithm is run |
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21 | until it terminates naturally: the fixed point reached is the {\em greatest} |
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22 | fixed point, not the least fixed point. |
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23 | |
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24 | The SDCC compiler, which has the MCS-51 among its target instruction sets, uses |
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25 | a least fix point algorithm, but does not take the presence of medium jumps |
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26 | into account. This makes the algorithm run in linear time with respect to the |
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27 | number of jumps in the program: starting out with every jump encoded as a |
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28 | short jump, each jump can be switched to a long jump once, but no jump ever |
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29 | goes back from long to short. This algorithm must be run until a fixed point |
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30 | is reached, because the intermediate solutions are not necessarily correct. |
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31 | |
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32 | This algorithm results in a least fixed point, which means its solution is |
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33 | potentially more optimal then the one reached by the {\tt gcc} algorithm. |
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34 | However, the solution is still not optimal, since there might be jumps whose |
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35 | destination is in the same segment. These jumps could be encoded as medium |
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36 | jumps, which are smaller than long jumps. |
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37 | |
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38 | Our first attempt at an algorithm was a least fixed point algorithm that took |
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39 | medium jumps into account. |
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40 | |
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41 | Here, we ran into a problem with proving termination: whereas the SDCC |
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42 | algorithm only switches jumps from short to long, when we add medium jumps, |
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43 | it is theoretically possible for a jump to switch from medium to long and back, |
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44 | as explained in the previous section. |
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45 | |
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46 | Proving termination then becomes difficult, because there is nothing that |
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47 | precludes a jump switching back and forth between medium and long indefinitely. |
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48 | |
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49 | In fact, this mirrors the argument from~\cite{Szymanski1978}. There, it is |
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50 | argued that for the problem to be NP-complete, it must be allowed to contain |
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51 | {\em pathological} jumps. These are jumps that can normally not be encoded as a |
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52 | short(er) jump, but gain this property when some other jumps are encoded as a |
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53 | long(er) jump. This is exactly what happens in figure~\ref{f:term_example}: by |
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54 | encoding the first jump as a long jump, another jump switches from long to |
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55 | medium (which is shorter). |
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56 | |
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57 | In order to keep the algorithm linear and more easily prove termination, we |
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58 | decided to explicitly enforce the `jumps must always increase' requirement: if |
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59 | a jump is encoded as a long jump in one step, it will also be encoded as a |
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60 | long jump in all the following steps. This means that any jump can change at |
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61 | maximum two times: once from short to medium (or long), and once from medium |
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62 | to long. |
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63 | |
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64 | There is one complicating factor: suppose that a jump is encoded in step $n$ |
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65 | as a medium jump, but in step $n+1$ it is determined that (because of changes |
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66 | elsewhere) it can now be encoded as a short jump. Due to the requirement that |
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67 | jumps must always increase, this means that the jump will be encoded as a |
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68 | medium jump in step $n+1$ as well. |
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69 | |
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70 | This is not necessarily correct, however: it is not the case that any short |
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71 | jump can correctly be encoded as a medium jump (a short jump can bridge |
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72 | segments, whereas a medium jump cannot). Therefore, in this situation |
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73 | we decide to encode the jump as a long jump, which is always correct. |
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74 | |
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75 | The resulting algorithm, while not optimal, is at least as good as the ones |
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76 | from {\tt gcc} and SDCC, and potentially better. Its complexity remains |
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77 | linear (though with a higher constant than SDCC). |
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78 | |
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79 | \subsection{The algorithm in detail} |
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80 | |
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81 | The branch displacement algorithm forms part of the translation from |
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82 | pseudo-code to assembler. More specifically, it is used by the function that |
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83 | translates pseudo-addresses (natural numbers indicating the position of the |
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84 | instruction in the program) to actual addresses in memory. |
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85 | |
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86 | The original intention was to have two different functions, one function |
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87 | $\mathtt{policy}: \mathbb{N} \rightarrow \{\mathtt{short}, \mathtt{medium}, |
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88 | \mathtt{long}\}$ to associate jumps to their intended translation, and a |
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89 | function $\sigma: \mathbb{N} \rightarrow \mathtt{Word}$ to associate |
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90 | pseudo-addresses to actual addresses. $\sigma$ would use $\mathtt{policy}$ to |
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91 | determine the size of jump instructions. |
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92 | |
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93 | This turned out to be suboptimal from the algorithmic point of view and |
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94 | impossible to prove correct. |
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95 | |
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96 | From the algorithmic point of view, in order to create the $\mathtt{policy}$ |
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97 | function, we must necessarily have a translation from pseudo-addresses |
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98 | to actual addresses (i.e. a $\sigma$ function): in order to judge the distance |
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99 | between a jump and its destination, we must know their memory locations. |
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100 | Conversely, in order to create the $\sigma$ function, we need to have the |
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101 | $\mathtt{policy}$ function, otherwise we do not know the sizes of the jump |
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102 | instructions in the program. |
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103 | |
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104 | Much the same problem appears when we try to prove the algorithm correct: the |
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105 | correctness of $\mathtt{policy}$ depends on the correctness of $\sigma$, and |
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106 | the correctness of $\sigma$ depends on the correctness of $\mathtt{policy}$. |
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107 | |
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108 | We solved this problem by integrating the $\mathtt{policy}$ and $\sigma$ |
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109 | algorithms. We now have a function |
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110 | $\sigma: \mathbb{N} \rightarrow \mathtt{Word} \times \mathtt{bool}$ which |
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111 | associates a pseudo-address to an actual address. The boolean denotes a forced |
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112 | long jump; as noted in the previous section, if during the fixed point |
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113 | computation a medium jump needs to be re-encoded as a short jump, the result |
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114 | is actually a long jump. It might therefore be the case that jumps are encoded |
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115 | as long jumps without this actually being necessary. |
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116 | |
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117 | The assembler function encodes the jumps by checking the distance between |
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118 | source and destination according to $\sigma$, so it could select a medium |
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119 | jump in a situation where there should be a long jump. The boolean is there |
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120 | to prevent this from happening by indicating the locations where a long jump |
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121 | should be encoded, even if a shorter jump is possible. This has no effect on |
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122 | correctness, since a long jump is applicable in any situation. |
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123 | |
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124 | \begin{figure} |
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125 | \begin{algorithmic} |
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126 | \Function{jump\_expansion\_step}{x} |
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127 | \EndFunction |
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128 | \end{algorithmic} |
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129 | \caption{The heart of the algorithm} |
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130 | \label{f:jump_expansion_step} |
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131 | \end{figure} |
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