1 % Chapter Template |
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2 |
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3 \chapter{Regular Expressions and POSIX Lexing} % Main chapter title |
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4 |
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5 \label{Chapter2} % In chapter 2 \ref{Chapter2} we will introduce the concepts |
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6 %and notations we |
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7 %use for describing the lexing algorithm by Sulzmann and Lu, |
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8 %and then give the algorithm and its variant, and discuss |
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9 %why more aggressive simplifications are needed. |
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10 |
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11 |
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12 \section{Basic Concepts and Notations for Strings, Languages, and Regular Expressions} |
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13 We have a primitive datatype char, denoting characters. |
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14 \[ char ::= a |
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15 \mid b |
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16 \mid c |
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17 \mid \ldots |
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18 \mid z |
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19 \] |
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20 (which one is better?) |
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21 \begin{center} |
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22 \begin{tabular}{lcl} |
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23 $\textit{char}$ & $\dn$ & $a | b | c | \ldots$\\ |
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24 \end{tabular} |
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25 \end{center} |
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26 They can form strings by lists: |
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27 \begin{center} |
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28 \begin{tabular}{lcl} |
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29 $\textit{string}$ & $\dn$ & $[] | c :: cs$\\ |
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30 & & $(c\; \text{has char type})$ |
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31 \end{tabular} |
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32 \end{center} |
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33 And strings can be concatenated to form longer strings: |
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34 \begin{center} |
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35 \begin{tabular}{lcl} |
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36 $[] @ s_2$ & $\dn$ & $s_2$\\ |
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37 $(c :: s_1) @ s_2$ & $\dn$ & $c :: (s_1 @ s_2)$ |
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38 \end{tabular} |
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39 \end{center} |
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40 |
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41 A set of strings can operate with another set of strings: |
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42 \begin{center} |
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43 \begin{tabular}{lcl} |
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44 $A @ B $ & $\dn$ & $\{s_A @ s_B \mid s_A \in A; s_B \in B \}$\\ |
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45 \end{tabular} |
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46 \end{center} |
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47 We also call the above "language concatenation". |
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48 The power of a language is defined recursively, using the |
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49 concatenation operator $@$: |
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50 \begin{center} |
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51 \begin{tabular}{lcl} |
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52 $A^0 $ & $\dn$ & $\{ [] \}$\\ |
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53 $A^{n+1}$ & $\dn$ & $A^n @ A$ |
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54 \end{tabular} |
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55 \end{center} |
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56 The union of all the natural number powers of a language |
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57 is denoted by the Kleene star operator: |
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58 \begin{center} |
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59 \begin{tabular}{lcl} |
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60 $\bigcup_{i \geq 0} A^i$ & $\denote$ & $A^*$\\ |
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61 \end{tabular} |
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62 \end{center} |
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63 |
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64 In Isabelle of course we cannot easily get a counterpart of |
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65 the $\bigcup_{i \geq 0}$ operator, so we instead define the Kleene star |
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66 as an inductive set: |
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67 \begin{center} |
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68 \begin{tabular}{lcl} |
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69 $[] \in A^*$ & &\\ |
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70 $s_1 \in A \land \; s_2 \in A^* $ & $\implies$ & $s_1 @ s_2 \in A^*$\\ |
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71 \end{tabular} |
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72 \end{center} |
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73 |
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74 We also define an operation of chopping off a character from |
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75 a language: |
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76 \begin{center} |
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77 \begin{tabular}{lcl} |
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78 $\textit{Der} \;c \;A$ & $\dn$ & $\{ s \mid c :: s \in A \}$\\ |
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79 \end{tabular} |
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80 \end{center} |
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81 |
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82 This can be generalised to chopping off a string from all strings within set $A$: |
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83 \begin{center} |
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84 \begin{tabular}{lcl} |
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85 $\textit{Ders} \;w \;A$ & $\dn$ & $\{ s \mid w@s \in A \}$\\ |
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86 \end{tabular} |
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87 \end{center} |
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88 |
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89 which is essentially the left quotient $A \backslash L'$ of $A$ against |
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90 the singleton language $L' = \{w\}$ |
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91 in formal language theory. |
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92 For this dissertation the $\textit{Ders}$ notation would suffice, there is |
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93 no need for a more general derivative definition. |
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94 |
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95 With the sequencing, Kleene star, and $\textit{Der}$ operator on languages, |
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96 we have a few properties of how the language derivative can be defined using |
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97 sub-languages. |
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98 \begin{lemma} |
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99 $\Der \; c \; (A @ B) = \textit{if} \; [] \in A \; \textit{then} ((\Der \; c \; A) @ B ) \cup \Der \; c\; B \quad \textit{else}\; (\Der \; c \; A) @ B$ |
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100 \end{lemma} |
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101 \noindent |
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102 This lemma states that if $A$ contains the empty string, $\Der$ can "pierce" through it |
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103 and get to $B$. |
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104 |
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105 The language $A^*$'s derivative can be described using the language derivative |
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106 of $A$: |
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107 \begin{lemma} |
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108 $\textit{Der} \;c \;A^* = (\textit{Der}\; c A) @ (A^*)$\\ |
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109 \end{lemma} |
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110 \begin{proof} |
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111 \begin{itemize} |
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112 \item{$\subseteq$} |
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113 The set |
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114 \[ \{s \mid c :: s \in A^*\} \] |
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115 is enclosed in the set |
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116 \[ \{s_1 @ s_2 \mid s_1 \, s_2. s_1 \in \{s \mid c :: s \in A\} \land s_2 \in A^* \} \] |
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117 because whenever you have a string starting with a character |
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118 in the language of a Kleene star $A^*$, then that character together with some sub-string |
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119 immediately after it will form the first iteration, and the rest of the string will |
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120 be still in $A^*$. |
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121 \item{$\supseteq$} |
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122 Note that |
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123 \[ \Der \; c \; A^* = \Der \; c \; (\{ [] \} \cup (A @ A^*) ) \] |
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124 and |
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125 \[ \Der \; c \; (\{ [] \} \cup (A @ A^*) ) = \Der\; c \; (A @ A^*) \] |
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126 where the $\textit{RHS}$ of the above equatioin can be rewritten |
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127 as \[ (\Der \; c\; A) @ A^* \cup A' \], $A'$ being a possibly empty set. |
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128 \end{itemize} |
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129 \end{proof} |
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130 Before we define the $\textit{Der}$ and $\textit{Ders}$ counterpart |
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131 for regular languages, we need to first give definitions for regular expressions. |
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132 |
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133 \section{Regular Expressions and Their Language Interpretation} |
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134 Suppose we have an alphabet $\Sigma$, the strings whose characters |
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135 are from $\Sigma$ |
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136 can be expressed as $\Sigma^*$. |
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137 |
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138 We use patterns to define a set of strings concisely. Regular expressions |
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139 are one of such patterns systems: |
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140 The basic regular expressions are defined inductively |
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141 by the following grammar: |
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142 \[ r ::= \ZERO \mid \ONE |
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143 \mid c |
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144 \mid r_1 \cdot r_2 |
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145 \mid r_1 + r_2 |
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146 \mid r^* |
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147 \] |
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148 |
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149 The language or set of strings defined by regular expressions are defined as |
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150 %TODO: FILL in the other defs |
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151 \begin{center} |
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152 \begin{tabular}{lcl} |
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153 $L \; (r_1 + r_2)$ & $\dn$ & $ L \; (r_1) \cup L \; ( r_2)$\\ |
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154 $L \; (r_1 \cdot r_2)$ & $\dn$ & $ L \; (r_1) \cap L \; (r_2)$\\ |
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155 \end{tabular} |
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156 \end{center} |
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157 Which are also called the "language interpretation". |
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158 |
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159 |
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160 |
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161 |
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162 % Derivatives of a |
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163 %regular expression, written $r \backslash c$, give a simple solution |
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164 %to the problem of matching a string $s$ with a regular |
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165 %expression $r$: if the derivative of $r$ w.r.t.\ (in |
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166 %succession) all the characters of the string matches the empty string, |
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167 %then $r$ matches $s$ (and {\em vice versa}). |
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168 |
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169 |
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170 \section{Brzozowski Derivatives of Regular Expressions} |
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171 Now with semantic derivatives of a language and regular expressions and |
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172 their language interpretations, we are ready to define derivatives on regexes. |
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173 |
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174 The Brzozowski derivative w.r.t character $c$ is an operation on the regex, |
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175 where the operation transforms the regex to a new one containing |
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176 strings without the head character $c$. |
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177 |
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178 The derivative of regular expression, denoted as |
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179 $r \backslash c$, is a function that takes parameters |
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180 $r$ and $c$, and returns another regular expression $r'$, |
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181 which is computed by the following recursive function: |
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182 |
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183 \begin{center} |
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184 \begin{tabular}{lcl} |
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185 $\ZERO \backslash c$ & $\dn$ & $\ZERO$\\ |
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186 $\ONE \backslash c$ & $\dn$ & $\ZERO$\\ |
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187 $d \backslash c$ & $\dn$ & |
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188 $\mathit{if} \;c = d\;\mathit{then}\;\ONE\;\mathit{else}\;\ZERO$\\ |
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189 $(r_1 + r_2)\backslash c$ & $\dn$ & $r_1 \backslash c \,+\, r_2 \backslash c$\\ |
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190 $(r_1 \cdot r_2)\backslash c$ & $\dn$ & $\mathit{if} \, nullable(r_1)$\\ |
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191 & & $\mathit{then}\;(r_1\backslash c) \cdot r_2 \,+\, r_2\backslash c$\\ |
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192 & & $\mathit{else}\;(r_1\backslash c) \cdot r_2$\\ |
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193 $(r^*)\backslash c$ & $\dn$ & $(r\backslash c) \cdot r^*$\\ |
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194 \end{tabular} |
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195 \end{center} |
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196 \noindent |
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197 The function derivative, written $r\backslash c$, |
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198 defines how a regular expression evolves into |
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199 a new regular expression after all the string it contains |
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200 is chopped off a certain head character $c$. |
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201 The most involved cases are the sequence |
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202 and star case. |
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203 The sequence case says that if the first regular expression |
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204 contains an empty string then the second component of the sequence |
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205 might be chosen as the target regular expression to be chopped |
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206 off its head character. |
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207 The star regular expression's derivative unwraps the iteration of |
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208 regular expression and attaches the star regular expression |
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209 to the sequence's second element to make sure a copy is retained |
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210 for possible more iterations in later phases of lexing. |
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211 |
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212 |
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213 The $\nullable$ function tests whether the empty string $""$ |
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214 is in the language of $r$: |
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215 |
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216 |
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217 \begin{center} |
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218 \begin{tabular}{lcl} |
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219 $\nullable(\ZERO)$ & $\dn$ & $\mathit{false}$ \\ |
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220 $\nullable(\ONE)$ & $\dn$ & $\mathit{true}$ \\ |
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221 $\nullable(c)$ & $\dn$ & $\mathit{false}$ \\ |
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222 $\nullable(r_1 + r_2)$ & $\dn$ & $\nullable(r_1) \vee \nullable(r_2)$ \\ |
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223 $\nullable(r_1\cdot r_2)$ & $\dn$ & $\nullable(r_1) \wedge \nullable(r_2)$ \\ |
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224 $\nullable(r^*)$ & $\dn$ & $\mathit{true}$ \\ |
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225 \end{tabular} |
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226 \end{center} |
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227 \noindent |
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228 The empty set does not contain any string and |
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229 therefore not the empty string, the empty string |
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230 regular expression contains the empty string |
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231 by definition, the character regular expression |
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232 is the singleton that contains character only, |
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233 and therefore does not contain the empty string, |
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234 the alternative regular expression (or "or" expression) |
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235 might have one of its children regular expressions |
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236 being nullable and any one of its children being nullable |
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237 would suffice. The sequence regular expression |
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238 would require both children to have the empty string |
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239 to compose an empty string and the Kleene star |
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240 operation naturally introduced the empty string. |
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241 |
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242 We have the following property where the derivative on regular |
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243 expressions coincides with the derivative on a set of strings: |
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244 |
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245 \begin{lemma} |
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246 $\textit{Der} \; c \; L(r) = L (r\backslash c)$ |
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247 \end{lemma} |
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248 |
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249 \noindent |
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250 |
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251 |
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252 The main property of the derivative operation |
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253 that enables us to reason about the correctness of |
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254 an algorithm using derivatives is |
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255 |
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256 \begin{center} |
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257 $c\!::\!s \in L(r)$ holds |
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258 if and only if $s \in L(r\backslash c)$. |
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259 \end{center} |
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260 |
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261 \noindent |
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262 We can generalise the derivative operation shown above for single characters |
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263 to strings as follows: |
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264 |
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265 \begin{center} |
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266 \begin{tabular}{lcl} |
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267 $r \backslash (c\!::\!s) $ & $\dn$ & $(r \backslash c) \backslash s$ \\ |
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268 $r \backslash [\,] $ & $\dn$ & $r$ |
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269 \end{tabular} |
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270 \end{center} |
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271 |
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272 \noindent |
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273 and then define Brzozowski's regular-expression matching algorithm as: |
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274 |
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275 \begin{definition} |
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276 $match\;s\;r \;\dn\; nullable(r\backslash s)$ |
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277 \end{definition} |
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278 |
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279 \noindent |
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280 Assuming the string is given as a sequence of characters, say $c_0c_1..c_n$, |
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281 this algorithm presented graphically is as follows: |
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282 |
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283 \begin{equation}\label{graph:successive_ders} |
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284 \begin{tikzcd} |
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285 r_0 \arrow[r, "\backslash c_0"] & r_1 \arrow[r, "\backslash c_1"] & r_2 \arrow[r, dashed] & r_n \arrow[r,"\textit{nullable}?"] & \;\textrm{YES}/\textrm{NO} |
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286 \end{tikzcd} |
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287 \end{equation} |
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288 |
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289 \noindent |
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290 where we start with a regular expression $r_0$, build successive |
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291 derivatives until we exhaust the string and then use \textit{nullable} |
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292 to test whether the result can match the empty string. It can be |
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293 relatively easily shown that this matcher is correct (that is given |
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294 an $s = c_0...c_{n-1}$ and an $r_0$, it generates YES if and only if $s \in L(r_0)$). |
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295 |
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296 Beautiful and simple definition. |
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297 |
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298 If we implement the above algorithm naively, however, |
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299 the algorithm can be excruciatingly slow. |
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300 |
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301 |
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302 \begin{figure} |
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303 \centering |
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304 \begin{tabular}{@{}c@{\hspace{0mm}}c@{\hspace{0mm}}c@{}} |
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305 \begin{tikzpicture} |
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306 \begin{axis}[ |
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307 xlabel={$n$}, |
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308 x label style={at={(1.05,-0.05)}}, |
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309 ylabel={time in secs}, |
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310 enlargelimits=false, |
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311 xtick={0,5,...,30}, |
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312 xmax=33, |
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313 ymax=10000, |
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314 ytick={0,1000,...,10000}, |
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315 scaled ticks=false, |
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316 axis lines=left, |
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317 width=5cm, |
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318 height=4cm, |
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319 legend entries={JavaScript}, |
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320 legend pos=north west, |
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321 legend cell align=left] |
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322 \addplot[red,mark=*, mark options={fill=white}] table {EightThousandNodes.data}; |
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323 \end{axis} |
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324 \end{tikzpicture}\\ |
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325 \multicolumn{3}{c}{Graphs: Runtime for matching $(a^*)^*\,b$ with strings |
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326 of the form $\underbrace{aa..a}_{n}$.} |
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327 \end{tabular} |
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328 \caption{EightThousandNodes} \label{fig:EightThousandNodes} |
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329 \end{figure} |
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330 |
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331 |
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332 (8000 node data to be added here) |
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333 For example, when starting with the regular |
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334 expression $(a + aa)^*$ and building a few successive derivatives (around 10) |
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335 w.r.t.~the character $a$, one obtains a derivative regular expression |
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336 with more than 8000 nodes (when viewed as a tree)\ref{EightThousandNodes}. |
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337 The reason why $(a + aa) ^*$ explodes so drastically is that without |
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338 pruning, the algorithm will keep records of all possible ways of matching: |
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339 \begin{center} |
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340 $(a + aa) ^* \backslash [aa] = (\ZERO + \ONE \ONE)\cdot(a + aa)^* + (\ONE + \ONE a) \cdot (a + aa)^*$ |
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341 \end{center} |
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342 |
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343 \noindent |
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344 Each of the above alternative branches correspond to the match |
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345 $aa $, $a \quad a$ and $a \quad a \cdot (a)$(incomplete). |
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346 These different ways of matching will grow exponentially with the string length, |
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347 and without simplifications that throw away some of these very similar matchings, |
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348 it is no surprise that these expressions grow so quickly. |
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349 Operations like |
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350 $\backslash$ and $\nullable$ need to traverse such trees and |
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351 consequently the bigger the size of the derivative the slower the |
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352 algorithm. |
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353 |
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354 Brzozowski was quick in finding that during this process a lot useless |
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355 $\ONE$s and $\ZERO$s are generated and therefore not optimal. |
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356 He also introduced some "similarity rules", such |
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357 as $P+(Q+R) = (P+Q)+R$ to merge syntactically |
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358 different but language-equivalent sub-regexes to further decrease the size |
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359 of the intermediate regexes. |
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360 |
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361 More simplifications are possible, such as deleting duplicates |
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362 and opening up nested alternatives to trigger even more simplifications. |
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363 And suppose we apply simplification after each derivative step, and compose |
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364 these two operations together as an atomic one: $a \backslash_{simp}\,c \dn |
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365 \textit{simp}(a \backslash c)$. Then we can build |
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366 a matcher with simpler regular expressions. |
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367 |
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368 If we want the size of derivatives in the algorithm to |
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369 stay even lower, we would need more aggressive simplifications. |
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370 Essentially we need to delete useless $\ZERO$s and $\ONE$s, as well as |
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371 delete duplicates whenever possible. For example, the parentheses in |
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372 $(a+b) \cdot c + b\cdot c$ can be opened up to get $a\cdot c + b \cdot c + b |
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373 \cdot c$, and then simplified to just $a \cdot c + b \cdot c$. Another |
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374 example is simplifying $(a^*+a) + (a^*+ \ONE) + (a +\ONE)$ to just |
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375 $a^*+a+\ONE$. These more aggressive simplification rules are for |
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376 a very tight size bound, possibly as low |
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377 as that of the \emph{partial derivatives}\parencite{Antimirov1995}. |
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378 |
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379 Building derivatives and then simplifying them. |
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380 So far, so good. But what if we want to |
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381 do lexing instead of just getting a YES/NO answer? |
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382 This requires us to go back again to the world |
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383 without simplification first for a moment. |
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384 Sulzmann and Lu~\cite{Sulzmann2014} first came up with a nice and |
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385 elegant(arguably as beautiful as the original |
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386 derivatives definition) solution for this. |
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387 |
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388 \subsection*{Values and the Lexing Algorithm by Sulzmann and Lu} |
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389 |
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390 |
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391 They first defined the datatypes for storing the |
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392 lexing information called a \emph{value} or |
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393 sometimes also \emph{lexical value}. These values and regular |
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394 expressions correspond to each other as illustrated in the following |
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395 table: |
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396 |
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397 \begin{center} |
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398 \begin{tabular}{c@{\hspace{20mm}}c} |
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399 \begin{tabular}{@{}rrl@{}} |
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400 \multicolumn{3}{@{}l}{\textbf{Regular Expressions}}\medskip\\ |
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401 $r$ & $::=$ & $\ZERO$\\ |
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402 & $\mid$ & $\ONE$ \\ |
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403 & $\mid$ & $c$ \\ |
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404 & $\mid$ & $r_1 \cdot r_2$\\ |
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405 & $\mid$ & $r_1 + r_2$ \\ |
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406 \\ |
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407 & $\mid$ & $r^*$ \\ |
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408 \end{tabular} |
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409 & |
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410 \begin{tabular}{@{\hspace{0mm}}rrl@{}} |
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411 \multicolumn{3}{@{}l}{\textbf{Values}}\medskip\\ |
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412 $v$ & $::=$ & \\ |
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413 & & $\Empty$ \\ |
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414 & $\mid$ & $\Char(c)$ \\ |
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415 & $\mid$ & $\Seq\,v_1\, v_2$\\ |
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416 & $\mid$ & $\Left(v)$ \\ |
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417 & $\mid$ & $\Right(v)$ \\ |
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418 & $\mid$ & $\Stars\,[v_1,\ldots\,v_n]$ \\ |
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419 \end{tabular} |
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420 \end{tabular} |
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421 \end{center} |
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422 |
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423 \noindent |
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424 |
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425 Building on top of Sulzmann and Lu's attempt to formalise the |
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426 notion of POSIX lexing rules \parencite{Sulzmann2014}, |
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427 Ausaf and Urban\parencite{AusafDyckhoffUrban2016} modelled |
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428 POSIX matching as a ternary relation recursively defined in a |
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429 natural deduction style. |
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430 With the formally-specified rules for what a POSIX matching is, |
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431 they proved in Isabelle/HOL that the algorithm gives correct results. |
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432 |
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433 But having a correct result is still not enough, |
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434 we want at least some degree of $\mathbf{efficiency}$. |
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435 |
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436 |
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437 |
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438 One regular expression can have multiple lexical values. For example |
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439 for the regular expression $(a+b)^*$, it has a infinite list of |
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440 values corresponding to it: $\Stars\,[]$, $\Stars\,[\Left(Char(a))]$, |
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441 $\Stars\,[\Right(Char(b))]$, $\Stars\,[\Left(Char(a),\,\Right(Char(b))]$, |
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442 $\ldots$, and vice versa. |
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443 Even for the regular expression matching a particular string, there could |
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444 be more than one value corresponding to it. |
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445 Take the example where $r= (a^*\cdot a^*)^*$ and the string |
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446 $s=\underbrace{aa\ldots a}_\text{n \textit{a}s}$. |
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447 If we do not allow any empty iterations in its lexical values, |
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448 there will be $n - 1$ "splitting points" on $s$ we can choose to |
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449 split or not so that each sub-string |
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450 segmented by those chosen splitting points will form different iterations: |
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451 \begin{center} |
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452 \begin{tabular}{lcr} |
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453 $a \mid aaa $ & $\rightarrow$ & $\Stars\, [v_{iteration \,a},\, v_{iteration \,aaa}]$\\ |
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454 $aa \mid aa $ & $\rightarrow$ & $\Stars\, [v_{iteration \, aa},\, v_{iteration \, aa}]$\\ |
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455 $a \mid aa\mid a $ & $\rightarrow$ & $\Stars\, [v_{iteration \, a},\, v_{iteration \, aa}, \, v_{iteration \, a}]$\\ |
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456 & $\textit{etc}.$ & |
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457 \end{tabular} |
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458 \end{center} |
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459 |
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460 And for each iteration, there are still multiple ways to split |
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461 between the two $a^*$s. |
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462 It is not surprising there are exponentially many lexical values |
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463 that are distinct for the regex and string pair $r= (a^*\cdot a^*)^*$ and |
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464 $s=\underbrace{aa\ldots a}_\text{n \textit{a}s}$. |
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465 |
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466 A lexer to keep all the possible values will naturally |
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467 have an exponential runtime on ambiguous regular expressions. |
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468 Somehow one has to decide which lexical value to keep and |
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469 output in a lexing algorithm. |
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470 In practice, we are usually |
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471 interested in POSIX values, which by intuition always |
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472 \begin{itemize} |
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473 \item |
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474 match the leftmost regular expression when multiple options of matching |
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475 are available |
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476 \item |
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477 always match a subpart as much as possible before proceeding |
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478 to the next token. |
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479 \end{itemize} |
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480 The formal definition of a $\POSIX$ value $v$ for a regular expression |
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481 $r$ and string $s$, denoted as $(s, r) \rightarrow v$, can be specified |
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482 in the following set of rules: |
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483 (TODO: write the entire set of inference rules for POSIX ) |
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484 \newcommand*{\inference}[3][t]{% |
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485 \begingroup |
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486 \def\and{\\}% |
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487 \begin{tabular}[#1]{@{\enspace}c@{\enspace}} |
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488 #2 \\ |
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489 \hline |
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490 #3 |
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491 \end{tabular}% |
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492 \endgroup |
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493 } |
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494 \begin{center} |
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495 \inference{$s_1 @ s_2 = s$ \and $(\nexists s_3 s_4 s_5. s_1 @ s_5 = s_3 \land s_5 \neq [] \land s_3 @ s_4 = s \land (s_3, r_1) \rightarrow v_3 \land (s_4, r_2) \rightarrow v_4)$ \and $(s_1, r_1) \rightarrow v_1$ \and $(s_2, r_2) \rightarrow v_2$ }{$(s, r_1 \cdot r_2) \rightarrow \Seq(v_1, v_2)$ } |
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496 \end{center} |
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497 |
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498 The reason why we are interested in $\POSIX$ values is that they can |
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499 be practically used in the lexing phase of a compiler front end. |
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500 For instance, when lexing a code snippet |
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501 $\textit{iffoo} = 3$ with the regular expression $\textit{keyword} + \textit{identifier}$, we want $\textit{iffoo}$ to be recognized |
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502 as an identifier rather than a keyword. |
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503 |
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504 For example, the above $r= (a^*\cdot a^*)^*$ and |
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505 $s=\underbrace{aa\ldots a}_\text{n \textit{a}s}$ example has the POSIX value |
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506 $ \Stars\,[\Seq(Stars\,[\underbrace{\Char(a),\ldots,\Char(a)}_\text{n iterations}], Stars\,[])]$. |
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507 The output of an algorithm we want would be a POSIX matching |
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508 encoded as a value. |
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509 |
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510 |
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511 |
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512 |
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513 The contribution of Sulzmann and Lu is an extension of Brzozowski's |
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514 algorithm by a second phase (the first phase being building successive |
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515 derivatives---see \eqref{graph:successive_ders}). In this second phase, a POSIX value |
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516 is generated if the regular expression matches the string. |
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517 How can we construct a value out of regular expressions and character |
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518 sequences only? |
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519 Two functions are involved: $\inj$ and $\mkeps$. |
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520 The function $\mkeps$ constructs a value from the last |
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521 one of all the successive derivatives: |
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522 \begin{ceqn} |
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523 \begin{equation}\label{graph:mkeps} |
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524 \begin{tikzcd} |
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525 r_0 \arrow[r, "\backslash c_0"] & r_1 \arrow[r, "\backslash c_1"] & r_2 \arrow[r, dashed] & r_n \arrow[d, "mkeps" description] \\ |
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526 & & & v_n |
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527 \end{tikzcd} |
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528 \end{equation} |
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529 \end{ceqn} |
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530 |
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531 It tells us how can an empty string be matched by a |
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532 regular expression, in a $\POSIX$ way: |
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533 |
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534 \begin{center} |
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535 \begin{tabular}{lcl} |
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536 $\mkeps(\ONE)$ & $\dn$ & $\Empty$ \\ |
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537 $\mkeps(r_{1}+r_{2})$ & $\dn$ |
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538 & \textit{if} $\nullable(r_{1})$\\ |
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539 & & \textit{then} $\Left(\mkeps(r_{1}))$\\ |
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540 & & \textit{else} $\Right(\mkeps(r_{2}))$\\ |
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541 $\mkeps(r_1\cdot r_2)$ & $\dn$ & $\Seq\,(\mkeps\,r_1)\,(\mkeps\,r_2)$\\ |
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542 $mkeps(r^*)$ & $\dn$ & $\Stars\,[]$ |
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543 \end{tabular} |
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544 \end{center} |
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545 |
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546 |
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547 \noindent |
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548 We favour the left to match an empty string if there is a choice. |
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549 When there is a star for us to match the empty string, |
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550 we give the $\Stars$ constructor an empty list, meaning |
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551 no iterations are taken. |
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552 |
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553 |
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554 After the $\mkeps$-call, we inject back the characters one by one in order to build |
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555 the lexical value $v_i$ for how the regex $r_i$ matches the string $s_i$ |
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556 ($s_i = c_i \ldots c_{n-1}$ ) from the previous lexical value $v_{i+1}$. |
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557 After injecting back $n$ characters, we get the lexical value for how $r_0$ |
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558 matches $s$. The POSIX value is maintained throughout the process. |
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559 For this Sulzmann and Lu defined a function that reverses |
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560 the ``chopping off'' of characters during the derivative phase. The |
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561 corresponding function is called \emph{injection}, written |
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562 $\textit{inj}$; it takes three arguments: the first one is a regular |
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563 expression ${r_{i-1}}$, before the character is chopped off, the second |
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564 is a character ${c_{i-1}}$, the character we want to inject and the |
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565 third argument is the value ${v_i}$, into which one wants to inject the |
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566 character (it corresponds to the regular expression after the character |
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567 has been chopped off). The result of this function is a new value. |
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568 \begin{ceqn} |
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569 \begin{equation}\label{graph:inj} |
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570 \begin{tikzcd} |
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571 r_1 \arrow[r, dashed] \arrow[d]& r_i \arrow[r, "\backslash c_i"] \arrow[d] & r_{i+1} \arrow[r, dashed] \arrow[d] & r_n \arrow[d, "mkeps" description] \\ |
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572 v_1 \arrow[u] & v_i \arrow[l, dashed] & v_{i+1} \arrow[l,"inj_{r_i} c_i"] & v_n \arrow[l, dashed] |
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573 \end{tikzcd} |
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574 \end{equation} |
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575 \end{ceqn} |
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576 |
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577 |
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578 \noindent |
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579 The |
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580 definition of $\textit{inj}$ is as follows: |
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581 |
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582 \begin{center} |
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583 \begin{tabular}{l@{\hspace{1mm}}c@{\hspace{1mm}}l} |
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584 $\textit{inj}\,(c)\,c\,Empty$ & $\dn$ & $Char\,c$\\ |
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585 $\textit{inj}\,(r_1 + r_2)\,c\,\Left(v)$ & $\dn$ & $\Left(\textit{inj}\,r_1\,c\,v)$\\ |
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586 $\textit{inj}\,(r_1 + r_2)\,c\,Right(v)$ & $\dn$ & $Right(\textit{inj}\,r_2\,c\,v)$\\ |
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587 $\textit{inj}\,(r_1 \cdot r_2)\,c\,Seq(v_1,v_2)$ & $\dn$ & $Seq(\textit{inj}\,r_1\,c\,v_1,v_2)$\\ |
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588 $\textit{inj}\,(r_1 \cdot r_2)\,c\,\Left(Seq(v_1,v_2))$ & $\dn$ & $Seq(\textit{inj}\,r_1\,c\,v_1,v_2)$\\ |
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589 $\textit{inj}\,(r_1 \cdot r_2)\,c\,Right(v)$ & $\dn$ & $Seq(\textit{mkeps}(r_1),\textit{inj}\,r_2\,c\,v)$\\ |
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590 $\textit{inj}\,(r^*)\,c\,Seq(v,Stars\,vs)$ & $\dn$ & $Stars((\textit{inj}\,r\,c\,v)\,::\,vs)$\\ |
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591 \end{tabular} |
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592 \end{center} |
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593 |
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594 \noindent This definition is by recursion on the ``shape'' of regular |
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595 expressions and values. |
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596 The clauses do one thing--identifying the ``hole'' on a |
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597 value to inject the character back into. |
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598 For instance, in the last clause for injecting back to a value |
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599 that would turn into a new star value that corresponds to a star, |
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600 we know it must be a sequence value. And we know that the first |
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601 value of that sequence corresponds to the child regex of the star |
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602 with the first character being chopped off--an iteration of the star |
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603 that had just been unfolded. This value is followed by the already |
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604 matched star iterations we collected before. So we inject the character |
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605 back to the first value and form a new value with this latest iteration |
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606 being added to the previous list of iterations, all under the $\Stars$ |
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607 top level. |
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608 |
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609 Putting all the functions $\inj$, $\mkeps$, $\backslash$ together, |
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610 we have a lexer with the following recursive definition: |
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611 \begin{center} |
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612 \begin{tabular}{lcr} |
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613 $\lexer \; r \; [] $ & $=$ & $\mkeps \; r$\\ |
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614 $\lexer \; r \;c::s$ & $=$ & $\inj \; r \; c (\lexer (r\backslash c) s)$ |
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615 \end{tabular} |
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616 \end{center} |
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617 |
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618 Pictorially, the algorithm is as follows: |
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619 |
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620 \begin{ceqn} |
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621 \begin{equation}\label{graph:2} |
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622 \begin{tikzcd} |
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623 r_0 \arrow[r, "\backslash c_0"] \arrow[d] & r_1 \arrow[r, "\backslash c_1"] \arrow[d] & r_2 \arrow[r, dashed] \arrow[d] & r_n \arrow[d, "mkeps" description] \\ |
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624 v_0 & v_1 \arrow[l,"inj_{r_0} c_0"] & v_2 \arrow[l, "inj_{r_1} c_1"] & v_n \arrow[l, dashed] |
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625 \end{tikzcd} |
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626 \end{equation} |
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627 \end{ceqn} |
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628 |
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629 |
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630 \noindent |
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631 For convenience, we shall employ the following notations: the regular |
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632 expression we start with is $r_0$, and the given string $s$ is composed |
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633 of characters $c_0 c_1 \ldots c_{n-1}$. In the first phase from the |
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634 left to right, we build the derivatives $r_1$, $r_2$, \ldots according |
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635 to the characters $c_0$, $c_1$ until we exhaust the string and obtain |
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636 the derivative $r_n$. We test whether this derivative is |
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637 $\textit{nullable}$ or not. If not, we know the string does not match |
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638 $r$, and no value needs to be generated. If yes, we start building the |
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639 values incrementally by \emph{injecting} back the characters into the |
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640 earlier values $v_n, \ldots, v_0$. This is the second phase of the |
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641 algorithm from right to left. For the first value $v_n$, we call the |
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642 function $\textit{mkeps}$, which builds a POSIX lexical value |
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643 for how the empty string has been matched by the (nullable) regular |
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644 expression $r_n$. This function is defined as |
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645 |
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646 |
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647 |
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648 We have mentioned before that derivatives without simplification |
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649 can get clumsy, and this is true for values as well--they reflect |
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650 the size of the regular expression by definition. |
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651 |
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652 One can introduce simplification on the regex and values but have to |
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653 be careful not to break the correctness, as the injection |
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654 function heavily relies on the structure of the regexes and values |
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655 being correct and matching each other. |
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656 It can be achieved by recording some extra rectification functions |
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657 during the derivatives step, and applying these rectifications in |
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658 each run during the injection phase. |
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659 And we can prove that the POSIX value of how |
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660 regular expressions match strings will not be affected---although it is much harder |
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661 to establish. |
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662 Some initial results in this regard have been |
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663 obtained in \cite{AusafDyckhoffUrban2016}. |
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664 |
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665 |
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666 |
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667 %Brzozowski, after giving the derivatives and simplification, |
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668 %did not explore lexing with simplification, or he may well be |
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669 %stuck on an efficient simplification with proof. |
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670 %He went on to examine the use of derivatives together with |
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671 %automaton, and did not try lexing using products. |
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672 |
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673 We want to get rid of the complex and fragile rectification of values. |
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674 Can we not create those intermediate values $v_1,\ldots v_n$, |
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675 and get the lexing information that should be already there while |
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676 doing derivatives in one pass, without a second injection phase? |
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677 In the meantime, can we make sure that simplifications |
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678 are easily handled without breaking the correctness of the algorithm? |
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679 |
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680 Sulzmann and Lu solved this problem by |
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681 introducing additional information to the |
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682 regular expressions called \emph{bitcodes}. |
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683 |
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684 |
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685 |
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686 |
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687 |
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688 |
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689 |
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