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1 % Chapter Template |
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2 |
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3 \chapter{A Better Bound and Other Extensions} % Main chapter title |
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4 |
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5 \label{Chapter5} %In Chapter 5\ref{Chapter5} we discuss stronger simplifications to improve the finite bound |
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6 %in Chapter 4 to a polynomial one, and demonstrate how one can extend the |
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7 %algorithm to include constructs such as bounded repetitions and negations. |
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8 |
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9 %---------------------------------------------------------------------------------------- |
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10 % SECTION strongsimp |
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11 %---------------------------------------------------------------------------------------- |
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12 \section{A Stronger Version of Simplification Inspired by Antimirov} |
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13 %TODO: search for isabelle proofs of algorithms that check equivalence |
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14 |
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15 |
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16 |
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17 %---------------------------------------------------------------------------------------- |
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18 % SECTION 1 |
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19 %---------------------------------------------------------------------------------------- |
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20 |
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21 \section{Adding Support for the Negation Construct, and its Correctness Proof} |
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22 We now add support for the negation regular expression: |
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23 \[ r ::= \ZERO \mid \ONE |
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24 \mid c |
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25 \mid r_1 \cdot r_2 |
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26 \mid r_1 + r_2 |
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27 \mid r^* |
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28 \mid \sim r |
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29 \] |
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30 The $\textit{nullable}$ function's clause for it would be |
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31 \[ |
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32 \textit{nullable}(~r) = \neg \nullable(r) |
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33 \] |
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34 The derivative would be |
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35 \[ |
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36 ~r \backslash c = ~ (r \backslash c) |
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37 \] |
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38 |
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39 The most tricky part of lexing for the $~r$ regex |
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40 is creating a value for it. |
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41 For other regular expressions, the value aligns with the |
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42 structure of the regex: |
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43 \[ |
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44 \vdash \Seq(\Char(a), \Char(b)) : a \cdot b |
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45 \] |
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46 But for the $~r$ regex, $s$ is a member of it if and only if |
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47 $s$ does not belong to $L(r)$. |
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48 That means when there |
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49 is a match for the not regex, it is not possible to generate how the string $s$ matched |
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50 with $r$. |
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51 What we can do is preserve the information of how $s$ was not matched by $r$, |
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52 and there are a number of options to do this. |
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53 |
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54 We could give a partial value when there is a partial match for the regex inside |
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55 the $\mathbf{not}$ construct. |
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56 For example, the string $ab$ is not in the language of $(a\cdot b) \cdot c$, |
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57 A value for it could be |
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58 \[ |
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59 \vdash \textit{Not}(\Seq(\Char(a), \Char(b))) : ~((a \cdot b ) \cdot c) |
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60 \] |
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61 The above example demonstrates what value to construct |
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62 when the string $s$ is at most a real prefix |
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63 of the strings in $L(r)$. When $s$ instead is not a prefix of any strings |
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64 in $L(r)$, it becomes unclear what to return as a value inside the $\textit{Not}$ |
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65 constructor. |
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66 |
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67 Another option would be to either store the string $s$ that resulted in |
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68 a mis-match for $r$ or a dummy value as a placeholder: |
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69 \[ |
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70 \vdash \textit{Not}(abcd) : ~(a^*) |
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71 \] |
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72 or |
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73 \[ |
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74 \vdash \textit{Not}(\textit{Dummy}) : ~(a^*) |
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75 \] |
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76 We choose to implement this as it is most straightforward: |
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77 \[ |
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78 \mkeps(~(r)) = \textit{if}(\nullable(r)) \; \textit{Error} \; \textit{else} \; \textit{Not}(\textit{Dummy}) |
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79 \] |
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80 |
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81 %---------------------------------------------------------------------------------------- |
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82 % SECTION 2 |
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83 %---------------------------------------------------------------------------------------- |
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84 |
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85 \section{Bounded Repetitions} |
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86 |
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87 |