62 This works builds on previous work by Ausaf and Urban using 

    62 This paper builds on previous work by Ausaf and Urban using 

    64 is both fast and satisfied the POSIX specification.

    64 is both fast and satisfies the POSIX specification.

    84 This might sound trivial in the case of producing a YES/NO answer,

    84 

    85 but once we require a lexing output to be produced (which is required

    85 

    86 in applications like compiler front-end, malicious attack domain extraction, 

    86 The reason that we do need to add a simplification phase

    87 etc.), it is not straightforward if we still extract what is needed according

    87 after each derivative step of  $\textit{blexer}$ is

    88 to the POSIX standard.

    88 because it produces intermediate

    89 

    89 regular expressions that can grow exponentially.

    90 By simplification, we mean specifically the following rules:

    90 For example, the regular expression $(a+aa)^*$ after taking

    91 

    91 derivative against just 10 $a$s will have size 8192.

    92 \begin{center}

    92 %TODO: add figure for this?

    93   \begin{tabular}{lcl}

    94   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ AZERO\ r\isactrlsub {\isadigit{2}}\ {\isasymleadsto}\ AZERO}}}\\

    95   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO}}}\\

    96   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ {\isacharparenleft}{\kern0pt}AONE\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymleadsto}\ fuse\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ bs\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\

    97   \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{1}}}}{\mbox{ASEQ\ bs\ bs\ r{\isadigit{3}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isasymleadsto}\ ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r{\isadigit{3}}{\isachardot}{\kern0pt}{\isadigit{0}}}}}\\

    98   \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{2}}}}{\mbox{ASEQ\ bs\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ bs\ {\isasymleadsto}\ ASEQ\ bs\ r{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ r\isactrlsub {\isadigit{2}}}}}\\

    99   \isa{\mbox{}\inferrule{\mbox{bs\ {\isasymleadsto}\ r\isactrlsub {\isadigit{1}}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rs{\isadigit{2}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}}}}\\

   100   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\ {\isacharat}{\kern0pt}\ AZERO\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}}}}\\

   101   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\mbox{$\,$}{\isacharcolon}{\kern0pt}{\isacharcolon}{\kern0pt}\mbox{$\,$}rsb{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharat}{\kern0pt}\ map\ {\isacharparenleft}{\kern0pt}fuse\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}{\isacharparenright}{\kern0pt}\ rs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharat}{\kern0pt}\ rsb{\isacharparenright}{\kern0pt}}}}\\

   102   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\ rs}}}\\

   103   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ {\isacharparenleft}{\kern0pt}bs\ {\isacharat}{\kern0pt}\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}map\ {\isacharparenleft}{\kern0pt}fuse\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ rs{\isacharparenright}{\kern0pt}}}}\\

   104   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO}}}\\

   105   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r\isactrlsub {\isadigit{1}}}}}\\

   106   \isa{\mbox{}\inferrule{\mbox{bs\mbox{$^\downarrow$}\ {\isacharequal}{\kern0pt}\ r\isactrlsub {\isadigit{2}}\mbox{$^\downarrow$}}}{\mbox{AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{2}}{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}\ {\isasymleadsto}\ AALTs\ bs\ {\isacharparenleft}{\kern0pt}rsa\ {\isacharat}{\kern0pt}\ {\isacharbrackleft}{\kern0pt}bs{\isacharbrackright}{\kern0pt}\ {\isacharat}{\kern0pt}\ rsb\ {\isacharat}{\kern0pt}\ rsc{\isacharparenright}{\kern0pt}}}}\\

   107 

   108 

   109   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Empty\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{1}}}}} & 

   110   \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Char\ c\ {\isacharcolon}{\kern0pt}\ c}}}\\[4mm]

   111   \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}{\mbox{Left\ v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}} &

   112   \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}{\mbox{Right\ v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}}}\\[4mm]

   113   \isa{\mbox{}\inferrule{\mbox{v\isactrlsub {\isadigit{1}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}}\\\ \mbox{v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{2}}}}{\mbox{Seq\ v\isactrlsub {\isadigit{1}}\ v\isactrlsub {\isadigit{2}}\ {\isacharcolon}{\kern0pt}\ r\isactrlsub {\isadigit{1}}\ {\isasymcdot}\ r\isactrlsub {\isadigit{2}}}}}  &

   114   \isa{\mbox{}\inferrule{\mbox{{\isasymforall}v{\isasymin}vs{\isachardot}{\kern0pt}\ v\ {\isacharcolon}{\kern0pt}\ r\ {\isasymand}\ {\isacharbar}{\kern0pt}v{\isacharbar}{\kern0pt}\ {\isasymnoteq}\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}}}{\mbox{Stars\ vs\ {\isacharcolon}{\kern0pt}\ r\isactrlsup {\isasymstar}}}}

   115 

   116 

   117   \end{tabular}

   118 \end{center}

   119 

   120 

   121 And these can be made compact by the following simplification function:

   122 

   134 remain so when we take derivative on both of them.

   141 remain "isomorphic" when we take the same sequence of

   142 derivatives on both of them.

  1748 %:%147=217%:%

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  2245 %:%654=698%:%

  2212 %:%652=699%:%

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  2215 %:%655=702%:%

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  2216 %:%656=703%:%

  2250 %:%657=703%:%

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  2279 %:%685=732%:%

  2246 %:%684=733%:%

  2280 %:%686=733%:%

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  2281 %:%687=734%:%

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  2282 %:%688=735%:%

  2249 %:%687=736%:%

  2283 %:%689=736%:%

  2250 %:%688=737%:%

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  2285 %:%691=738%:%

  2252 %:%689=739%:%

  2286 %:%692=739%:%

  2253 %:%690=740%:%

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  2291 %:%697=744%:%

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  2260 %:%697=747%:%

  2294 %:%700=747%:%

  2261 %:%698=748%:%

  2295 %:%701=748%:%

  2262 %:%699=749%:%

  2296 %:%702=749%:%

  2263 %:%700=750%:%

  2297 %:%703=750%:%

  2264 %:%701=751%:%

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  2265 %:%702=752%:%

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  2300 %:%706=753%:%

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  2309 %:%714=762%:%

  2276 %:%713=763%:%

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  2311 %:%715=764%:%

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  2279 %:%716=766%:%

  2313 %:%717=766%:%

  2280 %:%717=767%:%

  2314 %:%718=767%:%

  2281 %:%718=768%:%

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  2282 %:%719=769%:%

  2316 %:%720=769%:%

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  2323 %:%726=776%:%

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  2324 %:%727=777%:%

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  2325 %:%757=812%:%

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  2363 %:%765=816%:%

  2330 %:%762=817%:%

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  2343 %:%775=830%:%

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  2410 %:%808=863%:%

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  2414 %:%812=867%:%

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  2416 %:%814=869%:%

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  2417 %:%815=870%:%

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  2418 %:%816=871%:%

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  2419 %:%817=872%:%

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  2420 %:%818=873%:%

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  2422 %:%820=875%:%

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  2430 %:%828=883%:%

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  2437 %:%835=890%:%

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  2440 %:%838=893%:%

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  2424 %:%854=911%:%

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  2470 %:%867=923%:%

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  2470 %:%898=957%:%

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  2482 %:%910=969%:%

  2516 %:%911=969%:%

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  2524 %:%918=977%:%

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  2527 %:%921=980%:%

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  2528 %:%922=981%:%

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  2530 %:%924=983%:%

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  2532 %:%926=985%:%

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  2500 %:%923=987%:%

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  2540 %:%934=993%:%

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  2530 %:%952=1017%:%

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  2600 %:%988=1053%:%

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  2621 %:%1006=1074%:%

  2588 %:%1026=1081%:%

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