--- a/thys2/Journal/Paper.tex Mon Nov 01 10:52:44 2021 +0000
+++ b/thys2/Journal/Paper.tex Tue Nov 02 13:57:59 2021 +0000
@@ -47,7 +47,7 @@
%
\isatagdocument
%
-\isamarkupsection{Introduction%
+\isamarkupsection{Core of the proof%
}
\isamarkuptrue%
%
@@ -69,12 +69,93 @@
In the previous work, Ausaf and Urban established the below equality:
\begin{lemma}
-\isa{{\normalsize{}If\,}\ v\ {\isacharcolon}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}{\isacharbackslash}{\kern0pt}c\ {\normalsize \,then\,}\ retrieve\ {\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}c{\isacharparenright}{\kern0pt}\ v\ {\isacharequal}{\kern0pt}\ retrieve\ r\ {\isacharparenleft}{\kern0pt}inj\ {\isacharparenleft}{\kern0pt}r\mbox{$^\downarrow$}{\isacharparenright}{\kern0pt}\ c\ v{\isacharparenright}{\kern0pt}{\isachardot}{\kern0pt}}
+\isa{{\normalsize{}If\,}\ v\ {\isacharcolon}{\kern0pt}\ r{\isacharbackslash}{\kern0pt}s\ {\normalsize \,then\,}\ Some\ {\isacharparenleft}{\kern0pt}flex\ r\ id\ s\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ decode\ {\isacharparenleft}{\kern0pt}retrieve\ {\isacharparenleft}{\kern0pt}r\mbox{$^\uparrow$}\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ v{\isacharparenright}{\kern0pt}\ r{\isachardot}{\kern0pt}}
+\end{lemma}
+
+This lemma establishes a link with the lexer without bit-codes.
+
+With it we get the correctness of bit-coded algorithm.
+\begin{lemma}
+\isa{lexer\mbox{$_b$}\ r\ s\ {\isacharequal}{\kern0pt}\ lexer\ r\ s}
\end{lemma}
-This lemma links the derivative of a bit-coded regular expression with
-the regular expression itself before the derivative.
+However what is not certain is whether we can add simplification
+to the bit-coded algorithm, without breaking the correct lexing output.
+This might sound trivial in the case of producing a YES/NO answer,
+but once we require a lexing output to be produced (which is required
+in applications like compiler front-end, malicious attack domain extraction,
+etc.), it is not straightforward if we still extract what is needed according
+to the POSIX standard.
+
+By simplification, we mean specifically the following rules:
+
+\begin{center}
+ \begin{tabular}{lcl}
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ AZERO\ r\isactrlsub {\isadigit{2}}\ {\isasymleadsto}\ AZERO}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ AZERO\ {\isasymleadsto}\ AZERO}}}\\
+ \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}}}}}\\
+ \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}}}}}\\
+ \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}}}}}\\
+ \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}}}}\\
+ \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}}}}\\
+ \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}}}}\\
+ \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}}}\\
+ \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}}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ AZERO}}}\\
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{AALTs\ bs\ {\isacharbrackleft}{\kern0pt}r\isactrlsub {\isadigit{1}}{\isacharbrackright}{\kern0pt}\ {\isasymleadsto}\ fuse\ bs\ r\isactrlsub {\isadigit{1}}}}}\\
+ \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}}}}\\
+
+
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Empty\ {\isacharcolon}{\kern0pt}\ \isactrlbold {\isadigit{1}}}}} &
+ \isa{\mbox{}\inferrule{\mbox{}}{\mbox{Char\ c\ {\isacharcolon}{\kern0pt}\ c}}}\\[4mm]
+ \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}}}}} &
+ \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]
+ \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}}}}} &
+ \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}}}}
+
+
+ \end{tabular}
+\end{center}
+
+And these can be made compact by the following simplification function:
+
+\begin{center}
+ \begin{tabular}{lcl}
+ \isa{bsimp\ {\isacharparenleft}{\kern0pt}ASEQ\ bs\ r\isactrlsub {\isadigit{1}}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}ASEQ\ bs\ {\isacharparenleft}{\kern0pt}bsimp\ r\isactrlsub {\isadigit{1}}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}bsimp\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}}\\
+ \isa{bsimp\ {\isacharparenleft}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ rs{\isacharparenright}{\kern0pt}} & $\dn$ & \isa{bsimp{\isacharunderscore}{\kern0pt}AALTs\ bs{\isadigit{1}}{\isachardot}{\kern0pt}{\isadigit{0}}\ {\isacharparenleft}{\kern0pt}distinctBy\ {\isacharparenleft}{\kern0pt}flts\ {\isacharparenleft}{\kern0pt}map\ bsimp\ rs{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ erase\ {\isasymemptyset}{\isacharparenright}{\kern0pt}}\\
+ \isa{bsimp\ AZERO} & $\dn$ & \isa{AZERO}\\
+ \isa{{\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\ {\isacharplus}{\kern0pt}\ r\isactrlsub {\isadigit{2}}{\isacharparenright}{\kern0pt}\mbox{$^\uparrow$}} & $\dn$ & \isa{AALT\ {\isacharbrackleft}{\kern0pt}{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}Z{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{1}}\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}fuse\ {\isacharbrackleft}{\kern0pt}S{\isacharbrackright}{\kern0pt}\ {\isacharparenleft}{\kern0pt}r\isactrlsub {\isadigit{2}}\mbox{$^\uparrow$}{\isacharparenright}{\kern0pt}{\isacharparenright}{\kern0pt}}\\
+\end{tabular}
+\end{center}
+
+The core idea of the proof is that two regular expressions,
+if "isomorphic" up to a finite number of rewrite steps, will
+remain so when we take derivative on both of them.
+This can be expressed by the following rewrite relation lemma:
+\begin{lemma}
+\isa{{\isacharparenleft}{\kern0pt}r\mbox{$\bbslash$}s{\isacharparenright}{\kern0pt}\ {\isasymleadsto}{\isacharasterisk}{\kern0pt}\ bders{\isacharunderscore}{\kern0pt}simp\ r\ s}
+\end{lemma}%
+\end{isamarkuptext}\isamarkuptrue%
+%
+\isadelimdocument
+%
+\endisadelimdocument
+%
+\isatagdocument
+%
+\isamarkupsection{Introduction%
+}
+\isamarkuptrue%
+%
+\endisatagdocument
+{\isafolddocument}%
+%
+\isadelimdocument
+%
+\endisadelimdocument
+%
+\begin{isamarkuptext}%
Brzozowski \cite{Brzozowski1964} introduced the notion of the {\em
derivative} \isa{r{\isacharbackslash}{\kern0pt}c} of a regular expression \isa{r} w.r.t.\
a character~\isa{c}, and showed that it gave a simple solution to the
@@ -1586,1311 +1667,1311 @@
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\endinput
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%:%1561=1918%:%
%:%1562=1919%:%
@@ -3058,4 +3139,66 @@
%:%1567=1924%:%
%:%1568=1925%:%
%:%1569=1926%:%
-%:%1582=1932%:%
\ No newline at end of file
+%:%1570=1927%:%
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+%:%1573=1930%:%
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+%:%1575=1932%:%
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+%:%1609=1954%:%
+%:%1610=1955%:%
+%:%1611=1956%:%
+%:%1612=1957%:%
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+%:%1614=1959%:%
+%:%1615=1960%:%
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+%:%1617=1962%:%
+%:%1618=1963%:%
+%:%1619=1964%:%
+%:%1620=1965%:%
+%:%1621=1966%:%
+%:%1622=1967%:%
+%:%1623=1968%:%
+%:%1624=1969%:%
+%:%1625=1970%:%
+%:%1626=1971%:%
+%:%1627=1972%:%
+%:%1628=1973%:%
+%:%1629=1974%:%
+%:%1630=1975%:%
+%:%1631=1976%:%
+%:%1632=1977%:%
+%:%1633=1978%:%
+%:%1634=1979%:%
+%:%1635=1980%:%
+%:%1636=1981%:%
+%:%1637=1982%:%
+%:%1638=1983%:%
+%:%1639=1984%:%
+%:%1640=1985%:%
+%:%1641=1986%:%
+%:%1642=1987%:%
+%:%1643=1988%:%
+%:%1644=1989%:%
+%:%1645=1990%:%
+%:%1646=1991%:%
+%:%1647=1992%:%
+%:%1648=1993%:%
+%:%1649=1994%:%
+%:%1650=1995%:%
+%:%1663=2001%:%
\ No newline at end of file