1 \contentsline {section}{\numberline {1}Preliminary definitions}{1}{section.1}

     2 \contentsline {section}{\numberline {2}Direction \emph {\it finite\ partition\ {\emph {$\Rightarrow $}}\ regular\ language}}{5}{section.2}

     3 \contentsline {subsection}{\numberline {2.1}Proof for this direction}{9}{subsection.2.1}

     4 \contentsline {section}{\numberline {3}Direction: \emph {\it regular\ language\ {\emph {$\Rightarrow $}}finite\ partition}}{21}{section.3}

     5 \contentsline {subsection}{\numberline {3.1}The scheme for this direction}{21}{subsection.3.1}

     6 \contentsline {subsection}{\numberline {3.2}A small theory for list difference}{22}{subsection.3.2}

     7 \contentsline {subsection}{\numberline {3.3}Lemmas for basic cases}{23}{subsection.3.3}

     8 \contentsline {subsection}{\numberline {3.4}The case for \emph {\it SEQ}}{24}{subsection.3.4}

     9 \contentsline {subsection}{\numberline {3.5}The case for \emph {\it ALT}}{26}{subsection.3.5}

    10 \contentsline {subsection}{\numberline {3.6}The case for \emph {\it STAR}}{27}{subsection.3.6}

    11 \contentsline {subsection}{\numberline {3.7}The main lemma}{29}{subsection.3.7}