--- a/ChengsongTanPhdThesis/Chapters/Chapter1.tex	Sun May 01 11:50:46 2022 +0100
+++ b/ChengsongTanPhdThesis/Chapters/Chapter1.tex	Mon May 02 00:23:39 2022 +0100
@@ -13,13 +13,19 @@
 \newcommand{\file}[1]{\texttt{\bfseries#1}}
 \newcommand{\option}[1]{\texttt{\itshape#1}}
 
-
+\newcommand{\bderssimp}[2]{{#1} \backslash_{bsimp} {#2}}
 \newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}%
 \newcommand{\ZERO}{\mbox{\bf 0}}
 \newcommand{\ONE}{\mbox{\bf 1}}
 \def\lexer{\mathit{lexer}}
 \def\mkeps{\mathit{mkeps}}
 
+\def\AZERO{\textit{AZERO}}
+\def\AONE{\textit{AONE}}
+\def\ACHAR{\textit{ACHAR}}
+\def\ASEQ{\textit{ASEQ}}
+\def\AALTS{\textit{AALTS}}
+\def\ASTAR{\textit{ASTAR}}
 \def\DFA{\textit{DFA}}
 \def\bmkeps{\textit{bmkeps}}
 \def\retrieve{\textit{retrieve}}
@@ -37,11 +43,25 @@
 \def\Z{\mathit{Z}}
 \def\S{\mathit{S}}
 \def\rup{r^\uparrow}
+%\def\bderssimp{\mathit{bders}\_\mathit{simp}}
+\def\distinctWith{\textit{distinctWith}}
+
 \def\simp{\mathit{simp}}
 \def\simpALTs{\mathit{simp}\_\mathit{ALTs}}
 \def\map{\mathit{map}}
 \def\distinct{\mathit{distinct}}
 \def\blexersimp{\mathit{blexer}\_\mathit{simp}}
+\def\map{\textit{map}}
+\def\vsuf{\textit{vsuf}}
+\def\sflataux{\textit{sflat}\_\textit{aux}}
+\def\rrexp{\textit{rrexp}}
+\def\rsize{\textit{rsize}}
+\def\asize{\textit{asize}}
+\def\rerase{\textit{rerase}}
+\def\erase{\textit{erase}}
+\def\STAR{\textit{STAR}}
+\def\flts{\textit{flts}}
+
 %----------------------------------------------------------------------------------------
 %This part is about regular expressions, Brzozowski derivatives,
 %and a bit-coded lexing algorithm with proven correctness and time bounds.
@@ -50,13 +70,14 @@
 
 
 Regular expressions are widely used in computer science: 
-be it in IDEs with syntax hightlighting and auto completion, 
+be it in text-editors\parencite{atomEditor} with syntax hightlighting and auto completion, 
 command line tools like $\mathit{grep}$ that facilitates easy 
 text processing , network intrusion
 detection systems that rejects suspicious traffic, or compiler
-front ends--there is always an important phase of the
-task that involves matching a regular 
-exression with a string.
+front ends--the majority of the solutions to these tasks 
+involve lexing with regular 
+expressions.
+
 Given its usefulness and ubiquity, one would imagine that
 modern regular expression matching implementations
 are mature and fully-studied.