diff -r 0d1e68268d0f -r ba44144875b1 ChengsongTanPhdThesis/Chapters/Introduction.tex
--- a/ChengsongTanPhdThesis/Chapters/Introduction.tex	Mon Jul 10 14:32:48 2023 +0100
+++ b/ChengsongTanPhdThesis/Chapters/Introduction.tex	Mon Jul 10 19:29:22 2023 +0100
@@ -545,22 +545,71 @@
 
 %----------------------------------------------------------------------------------------
 \section{Contribution}
-{\color{red} \rule{\linewidth}{0.5mm}}
-\textbf{issue with this part: way too short, not enough details of what I have done.}
-{\color{red} \rule{\linewidth}{0.5mm}}
+%{\color{red} \rule{\linewidth}{0.5mm}}
+%\textbf{issue with this part: way too short, not enough details of what I have done.}
+ %{\color{red} \rule{\linewidth}{0.5mm}}
+\marginpar{\em Gerog: No sufficient context on related work making contribution hard to digest.}
+
 
-In this thesis,
-we propose a solution to catastrophic
-backtracking and error-prone matchers: a formally verified
-regular expression lexing algorithm
-that is both fast
-and correct. 
-{\color{red} \rule{\linewidth}{0.5mm}}
-\HandRight Added content:
+%In this thesis,
+%we propose a solution to catastrophic
+%backtracking and error-prone matchers: a formally verified
+%regular expression lexing algorithm
+%that is both fast
+%and correct. 
+%%{\color{red} \rule{\linewidth}{0.5mm}}
+%\HandRight Added content:
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+We have made mainly two contributions in this thesis: %the
+%lexer we developed based on Brzozowski derivatives and 
+%Sulzmanna and Lu's developments called 
+proving the lexer $\blexersimp$ is both correctness and fast.
+It is correct w.r.t a formalisation of POSIX lexing by Ausaf et al.\ref{AusafDyckhoffUrban2016}.
+It is fast compared with un-optimised implementations like Sulzmann and Lu's orignal
+development by our metric of internal data structures not growing unbounded.
+
+Our formalisation of complexity is unique among similar works in the sense that
+%is about the size of internal data structures.
+to our knowledge %we don't know of a 
+there is not a lexing/parsing algorithm with similar formalised correctness theorems.
+Common practices involve making some empirical analysis of the complexity of the algorithm
+in question (\ref{Verbatim}, \ref{Verbatimpp}), or relying 
+on prior (unformalised) complexity analysis of well-known algorithms (\ref{ValiantParsing}),
+making them prone to complexity bugs.
+%TODO: add citation
+%, for example in the Verbatim work \ref{Verbatim}
+
+%While formalised proofs have not included 
+%Issues known as "performance bugs" can
+Whilst formalised complexity theorems 
+have not yet appeared in other certified lexers/parsers,
+%while this work is done,
+they do find themselves in the broader theorem-proving literature:
+\emph{time credits} have been formalised for separation logic in Coq 
+\ref{atkey2010amortised}%not used in 
+to characterise the runtime complexity of an algorithm,
+where integer values are recorded %from the beginning of an execution
+as part of the program state
+and decremented in each step. 
+The idea is that the total number of decrements
+from the time credits during execution represents the complexity of an algorithm.
+%each time a basic step is executed. 
+%The way to use a time credit Time credit is an integer
+%is assigned to a program prior to execution, and each step in the program consumes
+%one credit.
+This has been further developed to include the big-O notation
+so one can prove both the functional correctness and asymptotic complexity
+of higher-order imperative algorithms \ref{bigOImperative}.
+%for example the complexity of
+%the Quicksort algorithm 
+%is $\Theta n\ln n$ 
+
+\marginpar{more work on formalising complexity}.
+\marginpar{new changes up to this point.}
+
 
 In particular, the main problem we solved on top of previous work was
 coming up with a formally verified algorithm called $\blexersimp$ based