structure that can be analyzed by computers for further processing.

This is because parsing algorithms are often ad hoc and their semantics

One new development in formal grammar is the introduction of Parsing Expression Grammar (PEG) as an extension of the standard Context Free Grammar (CFG)\cite{Ford04a}. The extension introduces new regular operators such as negation and conjunction to the right hand side of productions, as well as well as an priority ordering on productions. With these extensions, PEG becomes more powerful such that disambiguation formerly expressed using semantic filters can now be expressed directly using production expressions. This means a simpler and more systematic treatment of ambiguity and more concise grammar specification for programming languages.

However, one disadvantage of PEG is that it does not allow left recursion in grammar specification, because the accompanying algorithms of PEG\cite{Ford02b} can not deal with left recursions. Although some authors claimed new PEG parsing algorithm for left recursion\cite{conf/pepm/WarthDM08}, there is no correctness proof, not even in paper-and-pencil form. One aim of this research is to formalize a fixed point semantics of PEG, based on which an efficient, certified parsing algorithm is given.

There are several existing works we can draw upon:

\begin{enumerate}

    \item The works on PEG.

        \begin {enumerate}

            \item An operation semantics for PEG has already been given in \cite{Ford04a}, but it is not adequate to deal with left recursions. But this work gives at least a precise description of what the original PEG  meant for. This will serve an a basis to show the conservativeness of the fixed point semantics we are going to develop.

            \item The new algorithm\cite{conf/pepm/WarthDM08} which claimed to be able to deal with left recursions. Although there is no correctness proof yet, this may provide some useful inspirations for our new algorithm design.

        \end{enumerate}

    \item The works on Boolean Grammar\cite{Okhotin/04a}. Boolean Grammar is very closely related to PEG, because it also contains negative and conjunctive grammars. The main differences are: First, Boolean Grammar has no ordering on productions; Second: Boolean Grammar does not contain STAR operator. There are two works about Boolean Grammar which might be useful for this research:

        \begin{enumerate}

            \item A fixed point semantics for Boolean Grammar\cite{journals/iandc/KountouriotisNR09}. The idea to define the semantics of negative and conjunctive operators is certainly what we can borrow. Therefore, this work gives the basis on which we can add in production ordering and STAR operator.

            \item A parsing algorithm for Boolean Grammar based on CYK parsing\cite{journals/iandc/KountouriotisNR09}. The draw back of CYK parsing is that: the original grammar specification needs to be transformed into a normal form. This transformation may lead to grammar explosion and is undesirable. One aim of this research is to see whether this transformation can be avoided. For this purpose, other parsing style may provide useful inspirations, for example:

                    \begin{enumerate}

                        \item Derivative Parsing\cite{Brzozowski64,Almeidaetal10,OwensReppyTuron09,journals/corr/abs-1010-5023}. Christian Urban has used derivative methods to establish the correctness of a regular expression matcher, as well the the finite partition property of regular expression\cite{WuZhangUrban11}.  There are well founded envisage that the derivative methods may provide the foundation to the new parsing algorithms of PEG.

                        \item Early parsing\cite{Earley70,AycHor02}. It is a refinement of CYK parsing which does not require the transformation to normal forms, and therefore provide one possible direction to adapt the current CYK based parsing algorithm of Boolean Grammar for PEG grammar.

                        \item The new parsing algorithm proposed by Tom Ridge[???]. Recently, T. Ridge has proposed and certified an combinator style parsing algorithm for CFG, which borrows some ideas from Early parsing. The proposed algorithm is very simple and elegant. We are going to strive for a parsing algorithm as elegant as this one.

                    \end{enumerate}

                Which of the above possibilities will finally get into our final solutions is an interesting point about this current research.

        \end{enumerate}

\end{enumerate}

Based on these works, we are quite confident that our idea may lead to some concrete results.

\small

\bibliography{Journal/document/root}

\bibliographystyle{abbrv}