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The problem of parsing has been studied extensively for various formalgrammars. Given an input string and a grammar, the parsing problem is to checkif the input string belongs to the language generated by the grammar. A closelyrelated problem of great importance is one where the input are a string ${\calI}$ and a grammar $G$ and the task is to produce a string ${\cal I}'$ thatbelongs to the language generated by $G$ and the `distance' between ${\cal I}$and ${\cal I}'$ is the smallest (from among all the strings in the language).Specifically, if ${\cal I}$ is in the language generated by $G$, then theoutput should be ${\cal I}$. Any parser that solves this version of the problemis called an {\em error correcting parser}. In 1972 Aho and Peterson presenteda cubic time error correcting parser for context free grammars. Since then thisasymptotic time bound has not been improved under the (standard) assumptionthat the grammar size is a constant. In this paper we present an errorcorrecting parser for context free grammars that runs in $O(T(n))$ time, where$n$ is the length of the input string and $T(n)$ is the time needed to computethe tropical product of two $n\times n$ matrices.  In this paper we also present an $\frac{n}{M}$-approximation algorithm forthe {\em language edit distance problem} that has a run time of $O(Mn^\omega)$,where $O(n^\omega)$ is the time taken to multiply two $n\times n$ matrices. Tothe best of our knowledge, no approximation algorithms have been proposed forerror correcting parsing for general context free grammars.

An error correcting parser for context free grammars that takes less than cubic time