# Fast context-free grammar parsing requires fast boolean matrix multiplication

@article{Lee2002FastCG,
title={Fast context-free grammar parsing requires fast boolean matrix multiplication},
author={Lillian Lee},
journal={J. ACM},
year={2002},
volume={49},
pages={1-15}
}
• Lillian Lee
• Published 15 December 2001
• Computer Science
• J. ACM
In 1975, Valiant showed that Boolean matrix multiplication can be used for parsing context-free grammars (CFGs), yielding the asympotically fastest (although not practical) CFG parsing algorithm known. We prove a dual result: any CFG parser with time complexity <i>O</i>(<i>gn</i><sup>3-∈</sup>), where <i>g</i> is the size of the grammar and <i>n</i> is the length of the input string, can be efficiently converted into an algorithm to multiply <i>m</i> × <i>m</i> Boolean matrices in time <i>O</i…
139 Citations

## Figures from this paper

Approximating Language Edit Distance Beyond Fast Matrix Multiplication: Ultralinear Grammars Are Where Parsing Becomes Hard!
• Computer Science
ICALP
• 2017
Additive approximation algorithms for language edit distance are studied, providing two explicit combinatorial algorithms to obtain a string with minimum edit distance with performance dependencies on either the number of non-linear productions, k^*, or theNumber of nested non- linear production, k, used in the optimal derivation.
Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars
• Computer Science
CPM
• 2017
It is shown that, even on constant-size grammars, any improvement on Rajasekaran and Yooseph's parser would imply a breakthrough for the $k-Clique problem, and establishes tree-adjoining grammar parsing as a practically relevant problem with the unusual running time of$n^{2\omega}\$, up to lower order factors.
If the Current Clique Algorithms are Optimal, So is Valiant's Parser
• Computer Science
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
• 2015
It is proved that any improvement on Valiant' s algorithm, even for constant size grammars, would imply a breakthrough algorithm for the k-Clique problem: given a graph on n nodes, decide if there are k that form a clique.
CFG Parsing and Boolean Matrix Multiplication
It is shown that fast CFG parsing requires fast Boolean matrix multiplication, and vice versa, and the fundamental theorem that can be derived is that fastCFG parsing required fast Boolean Matrix Multiplication.
Language Edit Distance and Maximum Likelihood Parsing of Stochastic Grammars: Faster Algorithms and Connection to Fundamental Graph Problems
• B. Saha
• Computer Science
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
• 2015
This paper gives the first such algorithm that computes language edit distance almost optimally and designs the very first subcubic (Õ(nω)) algorithm that given an arbitrary stochastic context free grammar, and a string returns a nearly-optimal maximum likelihood parsing of that string.
An Error Correcting Parser for Context Free Grammars that Takes Less Than Cubic Time
• Computer Science
LATA
• 2016
An error correcting 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 compute the tropical product of two $$n\times n$$ matrices.
Reducing non-determinism in right nulled GLR parsers
• Computer Science
Acta Informatica
• 2004
It is shown that in certain cases conflicts can be removed from an RN table without compromising the correctness of the associated parser, which improves the performance of the parser and allows resolved RN tables for LR(1) grammars to be used with standard (deterministic) LR stack-based parsing algorithms.
Parsing expression grammars: a recognition-based syntactic foundation
• B. Ford
• Computer Science
POPL '04
• 2004
PEGs address frequently felt expressiveness limitations of CFGs and REs, simplifying syntax definitions and making it unnecessary to separate their lexical and hierarchical components, and are here proven equivalent in effective recognition power.
Syntax Analysers For EBNF Grammars
• Computer Science
• 2018
A formal specification for a parse tree representation of derivations which reflects the EBNF structure of the grammar, is worst case cubic size, and captures all derivations in the case of ambiguity is given.

## References

SHOWING 1-10 OF 63 REFERENCES
An efficient context-free parsing algorithm
A parsing algorithm which seems to be the most efficient general context-free algorithm known is described and appears to be superior to the top-down and bottom-up algorithms studied by Griffiths and Petrick.
An efficient context-free parsing algorithm
A parsing algorithm which seems to be the most efficient general context-free algorithm known is described and appears to be superior to the top-down and bottom-up algorithms studied by Griffiths and Petrick.
On the Complexity of General Context-Free Language Parsing and Recognition (Extended Abstract)
It is shown that parsing strings of length n is harder than recognizing such strings by a factor of only 0(log n), at most, and that these problems are solvable in linear time only if n×n Boolean matrix multiplication can be done in 0(n2).
An Efficient Recognition and Syntax-Analysis Algorithm for Context-Free Languages
It is shown in this paper that a context-free language is n cubed-recognizable in the sense of Hartmanis and Stearns and it is n to the 4th power- Recognizable by a single-head single-tape Turing machine.
Tree-Adjoining Grammar Parsing and Boolean Matrix Multiplication
• G. Satta
• Computer Science
Comput. Linguistics
• 1994
The computational problem of parsing a sentence in a tree-adjoining language is investigated and it is shown that any algorithm for the solution of the former problem can easily be converted into an algorithm to solve the latter problem.
On the Parsing of Deterministic Languages
• Computer Science
JACM
• 1974
Rigorous proofs that the families of deterministic, LR (k), and bounded right context languages are coextensive are presented for the first time.