Parsing Linear Context-Free Rewriting Systems with Fast Matrix Multiplication

TL;DR

This paper introduces a fast matrix multiplication-based recognition algorithm for a subset of binary linear context-free rewriting systems, improving parsing times for various mildly context-sensitive grammatical formalisms.

Contribution

It presents a novel matrix multiplication recognition algorithm for binary LCFRS with complexity depending on contact rank, extending to general LCFRS and improving parsing bounds for multiple grammar formalisms.

Findings

Recognition algorithm runs in $O(n^{ ext{omega} d})$ time for a subset of LCFRS.

General binary LCFRS recognition runs in $O(n^{ ext{omega} d + 1})$ time.

Parsing times for several formalisms are improved, e.g., $O(n^{4.76})$ for mildly context-sensitive grammars.

Abstract

We describe a matrix multiplication recognition algorithm for a subset of binary linear context-free rewriting systems (LCFRS) with running time $O(n^{\omega d})$ where $M(m) = O(m^{\omega})$ is the running time for $m \times m$ matrix multiplication and $d$ is the "contact rank" of the LCFRS -- the maximal number of combination and non-combination points that appear in the grammar rules. We also show that this algorithm can be used as a subroutine to get a recognition algorithm for general binary LCFRS with running time $O(n^{\omega d + 1})$. The currently best known $\omega$ is smaller than $2.38$. Our result provides another proof for the best known result for parsing mildly context sensitive formalisms such as combinatory categorial grammars, head grammars, linear indexed grammars, and tree adjoining grammars, which can be parsed in time $O(n^{4.76})$. It also shows that inversion transduction grammars can be parsed in time $O(n^{5.76})$. In addition, binary LCFRS subsumes many other formalisms and types of grammars, for some of which we also improve the asymptotic complexity of parsing.