Cocke--Younger--Kasami--Schwartz--Zippel algorithm and relatives

TL;DR

This paper introduces a polynomial-time algorithm for the limited equivalence problem of unambiguous grammars, significantly improving over the naive exponential approach and offering broader applicability.

Contribution

It presents a novel polynomial-time algorithm for the limited equivalence problem of unambiguous grammars, combining classic ideas and extending their utility.

Findings

Polynomial-time algorithm for limited equivalence problem

Reduction from exponential to polynomial complexity

Applicability to various other scenarios

Abstract

The equivalence problem for unambiguous grammars is an important, but very difficult open question in formal language theory. Consider the \emph{limited} equivalence problem for unambiguous grammars -- for two unambiguous grammars $G_1$ and $G_2$, tell whether or not they describe the same set of words of length $n$. Obviously, the naive approach requires exponential time with respect to $n$. By combining two classic algorithmic ideas, I introduce a $O({\rm poly}(n, |G_1|, |G_2|))$ algorithm for this problem. Moreover, the ideas behind the algorithm prove useful in various other scenarious.