On the Complexity of the Universality and Inclusion Problems for Unambiguous Context-Free Grammars

TL;DR

This paper investigates the computational complexity of universality and inclusion problems for unambiguous context-free grammars, proposing a PSPACE algorithm and connecting the problem to the SQRTSUM complexity class.

Contribution

It introduces a PSPACE algorithm for the universality problem and links the quantitative measure of unambiguous languages to the SQRTSUM problem, highlighting complexity challenges.

Findings

Decidable universality problem for unambiguous CFGs.

Proposed PSPACE algorithm based on recurrence equations.

Coin-flip measure computation is as hard as SQRTSUM.

Abstract

We study the computational complexity of universality and inclusion problems for unambiguous finite automata and context-free grammars. We observe that several such problems can be reduced to the universality problem for unambiguous context-free grammars. The latter problem has long been known to be decidable and we propose a PSPACE algorithm that works by reduction to the zeroness problem of recurrence equations with convolution. We are not aware of any non-trivial complexity lower bounds. However, we show that computing the coin-flip measure of an unambiguous context-free language, a quantitative generalisation of universality, is hard for the long-standing open problem SQRTSUM.