On the complexity of the universality and inclusion problems for unambiguous context-free grammars (technical report)

TL;DR

This paper investigates the computational complexity of universality and inclusion problems for unambiguous context-free grammars, proposing a PSPACE algorithm and exploring their relation to open problems in computational complexity.

Contribution

It introduces a PSPACE algorithm for the universality problem for unambiguous CFGs and connects the measure computation to the SQRTSUM problem, highlighting complexity challenges.

Findings

Proposes a PSPACE algorithm for universality in unambiguous CFGs.

Shows the measure computation problem is as hard as SQRTSUM.

Establishes the decidability of certain problems in this domain.

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.