The complexity of separation for levels in concatenation hierarchies

TL;DR

This paper studies the computational complexity of the separation problem for classes of regular languages, showing that for fixed alphabets, some levels are decidable in polynomial time while others are PSPACE-complete or harder.

Contribution

It proves that the complexity of separation is independent of input representation and provides new complexity bounds for specific levels of concatenation hierarchies.

Findings

Polynomial time algorithms for levels 1/2 and 1 with fixed alphabet

Separation is PSPACE-complete for level 3/2

Separation is between PSPACE-hard and EXPTIME for level 2

Abstract

We investigate the complexity of the separation problem associated to classes of regular languages. For a class C, C-separation takes two regular languages as input and asks whether there exists a third language in C which includes the first and is disjoint from the second. First, in contrast with the situation for the classical membership problem, we prove that for most classes C, the complexity of C-separation does not depend on how the input languages are represented: it is the same for nondeterministic finite automata and monoid morphisms. Then, we investigate specific classes belonging to finitely based concatenation hierarchies. It was recently proved that the problem is always decidable for levels 1/2 and 1 of any such hierarchy (with inefficient algorithms). Here, we build on these results to show that when the alphabet is fixed, there are polynomial time algorithms for both levels. Finally, we investigate levels 3/2 and 2 of the famous Straubing-Th\'erien hierarchy. We show that separation is PSPACE-complete for level 3/2 and between PSPACE-hard and EXPTIME for level 2.