Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages

TL;DR

This paper develops efficient algorithms to decide membership in various Boolean hierarchies of regular languages, establishing new decidability results and complexity bounds, including NL and PSPACE classifications.

Contribution

It introduces forbidden-chain characterizations and proves decidability and complexity results for Boolean hierarchies over regular language classes.

Findings

Boolean hierarchy over $oldsymbol{ extSigma_1}$ is decidable in NL.

Decidability extends to classes with modular predicates.

Membership problems are NL-hard and PSPACE-complete for certain classes.

Abstract

The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: - The classes of the Boolean hierarchy over level $\Sigma_1$ of the dot-depth hierarchy are decidable in $NL$ (previously only the decidability was known). The same remains true if predicates mod $d$ for fixed $d$ are allowed. - If modular predicates for arbitrary $d$ are allowed, then the classes of the Boolean hierarchy over level $\Sigma_1$ are decidable. - For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level $\Sigma_2$ of the Straubing-Th\'erien hierarchy are decidable in $NL$. This is the first decidability result for this hierarchy. - The membership problems for all mentioned Boolean-hierarchy classes are logspace many-one hard for $NL$. - The membership problems for quasi-aperiodic languages and for $d$-quasi-aperiodic languages are logspace many-one complete for $PSPACE$.