Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages

Christian Glasser (1) , Heinz Schmitz (2) , Victor Selivanov (3)

1 Theoretische Informatik [Würzburg]
2 Fachbereich Informatik
3 Laboratory of Theoretical Programming

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-Thérien 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 $PSPACE$.

Mots clés: automata and formal languages; computational complexity; dot-depth hierarchy; Boolean hierarchy; decidability; efficient algorithms

Domaines: Complexité [cs.CC]; Théorie de l'information [cs.IT]

Christian Glasser, Heinz Schmitz, Victor Selivanov. Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages. STACS 2008, Feb 2008, Bordeaux, France. pp.337-348. ⟨hal-00227550⟩