Efficient algorithms for membership in boolean hierarchies of regular languages

Christian Glaßer, Heinz Schmitz, Victor Selivanov
2016 Theoretical Computer Science  
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 Σ1 of the dot-depth hierarchy are decidable in NL (previously only the decidability was known). The same remains true if
more » ... cates mod d for fixed d are allowed. • If modular predicates for arbitrary d are allowed, then the classes of the Boolean hierarchy over level Σ1 are decidable. • For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level Σ2 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.
doi:10.1016/j.tcs.2016.07.017 fatcat:tqu5qwscmbbbvebstf2lkexy2m