Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages

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{'\e}rien 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
   $PSPACE$.