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

$PSPACE$.