Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages
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
$PSPACE$.
Automata and formal languages
computational complexity
dot-depth hierarchy
Boolean hierarchy
decidability
efficient algorithms
337-348
Regular Paper
Christian
Glasser
Christian Glasser
Heinz
Schmitz
Heinz Schmitz
Victor
Selivanov
Victor Selivanov
10.4230/LIPIcs.STACS.2008.1355
Creative Commons Attribution-NoDerivs 3.0 Unported license
https://creativecommons.org/licenses/by-nd/3.0/legalcode