3,798 research outputs found
The Complexity of Separation for Levels in Concatenation Hierarchies
We investigate the complexity of the separation problem associated to classes
of regular languages. For a class C, C-separation takes two regular languages
as input and asks whether there exists a third language in C which includes the
first and is disjoint from the second. First, in contrast with the situation
for the classical membership problem, we prove that for most classes C, the
complexity of C-separation does not depend on how the input languages are
represented: it is the same for nondeterministic finite automata and monoid
morphisms. Then, we investigate specific classes belonging to finitely based
concatenation hierarchies. It was recently proved that the problem is always
decidable for levels 1/2 and 1 of any such hierarchy (with inefficient
algorithms). Here, we build on these results to show that when the alphabet is
fixed, there are polynomial time algorithms for both levels. Finally, we
investigate levels 3/2 and 2 of the famous Straubing-Th\'erien hierarchy. We
show that separation is PSPACE-complete for level 3/2 and between PSPACE-hard
and EXPTIME for level 2
Separating regular languages with two quantifier alternations
We investigate a famous decision problem in automata theory: separation.
Given a class of language C, the separation problem for C takes as input two
regular languages and asks whether there exists a third one which belongs to C,
includes the first one and is disjoint from the second. Typically, obtaining an
algorithm for separation yields a deep understanding of the investigated class
C. This explains why a lot of effort has been devoted to finding algorithms for
the most prominent classes.
Here, we are interested in classes within concatenation hierarchies. Such
hierarchies are built using a generic construction process: one starts from an
initial class called the basis and builds new levels by applying generic
operations. The most famous one, the dot-depth hierarchy of Brzozowski and
Cohen, classifies the languages definable in first-order logic. Moreover, it
was shown by Thomas that it corresponds to the quantifier alternation hierarchy
of first-order logic: each level in the dot-depth corresponds to the languages
that can be defined with a prescribed number of quantifier blocks. Finding
separation algorithms for all levels in this hierarchy is among the most famous
open problems in automata theory.
Our main theorem is generic: we show that separation is decidable for the
level 3/2 of any concatenation hierarchy whose basis is finite. Furthermore, in
the special case of the dot-depth, we push this result to the level 5/2. In
logical terms, this solves separation for : first-order sentences
having at most three quantifier blocks starting with an existential one
Advances and applications of automata on words and trees : abstracts collection
From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
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