2,558 research outputs found
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
of the dot-depth hierarchy are decidable in (previously only the
decidability was known). The same remains true if predicates mod for fixed
are allowed. - If modular predicates for arbitrary are allowed, then
the classes of the Boolean hierarchy over level are decidable. - For
the restricted case of a two-letter alphabet, the classes of the Boolean
hierarchy over level of the Straubing-Th\'erien hierarchy are
decidable in . This is the first decidability result for this hierarchy. -
The membership problems for all mentioned Boolean-hierarchy classes are
logspace many-one hard for . - The membership problems for quasi-aperiodic
languages and for -quasi-aperiodic languages are logspace many-one complete
for
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
Separation for dot-depth two
The dot-depth hierarchy of Brzozowski and Cohen classifies the star-free
languages of finite words. By a theorem of McNaughton and Papert, these are
also the first-order definable languages. The dot-depth rose to prominence
following the work of Thomas, who proved an exact correspondence with the
quantifier alternation hierarchy of first-order logic: each level in the
dot-depth hierarchy consists of all languages that can be defined with a
prescribed number of quantifier blocks. One of the most famous open problems in
automata theory is to settle whether the membership problem is decidable for
each level: is it possible to decide whether an input regular language belongs
to this level?
Despite a significant research effort, membership by itself has only been
solved for low levels. A recent breakthrough was achieved by replacing
membership with a more general problem: separation. Given two input languages,
one has to decide whether there exists a third language in the investigated
level containing the first language and disjoint from the second. The
motivation is that: (1) while more difficult, separation is more rewarding (2)
it provides a more convenient framework (3) all recent membership algorithms
are reductions to separation for lower levels.
We present a separation algorithm for dot-depth two. While this is our most
prominent application, our result is more general. We consider a family of
hierarchies that includes the dot-depth: concatenation hierarchies. They are
built via a generic construction process. One first chooses an initial class,
the basis, which is the lowest level in the hierarchy. Further levels are built
by applying generic operations. Our main theorem states that for any
concatenation hierarchy whose basis is finite, separation is decidable for
level one. In the special case of the dot-depth, this can be lifted to level
two using previously known results
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
Languages of Dot-depth One over Infinite Words
Over finite words, languages of dot-depth one are expressively complete for
alternation-free first-order logic. This fragment is also known as the Boolean
closure of existential first-order logic. Here, the atomic formulas comprise
order, successor, minimum, and maximum predicates. Knast (1983) has shown that
it is decidable whether a language has dot-depth one. We extend Knast's result
to infinite words. In particular, we describe the class of languages definable
in alternation-free first-order logic over infinite words, and we give an
effective characterization of this fragment. This characterization has two
components. The first component is identical to Knast's algebraic property for
finite words and the second component is a topological property, namely being a
Boolean combination of Cantor sets.
As an intermediate step we consider finite and infinite words simultaneously.
We then obtain the results for infinite words as well as for finite words as
special cases. In particular, we give a new proof of Knast's Theorem on
languages of dot-depth one over finite words.Comment: Presented at LICS 201
The Covering Problem
An important endeavor in computer science is to understand the expressive
power of logical formalisms over discrete structures, such as words. Naturally,
"understanding" is not a mathematical notion. This investigation requires
therefore a concrete objective to capture this understanding. In the
literature, the standard choice for this objective is the membership problem,
whose aim is to find a procedure deciding whether an input regular language can
be defined in the logic under investigation. This approach was cemented as the
right one by the seminal work of Sch\"utzenberger, McNaughton and Papert on
first-order logic and has been in use since then. However, membership questions
are hard: for several important fragments, researchers have failed in this
endeavor despite decades of investigation. In view of recent results on one of
the most famous open questions, namely the quantifier alternation hierarchy of
first-order logic, an explanation may be that membership is too restrictive as
a setting. These new results were indeed obtained by considering more general
problems than membership, taking advantage of the increased flexibility of the
enriched mathematical setting. This opens a promising research avenue and
efforts have been devoted at identifying and solving such problems for natural
fragments. Until now however, these problems have been ad hoc, most fragments
relying on a specific one. A unique new problem replacing membership as the
right one is still missing. The main contribution of this paper is a suitable
candidate to play this role: the Covering Problem. We motivate this problem
with 3 arguments. First, it admits an elementary set theoretic formulation,
similar to membership. Second, we are able to reexplain or generalize all known
results with this problem. Third, we develop a mathematical framework and a
methodology tailored to the investigation of this problem
The FO^2 alternation hierarchy is decidable
We consider the two-variable fragment FO^2[<] of first-order logic over
finite words. Numerous characterizations of this class are known. Th\'erien and
Wilke have shown that it is decidable whether a given regular language is
definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<]
is interesting since its satisfiability problem is in NP. Restricting the
number of quantifier alternations yields an infinite hierarchy inside the class
of FO^2[<]-definable languages. We show that each level of this hierarchy is
decidable. For this purpose, we relate each level of the hierarchy with a
decidable variety of finite monoids. Our result implies that there are many
different ways of climbing up the FO^2[<]-quantifier alternation hierarchy:
deterministic and co-deterministic products, Mal'cev products with definite and
reverse definite semigroups, iterated block products with J-trivial monoids,
and some inductively defined omega-term identities. A combinatorial tool in the
process of ascension is that of condensed rankers, a refinement of the rankers
of Weis and Immerman and the turtle programs of Schwentick, Th\'erien, and
Vollmer
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