885 research outputs found
From algebra to logic: there and back again -- the story of a hierarchy
This is an extended survey of the results concerning a hierarchy of languages
that is tightly connected with the quantifier alternation hierarchy within the
two-variable fragment of first order logic of the linear order.Comment: Developments in Language Theory 2014, Ekaterinburg : Russian
Federation (2014
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
Adding modular predicates to first-order fragments
We investigate the decidability of the definability problem for fragments of
first order logic over finite words enriched with modular predicates. Our
approach aims toward the most generic statements that we could achieve, which
successfully covers the quantifier alternation hierarchy of first order logic
and some of its fragments. We obtain that deciding this problem for each level
of the alternation hierarchy of both first order logic and its two-variable
fragment when equipped with all regular numerical predicates is not harder than
deciding it for the corresponding level equipped with only the linear order and
the successor. For two-variable fragments we also treat the case of the
signature containing only the order and modular predicates.Relying on some
recent results, this proves the decidability for each level of the alternation
hierarchy of the two-variable first order fragmentwhile in the case of the
first order logic the question remains open for levels greater than two.The
main ingredients of the proofs are syntactic transformations of first order
formulas as well as the algebraic framework of finite categories
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
On FO2 quantifier alternation over words
We show that each level of the quantifier alternation hierarchy within
FO^2[<] -- the 2-variable fragment of the first order logic of order on words
-- is a variety of languages. We then use the notion of condensed rankers, a
refinement of the rankers defined by Weis and Immerman, to produce a decidable
hierarchy of varieties which is interwoven with the quantifier alternation
hierarchy -- and conjecturally equal to it. It follows that the latter
hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one
can effectively compute an integer m such that alpha is equivalent to a formula
with at most m+1 alternating blocks of quantifiers, but not to a formula with
only m-1 blocks. This is a much more precise result than what is known about
the quantifier alternation hierarchy within FO[<], where no decidability result
is known beyond the very first levels
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
- …