1,797 research outputs found
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
Three hierarchies of transducers
Composition of top-down tree transducers yields a proper hierarchy of transductions and of output languages. The same is true for ETOL systems (viewed as transducers) and for two-way generalized sequential machines
The dot-depth and the polynomial hierarchy correspond on the delta levels
It is well-known that the Sigma_k- and Pi_k-levels of the dot-depth hierarchy and the polynomial hierarchy correspond via leaf languages. In this paper this correspondence will be extended to the Delta_k-levels of these hierarchies: Leaf^P(Delta_k^L) = Delta_k^p
On All Things Star-Free
We investigate the star-free closure, which associates to a class of languages its closure under Boolean operations and marked concatenation. We prove that the star-free closure of any finite class and of any class of groups languages with decidable separation (plus mild additional properties) has decidable separation. We actually show decidability of a stronger property, called covering. This generalizes many results on the subject in a unified framework. A key ingredient is that star-free closure coincides with another closure operator where Kleene stars are also allowed in restricted contexts
A method of encoding generalized link diagrams
We describe a method of encoding various types of link diagrams, including
those with classical, flat, rigid, welded, and virtual crossings. We show that
this method may be used to encode link diagrams, up to equivalence, in a
notation whose length is a cubic function of the number of 'riser marks'. For
classical knots, the minimal number of such marks is twice the bridge index,
and a classical knot diagram in minimal bridge form with bridge index may
be encoded in space . A set of moves on the notation is
defined. As a demonstration of the utility of the notation we give another
proof that the Kishino virtual knot is non-classical.Comment: 17 pages, 13 figures; to appear in the Journal of Knot Theory & Its
Ramification
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