1,454 research outputs found
A Proof of the Factorization Forest Theorem
We show that for every homomorphism where is a finite
semigroup there exists a factorization forest of height \leq 3 \abs{S}. The
proof is based on Green's relations.Comment: 4 page
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
The separation problem for regular languages by piecewise testable languages
Separation is a classical problem in mathematics and computer science. It
asks whether, given two sets belonging to some class, it is possible to
separate them by another set of a smaller class. We present and discuss the
separation problem for regular languages. We then give a direct polynomial time
algorithm to check whether two given regular languages are separable by a
piecewise testable language, that is, whether a sentence can
witness that the languages are indeed disjoint. The proof is a reformulation
and a refinement of an algebraic argument already given by Almeida and the
second author
Going higher in the First-order Quantifier Alternation Hierarchy on Words
We investigate the quantifier alternation hierarchy in first-order logic on
finite words. Levels in this hierarchy are defined by counting the number of
quantifier alternations in formulas. We prove that one can decide membership of
a regular language to the levels (boolean combination of
formulas having only 1 alternation) and (formulas having only 2
alternations beginning with an existential block). Our proof works by
considering a deeper problem, called separation, which, once solved for lower
levels, allows us to solve membership for higher levels
A Characterization for Decidable Separability by Piecewise Testable Languages
The separability problem for word languages of a class by
languages of a class asks, for two given languages and
from , whether there exists a language from that
includes and excludes , that is, and . In this work, we assume some mild closure properties for
and study for which such classes separability by a piecewise
testable language (PTL) is decidable. We characterize these classes in terms of
decidability of (two variants of) an unboundedness problem. From this, we
deduce that separability by PTL is decidable for a number of language classes,
such as the context-free languages and languages of labeled vector addition
systems. Furthermore, it follows that separability by PTL is decidable if and
only if one can compute for any language of the class its downward closure wrt.
the scattered substring ordering (i.e., if the set of scattered substrings of
any language of the class is effectively regular).
The obtained decidability results contrast some undecidability results. In
fact, for all (non-regular) language classes that we present as examples with
decidable separability, it is undecidable whether a given language is a PTL
itself.
Our characterization involves a result of independent interest, which states
that for any kind of languages and , non-separability by PTL is
equivalent to the existence of common patterns in and
- …