32 research outputs found
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
On factorisation forests
The theorem of factorisation forests shows the existence of nested
factorisations -- a la Ramsey -- for finite words. This theorem has important
applications in semigroup theory, and beyond. The purpose of this paper is to
illustrate the importance of this approach in the context of automata over
infinite words and trees. We extend the theorem of factorisation forest in two
directions: we show that it is still valid for any word indexed by a linear
ordering; and we show that it admits a deterministic variant for words indexed
by well-orderings. A byproduct of this work is also an improvement on the known
bounds for the original result. We apply the first variant for giving a
simplified proof of the closure under complementation of rational sets of words
indexed by countable scattered linear orderings. We apply the second variant in
the analysis of monadic second-order logic over trees, yielding new results on
monadic interpretations over trees. Consequences of it are new caracterisations
of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page
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
Cliquewidth and dimension
We prove that every poset with bounded cliquewidth and with sufficiently
large dimension contains the standard example of dimension as a subposet.
This applies in particular to posets whose cover graphs have bounded treewidth,
as the cliquewidth of a poset is bounded in terms of the treewidth of the cover
graph. For the latter posets, we prove a stronger statement: every such poset
with sufficiently large dimension contains the Kelly example of dimension
as a subposet. Using this result, we obtain a full characterization of the
minor-closed graph classes such that posets with cover graphs in
have bounded dimension: they are exactly the classes excluding
the cover graph of some Kelly example. Finally, we consider a variant of poset
dimension called Boolean dimension, and we prove that posets with bounded
cliquewidth have bounded Boolean dimension.
The proofs rely on Colcombet's deterministic version of Simon's factorization
theorem, which is a fundamental tool in formal language and automata theory,
and which we believe deserves a wider recognition in structural and algorithmic
graph theory
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
Algebraic Characterization of FO for Scattered Linear Orderings
We prove that for the class of sets of words indexed by countable scattered linear orderings, there is an equivalence between definability in first-order logic, star-free expressions with marked product, and recognizability by finite aperiodic semigroups which satisfy some additional equation
Complementation of Rational Sets on Countable Scattered Linear Orderings
In a preceding paper (Bruyère and Carton, automata on linear orderings, MFCS'01), automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite and even transfinite words studied by Büchi. Kleene's theorem has been generalized to these words. We prove that rational sets of words on countable scattered linear orderings are closed under complementation using an algebraic approach
Locally countable pseudovarieties
The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressible in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J -classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality 1 6 x n is locally countable if and only if n = 1