92 research outputs found
Iterated periodicity over finite aperiodic semigroups
This paper provides a characterization of pseudowords over the pseudovariety of all finite aperiodic semigroups that can be described from the free generators using only the operations of multiplication and omega-power. A necessary and sufficient condition for this property to hold turns out to be given by the conjunction of two rather simple finiteness conditions: the nonexistence of infinite anti-chains of factors and the rationality of the language of McCammond normal forms of omega-terms that define factors of the given pseudoword. The relationship between pseudowords with this property and arbitrary pseudowords is also investigated.Projeto PTDC/MAT/65481/2006 financiado em parte pelo European Community
Fund FEDERPrograma AutoMathA da European Science Fundation (ESF)Projeto Egide-Grices 11113YMFundação para a Ciência e a Tecnologia (FCT
The omega-inequality problem for concatenation hierarchies of star-free languages
The problem considered in this paper is whether an inequality of omega-terms
is valid in a given level of a concatenation hierarchy of star-free languages.
The main result shows that this problem is decidable for all (integer and half)
levels of the Straubing-Th\'erien hierarchy
ω-terms over finite aperiodic semigroups
This paper provides a characterization of pseudowords over the pseudovariety of all finite aperiodic semigroups that are given by w-terms, that is that can be obtained from the free generators using only multiplication and the w-power. A necessary and sufficient condition for this property to hold turns out to be given by the conjunction of two rather simple finiteness conditions: the nonexistence of infinite anti-chains of factors and the rationality of the language of McCammond normal forms of w-terms that define factors.FCT through the Centro de Matemática da Universidade do Minho and the Centro de Matemática da Universidade do PortoEuropean Community Fund FEDE
Automaton Semigroups and Groups: On the Undecidability of Problems Related to Freeness and Finiteness
In this paper, we study algorithmic problems for automaton semigroups and
automaton groups related to freeness and finiteness. In the course of this
study, we also exhibit some connections between the algebraic structure of
automaton (semi)groups and their dynamics on the boundary. First, we show that
it is undecidable to check whether the group generated by a given invertible
automaton has a positive relation, i.e. a relation p = 1 such that p only
contains positive generators. Besides its obvious relation to the freeness of
the group, the absence of positive relations has previously been studied and is
connected to the triviality of some stabilizers of the boundary. We show that
the emptiness of the set of positive relations is equivalent to the dynamical
property that all (directed positive) orbital graphs centered at non-singular
points are acyclic.
Gillibert showed that the finiteness problem for automaton semigroups is
undecidable. In the second part of the paper, we show that this undecidability
result also holds if the input is restricted to be bi-reversible and invertible
(but, in general, not complete). As an immediate consequence, we obtain that
the finiteness problem for automaton subsemigroups of semigroups generated by
invertible, yet partial automata, so called automaton-inverse semigroups, is
also undecidable.
Erratum: Contrary to a statement in a previous version of the paper, our
approach does not show that that the freeness problem for automaton semigroups
is undecidable. We discuss this in an erratum at the end of the paper
Complete reducibility of pseudovarieties
The notion of reducibility for a pseudovariety has been introduced as an abstract property which may be used to prove decidability results for various pseudovariety constructions. This paper is a survey of recent results
establishing this and the stronger property of complete reducibility for specific pseudovarieties.FCT through the Centro de Matemática da Universidade do Minho and Centro de Matemática
da Universidade do Port
Closures of regular languages for profinite topologies
The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology. A similar descriptive procedure is shown to hold for the
pseudovariety A of aperiodic semigroups, where the closure is taken in the free aperiodic omega-semigroup. It is inherited by a
subpseudovariety of a given pseudovariety if both of them enjoy the property of being full. The pseudovariety A, as well as some of
its subpseudovarieties are shown to be full. The interest in such descriptions stems from the fact that, for each of the main pseudovarieties V in our examples, the closures of two regular languages are disjoint if and only if the languages can be separated by a language whose syntactic semigroup lies in V. In the cases of A and of the pseudovariety DA of semigroups in which all regular elements are idempotents, this is a new result.PESSOA French-Portuguese project
Egide-Grices 11113YM, "Automata, profinite semigroups and symbolic dynamics".FCT -- Fundação para a Ciência e a Tecnologia, respectively under the projects
PEst-C/MAT/UI0144/2011 and PEst-C/MAT/UI0013/2011.ANR 2010 BLAN 0202 01 FREC.AutoMathA programme of the European Science Foundation.FCT and the project PTDC/MAT/65481/2006 which was partly funded by the European Community Fund FEDER
Representations of the free profinite object over DA
In this paper, we extend to DA some techniques developed by Almeida and Weil, and Almeida and Zeitoun for the pseudovariety R to obtain representations of the implicit operations on DA: by labeled trees of finite height, by quasi-ternary labeled trees, and by labeled linear orderings. We prove that two implicit operations are equal over DA if and only if they have the same representation, for any of the three representations. We end the paper by relating these representations.info:eu-repo/semantics/publishedVersio
McCammond's normal forms for free aperiodic semigroups revisited
This paper revisits the solution of the word problem for omega-terms interpreted over finite aperiodic
semigroups, obtained by J. McCammond. The original proof of correctness of McCammond's
algorithm, based on normal forms for such terms, uses McCammond's solution of the word
problem for certain Burnside semigroups. In this paper, we establish a new, simpler, correctness
proof of McCammond's algorithm, based on properties of certain regular languages associated
with the normal forms. This method leads to new applications.Pessoa French-Portuguese project Egide-Grices 11113YMEuropean Regional Development Fund, through the programme COMPETEEuropean Community Fund FEDERANR 2010 BLAN 0202 01 FRE
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