Two-variable first order logic with modular predicates over words

We consider first order formulae over the signature consisting of the symbols of the alphabet, the symbol < (interpreted as a linear order) and the set MOD of modular numerical predicates. We study the expressive power of FO 2 [<, MOD], the two-variable first order logic over this signature, interpreted over finite words. We give an algebraic characterization of the corresponding regular languages in terms of their syntactic morphisms and we also give simple unambiguous regular expressions for them. It follows that one can decide whether a given regular language is captured by FO 2 [<, MOD]. Our proofs rely on a combination of arguments from semigroup theory (stamps), model theory (Ehrenfeucht-Fraïssé games) and combinatorics

Reducibility of joins involving some locally trivial pseudovarieties

In this paper, we show that sigma-reducibility is preserved under joins with K, where K is the pseudovariety of semigroups in which idempotents are left zeros. Reducibility of joins with D, the pseudovariety of semigroups in which idempotents are right zeros, is also considered. In this case, we were able to prove that sigma-reducibility is preserved for joins with pseudovarieties verifying a certain property of cancellation. As an example involving the semidirect product, we prove that Sl*K is k-tame, where Sl stands for the pseudovariety of semilattices.FCT through the Centro de Matemática da Universidade do MinhoEuropean Community Fund FEDE

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

Semidirect product with an order-computable pseudovariety and tameness

Dedicated to the memory of Walter Douglas Munn.The semidirect product of pseudovarieties of semigroups with an ordercomputable pseudovariety is investigated. The essential tool is the natural representation of the corresponding relatively free profinite semigroups and how it transforms implicit signatures. Several results concerning the behavior of the operation with respect to various kinds of tameness properties are obtained as applications.ESF programme “Automata: from Mathematics to Applications (AutoMathA)”.Project PTDC/MAT/65481/2006, which is partly funded by the European Community Fund FEDER.Fundação para a Ciência e a Tecnologia (FCT) through the Centro de Matemática da Universidade do Porto and Centro de Matemática da Universidade do Minh

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