The FO^2 alternation hierarchy is decidable

We consider the two-variable fragment FO^2[<] of first-order logic over finite words. Numerous characterizations of this class are known. Th\'erien and Wilke have shown that it is decidable whether a given regular language is definable in FO^2[<]. From a practical point of view, as shown by Weis, FO^2[<] is interesting since its satisfiability problem is in NP. Restricting the number of quantifier alternations yields an infinite hierarchy inside the class of FO^2[<]-definable languages. We show that each level of this hierarchy is decidable. For this purpose, we relate each level of the hierarchy with a decidable variety of finite monoids. Our result implies that there are many different ways of climbing up the FO^2[<]-quantifier alternation hierarchy: deterministic and co-deterministic products, Mal'cev products with definite and reverse definite semigroups, iterated block products with J-trivial monoids, and some inductively defined omega-term identities. A combinatorial tool in the process of ascension is that of condensed rankers, a refinement of the rankers of Weis and Immerman and the turtle programs of Schwentick, Th\'erien, and Vollmer

Commutative positive varieties of languages

We study the commutative positive varieties of languages closed under various operations: shuffle, renaming and product over one-letter alphabets

Algebraic Characterization of the Alternation Hierarchy in FO^2[<] on Finite Words

We give an algebraic characterization of the quantifier alternation hierarchy in first-order two-variable logic on finite words. As a result, we obtain a new proof that this hierarchy is strict. We also show that the first two levels of the hierarchy have decidable membership problems, and conjecture an algebraic decision procedure for the other levels

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

Dual Space of a Lattice as the Completion of a Pervin Space

16th International Conference, RAMiCS 2017, Lyon, France, May 15-18, 2017, ProceedingsInternational audienceThis survey paper presents well-known results from a new angle. A Pervin space is a set X equipped with a set of subsets,called the blocks of the Pervin space. Blocks are closed under finite intersections and finite unions and hence form a lattice of subsets of X. Pervin spaces are thus easier to define than topological spaces or (quasi)-uniform spaces. As a consequence, most of the standard topological notions, like convergence and cluster points, specialisation order, filtersand Cauchy filters, complete spaces and completion are much easier to define for Pervin spaces. In particular, the completion of a Pervin space turns out to be the dual space (in the sense of Stone) of the original lattice.We show that any lattice of subsets can be described by a set of inequations of the form u ≤ v, where u and v are elements of its dual space. Applications to formal languages and complexity classes are given.Cet article de synthèse présente des résultats bien connus sous un nouvel angle. Un espace de Pervin est unensemble X équipé d'un ensemble de parties, appelé les blocs de l'espace de Pervin. Les blocs sont fermés par intersection finie et union finie et forment ainsi un treillis de parties de X. Les espaces de Pervin sont doncplus faciles à définir que les espaces topologiques ou les espaces (quasi-)uniformes. Par conséquent, la plupart des notions topologiques, comme la convergence et les points d'adhérence, l'ordre de spécialisation, les filtres de Cauchy, les espaces complets et la complétion sont beaucoup plus faciles à définir pour les espaces Pervin. En particulier, la complétion d'un espace Pervin s'avère être l'espace dual (au sens de Stone) du treillis de départ.Nous montrons que tout treillis de parties peut être décrit par un ensemble d'inéquations de la forme u ≤ v, où u et v sont des éléments de son espace dual. On donne des applications aux langages formels et aux classes de complexité

Commutative positive varieties of languages

We study the commutative positive varieties of languages closed under various operations: Shuffle, renaming and product over one-letter alphabets

Complete reducibility of systems of equations with respect to R

It is shown that the pseudovariety R of all finite R-trivial semigroups is completely reducible with respect to the canonical signature. Informally, if the variables in a finite system of equations with rational constraints may be evaluated by pseudowords so that each value belongs to the closure of the corresponding rational constraint and the system is verified in R, then there is some such evaluation which is “regular”, that is one in which, additionally, the pseudowords only involve multiplications and ω-powers.Fundação para a Ciência e a Tecnologia (FCT)Pessoa bilateral project Egide/Grices 11113Y

On fixed points of the lower set operator

Lower subsets of an ordered semigroup form in a natural way an ordered semigroup. This lower set operator gives an analogue of the power operator already studied in semigroup theory. We present a complete description of the lower set operator applied to varieties of ordered semigroups. We also obtain large families of fixed points for this operator applied to pseudovarieties of ordered semigroups, including all examples found in the literature. This is achieved by constructing six types of inequalities that are preserved by the lower set operator. These types of inequalities are shown to be independent in a certain sense. Several applications are also presented, including the preservation of the period for a pseudovariety of ordered semigroups whose image under the lower set operator is proper

The linear nature of pseudowords

Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.The work of the first, third, and fourth authors was partly supported by the Pessoa French-Portuguese project “Separation in automata theory: algebraic, logical, and combinatorial aspects”. The work of the first three authors was also partially supported respectively by CMUP (UID/MAT/ 00144/2019), CMUC (UID/MAT/00324/2019), and CMAT (UID/MAT/ 00013/2013), which are funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020. The work of the fourth author was partly supported by ANR 2010 BLAN 0202 01 FREC and by the DeLTA project ANR-16-CE40-000