EF+EX Forest Algebras

We examine languages of unranked forests definable using the temporal operators EF and EX. We characterize the languages definable in this logic, and various fragments thereof, using the syntactic forest algebras introduced by Bojanczyk and Walukiewicz. Our algebraic characterizations yield efficient algorithms for deciding when a given language of forests is definable in this logic. The proofs are based on understanding the wreath product closures of a few small algebras, for which we introduce a general ideal theory for forest algebras. This combines ideas from the work of Bojanczyk and Walukiewicz for the analogous logics on binary trees and from early work of Stiffler on wreath product of finite semigroups

On a conjecture concerning dot-depth two languages

AbstractIn this paper, we study the second level of the dot-depth hierarchy for star-free regular languages. We investigate a necessary condition stated by Straubing for a language to have dot-depth two, and prove that it is sufficient for languages whose syntactic monoid is inverse with three inverse generators. Also we disprove a conjecture according to which Straubing's condition would be equivalent to both dot-depth two and another condition expressed in terms of two-sided semidirect product

From algebra to logic: there and back again -- the story of a hierarchy

This is an extended survey of the results concerning a hierarchy of languages that is tightly connected with the quantifier alternation hierarchy within the two-variable fragment of first order logic of the linear order.Comment: Developments in Language Theory 2014, Ekaterinburg : Russian Federation (2014

Vectorial Languages and Linear Temporal Logic

International audienceDetermining for a given deterministic complete automaton the sequence of visited states while reading a given word is the core of important problems with automata-based solutions, such as approximate string matching. The main difficulty is to do this computation efficiently, especially when dealing with very large texts. Considering words as vectors and working on them using vectorial (parallel) operations allows to solve the problem faster than in linear time using sequential computations. In this paper, we show first that the set of vectorial operations needed by an algorithm representing a given automaton depends only on the language accepted by the automaton. We give precise characterizations of vectorial algorithms for star-free, solvable and regular languages in terms of the vectorial operations allowed. We also consider classes of languages associated with restricted sets of vectorial operations and relate them with languages defined by fragments of linear temporal logic. Finally, we consider the converse problem of constructing an automaton from a given vectorial algorithm. As a byproduct, we show that the satisfiability problem for some extensions of linear-time temporal logic characterizing solvable and regular languages is PSPACE-complete

On FO2 quantifier alternation over words

We show that each level of the quantifier alternation hierarchy within FO^2[<] -- the 2-variable fragment of the first order logic of order on words -- is a variety of languages. We then use the notion of condensed rankers, a refinement of the rankers defined by Weis and Immerman, to produce a decidable hierarchy of varieties which is interwoven with the quantifier alternation hierarchy -- and conjecturally equal to it. It follows that the latter hierarchy is decidable within one unit: given a formula alpha in FO^2[<], one can effectively compute an integer m such that alpha is equivalent to a formula with at most m+1 alternating blocks of quantifiers, but not to a formula with only m-1 blocks. This is a much more precise result than what is known about the quantifier alternation hierarchy within FO[<], where no decidability result is known beyond the very first levels

Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which form a basis of a free submonoid in the free monoid of all words (over a given alphabet) and local Rees extensions provide structural insight into regular languages over words. So, it seems to be a legitimate title for an extended abstract presented at the conference WORDS 2017. However, this work is more ambitious, it outlines some less obvious but much more interesting link between these topics. This link is based on a structure theory of finite monoids with varieties of groups and the concept of local divisors playing a prominent role. Parts of this work appeared in a similar form in conference proceedings where proofs and further material can be found.Comment: Extended abstract of an invited talk given at WORDS 201

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 $\mathcal{B}\Sigma_2$ (boolean combination of formulas having only 1 alternation) and $\Sigma_3$ (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

On Varieties of Ordered Automata

The Eilenberg correspondence relates varieties of regular languages to pseudovarieties of finite monoids. Various modifications of this correspondence have been found with more general classes of regular languages on one hand and classes of more complex algebraic structures on the other hand. It is also possible to consider classes of automata instead of algebraic structures as a natural counterpart of classes of languages. Here we deal with the correspondence relating positive $\mathcal C$-varieties of languages to positive $\mathcal C$-varieties of ordered automata and we present various specific instances of this correspondence. These bring certain well-known results from a new perspective and also some new observations. Moreover, complexity aspects of the membership problem are discussed both in the particular examples and in a general setting

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