Opérations polynomiales et hiérarchies de concaténation

RésuméSoit C une classe de langages. Notons Pol(C) la fermeture polynomiale de C. Pol(C) est la plus petite classe de langages contenant C et fermée par union finie et produit marqué LaL' où a est une lettre. Nous déterminons les clôtures polynomiales de diverses classes de langages rationnels puis nous étudions les propriétés des fermetures polynomiales. Par exemple, si C est fermée par quotients (resp. quotients et morphisme inverse), alors il en est de même de Pol(C). Notre résultat principal montre que si C est une algèbre de Boole fermée par résiduels alors Pol(C) est fermée par intersection. Comme application, nous affinons la hiérarchie de concaténation introduite par Straubing et nous prouvons la décidabilité des niveaux 12 et 32 de cette hiérarchie.AbstractGiven a class C of languages, let Pol(C) be the polynomial closure of C, that is, the smallest class of languages containing C and closed under the operations union and marked product LaL', where a is a letter. We determine the polynomial closure of various classes of rational languages and we study the properties of polynomial closures. For instance, if C is closed under quotients (resp. quotients and inverse morphism) then Pol(C) has the same property. Our main result shows that if C is a boolean algebra closed under quotients then Pol(C) is closed under intersection. As an application, we refine the concatenation hierarchy introduced by Straubing and we show that the levels 12 and 32 of this hierarcy are decidable

Regular Cost Functions, Part I: Logic and Algebra over Words

The theory of regular cost functions is a quantitative extension to the classical notion of regularity. A cost function associates to each input a non-negative integer value (or infinity), as opposed to languages which only associate to each input the two values "inside" and "outside". This theory is a continuation of the works on distance automata and similar models. These models of automata have been successfully used for solving the star-height problem, the finite power property, the finite substitution problem, the relative inclusion star-height problem and the boundedness problem for monadic-second order logic over words. Our notion of regularity can be -- as in the classical theory of regular languages -- equivalently defined in terms of automata, expressions, algebraic recognisability, and by a variant of the monadic second-order logic. These equivalences are strict extensions of the corresponding classical results. The present paper introduces the cost monadic logic, the quantitative extension to the notion of monadic second-order logic we use, and show that some problems of existence of bounds are decidable for this logic. This is achieved by introducing the corresponding algebraic formalism: stabilisation monoids.Comment: 47 page

Some results on the generalized star-height problem

We prove some results related to the generalized star-height problem. In this problem, as opposed to the restricted star-height problem, complementation is considered as a basic operator. We first show that the class of languages of star-height ? n is closed under certain operations (left and right quotients, inverse alphabetic morphisms, injective star-free substitutions). It is known that languages recognized by a commutative group are of star-height 1. We extend this result to nilpotent groups of class 2 and to the groups that divide a semidirect product of a commutative group by (Z/2Z)n. In the same direction, we show that one of the languages that was conjectured to be of star height 2 during the past ten years, is in fact of star height 1. Next we show that if a rational language L is recognized by a monoid of the variety generated by wreath products of the form M o (G o N), where M and N are aperiodic monoids, and G is a commutative group, then L is of star-height ? 1. Finally we show that every rational language is the inverse image, under some morphism between free monoids, of a language of (restricted) star-height 1

Algorithms for determining relative star height and star height

AbstractLet C = {R1, …, Rm} be a finite class of regular languages over a finite alphabet Σ. Let Δ = {b1, …, bm} be an alphabet, and δ be the substitution from Δ∗ into Σ∗ such that δ(bi) = Ri for all i (1 ≤ i ≤ m). Let R be a regular language over Σ which can be defined from C by a finite number of applications of the operators union, concatenation, and star. Then there exist regular languages over Δ which can be transformed onto R by δ. The relative star height of R w.r.t. C is the minimum star height of regular languages over Δ which can be transformed onto R by δ. This paper proves the existence of an algorithm for determining relative star height. This result obviously implies the existence of an algorithm for determining the star height of any regular language

Closure properties and complexity of rational sets of regular languages

This work received funding in part by the National Research Network RiSE on Rigorous Systems Engineering (Austrian Science Fund (FWF): S11403-N23), by the Vienna Science and Technology Fund (WWTF) through grant PROSEED, by an Erwin Schrödinger Fellowship (Austrian Science Fund (FWF): J3696-N26), and by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement DIADEM no. 246858

Syntactic semigroups

This chapter gives an overview on what is often called the algebraic theory of finite automata. It deals with languages, automata and semigroups, and has connections with model theory in logic, boolean circuits, symbolic dynamics and topology