Une Propriété de Certaines Parties d'un Monoöde Libre

Une propriété de certaines parties d'un monoïde libre permet de répondre à une question posée par Eggan

The star height of reset-free events and strictly locally testable events

An algorithm is presented for determining the star height of reset-free events and strictly locally testable events

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

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 algorithms and the star-height problem of regular languages

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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