Varieties of Cost Functions.

Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring

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

Une application de la representation matricielle des transductions

RésuméOn étudie le problème suivant, fréquemment rencontré en théorie des langages: soient n langages L1,…,Ln reconnus par les monoïdes M1,…,Mn respectivement. Etant donné une opération ϕ, on cherche à construire un monoïde M, fonction de M1,…,Mn, qui reconnaisse le langage (L1,…,Ln)ϕ. Nous montrons que la plupart des constructions proposées dans la littérature pour ce type de problème sont en fait des cas particuliers d'une méthode générale que nous exposons ici. Cette méthode s'applique également à certains problèmes moins classiques relatifs par exemple à la réduction du groupe libre ou aux opérations de contrôle sur les T0L-systèmes.AbstractWe study the following classical problem of formal language theory: let L1,…,Ln be n languages recognized by the monoids M1,…,Mn respectively. Given an operation ϕ, we want to build a monoid M, function of M1,…,Mn, which recognizes the language (L1,…,Ln)ϕ. We show that most of the constructions given in the literature for this kind of problem are particular cases of a general method. This method can also be applied to some less classical problems related for example to the Dyck-reduction of the free-group or to control operations on T0L-systems

A maxmin problem on finite automata

AbstractWe solve the following problem proposed by Straubing. Given a two-letter alphabet A, what is the maximal number of states f(n) of the minimal automaton of a subset of An, the set of all words of length n. We give an explicit formula to compute f(n) and we show that 1= lim infn→∞nƒ(n)/2n≤lim supn→∞nƒ(n)/2n=2

Complexity of checking whether two automata are synchronized by the same language

A deterministic finite automaton is said to be synchronizing if it has a reset word, i.e. a word that brings all states of the automaton to a particular one. We prove that it is a PSPACE-complete problem to check whether the language of reset words for a given automaton coincides with the language of reset words for some particular automaton.Comment: 12 pages, 4 figure

Synchronizing automata with random inputs

We study the problem of synchronization of automata with random inputs. We present a series of automata such that the expected number of steps until synchronization is exponential in the number of states. At the same time, we show that the expected number of letters to synchronize any pair of the famous Cerny automata is at most cubic in the number of states

Reset thresholds of automata with two cycle lengths

We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, q > p. We describe reset thresholds of the colorings of primitive digraphs with exactly one cycle of length p and one cycle of length q. Also, we study reset thresholds of the colorings of primitive digraphs with exactly one cycle of length q and two cycles of length p.Comment: 11 pages, 5 figures, submitted to CIAA 201

Towards A Holographic Model of D-Wave Superconductors

The holographic model for S-wave high T_c superconductors developed by Hartnoll, Herzog and Horowitz is generalized to describe D-wave superconductors. The 3+1 dimensional gravitational theory consists a symmetric, traceless second-rank tensor field and a U(1) gauge field in the background of the AdS black hole. Below T_c the tensor field which carries the U(1) charge undergoes the Higgs mechanism and breaks the U(1) symmetry of the boundary theory spontaneously. The phase transition characterized by the D-wave condensate is second order with the mean field critical exponent beta = 1/2. As expected, the AC conductivity is isotropic below T_c and the system becomes superconducting in the DC limit but has no hard gap.Comment: 14 pages, 2 figures, Some typos corrected, Matched with the published versio