Newton’s Forward Difference Equation for Functions from Words to Words

International audienceNewton's forward difference equation gives an expression of a function from ℕ to ℤ in terms of the initial value of the function and the powers of the forward difference operator. An extension of this formula to functions from A* to ℤ was given in 2008 by P. Silva and the author. In this paper, the formula is further extended to functions from A* into the free group over B.L'équation aux différences avant de Newton donne une expression d'une fonction de ℕ dans ℤ en termes de la valeur initiale de la fonction et des puissances de l'opérateur de difference avant. Une extension de cette formule aux fonctions de A* dans ℤ a été donnée en 2008 par P. Silva et l'auteur. Dans cet article, cette formule est à nouveau généralisée, cette fois aux fonctions de A* dans le groupe libre sur B

Sur un cas particulier de la conjecture de Cerný

International audienceLet A be a finite automaton. We are concerned with the minimal length of the words that send all states on a unique state (synchronizing words). J. Cerný has conjectured that, if there exists a synchronizing word in A, then there exists such a word with length ≤ (n-1)^2 where n is the number of states of A. As a generalization, we conjecture that, if there exists a word of rank ≤ k in A, there exists such a word with length ≤ (n-k)^2. In this paper we deal only with automata in which a letter induces a circular permutation and prove the following results: (1) the second conjecture is true for (n-1)/2 ≤ k ≤ n, (2) if n is prime, the first conjecture is true, (3) if n is prime and if there exists a letter of rank n - 1, the second conjecture is true

Algorithmique et Programmation. Automates finis. Chap. I/9

This is the introduction to Section I/9, Algorithms and Programming, of the Encyclopaedia of Computer Science and Information Systems. This section is devoted to three fundamental tools in computer science: algorithms, machine models and programming languages. Introduced around 1950, finite automata form the most elementary model of machine. This chapter presents automata in the usual sense, that accept or reject a given word and sequential automata, that produce an output. After a brief presentation of Kleene's theorem, the cornerstone of the theory of automata, we describe applications in various domains, notably their use as a verification model and their applications to natural languages

Sur la longueur des mots de rang donné d'un automate fini

Let A be a finite automaton with n states. We prove that if there exists a word of rank ? (n-k) in A, then there exists such a word with length ? P(k), where P is a polynomial of degree 4

Logic on words

First published in the Bulletin of the European Association of Theoretical Computer Science 54 (1994), 145-165.This dialog between Quisani, Yuri Gurevich's imaginary student, and the author, was published in the "Logic in Computer Science Column" of the EATCS Bulletin. It is first addressed to logicians. This dialog is an occasion to present the connections between Büchi's sequential calculus and the theory of finite automata. In particular, the essential role of first order formulæ is emphasized. The quantifier hierarchies on these formulæ are an occasion to present open problems

The expressive power of existential first order sentences of Büchi's sequential calculus

The aim of this paper is to study the first order theory of the successor, interpreted on finite words. More specifically, we complete the study of the hierarchy based on quantifier alternations (or ?n-hierarchy). It was known (Thomas, 1982) that this hierarchy collapses at level 2, but the expressive power of the lower levels was not characterized effectively. We give a semigroup theoretic description of the expressive power of B?1-, the boolean combinations of existential formulas. Our characterization is algebraic and makes use of the syntactic semigroup, but contrary to a number of results in this field, is not in the scope of Eilenberg's variety theorem, since B?1-definable languages are not closed under residuals. An important consequence is the following: given one of the levels of the hierarchy, there is polynomial time algorithm to decide whether the language accepted by a deterministic n-state automaton is expressible by a sentence of this level

On two combinatorial problems arising from automata theory

International audienceWe present some partial results on the following conjectures arising from automata theory. The first conjecture is the triangle conjecture due to Perrin and Schützenberger. Let A = {a, b} be a two-letter alphabet, d a positive integer and let B_d = {a^iba^j | 0 <= i+j <= d}. If X \subset B_d is a code, then |X| <= d+1. The second conjecture is due to Cerný and the author. Let A be an automaton with n states. If there exists a word of rank <= k in A, there exists such a word with length <= (n-k)^2

Algebraic tools for the concatenation product

This paper is a contribution to the algebraic study of the concatenation product. In the first part of the paper, we extend to the ordered case standard algebraic tools related to the concatenation product, like the Schützenberger product and the relational morphisms. We show in a precise way how the ordered Schützenberger product corresponds to polynomial operations on languages. In the second part of the paper, we apply these results to establish a bridge between the three standard concatenation hierarchies, namely the Straubing-Thérien's hierarchy, the Brzozowski's (or dot-depth) hierarchy and the group hierarchy

Algorithmique et Programmation. Introduction

This is the introduction to Section I/9, Algorithms and Programming, of the Encyclopaedia of Computer Science and Information Systems. This section is devoted to three fundamental tools in computer science: algorithms, machine models and programming languages

Automates réversibles: combinatoire, algèbre et topologie

Leçons de mathématiques d'aujourd'hui, édité par E.Charpentier, à paraitre chez Cassini.A reversible automaton is a finite (possibly incomplete) automaton in which each letter induces a partial one-to-one map from the set of states into itself. We give four non-trivial characterizations of the languages accepted by a reversible automaton equipped with a set of initial and final states and we show that one can effectively decide whether a given rational (or regular) language can be accepted by a reversible automaton. The first characterization gives a description of the subsets of the free group accepted by a reversible automaton that is somewhat reminiscent of Kleene's theorem. The second characterization is more combinatorial in nature. The decidability follows from the third -- algebraic -- characterization. The last and somewhat unexpected characterization is a topological description of our languages that solves an open problem about the finite-group topology of the free monoid