Georges Hansel

Bertrand numeration systems and recognizability

Abstract There exist various well-known characterizations of sets of numbers recognizable by a finite automaton, when they are represented in some integer base p ⩾2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number. We also prove some related interesting properties.

AUTOMATE, a computing package for automata and finite semigroups

AUTOMATE is a package for symbolic computation on finite automata, extended rational expressions and finite semigroups. On the one hand, it enables one to compute the deterministic minimal automaton of the language represented by a rational expression or given by its table. On the other hand, given the transition table of a deterministic automaton, AUTOMATE computes the associated transition monoid. The regular D-classes structure, and many properties of the elements in the monoid are provided. The program AUTOMATE has been written in C and is quite portable. The user interface includes specialized editors for easy displaying of the computed results.

Compression and Entropy

The connection between text compression and the measure of entropy of a source seems to be well known but poorly documented. We try to partially remedy this situation by showing that the topological entropy is a lower bound for the compression ratio of any compressor. We show that for factorial sources the 1978 version of the Ziv-Lempel compression algorithm achieves this lower bound.

UNAVOIDABLE SETS OF CONSTANT LENGTH

A set of words X is called unavoidable on a given alphabet A if every infinite word on A has a factor in X. For k,q≥1, let c(k,q) be the number of conjugacy classes of words of length k on q letters. An unavoidable set of words of length k on q symbols has at least c(k,q) elements. We show that for any k,q≥1, there exists an unavoidable set of words of length k on q symbols having c(k,q) elements.

A simple proof of the Skolem-Mahler-Lech Theorem

Abstract A simple proof of the Skolem-Mahler-Lech theorem is given, which does not make the use of the p -adic number theory. It mainly relies on the equivalence between rational and recognizable series. In particular, this equivalence allows to deduce easily the general case of a commutative field of characteristic zero from the rational case.

Rational probability measures

Abstract The theory of finite automata and coding has been linked to probability theory since its foundation by the work of Shannon, and its further enrichment, especially by Schutzenberger. In a parallel development, symbolic dynamical systems whose behaviour is governed by a finite automaton have been studied under several aspects, especially by Hell, Robertson, Binkowska and Kaminski [4, 13, 17]. Weiss has given to them the name of sofic systems and he has shown that they constitute a generalization of topological Markov chains [20]. Since then, Adler and his associates have shown how theoretical development in this field could lead to a remarkable treatment of engineering problems in coding [1]. In this paper, we continue the exploration of this borderline area and present some new results. Probability measures on words are studied in their relation with coding and finite automata. We particularly insist on the notion of a rational probability measure which, although perhaps not quite new, has not yet received all the attention it deserves. In a first section, we fix the notation and prove some preliminary results. In particular the basic properties of rational probability measures are established. The second section deals with probability measures for which the sequences obtained after some coding are mutually independent. It contains two main results. The first one gives conditions under which the restriction of a probability measure to a given free submonoid is a Bernoulli measure. It extends a result due to Schutzenberger. The second result is due to Langlois and is published here for the first time. In the two remaining sections, we consider the problem of transferring a probability measure from one free monoid into another one. Several new results are proved which complement the ones previously obtained in collaboration with Blanchard [5, 6].

Independent numeration systems

Ce travail est dédié au sourire de M. P. Schützenberger, s ou ire incomparable, inoubliable, expression de qualités rarement rencontrées en semble, génie et gentillesse, humour et rigueur intellectuelle, le tout appuyé sur un cou rage indomptable face aux épreuves de l’histoire ou de la vie. Dans les lignes qui v ont suivre, il aurait reconnu – ou peut-être reconnaı̂t-il – la trace de quelques -un de ses innombrables et profondes idées.

Recognizable Sets of Numbers in Nonstandard Bases

There are several characterizations of sets of integers recognizable by automata, when they are written in p-ary representations, p≥2. We prove that most of them can be adapted to sets of integers written in nonstandard numeration systems, like Fibonacci numeration system.

Similarity relations and cover automata

Cover automata for finite languages have been much studied a few years ago. It turns out that a simple mathematical structure, namely similarity relations over a finite set of words, is underlying these studies. In the present work, we investigate in detail for themselves the properties of these relations beyond the scope of finite languages. New results with straightforward proofs are obtained in this generalized framework, and previous results concerning cover automata are obtained as immediate consequences.

Quasicontinuity and Namioka's theorem

Abstract First a simple proof of Namiokas theorem on separate and joint continuity is given. Then we show a multivariate Namiokas theorem whose all component spaces are countably Cech complete regular. The proof relies on a surprising result about the quasicontinuity of a separately continuous function.