arXiv subjects

Michel Rigo

Publications and source records attributed to Michel Rigo.

42 records · Page 3Linked to original sources

arXiv2008

Abstract numeration systems on bounded languages and multiplication by a constant

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an integer $λ\ge2$ does not preserve $S$-recognizability, meaning that there always exists a $S$-recognizable set $X$ such that $λX$ is not $S$-recognizable. The main tool is a bijection between the representation of an integer over a bounded language and its decomposition as a sum of binomial coefficients with certain properties, the so-called combinatorial numeration system.

cs.DM↗

arXiv2001

Towards a characterization of the star-free sets of integers

Let U be a numeration system, a set X of integers is U-star-free if the set made up of the U-representations of the elements in X is a star-free regular language. Answering a question of A. de Luca and A. Restivo, we obtain a complete logical characterization of the U-star-free sets of integers for suitable numeration systems related to a Pisot number and in particular for integer base systems. For these latter systems, we study as well the problem of the base dependence. Finally, the case of k-adic systems is also investigated.

cs.CC↗

arXiv2000

Numeration systems on a regular language: Arithmetic operations, Recognizability and Formal power series

Generalizations of numeration systems in which N is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language L over a finite alphabet A. For these systems, we obtain a characterization of recognizable sets of integers in terms of rational formal series. We also show that, if the complexity of L is Theta (n^q) (resp. if L is the complement of a polynomial language), then multiplication by an integer k preserves recognizability only if k=t^{q+1} (resp. if k is not a power of the cardinality of A) for some integer t. Finally, we obtain sufficient conditions for the notions of recognizability and U-recognizability to be equivalent, where U is some positional numeration system related to a sequence of integers.

cs.CC↗

arXiv1999

Construction of regular languages and recognizability of polynomials

A generalization of numeration system in which the set N of the natural numbers is recognizable by finite automata can be obtained by describing a lexicographically ordered infinite regular language. Here we show that if P belonging to Q[x] is a polynomial such that P(N) is a subset of N then we can construct a numeration system in which the set of representations of P(N) is regular. The main issue in this construction is to setup a regular language with a density function equals to P(n+1)-P(n) for n large enough.

cs.CC↗

arXiv1999

Generalization of automatic sequences for numeration systems on a regular language

Let L be an infinite regular language on a totally ordered alphabet (A,<). Feeding a finite deterministic automaton (with output) with the words of L enumerated lexicographically with respect to < leads to an infinite sequence over the output alphabet of the automaton. This process generalizes the concept of k-automatic sequence for abstract numeration systems on a regular language (instead of systems in base k). Here, I study the first properties of these sequences and their relations with numeration systems.

cs.CC↗

arXiv1999

Numeration systems on a regular language

Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of numeration, we show that ultimately periodic sets are recognizable. We also study the translation and the multiplication by constants as well as the order dependence of the recognizability.

cs.OH↗