# Christiane Frougny

Numeration systems, the basis of which is defined by a linear recurrence with integer coefficients, are considered. We give conditions on the recurrence under which the function of normalization which transforms any representation of an integer into the normal one—obtained by the usual algorithm—can be realized by a finite automaton. Addition is a(More)
• 3
• STACS
• 1991
The purpose of this paper is the s tudy of the rational relations tha t is relations on words that are computable by finite 2-tape au tomata with the Proper ty that the difference of the length of two words in relation is bounded. The first motivat ion for this work has been the problem of representations of integers by means of non classical numerat ion(More)
• 3
• Mathematical systems theory
• 1994
We prove that the function of normalization in base θ, which maps any θ-representation of a real number onto its θ-development, obtained by a greedy algorithm, is a function computable by a finite automaton over any alphabet if and only if θ is a Pisot number.
• 2
• A new method for representing positive integers and real numbers in a rational base is considered. It amounts to computing the digits from right to left, least significant first. Every integer has a unique such expansion. The set of expansions of the integers is not a regular language but nevertheless addition can be performed by a letter-to-letter finite(More)
Let U be a strictly increasing sequence of integers. By a greedy algorithm, every nonnegative integer has a greedy U -representation. The successor function maps the greedy U -representation of N onto the greedy U -representation of N+1. We characterize the sequences U such that the successor function associated to U is a left, resp. a right sequential(More)
• IJAC
• 1999
Every positive integer can be written as a sum of Fibonacci numbers; it can also be written as a ((nite) sum of (positive and negative) powers of the golden mean '. We show that there exists a letter-to-letter nite two-tape automaton that maps the Fibonacci representation of any positive integer onto its '-expansion, provided the latter is folded around the(More)
• Developments in Language Theory
• 2009
We study expansions in non-integer negative base −β introduced by Ito and Sadahiro [7]. Using countable automata associated with (−β)-expansions, we characterize the case where the (−β)-shift is a system of finite type. We prove that, if β is a Pisot number, then the (−β)-shift is a sofic system. In that case, addition (and more generally normalization on(More)