Abstract

Linear numeration systems defined by a linear recurrence relation with integer coefficients are considered. The normalization function maps any representation of a positive integer with respect to a linear numeration system onto the normal one, obtained by the greedy algorithm. Addition is a particular case of normalization. We show that if the characteristic polynomial of the linear recurrence is the minimal polynomial of a Pisot number, then normalization is a function computable by a finite 2-tape automaton on any finite alphabet of integers. Conversely, if the characteristic polynomial is the minimal polynomial of a Perron number which is not a Pisot number, then there exist alphabets on which normalization is not computable by a finite 2-tape automaton.

Introduction

In this paper we study numeration systems defined by a linear recurrence relation with integer coefficients. These numeration systems have also been considered in [Fra] and [PT]. The best known example is the Fibonacci numeration system defined from the sequence of Fibonacci numbers. In the Fibonacci numeration system, every integer can be represented using digits 0 and 1. The representation is not unique, but one of them is distinguished : the one which does not contain two consecutive l's.

Let U be an integer sequence satisfying a linear recurrence. By a greedy algorithm, every positive integer has a representation in the system U that we call the normal representation. The normalization is the function which transforms any representation on any finite alphabet of integers into the normal one. From now on, by alphabet we mean finite alphabet of integers.