3 - Abstract numeration systems

Summary

The primary motivation for the introduction of the abstract numeration systems stems from the celebrated theorem of Cobham dating back to 1969 about the so-called recognisable sets of integers in any integer base numeration system. Representations of numbers are words over a finite alphabet. There is a one-to-one correspondence between the sets of numbers and the languages made of the corresponding representations. Hence it is natural to consider questions related to formal language theory. In particular, we study sets of integers corresponding to regular languages. The different sections of this chapter are largely independent. However, Section 3.2 presents basic concepts and notation used in all later sections. The main focus is on the representation of integers. Extension to abstract numeration systems of the notion of recognisable sets of integers is studied in Section 3.3. In particular, we present some results about the stability of recognisability after multiplication by a constant. This requires us to discuss the complexity (or counting) function of regular languages. Section 3.4 is about the extension – to any substitutive sequence – of Cobham's theorem from 1972 about the equality of the set of infinite k-automatic words and the set of images under codings of the fixed points of substitutions of constant length k. The notion of an S-automatic sequence is then introduced and various applications to S-recognisability are considered. This chapter ends with a discussion about the representation of real numbers using abstract numeration systems.