Numeration Systems: a Link between Number Theory and Formal Language Theory

TL;DR

This paper surveys the connections between numeration systems, number theory, and formal language theory, highlighting key results, interpretations, and open problems in the field.

Contribution

It provides a personal perspective on Cobham's seminal results, exploring recognizability, automatic sequences, transcendence, and applications to game theory and system verification.

Findings

Overview of Cobham's key results from the 1960s and 70s

Discussion of recognizability and automatic sequences

Identification of open problems and research directions

Abstract

We survey facts mostly emerging from the seminal results of Alan Cobham obtained in the late sixties and early seventies. We do not attempt to be exhaustive but try instead to give some personal interpretations and some research directions. We discuss the notion of numeration systems, recognizable sets of integers and automatic sequences. We briefly sketch some results about transcendence related to the representation of real numbers. We conclude with some applications to combinatorial game theory and verification of infinite-state systems and present a list of open problems.