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

Abstract

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.

Logical Characterization of Recognizable Sets of Polynomials Over a Finite Field

“…This result motivates by itself the study of k-recognizable sets and more generally the introduction of non-standard numeration systems. See for instance [3,5] and the talk given by the first author during the 14th DLT conference [17]. Moreover, the Büchi-Bruyère's theorem [6] states that a set of integers is k-recognizable if and only if it can be defined by a first-order formula in the structure N, +, V k where the Presburger arithmetic N, + is extended with a unary function V k defined by V k (0) = 1 and V k (n) is the largest power of k diving n, n ≥ 1.…”