Bertrand numeration systems and recognizability

Bertrand Numeration Systems and Recognizability

There exist various well-known characterizations of sets of numbers recognizable by a nite automaton, when they are represented in some integer base p 2. We show how to modify these characterizations, when integer bases p are replaced by linear numeration systems whose characteristic polynomial is the minimal polynomial of a Pisot number. We also prove some related interesting properties.

Numeration systems on a regular language: arithmetic operations, recognizability and formal power series

Generalizations of numeration systems in which N is recognizable by a finite automaton are obtained by describing a lexicographically ordered infinite regular language L ⊂ Σ∗. For these systems, we obtain a characterization of recognizable sets of integers in terms of rational formal series. We also show that, if the complexity of L is Θ(n) (resp. if L is the complement of a polynomial language...

Numeration Systems

Automata and Numeration Systems ∗

This article is a short survey on the following problem: given a set X ⊆ N, find a “simple algorithm” accepting X and rejecting N \ X. By simple algorithm, we mean a finite automaton, a substitution, a logical formula, . . . We will see that these algorithms strongly depend on the way one represents the integers. However, once the base of representation is fixed, these models are all equivalent...

Arrays, Numeration Systems and Games

We deene an innnite array A of nonnegative integers based on a linear recurrence, which produces basis elements of an exotic ternary numeration system. Using the numeration system we explore many properties of A. Further, we propose and analyze a family of 2-player games on a variable number of heaps of tokens, and present a winning strategy based on certain subarrays of A. Though the strategy ...