Numeration and enumeration

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 ...

Arithmetics in ß-Numeration

The β-numeration, born with the works of Rényi and Parry, provides a generalization of the notions of integers, decimal numbers and rational numbers by expanding real numbers in base β, where β > 1 is not an integer. One of the main differences with the case of numeration in integral base is that the sets which play the role of integers, decimal numbers and rational numbers in base β are not st...

Numeration-automatic sequences

We present a base class of automata that induce a numeration system and we give an algorithm to give the n-th word in the language of the automaton when the expansion of n in the induced numeration system is feeded to the automaton. Furthermore we give some algorithms for reverse reading of this expansion and a way to combine automata to other automata having the same properties.