On the cost and complexity of the successor function
Valérie Berthé, Christiane Frougny, Michel Rigo, Jacques Sakarovitch · 2007
Abstract. For a given numeration system, the successor function maps the representation of an integer n onto the representation of its successor n+1. In a general setting, the successor function maps the n-th word of a genealogically ordered language L onto the (n+1)-th word of L. We show that, if the ratio of the number of elements of length n + 1 over the number of elements of length n of the language has a limit β> 1, then the amortized cost of the successor function is equal to β/(β − 1). From this, we deduce the value of the amortized cost for several classes of numeration systems (integer base systems, canonical numeration systems associated with a Parry number, abstract numeration systems built on a rational language, and rational base numeration systems). 1