Abstract not found

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 +1overthenumber 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

We consider biin nite sequences on a nite alphabet of digits which satisfy a constraint of nite type. Such sequences are perturbed by adding a 1 in position 0. The odometer is the function which transforms the initial sequence into an admissible sequence equivalent to the perturbed one. It is shown that the odometer can be realized by an on-line nite automaton when the constraint is linked to numeration in base , where is a Pisot number satisfying the equation = t1 + +tm , where t1 t2 tm 1 are integers.

Abstract. Odometers or “adding machines ” are usually introduced in the context of positional numeration systems built on a strictly increasing sequence of integers. We generalize this notion to systems defined on an arbitrary infinite regular language. In this latter situation, if (A, <) is a totally ordered alphabet, then enumerating the words of a regular language L over A with respect to the induced genealogical ordering gives a one-to-one correspondence between N and L. In this general setting, the odometer is not defined on a set of sequences of digits but on a set of pairs of sequences where the first (resp. the second) component of the pair is an infinite word over A (resp. an infinite sequence of states of the minimal automaton of L). We study some properties of the odometer like continuity, injectivity, surjectivity, minimality,... We then study some particular cases: we show the equivalence of this new function with the classical odometer built upon a sequence of integers whenever the set of greedy representations of all the integers is a regular language; we also consider substitution numeration systems as well as the connection with β-numerations. 1.

Abstract. Consider a non-standard numeration system like the one built over the Fibonacci sequence where nonnegative integers are represented by words over {0, 1} without two consecutive 1. Given a set X of integers such that the language of their greedy representations in this system is accepted by a finite automaton, we consider the problem of deciding whether or not X is a finite union of arithmetic progressions. We obtain a decision procedure for this problem, under some hypothesis about the considered numeration system. In a second part, we obtain an analogous decision result for a particular class of abstract numeration systems built on an infinite regular language. 1.

We define and study here the class of rational functions that are finite union of sequential functions. These functions can be realized by cascades of sequential transducers. After showing that cascades of any height are equivalent to cascades of height at most two and that this class strictly contains sequential functions and is strictly contained in the class of rational functions, we prove the result whose statement gives the paper its title.

Abstract not found

Noname manuscript No. (will be inserted by the editor) Numeration and discrete dynamical systems