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Regular maps in generalized number systems J.P. Allouche, K. Scheicher and R. F. Tichy
 

Summary: Regular maps in generalized number systems
J.­P. Allouche, K. Scheicher and R. F. Tichy
Abstract
This paper extends some results of Allouche and Shallit for q­regular
sequences to numeration systems in algebraic number fields and to linear
numeration systems. We also construct automata that perform addition
and multiplication by a fixed number.
1 Introduction
A sequence is called q­automatic if its n­th term can be generated by a finite
state machine from the q­ary digits of n. The concept of automatic sequences
was introduced in 1969 and 1972 by Cobham [8, 9]. In 1979 Christol [6] (see also
Christol, Kamae, Mend`es France and Rauzy [7]) discovered a nice arithmetic
property of automatic sequences: a sequence with values in a finite field of
characteristic p is p­automatic if and only if the corresponding power series is
algebraic over the field of rational functions over this finite field. A brief survey
on this subject is given in [2], see also [10]. Some generalizations of this concept
were studied in [27, 23, 24, 3], see also the survey [1]. An automatic sequence
has to take its values in a finite set. To relax this condition, Allouche and
Shallit [5] introduced the notion of q­regular sequences. To give a hint of what
q­regularity is, let us consider the following example. If S(n) is the sum of the

  

Source: Allouche, Jean-Paul - Laboratoire de Recherche en Informatique, Université de Paris-Sud 11

 

Collections: Computer Technologies and Information Sciences