Finite Automata and the Writing of Numbers

Abstract

Numbers do exist, independently of the way we represent them, of the way we write them. And there are many ways to write them: integers as finite sequence of digits once a base is fixed, rational numbers as a pair of integer or as an ultimately periodic infinite sequence of digits, or reals as an infinite sequence of digits but also as a continued fraction, just to quote a few. Operations on numbers are defined, independently of the way they are computed. But when they are computed they amounts to be algorithms that work on the representations of numbers.

Here, numbers will be represented by their development in a base, hence by words over an alphabet of digits and the algorithms we shall consider are those that can be performed by finite state machines, that is, by the simplest machines one can think of. Which operations can be thus defined? which set of numbers can be thus described? how this is related to the chosen base? how the choice of the alphabet of digits may influence the way the operations may be computed? These are the questions that will be asked and, hopefully and to a certain extent, answered in this conference.

We shall begin with the example of divisibility by a given integer in a given base p, the generalization — due to Blaise Pascal — of the casting out nines and, more seriously, with the beautiful Cobham’s Theorem [1,2,3]. This result leads to the distinction between recognizable and p-recognizable sets of integers that generalizes to set of tuples of integers and sets the problem of the decidability of the former among the latter, answered positively by Honkala, Muchnik and Leroux [4,5,6].

Another obvious appearance of finite automata, of finite transducers indeed, in the processing of written numbers occurs when signed digits are used, as has been popularized in the field of computer arithmetics by Avizienis for instance [7]. In this framework arises the interesting problem of the trade-off between the redundancy of a number system and the “compexity” of the operations performed on numbers written in that system.

The next case that will retain our attention is the one of non standard number systems; here, a non integer real β is taken as a base and the (real) numbers are written in this base. We put into correspondance the so-called arithmetic properties of β — that is, which kind of algebraic integer β is — the rationality of the set of expansions in such a base and the possibility of defining a linear recurrence that yields a system for representing the integers (cf. [8, Ch. VII]). A striking result is the fact that the addition is realized by a finite transducer if, and only if, β is a Pisot number [9,10]. In all these systems, the algorithm for computing the expansions is the greedy algorithm and produces the most significant digit first.

In a last part we shall touch on a more recent topic: rational base number systems (cf. [11]). In these systems, every integer has a unique finite expansion, which is not computed by a greedy algorithm but by a right to left algorithm, that is, by an algorithm which computes the least significant digit first. The set of all expansions is not a rational language, a very intriguing set of words indeed, but a finite transducer exists which converts a representation written on any alphabet of digits into a representation of the same number written on the canonical alphabet.