Radix enumeration of rational languages is almost co-sequential

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. Introduction We define and study here the class of rational functions that are finite union of sequential functions (of pairwise disjoint domains). This class appeared rather naturally in the study of the concrete complexity of the successor function in some non standard numeration systems (cf. [2, 3]). Without going into details of that work, one of the problems which is tackled there is the definition of a computation model that is powerful enough to describe successor functions and that allows the definition of complexity with a sufficient degree of abstraction. As we shall see, the successor function is not necessarily a sequential function and, as we want to deal with deterministic automata only, we consider cascades of sequential transducers, that is, functions that compute the value of a word u by the following procedure : u is read by a first sequential transducer τ which outputs a word v; then v is read by another sequential transducer σq that depends on the state q reached by τ at the end of the reading of u, and so on, for a fixed number of steps h. In this paper we first characterize these functions: namely we prove that they all are of the kind we just said: finite union of sequential functions with pairwise disjoint domains, independently of the parameter h and by that they strictly contain the class of sequential functions, and that they form a proper subfamily of rational functions. Then, we prove the main result of this paper, namely :