The Theory of Rational Relations on Transfinite Strings

In order to explain the purpose of this paper, we recall briefly how the theory of “rationality” developed in the last fourty years. The theory of sets of finite strings recognized by finite automata, also known as regular or rational sets was developed in the fifties. It rapidly extended in two directions. Indeed, by solving the decidability problem of the second order monadic theory of one successor, Büchi was led naturally to introduce the notion of finite automata working on infinite strings. He further extended this result to the monadic theory of all denumerable ordinals, and by doing so he again modified the original notion of finite automata to suit his new purpose, [9]. At this point, the equivalence between the notions of recognizability (by “finite automata”), rationality (by “rational expressions”) and definability (by “monadic second order logics”) was achieved as far as strings of denumerable lengths were concerned. In the late sixties, Elgot and Mezei wrote an historical paper on rational relations [15] which was a successful attempt to construct the theory of relations between free monoids that could be recognized by so-called n-tape automata. Though hard to read, it contained the basic results of the theory. In the mid eighties Gire and Nivat showed that the theory of rational relations on finite strings carries over to