Rational Functions

Abstract

The present chapter deals with rational functions, i.e. rational transductions which are partial functions. Rational functions have remarkable properties. First, several decision problems become solvable. This is shown in Section 1. Then there exist special representations, called unambiguous representations for rational functions. They are defined by the property that there is at most one successful path for each input word. Two different methods for constructing unambiguous representations are given in Section 3 and 4, the first by means of a cross-section theorem due to Eilenberg, the second through so-called semimonomial representations and due to Schützenberger. Section 2 is concerned with sequential functions which are a particular case of rational functions. In Section 5, bimachines are defined and are used to show that any rational function can be obtained as a composition of a left sequential followed by a right sequential function. In Section 6, we prove that it is decidable whether a rational function is sequential.