Rational Relations as Rational Series

Abstract

A rational relation is a rational subset of the direct product of two free monoids: R C A* x B*. Consider R as a function of A* into the family of subsets of B* by posing for all u E A*, R(u) = {v E B* ] (u, v) E R}. Assume R(u) is a finite set for all u E A*. We study how the cardinality of R(u) behaves as the length of u tends to infinity and we show that there exists an infinite hierachy of growth functions.