  Presentations of Kan extensions of category actions provide a natural
framework for expressing induced actions, and therefore a range of different
combinatorial problems. Rewrite systems for Kan extensions have been defined
and a variation on the Knuth-Bendix completion procedure can be used to
complete them -- when possible. Regular languages and automata are a useful way
of expressing sets and actions, and in this paper we explain how to use rewrite
systems for Kan extensions to construct automata expressing the induced action
and how sets of normal forms can be calculated by obtaining language equations
from the automata.
