Representing Small Ordinals by Finite Automata

It is known that an ordinal is the order type of the lexicographic ordering of a regular language if and only if it is less than ww. We design a polynomial time algorithm that constructs, for each well-ordered regular language L with respect to the lexicographic ordering, given by a deterministic finite automaton, the Cantor Normal Form of its order type. It follows that there is a polynomial time algorithm to decide whether two deterministic finite automata accepting well-ordered regular languages accept isomorphic languages. We also give estimates on the size of the smallest automaton representing an ordinal less than ww, together with an algorithm that translates each such ordinal to an automaton