Representing Small Ordinals by Finite Automata

TL;DR

This paper presents a polynomial time algorithm to compute the Cantor Normal Form of the order type of well-ordered regular languages recognized by finite automata, enabling efficient comparison and representation of small ordinals.

Contribution

It introduces a polynomial time method to construct automata representing small ordinals and to decide isomorphism between such automata, advancing the understanding of ordinal representations in automata theory.

Findings

Polynomial time algorithm for Cantor Normal Form of automata-accepted ordinals

Efficient automaton construction for ordinals less than omega^omega

Estimates on automaton sizes representing small ordinals

Abstract

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 omega^omega. 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 omega^omega, together with an algorithm that translates each such ordinal to an automaton.