Isomorphism of regular trees and words

TL;DR

This paper analyzes the computational complexity of the isomorphism problem for regular trees, linear orders, and words, providing complexity classifications and a polynomial-time algorithm for regular linear orders.

Contribution

It establishes the complexity of the isomorphism problem for regular trees and linear orders, and introduces a polynomial-time algorithm for regular linear orders from DFAs.

Findings

Regular trees are EXPTIME-complete with NFAs

Finite trees are PSPACE-complete with acyclic NFAs

Polynomial-time algorithm for regular linear orders from DFAs

Abstract

The computational complexity of the isomorphism problem for regular trees, regular linear orders, and regular words is analyzed. A tree is regular if it is isomorphic to the prefix order on a regular language. In case regular languages are represented by NFAs (DFAs), the isomorphism problem for regular trees turns out to be EXPTIME-complete (resp. P-complete). In case the input automata are acyclic NFAs (acyclic DFAs), the corresponding trees are (succinctly represented) finite trees, and the isomorphism problem turns out to be PSPACE-complete (resp. P-complete). A linear order is regular if it is isomorphic to the lexicographic order on a regular language. A polynomial time algorithm for the isomorphism problem for regular linear orders (and even regular words, which generalize the latter) given by DFAs is presented. This solves an open problem by Esik and Bloom.