The Isomorphism Problem On Classes of Automatic Structures

TL;DR

This paper investigates the complexity of the isomorphism problem across various classes of automatic structures, revealing new completeness results and resolving open questions in the field.

Contribution

It provides new complexity classifications for the isomorphism problem in automatic equivalence relations, trees, and linear orders, advancing understanding of their computational complexity.

Findings

Isomorphism problem for automatic equivalence relations is $ ext{Pi}^0_1$-complete.

Isomorphism problem for automatic trees of height n is $ ext{Pi}^0_{2n-3}$-complete.

Isomorphism problem for automatic linear orders is non-arithmetical.

Abstract

Automatic structures are finitely presented structures where the universe and all relations can be recognized by finite automata. It is known that the isomorphism problem for automatic structures is complete for $\Sigma^1_1$; the first existential level of the analytical hierarchy. Several new results on isomorphism problems for automatic structures are shown in this paper: (i) The isomorphism problem for automatic equivalence relations is complete for $\Pi^0_1$ (first universal level of the arithmetical hierarchy). (ii) The isomorphism problem for automatic trees of height $n \geq 2$ is $\Pi^0_{2n-3}$-complete. (iii) The isomorphism problem for automatic linear orders is not arithmetical. This solves some open questions of Khoussainov, Rubin, and Stephan.