The isomorphism problem on classes of automatic structures with transitive relations

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 Σ1/1, the first existential level of the analytical hierarchy. Positive results on ordinals and on Boolean algebras raised hope that the isomorphism problem is simpler for transitive relations. We prove that this is not the case. More precisely, this paper shows: (i) The isomorphism problem for automatic equivalence relations is complete for Π0/1 (the first universal level of the arithmetical hierarchy). (ii) The isomorphism problem for automatic trees of height n ≥ 2 is Π0/2n-3-complete. (iii) The isomorphism problem for well-founded automatic order trees is recursively equivalent to true first-order arithmetic. (iv) The isomorphism problem for automatic order trees is Σ1/1-complete. (v) The isomorphism problem for automatic linear orders is Σ1/1-complete. We also obtain Π0/1-completeness of the elementary equivalence problem for several classes of automatic structures and Σ1/1-completeness of the isomorphism problem for trees (resp., linear orders) consisting of a deterministic context-free language together with the prefix order (resp., lexicographic order). This solves several open questions of ́Esik, Khoussainov, Nerode, Rubin, and Stephan.