The Isomorphism Problem on Classes of Automatic Structures

Abstract

Several undecidability results on isomorphism problems for automatic structures are shown: (i) The isomorphism problem for automatic equivalence relations is Π

The Isomorphism Problem for ω-Automatic Trees

“…A more precise analysis moreover reveals at which height the complexity jump for ω-automatic trees occurs: For automatic as well as for ω-automatic trees of height 2, the isomorphism problem is Π 0 1 -complete and hence arithmetical. But the isomorphism problem for ω-automatic trees of height 3 is hard for Π 1 1 (and therefore outside of the arithmetical hierarchy) while the isomorphism problem for automatic trees of height 3 is Π 0 3 -complete [KLL10]. Our lower bounds for ω-automatic trees even hold for the smaller class of injectively ω-automatic trees.…”

“…In our recent paper [KLL10], we studied the isomorphism problem for restricted classes of automatic graphs. Among other results, we proved that (i) the isomorphism problem for automatic trees of height at most n ≥ 2 is complete for the level Π 0 2n−3 of the arithmetical hierarchy and (ii) that the isomorphism problem for automatic trees of finite height is recursively equivalent to true arithmetic.…”

“…Among other results, we proved that (i) the isomorphism problem for automatic trees of height at most n ≥ 2 is complete for the level Π 0 2n−3 of the arithmetical hierarchy and (ii) that the isomorphism problem for automatic trees of finite height is recursively equivalent to true arithmetic. In this paper, we extend our techniques from [KLL10] to ω-automatic trees. The class of ω-automatic structures was introduced in [Blu99], it generalizes automatic structures by replacing ordinary finite automata by Büchi automata on ω-words.…”

“…Using Schoenfield's absoluteness theorem, they infer that isomorphism of ω-automatic structures does not belong to Σ 1 2 . The extension of our elementary techniques from [KLL10] to ω-automatic trees allows us to show directly (without a "detour" through set theory) that the isomorphism problem for ω-automatic trees of finite height is not analytical (i.e., does not belong to any of the levels Σ 1 n ). For this, we prove that the isomorphism problem for ω-automatic trees of height n ≥ 4 is hard for both levels Σ 1 n−3 and Π 1 n−3 of the analytical hierarchy (our proof is uniform in n).…”

“…The basic idea of the reduction then is that a subset X ⊆ N can be encoded by an ω-word w X over {0, 1}, where the i-th symbol is 1 if and only if i ∈ X. The combination of this basic observation with our techniques from [KLL10] allows us to encode monadic second-order formulas over (N, +, ×) by ω-automatic trees of finite height. This yields the lower bounds mentioned above.…”