Computable categoricity of trees of finite height

Abstract

We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ 0 3 -condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆ 0 n+1 -categorical but not ∆ 0 n -categorical.

On limitwise monotonicity and maximal block functions

“…Lempp et al [LMMS05] proved that, for every n ≥ 1 there is a computable tree of finite height which is ∆ 0 n+1 -categorical but not ∆ 0 n -categorical. The question that we address here is that of how we obtain this sort of result-for small n at least-in the context of η-like computable linear orderings.…”

On the Effectiveness of Symmetry Breaking

“…Trees which are computable as partial orders (but for which the predecessor function is not necessarily computable) are considered in a different context in [3,4], which may provide useful background for readers interested in investigating these questions. In such a tree, it is not generally possible to compute the level of a node, and this makes it substantially more difficult to determine which nodes lie in the same orbit under automorphisms of the tree.…”

The complexity of computable categoricity

“…For many classes of structures, there is a concise syntactic definition of the computably categorical structures: A computable linear order is computably categorical if and only if it has finitely many adjacencies (Dzgoev and Goncharov [GD80]); a computable Boolean algebra is computably categorical if and only if it has finitely many atoms (Goncharov, and independently La Roche [LR78]); a computable ordered abelian group is computably categorical if and only if it has finite rank (Goncharov, Lempp, and Solomon [GLS03]); a computable tree of finite height is computably categorical if and only if it is of finite type (Lempp, McCoy, R. Miller, and Solomon [LMMS05]); a computable torsion-free abelian group is computably categorical iff it has finite rank (Nurtazin [Nur74]); a computable p-group is computably categorical iff it can be written in one of the following forms: (i) (Z(p ∞ )) ⊕ G for ∈ ω ∪ {∞} and G finite, or (ii) (Z(p ∞ )) n ⊕ (Z p k ) ∞ ⊕ G where G is finite, and n, k ∈ ω (Goncharov [Gon80] and Smith [Smi81]); and so on.…”