Omega-自动树的同构问题
The Isomorphism Problem for omega-Automatic Trees
摘要：
本文的主要结果是有限高的ω-自动树的同构至少具有二阶算术的困难性，因此不是解析的。这加强了Hjorth、Khoussainov、Montalban和Nies最近的一个结果，该结果表明Ω-自动结构的同构问题不是$Σ^1_2$。此外，假设连续统假设CH，我们可以证明有限高度的ω-自动树的同构问题与二阶算术递归等价。在得到主要结果的过程中，我们给出了任意有限高度的omega-自动树同构问题的上、下界：（i）对于高度1（2），它是可判定的（分别为$Π^0_1$ -完全的），(ii)对于高度3，$Π^1_1$ -硬且在$Π^1_2$中，以及（iii）对于所有n > 3，$Π^1_{n-3}$ -和$Σ^1_{n-3}$ -硬且在$Π^1_{2n-4}$中（假设CH）。所有的证明都是基本的，不依赖于集合论的定理。
英文摘要：
The main result of this paper is that the isomorphism for omega-automatic trees of finite height is at least has hard as second-order arithmetic and therefore not analytical. This strengthens a recent result by Hjorth, Khoussainov, Montalban, and Nies showing that the isomorphism problem for omega-automatic structures is not $Σ^1_2$. Moreover, assuming the continuum hypothesis CH, we can show that the isomorphism problem for omega-automatic trees of finite height is recursively equivalent with second-order arithmetic. On the way to our main results, we show lower and upper bounds for the isomorphism problem for omega-automatic trees of every finite height: (i) It is decidable ($Π^0_1$-complete, resp,) for height 1 (2, resp.), (ii) $Π^1_1$-hard and in $Π^1_2$ for height 3, and (iii) $Π^1_{n-3}$- and $Σ^1_{n-3}$-hard and in $Π^1_{2n-4}$ (assuming CH) for all n > 3. All proofs are elementary and do not rely on theorems from set theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
