自动结构类的同构问题
The Isomorphism Problem On Classes of Automatic Structures
摘要：
自动结构是一种抽象的结构，其中的论域和所有关系都可以由有限自动机来识别。已知自动结构的同构问题对于$Σ^1_1$是完备的，$Σ^1_1$是层次分析的第一存在级。本文给出了关于自动结构同构问题的几个新结果：（i）自动等价关系同构问题对于$Π^0_1$（算术体系的第一通用层）是完全的。(ii)高度为$n \geq 2$的自动树的同构问题是$Π^0_{2n-3}$ -完全的. (iii)自动线性订单的同构问题不是算术问题。这解决了Khoussainov，Rubin和Stephan的一些未决问题。
英文摘要：
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 $Σ^1_1$; 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 $Π^0_1$ (first universal level of the arithmetical hierarchy). (ii) The isomorphism problem for automatic trees of height $n \geq 2$ is $Π^0_{2n-3}$-complete. (iii) The isomorphism problem for automatic linear orders is not arithmetical. This solves some open questions of Khoussainov, Rubin, and Stephan.
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