给定度下的切换范畴性行为
Switching Categoricity Behavior with a Given Degree
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中文总结 AI 辅助
针对给定不可计算可枚举度，构造结构实现计算范畴性与相对范畴性的切换，否定相关猜想。
中文摘要 AI 辅助
给定一个不可计算的、计算可枚举的集合 $D$，我们构造一个可计算结构 $G$，使得 $G$ 的任意两个可计算副本都是计算同构的，而存在 $G$ 的两个 $D$-可计算副本不是 $D$-计算同构的。换言之，$G$ 是计算范畴的，但相对于 $D$ 不是计算范畴的。相反，我们构造一个可计算结构，它不是计算范畴的，但相对于 $D$ 是计算范畴的。这些结果与文献中关于计算范畴性及其相对化的结果不同，因为这里的度 $D$ 是给定的而非构造的。我们的工作否定了 Downey、Harrison-Trainor 和 Melnikov 在计算可枚举情形下的一个问题。
英文摘要
Given a noncomputable, computably enumerable set $D$, we construct a computable structure $G$ such that any two computable copies of $G$ are computably isomorphic while there are two $D$-computable copies of $G$ that are not $D$-computably isomorphic. In other words, $G$ is computably categorical, but not computably categorical relative to $D$. Conversely, we construct a computable structure that is not computably categorical, but is computably categorical relative to $D$. These results differ from those in the literature regarding computable categoricity and its relativiziations because the degree $D$ is given instead of constructed. Our work negatively answers a question of Downey, Harrison-Trainor and Melnikov in the computably enumerable case.
