Abstract

We are studying the punctual structures, i.e., the primitive recursive structures on the whole set of integers. The punctual categoricity relative to a computable oracle \(f\) means that between any two punctual copies of a structure there is an isomorphism which togeteher with its inverse can be derived via primitive recursive schemes augmented with \(f\). We will prove that the punctual categoricity relative to a computable oracle can hold only for finitely generated or locally finite structures. We will show that the punctual categoricity of finitely generated structures is exhaused by the computable oracles with primitive recursive graph. We also present an example of locally finite structure where the punctual categoricity is provided by a primitive recursively bounded computable oracle.

中文翻译：

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相对于可计算 Oracle 的准时分类

摘要

我们正在研究准点结构，即整个整数集上的原始递归结构。相对于可计算预言机\(f\)的准时范畴性意味着在结构的任何两个准时副本之间存在同构，该同构与其逆可以通过用\(f\)增强的原始递归方案导出. 我们将证明相对于可计算预言机的准时范畴性仅适用于有限生成或局部有限结构。我们将证明有限生成结构的准时范畴被具有原始递归图的可计算预言机耗尽。我们还展示了一个局部有限结构的例子，其中准时类别由原始递归有界可计算预言机提供。