可计算逐点约简

摘要：

我们认为等价关系和前序完全的算术层次结构的各个级别下可计算的，组件明智的reductionary。证明了一阶逻辑中的蕴涵是$\SI 1$的完全预序，$\SI 2$的EXPTIME集上的$\le^P_m$关系以及$\SI 3$的$(\QQ,+)$的可计算子群的可嵌入性.在所有情况下，前序的对称片段对于同一水平上的等价关系是完备的。我们提出了一个表征$\PI 1$等价关系，使我们能够建立多项式时间函数的平等和包含多项式时间集是完全的$\PI 1$等价关系和前序分别。我们还表明，这是调查的限制：对于$n\geq 2$有没有$\PI n$，也没有$\DE n$ -完全等价关系。

英文摘要：

We consider equivalence relations and preorders complete for various levels of the arithmetical hierarchy under computable, component-wise reducibility. We show that implication in first order logic is a complete preorder for $\SI 1$, the $\le^P_m$ relation on EXPTIME sets for $\SI 2$ and the embeddability of computable subgroups of $(\QQ,+)$ for $\SI 3$. In all cases, the symmetric fragment of the preorder is complete for equivalence relations on the same level. We present a characterisation of $\PI 1$ equivalence relations which allows us to establish that equality of polynomial time functions and inclusion of polynomial time sets are complete for $\PI 1$ equivalence relations and preorders respectively. We also show that this is the limit of the enquiry: for $n\geq 2$ there are no $\PI n$ nor $\DE n$-complete equivalence relations.

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。