度量图灵机输出计算的复杂性
On the Complexity of Computing Outputs of a Metric Turing Machine
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中文总结 AI 辅助
该研究针对MidP、MedP等函数类引入依赖输入的阶函数的第k个解计算函数类，证明其与对应类的多项式时间1-图灵互归约性，建立MaxP与中位数类的包含关系及NPSV_t、MaxP的包含关系的真包含条件。
中文摘要 AI 辅助
MidP、MedP和$\boldsymbol{\text{MedP}}$类包含针对特定问题计算中位数解的函数。本文针对这些类引入了类似的函数类，用于计算第k个解，其中k是依赖于输入的阶函数。我们证明，MidP、MedP和$\boldsymbol{\text{MedP}}$类与对应的函数类是多项式时间1-图灵可互归约的，其中阶函数来自FP或FP$^{\text{#P}}$。对于MedP，我们还证明它与对应的函数类重合，其中阶函数来自FP或#P。对于函数类之间的若干包含关系，我们给出了语言类之间的等价包含关系，特别是建立了MaxP与中位数类MidP、MedP和$\boldsymbol{\text{MedP}}$之间的包含关系。我们还证明NPSV$_{\text{t}} \boldsymbol{\text{⊆}}$ MaxP $\boldsymbol{\text{⊆}}$ FP$^{\text{NP}}$，且这两个包含关系均是真包含当且仅当NP $\boldsymbol{\text{≠}}$ coNP。
英文摘要
The classes MidP, MedP, and $\small{\overline{\text{MedP}}}$ contain functions that compute the median solution for certain types of problems. In this paper, for these classes we introduce analogous classes of functions that compute the k-th solution, where k is an order function that depends on the input. We prove that the classes MidP, MedP, and $\small{\overline{\text{MedP}}}$ are polynomial-time 1-Turing inter-reducible with the corresponding classes, where the order function is from FP or FP$^{\text{#P}}$. For MedP we also prove that it coincides with the corresponding classes, where the order function is from FP or #P. For several inclusions between function classes we give equivalent inclusions between language classes. In particular, we establish inclusion relations between MaxP and median classes MidP, MedP, and $\small{\overline{\text{MedP}}}$. We also prove that NPSV$_{\text{t}} \subseteq$ MaxP $\subseteq$ FP$^{\text{NP}}$ and both inclusions are proper if and only if NP $\neq$ coNP.
