AI 中文总结
研究奇偶性基于的位计数复杂性类的闭包性质及包含关系，证明其与US和⊕P的关系，并展示其包含PH且被CH所包含。
AI 中文摘要
我们研究了一些基于奇偶性的位计数复杂性类B_{|0|⊕}P和B_{|1|⊕}P的性质。我们首先证明这两个复杂性类在补集下封闭，并证明B_{|1|⊕}P⊆B_{|0|⊕}P。接着证明US⊆P^{B_{|1|⊕}P}和US⊆P^{B_{|0|⊕}P}。然后研究这些类的特征函数，其中B_{|1|⊕}P的特征函数输出普鲁het-图尔-莫尔斯序列。接着证明这些序列的有限连续块给出起始数的奇偶性，并证明⊕P⊆P^{B_{|0|⊕}P}和⊕P⊆P^{B_{|1|⊕}P}。最后利用这些类定义各种层次，并展示它们都包含PH且被CH所包含。
英文摘要
We study some properties of the parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We first prove that both of these complexity classes are closed under complement and ${\bf B_{|1|\oplus}P}\subseteq {\bf B_{|0|\oplus}P}$. We then prove that ${\bf US}\subseteq {\bf P}^{{\bf B_{|1|\oplus}P}}$ and ${\bf US}\subseteq {\bf P}^{{\bf B_{|0|\oplus}P}}$. We then study the class defining characteristic functions of the parity based bit-counting complexity classes, where the one associated with ${\bf B_{|1| \oplus}P}$ produces the Prouhet-Thue-Morse sequence. We then prove that a contiguous block of four values from either sequence determines the parity of its starting index and use this fact to show that ${\bf \oplus P}\subseteq {\bf P}^{{\bf B_{|0|\oplus}P}}$ and ${\bf \oplus P}\subseteq {\bf P}^{{\bf B_{|1|\oplus}P}}$. We then use the parity based bit-counting complexity classes to define various hierarchies and show that they all contain ${\bf PH}$ and are contained in ${\bf CH}$.
