基于奇偶性的位计数复杂性类的更多性质
Even more properties of parity based bit-counting complexity classes
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中文总结 AI 辅助
本研究针对基于奇偶性的位计数复杂性类B_{|0|⊕}P和B_{|1|⊕}P，通过证明MNS包含于对应多项式时间类、PP与#P的包含关系，得出相关复杂性类等价及位计数层次包含CH的结论。
中文摘要 AI 辅助
我们研究基于奇偶性的位计数复杂性类 $\bf B_{|0| \oplus}P$ 和 $\bf B_{|1| \oplus}P$ 的若干额外性质。首先证明 $\bf MNS \subseteq P^{\bf B_{|1| \oplus}P} = P^{\bf B_{|0| \oplus}P}$，结合已知的 $\bf C_=P = ES = MNS$，得到 $\bf C_=P = ES = MNS \subseteq P^{\bf B_{|1| \oplus}P} = P^{\bf B_{|0| \oplus}P}$。接着证明 $\bf PP \subseteq P^{\bf B_{|1| \oplus}P}$ 且 $\bf PP \subseteq P^{\bf B_{|0| \oplus}P}$，由此推出 $\bf P^{\bf PP} = P^{\bf B_{|0| \oplus}P} = P^{\bf B_{|1| \oplus}P}$。随后证明相同方法可用于得到 $\bf \\#P \subseteq FP^{\bf B_{|1| \oplus}P}$ 和 $\bf \\#P \subseteq FP^{\bf B_{|0| \oplus}P}$，还表明基于奇偶性的位计数层次包含 $\bf CH$。
英文摘要
We study several additional properties of parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We first prove that ${\bf MNS}\subseteq{\bf P}^{{\bf B_{|1|\oplus}P}}={\bf P}^{{\bf B_{|0|\oplus}P}}$ and since ${\bf C_=P}={\bf ES}={\bf MNS}$ is already known, we establish that ${\bf C_=P}={\bf ES}={\bf MNS}\subseteq{\bf P}^{{\bf B_{|1|\oplus}P}}={\bf P}^{{\bf B_{|0|\oplus}P}}$. We then prove that ${\bf PP}\subseteq{\bf P}^{{\bf B_{|1|\oplus}P}}$ and ${\bf PP}\subseteq{\bf P}^{{\bf B_{|0|\oplus}P}}$, which consequently yields ${\bf P}^{\bf PP}={\bf P}^{\bf B_{|0|\oplus}P}={\bf P}^{\bf B_{|1|\oplus}P}$. We then demonstrate that the same method can be used to prove ${\bf \# P}\subseteq{\bf FP}^{{\bf B_{|1|\oplus}P}}$ and ${\bf \# P}\subseteq{\bf FP}^{{\bf B_{|0|\oplus}P}}$. We also show that the parity based bit-counting hierarchies contain ${\bf CH}$.
