Even more properties of parity based bit-counting complexity classes

Abstract

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 ${\bf CH}$.