We define a new family of complexity classes called bit-counting complexity classes, since membership depends not merely on the number of accepting paths, but also on the binary profile of that number. We study the relationship between this new family of complexity classes and the classical complexity classes. We prove that the classical complexity class ${\bf PP}$ is contained in our comparison based bit-counting complexity classes ${\bf B_{|0|=|1|}P}$, ${\bf B_{|0|<|1|}P}$ and ${\bf B_{|0|>|1|}P}$. We further show that all of these complexity classes are Turing equivalent ${\bf P}^{\bf PP} = {\bf P}^{{\bf B_{|0|=|1|}P}}={\bf P}^{{\bf B_{|0|>|1|}P}}={\bf P}^{{\bf B_{|0|<|1|}P}}$. We also prove that classical complexity classes ${\bf NP}$ and ${\bf CoNP}$ are contained in both of our parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$.

翻译：我们定义了一类新的复杂度类族，称为比特计数复杂度类，其成员关系不仅取决于接受路径的数量，还取决于该数量的二进制特征。我们研究了这一新复杂度类族与经典复杂度类之间的关系。我们证明经典复杂度类${\bf PP}$包含于基于比较的比特计数复杂度类${\bf B_{|0|=|1|}P}$、${\bf B_{|0|<|1|}P}$和${\bf B_{|0|>|1|}P}$中。进一步，我们证明所有这些复杂度类在图灵意义下等价：${\bf P}^{\bf PP} = {\bf P}^{{\bf B_{|0|=|1|}P}}={\bf P}^{{\bf B_{|0|>|1|}P}}={\bf P}^{{\bf B_{|0|<|1|}P}}$。我们还证明经典复杂度类${\bf NP}$和${\bf CoNP}$均包含于基于奇偶性的比特计数复杂度类${\bf B_{|0| \oplus}P}$和${\bf B_{|1| \oplus}P}$中。