Bit-counting complexity classes

We define bit-counting complexity classes, where the membership depends on the binary profile of the number of accepting paths of non-deterministic polynomial time Turing machines. We study the relationship between this new family of complexity classes and the classical complexity classes. We prove that the complexity class $\mathbf{P}\mathbf{P}$ is contained in our comparison based bit-counting complexity classes ${\mathbf{B}}_{\mathbf{\mid }\mathbf{0}\mathbf{\mid }=\mathbf{\mid }\mathbf{1}\mathbf{\mid }}\mathbf{P}$, ${\mathbf{B}}_{\mathbf{\mid }\mathbf{0}\mathbf{\mid }<\mathbf{\mid }\mathbf{1}\mathbf{\mid }}\mathbf{P}$ and ${\mathbf{B}}_{\mathbf{\mid }\mathbf{0}\mathbf{\mid }>\mathbf{\mid }\mathbf{1}\mathbf{\mid }}\mathbf{P}$. We further show that all of these complexity classes are Turing equivalent ${\mathbf{P}}^{\mathbf{P}\mathbf{P}}={\mathbf{P}}^{{\mathbf{B}}_{\mathbf{\mid }\mathbf{0}\mathbf{\mid }=\mathbf{\mid }\mathbf{1}\mathbf{\mid }}\mathbf{P}}={\mathbf{P}}^{{\mathbf{B}}_{\mathbf{\mid }\mathbf{0}\mathbf{\mid }>\mathbf{\mid }\mathbf{1}\mathbf{\mid }}\mathbf{P}}={\mathbf{P}}^{{\mathbf{B}}_{\mathbf{\mid }\mathbf{0}\mathbf{\mid }<\mathbf{\mid }\mathbf{1}\mathbf{\mid }}\mathbf{P}}$. We also prove that complexity classes $\mathbf{N}\mathbf{P}$ and $\mathbf{C}\mathbf{o}\mathbf{N}\mathbf{P}$ are contained in both of our parity based bit-counting complexity classes ${\mathbf{B}}_{\mathbf{\mid }\mathbf{0}\mathbf{\mid }\oplus }\mathbf{P}$ and ${\mathbf{B}}_{\mathbf{\mid }\mathbf{1}\mathbf{\mid }\oplus }\mathbf{P}$.