# On clarification of intersection of classes definition

How do you define $\oplus P\cap PP$?

1. $L\in\oplus P$ iff $\exists\mbox{ NTM }M:\forall x,\#acc_M(x)\mod2\equiv0$.
2. $L\in PP$ iff $\exists\mbox{ NTM }M:\forall x,\#acc_M(x)>\#rej_M(x)$.

Consider the classes $C_1,C_2,C_3$.

$L\in C_1$ iff $\exists\mbox{ NTM }M:\forall x\#acc_M(x)>\#rej_M(x)\mbox{ and }\#acc_M(x)\mod2\equiv0$?

$L\in C_2$ if $\exists\mbox{ NTM }M_1:\forall x\#acc_{M_1}(x)>\#rej_{M_1}(x)\mbox{ and }\#acc_{M_1}(x)\mod2\equiv0$ and $L\not\in C_2$ if $\exists\mbox{ NTM }{M_2}:\forall x\#acc_{M_2}(x)\leq \#rej_{M_2}(x)\mbox{ and }\#acc_{M_2}(x)\mod2\equiv1$.

$\exists\mbox{ NTM }M'$ such that $L\in C_3$ if $\forall x\#acc_{M_1}(x)>\#rej_{M'}(x)\mbox{ and }\#acc_{M'}(x)\mod2\equiv0$ and $L\not\in C_3$ if $\forall x\#acc_{M'}(x)\leq \#rej_{M'}(x)\mbox{ and }\#acc_{M'}(x)\mod2\equiv1$.

Which of $C_1,C_2,C_3$ defines $\oplus P\cap PP$ and which one $\oplus P\wedge PP$? What about last one?

• Which $\mathcal C_i$ defines $\oplus P\wedge PP$? – T.... Jan 3 '18 at 23:48