We do not know if $P=NP$ has an impact on $VP=VNP$ however how about $P=PP$?

  1. Is $PP$ and $VNP$ related and would $P=PP$ or $P=PSPACE$ imply $VP=VNP$?

  2. Is there a way to show $\#P$ is in $FP^{PP}$?


Note that $\#P\in FP^{PP}$ follows from a simple binary search. Given some non-deterministic machine $M$, the language $\{(x,k) |\#M(x)\ge k\}$, where $\#M(x)$ is the number of accepting paths on input $x$, is in PP (you can change the constant in the definition of PP to any FP function). Thus, using a PP oracle we can count the number of accepting paths via binary search.

This implies that $\mathsf{P=PP}$ puts permanent in $VP$, as the problem of counting the number of perfect matchings in a bipartite graph (or equivalently, computing the permanent of the adjacency matrix) is in $\#P$.


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