Boundaries of VP and VNP

Abstract

One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = $\overline{\textrm{VP}}$, where VP is the class of families of polynomials that are of polynomial degree and can be computed by arithmetic circuits of polynomial size, and $\overline{\textrm{VP}}$ is the class of families of polynomials that are of polynomial degree and can be approximated infinitesimally closely by arithmetic circuits of polynomial size. The goal of this article is to study the conjecture in (Mulmuley, FOCS 2012) that $\overline{\textrm{VP}}$ is not contained in VP. Towards that end, we introduce three degenerations of VP (i.e., sets of points in $\overline{\textrm{VP}}$), namely the stable degeneration Stable-VP, the Newton degeneration Newton-VP, and the p-definable one-parameter degeneration VP*. We also introduce analogous degenerations of VNP. We show that Stable-VP $\subseteq$ Newton-VP $\subseteq$ VP* $\subseteq$ VNP, and Stable-VNP = Newton-VNP = VNP* = VNP. The three notions of degenerations and the proof of this result shed light on the problem of separating $\overline{\textrm{VP}}$ from VP. Although we do not yet construct explicit candidates for the polynomial families in $\overline{\textrm{VP}}\setminus$VP, we prove results which tell us where not to look for such families. Specifically, we demonstrate that the families in Newton-VP $\setminus$ VP based on semi-invariants of quivers would have to be non-generic by showing that, for many finite quivers (including some wild ones), any Newton degeneration of a generic semi-invariant can be computed by a circuit of polynomial size. We also show that the Newton degenerations of perfect matching Pfaffians, monotone arithmetic circuits over the reals, and Schur polynomials have polynomial-size circuits.