VPSPACE and a Transfer Theorem over the Reals

TL;DR

This paper introduces VPSPACE, a class of polynomials with coefficients computable in polynomial space, and establishes a transfer theorem linking efficient evaluation of VPSPACE to a collapse of complexity classes over the reals.

Contribution

It defines VPSPACE and proves a transfer theorem that connects polynomial space evaluation difficulty with class separations over the reals.

Findings

VPSPACE is a new class of polynomial families with polynomial space coefficient computation.

Efficient evaluation of VPSPACE families implies PAR collapses to P over the reals.

Separating P from NP over the reals requires VPSPACE families to be hard to evaluate.

Abstract

We introduce a new class VPSPACE of families of polynomials. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main theorem is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class PAR of decision problems that can be solved in parallel polynomial time over the real numbers collapses to P. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate over the reals P from NP, or even from PAR.