PosSLP vs ER

Establish whether NP^{PosSLP} equals ER, i.e., determine if polynomial-time computation with oracle access to the Positive Straight-Line Program (PosSLP) decision problem has exactly the power of the existential theory of the reals complexity class.

Background

PosSLP asks whether the final value computed by a straight-line program over {+,−,·} is positive. It captures the gap between word RAM and real RAM models and is central to understanding the power of continuous computation.

The survey highlights deep connections between PosSLP, convex programming, and ER. The authors explicitly state a conjecture relating NP augmented with PosSLP-oracle access to ER, which, if true, would have major implications for the complexity of many problems expressible via real algebraic constraints.