Complexity classification of PosSLP

Determine the exact classical complexity of PosSLP, the problem of deciding whether a given arithmetic straight‑line program over {+, −, ×} with starting constant 1 evaluates to a positive integer, including whether PosSLP is NP‑hard or belongs to a lower complexity class such as P.

Background

PosSLP plays a central role in relating real and discrete computation: P with a PosSLP oracle coincides with polynomial‑time computations on a real RAM with basic arithmetic. Despite this importance, the precise classical complexity of PosSLP is not settled.

The authors note that little is known about its complexity beyond decidability and some conditional NP‑hardness results, emphasizing that its precise placement in standard complexity classes remains open.

References

As a matter of fact, we do not know much of the complexity of PosSLP and it seems very hard to understand the precise nature of this algorithmic problem.

— Beyond Bits: An Introduction to Computation over the Reals  (2603.29427 - Miltzow, 31 Mar 2026) in Subsection “Relation between Real and Discrete Computations,” paragraph on Straight‑line programs (PosSLP)