Strictness of inclusions P ⊆ NP ⊆ PP ⊆ PSPACE

Determine whether each containment in the hierarchy P ⊆ NP ⊆ PP ⊆ PSPACE is strict; specifically, prove or refute the equalities P = NP, NP = PP, and PP = PSPACE.

Background

In discussing the complexity hierarchy and its relevance to epistemic limits, the essay notes that while the inclusions P ⊆ NP ⊆ PP ⊆ PSPACE are provably true, it is unknown whether any of these containments are proper. This frames a family of open problems about the strictness of these class separations.

The footnote explicitly states the absence of known strictness results, highlighting that despite strong evidence, no theorems currently establish whether any of the adjacent classes in this chain are equal or distinct.

References

Each inclusion is known to hold (provably). None of the inclusions is known to be strict, but we have strong evidence that they are.

— From Heisenberg and Schrödinger to the P vs. NP Problem (2511.07502 - Weinstein, 10 Nov 2025) in Subsection 4.3. The White Raven and a Needle in the Haystack, Footnote labeled “class”