Resolve the unordered fixed-point equivalence

Determine whether inflationary fixed-point definability and partial fixed-point definability coincide over unordered finite structures, equivalently whether PTIME equals PSPACE.

Background

The paper formalizes the Abiteboul–Vianu theorem, which relates two descriptive-complexity formalisms over unordered finite structures. The theorem states that equality of inflationary and partial fixed-point definability is equivalent to the equality PTIME = PSPACE.

Because the complexity-class equality is unresolved, the corresponding equality of the two fixed-point formalisms is also unresolved. The paper proves the equivalence but does not settle either side.

References

This result is of a different nature, an equivalence with an open question on either side: $()$ and $()$ coincide over unordered finite structures exactly when $PTIME = PSPACE$, \decl{ifpDefinableFree_eq_pfpDefinableFree_iff_ptime_eq_pspace}.

— Descriptive Complexity in Lean: Completeness by First-Order Reductions  (2609.18261 - Senellart et al., 16 Sep 2026) in Section 3.3, “Relations between the classes,” paragraph “Abiteboul--Vianu theorem” (subsection label sec:classes:relations)