Exact composition complexity of the query Ψ_k

Determine whether the query Ψ_k(x,y) = ∃z ∧_{i=1}^k(R_i(x,z) ∨ R_i(z,y)) requires 2^k compositions in every equivalent calculus-of-relations term or circuit, rather than only the established 2^{k}/poly(k)-scale lower bound for this query.

Background

The query Ψ_k has a straightforward equivalent relational term using 2k compositions. The paper proves a lower bound of roughly 2k/√k compositions for equivalent terms and circuits, arising from the central binomial coefficient.

The authors show that this loss is inherent to the particular detectable-color construction and its type-constant interpretations, but explicitly do not determine whether Ψ_k itself requires the full 2k compositions. Thus the exact composition complexity of this query remains unresolved.

References

The resulting gap of a factor $\Theta(\sqrt k)$ against the $2k$ compositions of eq:omega-upper is not closed in this paper; by the same proposition, it is inherent to $\Psi_k$ and to this construction, and not an artifact of the particular relations we chose. The formula $\Phi_k$ avoids this loss by requiring agreement rather than inclusion, which allows all $2k$ subsets of $[k]$ to serve as labels (Section~\ref{sec:phi}). Finally, the only property of conjunction used in the argument is the one isolated in Section~\ref{ov:twocolors}, namely that two colors fix its three bits. Section~\ref{sec:matrix} replaces conjunction by a semiring product, and the same argument goes through with three colors in place of two.

eq:omega-upper:

Ψk:=⋃I⊆[k]((⋂i∈IRi);(⋂i∉IRi)),_{\Psi_k} := \bigcup_{I\subseteq[k]} \biggl(\Bigl(\bigcap_{i\in I}R_i\Bigr);\Bigl(\bigcap_{i\notin I}R_i\Bigr)\biggr),

— An Exponential Succinctness Gap between Three-Variable Logic and the Calculus of Relations  (2609.26778 - Uezato, 22 Sep 2026) in Section 3, subsection 'The two formulas', paragraph 'Why the middle layer'; see also Appendix 'The middle layer is forced'