Symmetry of Information for Kt with nonadaptive SAT access

Determine whether Symmetry of Information fails for Levin complexity Kt with nonadaptive access to a SAT oracle, denoted Kt^{||SAT}.

Background

The paper observes that the known failure of Symmetry of Information for Kt relativizes to Kt with adaptive SAT-oracle access. It also relates nKt, up to multiplicative overhead, to Kt with nonadaptive SAT queries.

Whether the same failure persists under nonadaptive access is explicitly left unresolved, highlighting the role of query adaptivity in the nondeterministic setting.

References

However, similarly to the situation for $nKt$ and related measures, we currently do not know if SoI fails for $Kt{|| SAT}$.

— Asymmetry and Complexity of Nondeterministic Computations  (2610.20188 - Goldberg et al., 14 Oct 2026) in Section 1, subsection “A Correspondence Between Asymmetry, Hardness, and Explicit Constructions,” paragraph “More on Adaptive vs Nonadaptive Queries”

Unfortunately, it is not clear how to prove \Cref{eq:intro_techniques_ideal} exactly.

— The Gap Between Average-Case and Worst-Case Complexity  (2610.19162 - Goldberg et al., 13 Oct 2026) in Section 3, subsection “Characterizing Easiness of NP via Easiness of Mild-Gap-MINpKT^PH,” proof of Lemma 3.4, immediately following equation (24)