Worst-case Symmetry of Information for probabilistic nondeterministic complexity

Establish whether the average-case Symmetry of Information inequality proved for pnKt and rnKt extends to the worst case, namely whether the corresponding inequality holds for every pair of n-bit strings.

Background

The paper proves Symmetry of Information on average for probabilistic extensions of nKt over efficiently samplable distributions. It then contrasts this result with the unresolved worst-case question, in which the inequality must hold simultaneously for all strings x and y of a given length.

The authors show that such a worst-case theorem would have major complexity-theoretic consequences, including strong separations involving coNP and Arthur–Merlin classes.

References

It is unclear how to extend the proof of \Cref{t:soi-pKt-avg} to the worst case, i.e., to prove that the relevant inequality holds for all strings $x, y \in {0,1}n$.

— Asymmetry and Complexity of Nondeterministic Computations  (2610.20188 - Goldberg et al., 14 Oct 2026) in Section 1, subsection “The Intricate Setting of Nondeterministic Computation”