Power of non-adaptive downward self-reductions

Determine whether, for any TFΣ_i^P problem, the existence of a non-adaptive downward self-reduction implies membership in a class strictly smaller than PLS with a Σ_{i-1}^P oracle.

Background

The paper’s collapses from TFΣi^P to PLS^Σ{i-1}^P rely on downward self-reductions. Several showcased reductions (e.g., for graph games, P-LCP, and LOP) are non-adaptive, meaning the recursive queries are fixed independently of answers.

It is natural to ask whether non-adaptivity provides additional structure enabling placement in even smaller TFNP subclasses than PLS^{Σ_{i-1}^P.}

References

Below we list a few open questions that arise from our work. Does a non-adaptive downward self-reduction for a \cc{TF\Sigma_i^P} problem imply membership in a smaller class than \cc{PLS^{{\Sigma_{i-1}^P}?} This is a particularly important question since our downward self-reductions for graph games (\cref{subsubsec: graph-games}), #1{P-LCP} (\cref{subsubsec: p-lcp}), and #1{LOP} (\cref{sec:lop}) are all non-adaptive.

— Downward self-reducibility in the total function polynomial hierarchy (2507.19108 - Gajulapalli et al., 25 Jul 2025) in Discussion and Open questions, Item 3

Power of non-adaptive downward self-reductions

Background

References

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