Conditions on μ-d.s.r. guaranteeing membership between PLS and UEOPL (e.g., CLS for Tarski)

Identify sufficient structural conditions on a μ-downward self-reduction that imply membership in TFNP subclasses strictly between PLS and UEOPL, and in particular extend the μ-d.s.r framework to show that the Tarski fixed-point problem lies in CLS.

Background

While the μ-d.s.r framework yields PLS and UEOPL containments, the landscape of intermediate TFNP classes (e.g., CLS) remains less understood in this setting.

Characterizing conditions under which μ-d.s.r. implies membership in these intermediate classes would sharpen the framework’s granularity; the Tarski problem is a concrete target, since the paper gives a PLS containment but leaves the question of CLS-membership open.

References

Below we list a few open questions that arise from our work. What conditions can we put on a $\mu$-d.s.r to show membership in \cc{TFNP} classes between \cc{PLS} and \cc{UEOPL}? For example, how should we extend the $\mu$-d.s.r framework in a way that enables us to show that #1{Tarski} is in \cc{CLS}?

— Downward self-reducibility in the total function polynomial hierarchy (2507.19108 - Gajulapalli et al., 25 Jul 2025) in Discussion and Open questions, Item 6