Deciding reducibility of quantifier patterns

Develop a decision procedure that, given two quantifier patterns, decides whether the complete formula for the first is realizability-theoretically many-one reducible to the complete formula for the second.

Background

The problem asks for an effective classification of the many-one ordering on all finite patterns drawn from the four quantifiers ordinary and almost-everywhere existential or universal.

References

Is there a decision procedure which, given quantifier patterns $\overline{P},\overline{Q} \in {\exists,\forall,\exists\infty,\forall\infty}{<\omega},$ decides whether $\langle\overline{P}\rangle \leq_m \langle\overline{Q}\rangle$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 3