Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem

We give a new characterization of the strict ∀Σj sentences provable using Σk induction, for 1 ≤ j ≤ k. As a small application we show that Buss’s witnessing theorem for strict Σk formulas already holds over the relatively weak theory PV. We exhibit a combinatorial principle with the property that a lower bound for it in constant-depth Frege would imply that the narrow CNFs with short depth j Frege refutations form a strict hierarchy with j, and hence that the relativized bounded arithmetic hierarchy can be separated by a family of ∀Σ1 sentences.