The polynomial and linear hierarchies in models where the weak pigeonhole principle fails

Abstract

We show, under the assumption that factoring is hard, that a model of PV exists in which the polynomial hierarchy does not collapse to the linear hierarchy; that a model of S₂¹ exists in which NP is not in the second level of the linear hierarchy; and that a model of S₂¹ exists in which the polynomial hierarchy collapses to the linear hierarchy.

Our methods are model-theoretic. We use the assumption about factoring to get a model in which the weak pigeonhole principle fails in a certain way, and then work with this failure to obtain our results.

As a corollary of one of the proofs, we also show that in S₂¹ the failure of WPHP (for Σ^b₁ definable relations) implies that the strict version of PH does not collapse to a finite level.