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

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 S2 exists in which NP is not in the second level of the linear hierarchy; and that a model of S2 exists in which the polynomial hierarchy collapses to the linear hierarchy and in which the strict version of PH does not collapse to a finite level. 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. One of the main goals of the research into fragments of bounded arithmetic is to understand which relations between computational complexity classes ∗Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland, [email protected]. This work was carried out while the author was visiting the Mathematical Institute of the Academy of Sciences of the Czech Republic in Prague. †Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, CZ115 67 Praha 1, Czech Republic, [email protected]. Supported in part by grant AV0Z10190503 and by the Eduard Čech Center grant LC505.