The strong exponential hierarchy collapses

Abstract

The polynomial hierarchy, composed of the levels P, NP, PNP, NPNP, etc., plays a central role in classifying the complexity of feasible computations. It is not known whether the polynomial hierarchy collapses. We resolve the question of collapse for an exponential-time analogue of the polynomial-time hierarchy. Composed of the levels E (i.e., ⋓c DTIME[2cn]), NE, PNE, NPNE, etc., the strong exponential hierarchy collapses to its &Dgr;2 level. E ≠ PNE = NPNE ⋓ NPNPNE ⋓ ··· Our proof stresses the use of partial census information and the exploitation of nondeterminism. Extending our techniques, we also derive new quantitative relativization results. We show that if the (weak) exponential hierarchy's &Dgr;j+1 and &Sgr;j+1 levels, respectively E&Sgr;pj and NE&Sgr;pj, do separate, this is due to the large number of queries NE makes to its &Sgr;pj database.1 Our technique provide a successful method of proving the collapse of certain complexity classes.