The Strong Exponential Hierarchy Collapses

The polynomial hierarchy, composed of the levels P, NP, P
^NP
, NP
^NP
, 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., U
c
DTIME[2
^cn
]), NE, P
^NE
, NP, etc., the strong exponential hierarchy collapses to its Δ2 level.
$\text{E}\, \ne \text{ P}^{\text{NE}} \, = \,\text{NP}^{\text{NE}} \,\text{U}\,\text{NP}^{\text{NP}} ^{\text{NE}} \,\text{U} ···$
Our proof stresses the use of partial census information and the exploitation of nonde-terminism. Extending our techniques, we also derive new quantitative relativization results. We show that if the (weak) exponential hierarchy's Δ
j
+
1
and Σ
j
+
1
levels, respectively
$\text{E}^\Sigma _j^p and \text{NE}^\Sigma _j^p$
, do separate, this is due to the large number of queries NE makes to its
$\Sigma _j^p$
database. Our techniques provide a successful method of proving the collapse of certain complexity classes.