The Sky is Falling: The Strong Exponential Hierarchy Collapses

This paper investigates the complexity of the high levels of the exponential hierarchy [HY84,HIS85]: Are they hard, and if so why are they hard? We show that $P^{NE} = NP^{NE}$. From this we conclude that the strong exponential hierarchy collapses: $P^{NE} = NP^{NE} \bigcup NP^{NP^{NE}} \bigcup NP^{NP^{NP^{NE}}} \bigcup \cdots,$ where NE is nondeterministic exponential time. This suprising result, a nontrivial hierarchy collapse, is based on $P^{NE}$ overmastering the $NP^{NE}$ computation tree by computing better and better partial census information. We note why the combinatorics involved prevents us from similarly proving that the polynomial hierarchy collapses. Next we look at the exponential hierarchy, which is NE given a rich database: $NE \bigcup NE^{NP} \bigcup NE^{NP^{NP}} \bigcup \cdots$. We show that if the exponential hierarchy's $\Delta_{i}$ and $\Sigma_{i}$ levels do separate, this is due not to the power of the database but to the extravagant number of queries NE makes to the database. Thus the high levels of the strong exponential hierarchy are no harder than the low levels. The high levels of the exponential hierarchy separate completely only if NE floods its database with queries. Extending our techniques, we derive sufficient conditions for collapsing complexity classes, and use them to generate strong new quantitative relativization results