The strong exponential hierarchy collapses

Abstract Composed of the levels E (i.e., ∪ c DTIME[2 cn ]), NE, P NE , NP NE , etc., the strong exponential hierarchy is an exponential-time analogue of the polynomial-time hierarchy. This paper shows that the strong exponential hierarchy collapses to P NE , its Δ 2 level. E ≠ p NE = NP NE ∪ NP NP NE ∪ … The proof stresses the use of partial census information and the exploitation of nondeterminism. Extending these techniques, we derive new quantitative relativization results: if the weak exponential hierarchy's Δ J + 1 and Σ j + 1 levels, respectively E Σ j p and NE Σ j p , do separate, this is due to the large number of queries NE makes to its Σ j p database. Our techniques provide a successful method of proving the collapse of certain complexity classes.

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