Sparse sets lowness and highness
SIAM Journal on Computing
Structural complexity 1
On sparse oracles separating feasible complexity classes
Information Processing Letters
Structural complexity 2
PP is as hard as the polynomial-time hierarchy
SIAM Journal on Computing
Lower bounds for the low hierarchy
Journal of the ACM (JACM)
Boolean operations, joins, and the extended low hierarchy
Theoretical Computer Science
The complexity theory companion
The complexity theory companion
Separating, Strongly Separating, and Collapsing Relativized Complexity Classes
Proceedings of the Mathematical Foundations of Computer Science 1984
| Hi-index | 0.89 |
Denote by C the class of oracles relative to which P=NP (collapsing oracles), and by S the class of oracles relative to which PNP (separating oracles). We present structural results on C and S. Using a diagonalization argument, we show that neither C nor S is closed under disjoint union, also known as join. We show that this implies that neither C nor S is closed under union, intersection, or symmetric difference. Consequently EL"1, the first level of the extended low hierarchy, is not closed under join.