Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
The polynomial-time hierarchy and sparse oracles
Journal of the ACM (JACM)
Relativizing complexity classes with sparse oracles
Journal of the ACM (JACM)
Sparse sets lowness and highness
SIAM Journal on Computing
Relativized polynomial time hierarchies having exactly K levels
SIAM Journal on Computing
Probalisitic complexity classes and lowness
Journal of Computer and System Sciences
Separating and collapsing results on the relativized probabilistic polynomial-time hierarchy
Journal of the ACM (JACM)
Theoretical Computer Science
Lower bounds for the low hierarchy
Journal of the ACM (JACM)
The Extended Low Hierarchy Is an Infinite Hierarchy
SIAM Journal on Computing
Randomness and the density of hard problems
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
The equivalence problem for regular expressions with squaring requires exponential space
SWAT '72 Proceedings of the 13th Annual Symposium on Switching and Automata Theory (swat 1972)
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Certain recursive oracles can force the polynomial hierarchy to collapse to any fixed level. All collections of such oracles associated with each collapsing level form an infinite hierarchy, called the collapsing recursive oracle polynomial (CROP) hierarchy. This CROP hierarchy is a significant part of the extended low hierarchy (note that the assumption P = NP makes the both hierarchies coincide). We prove that all levels of the CROP hierarchy are distinct by showing “strong” types of separation. First, we prove that each level of the hierarchy contains a set that is immune to its lower level. Second, we show that any two adjacent levels of the CROP hierarchy can be separated by another level of the CROBPP hierarchy—a bounded-error probabilistic analogue of the CROP hierarchy. Our proofs extend the circuit lower-bound techniques of Yao, Håstad, and Ko.