Quantitative relativizations of complexity classes
SIAM Journal on Computing
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)
The polynomial-time hierarchy and sparse oracles
Journal of the ACM (JACM)
The ismorphism conjecture fails relative to a random oracle
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Separating and collapsing results on the relativized probabilistic polynomial-time hierarchy
Journal of the ACM (JACM)
COCOON '98 Proceedings of the 4th Annual International Conference on Computing and Combinatorics
Competing provers yield improved Karp-Lipton collapse results
Information and Computation
Competing provers yield improved Karp-Lipton collapse results
Information and Computation
Distributionally-hard languages
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
Collapsing recursive oracles for relativized polynomial hierarchies
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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Baker, Gill, and Solovay constructed sparse sets A and B such that P(A) ≠ NP(A) and NP(B) ≠ co-NP(B). In contrast to their results, we prove that P = NP if and only if for every tally language T, P(T) = NP( T), and that NP = co-NP if and only if for every tally language T, NP(T) = co-NP(T). We show that the polynomial hierarchy collapses if and only if there is a sparse set S such that the polynomial hierarchy relative to S collapses. Similar results are obtained for several other complexity classes.