Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?

0352519 - MÚ 2011 RIV US eng J - Journal Article
Buhrman, H. - Fortnow, L. - Koucký, Michal - Rogers, J.D. - Vereshchagin, N.K.
Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?
Theory of Computing Systems. Roč. 46, č. 1 (2010), s. 143-156. ISSN 1432-4350. E-ISSN 1433-0490.
[2nd International Computer Science Symposium in Russia (CSR 2007). Ekaterinburg, 03.09.2007-07.09.2007]
R&D Projects: GA ČR GP201/07/P276; GA MŠMT(CZ) 1M0545
Institutional research plan: CEZ:AV0Z10190503
Keywords : one-way functions * polynomial hierarchy * Kolmogorov generic oracles
Subject RIV: BA - General Mathematics
Impact factor: 0.600, year: 2010
http://link.springer.com/article/10.1007%2Fs00224-008-9160-8

The class TFNP, defined by Megiddo and Papadimitriou, consists of multivalued functions with values that are polynomially verifiable and guaranteed to exist. Do we have evidence that such functions are hard, for example, if TFNP is computable in polynomial-time does this imply the polynomial-time hierarchy collapses? By computing a multivalued function in deterministic polynomial-time we mean on every input producing one of the possible values of the function on that input. We give a relativized negative answer to this question by exhibiting an oracle under which TFNP functions are easy to compute but the polynomial-time hierarchy is infinite. We also show that relative to this same oracle, P/not=UP and TFNP^NP functions are not computable in polynomial-time with an NP oracle.
Permanent Link: http://hdl.handle.net/11104/0192010