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Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?

Harry Buhrman, Lance Fortnow, Michal Koucký, John D. Rogers, Nikolay Vereshchagin
2008 Theory of Computing Systems  
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?  ...  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.  ...  , and reproduction in any medium, provided the original author(s) and source are credited.  ... 
doi:10.1007/s00224-008-9160-8 fatcat:pxvzqb3xnvemjcipo54baee7ky

Inverting onto functions

Stephen A. Fenner, Lance Fortnow, Ashish V. Naik, John D. Rogers
2003 Information and Computation  
We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse.  ...  * . • Every total multivalued nondeterministic function has a polynomial-time computable refinement. • In polynomial time, one can compute satisfying assignments for any polynomial-time computable set  ...  Acknowledgments The authors would like to thank Lane Hemaspaandra, Stuart Kurtz, and Alan Selman for their insightful comments on this work.  ... 
doi:10.1016/s0890-5401(03)00119-6 fatcat:vlvq2munznb3npgbtw2da5iivi

Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup [article]

Rawad Mezher, Joe Ghalbouni, Joseph Dgheim, Damian Markham
2020 arXiv   pre-print
Phys. (2016).) required choosing gates from ensembles which contained inverses for all elements, and that the entries of the unitaries are algebraic.  ...  We reduce these requirements, to sets that contain elements without inverses in the set, and non-algebraic entries, which we dub partially invertible universal sets.  ...  Acknowledgements We thank Juan Bermejo-Vega for useful discussions, and for pointing out that our graph gadgets are hard to sample from classically.  ... 
arXiv:1905.01504v4 fatcat:k477biqocbgbtgdjmrtgf2c44m

Non-uniform proof systems: A new framework to describe non-uniform and probabilistic complexity classes [chapter]

Jürgen Kämper
1988 Lecture Notes in Computer Science  
Relatedly, we prove that if A 2 NP is NPSV-selective, then A is Low 2 . We prove that the polynomial hierarchy collapses even further, namely to NP, if all coNP sets are NPMV-selective.  ...  We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to ZPP NP (and thus, in particular, to NP NP ).  ...  Acknowledgments The authors would like to thank S. Biswas, H. Buhrman, L. Fortnow, Y. Han, E. Hemaspaandra, and M. Zimand for many helpful comments and suggestions.  ... 
doi:10.1007/3-540-50517-2_81 fatcat:xrsn7c5lyzfs7cszkernbfwxom

Tally NP Sets and Easy Census Functions [article]

Judy Goldsmith, Mitsunori Ogihara, Joerg Rothe
1998 arXiv   pre-print
We prove that every #P_1^PH function can be computed in FP^#P_1^#P_1. Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does.  ...  We characterize this question in terms of unlikely collapses of language and function classes such as the containment of #P_1 in FP, where #P_1 is the class of functions that count the witnesses for tally  ...  pointers to the literature.  ... 
arXiv:cs/9809002v1 fatcat:i5htntseirblroresznnlwz65m

Inverting onto functions

S.A. Fenner, L. Fortnow, A.V. Naik, J.D. Rogers
Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)  
We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse.  ...  * . • Every total multivalued nondeterministic function has a polynomial-time computable refinement. • In polynomial time, one can compute satisfying assignments for any polynomial-time computable set  ...  Acknowledgments The authors would like to thank Lane Hemaspaandra, Stuart Kurtz, and Alan Selman for their insightful comments on this work.  ... 
doi:10.1109/ccc.1996.507683 dblp:conf/coco/FennerFNR96 fatcat:kijgls2gcncp3ib36p6ixorqv4

Human balancing of an inverted pendulum: is sway size controlled by ankle impedance?

Ian D. Loram, Sue M. Kelly, Martin Lakie
2001 Journal of Physiology  
Increasing ankle impedance, stiffness or viscosity are not the only methods by which sway size could be reduced.  ...  Using the ankle musculature, subjects balanced a large inverted pendulum. The equilibrium of the pendulum is unstable and quasi-regular sway was observed like that in quiet standing.  ...  Acknowledgements The apparatus was funded by a grant from the Sir Jules Thorn Charitable Trusts and constructed by Mr Steve Allen.  ... 
doi:10.1111/j.1469-7793.2001.0879e.x pmid:11313453 fatcat:xvrjdv4g5fgaver2kjkrxgkq6u

A hierarchy based on output multiplicity

Ashish V. Naik, John D. Rogers, James S. Royer, Alan L. Selman
1998 Theoretical Computer Science  
If for any k > 1, the class NPkV collapses into the class NP(k -1 )V, then the polynomial hierarchy collapses to C,'. 2.  ...  We exhibit an oracle relative to which the polynomial hierarchy collapses to PNP, but the output-multiplicity hierarchy is strict. 3.  ...  Acknowledgements We would like to thank Lance Fortnow for his quick and useful comments on an early draft of the main proof of Section 3 and Ken Regan for helpful discussions on random oracles.  ... 
doi:10.1016/s0304-3975(98)00060-7 fatcat:wuughxj625f63j5uslrblmvtqu

Beautiful Structures: An Appreciation of the Contributions of Alan Selman [article]

Lane A. Hemaspaandra
2014 arXiv   pre-print
This article is an appreciation, on the occasion of his retirement, of some of the most lovely concepts and results that Alan has contributed to the field.  ...  Professor Alan Selman has been a giant in the field of computational complexity for the past forty years.  ...  Figure 1 : 1 Alan Selman The polynomial hierarchy collapses if and only if there is a sparse oracle relative to which the polynomial hierarchy collapses.  ... 
arXiv:1406.4106v2 fatcat:jdychuy42fdg7gtpturzlmtt3a

Tally NP Sets and Easy Census Functions

Judy Goldsmith, Mitsunori Ogihara, Jörg Rothe
2000 Information and Computation  
We prove that every *P 1 PH function can be computed in FP *P 1 *P 1 . Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does.  ...  We characterize this question in terms of unlikely collapses of language and function classes such as *P 1 FP, where *P 1 is the class of functions that count the witnesses for tally NP sets.  ...  against P having easy census functions, in light of the results of Impagliazzo and Wigderson [IW97] , showing that very reasonable hypotheses imply P=BPP.  ... 
doi:10.1006/inco.1999.2810 fatcat:6pdom6ck2javbhwsuito7kyhlq

Tally NP sets and easy census functions [chapter]

Judy Goldsmith, Mitsunori Ogihara, Jörg Rothe
1998 Lecture Notes in Computer Science  
We prove that every *P 1 PH function can be computed in FP *P 1 *P 1 . Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does.  ...  We characterize this question in terms of unlikely collapses of language and function classes such as *P 1 FP, where *P 1 is the class of functions that count the witnesses for tally NP sets.  ...  against P having easy census functions, in light of the results of Impagliazzo and Wigderson [IW97] , showing that very reasonable hypotheses imply P=BPP.  ... 
doi:10.1007/bfb0055798 fatcat:q4723pfln5bb5drp57hlw7ecmy

On sets polynomially enumerable by iteration

Lane A. Hemachandra, Albrecht Hoene, Dirk Siefkes, Paul Young
1991 Theoretical Computer Science  
In the final section of the paper we show that ato NP.complete set can be iteratively enumerated in lexicographically increasing order unless the polynomial time hierarchy collapses to N P.  ...  ,-self-reduction is via a function whose inverse is computable in polynomial time, then the above results hold with the polynomial enumeration gi"en by a function whose inverse is computable in pol.  ...  We are particularly grateful to Jos6 Balc/~xar for pointing out that the oienumerability results apply to all recursive P-cylinders, and to Richard Chang for simplifying the proof of Corullary 6.7.  ... 
doi:10.1016/0304-3975(91)90388-i fatcat:rd2srjzg4zb5vnhgm2vnq4237a

NP might not be as easy as detecting unique solutions

Richard Beigel, Harry Buhrman, Lance Fortnow
1998 Proceedings of the thirtieth annual ACM symposium on Theory of computing - STOC '98  
The oracle A is the first where The construction gives a much simpler proof than that of Fenner, Fortnow and Kurtz of a relativized world where all the NP-complete sets are polynomial-time isomorphic.  ...  It is the first such computable oracle. Relative to A we have a collapse of EXP A ZPP A P A /poly.  ...  Acknowledgments We thank Alexis Maciel for helpful discussions and Dieter van Melkebeek for helpful comments on the write-up.  ... 
doi:10.1145/276698.276737 dblp:conf/stoc/BeigelBF98 fatcat:xjmozzrylbd5day56z4q3zfx7m

The complexity of perfect zero-knowledge

L. Fortnow
1987 Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87  
This result implies that there are not any perfect zero-knowledge protocols for NP-complete languages unless the polynomial time hierarchy collapses.  ...  Acknowledgements The author would like to express his gratitude to his advisor, Mike Sipser, for his support and encouragement.  ...  The author would also like to thank Mike, Silvio Micali, Oded Goldreich, Joan Feigenbaum, Paul Beame, Eric Schwabe and Su-Ming Wu for their useful comments on this paper.  ... 
doi:10.1145/28395.28418 dblp:conf/stoc/Fortnow87 fatcat:aom3t3xkdben5m3fipabnf6fhi

On the structure and complexity of infinite sets with minimal perfect hash functions [chapter]

Judy Goldsmith, Lane A. Hemachandra, Kenneth Kunen
1991 Lecture Notes in Computer Science  
This paper studies the class of infinite sets that have minimal perfect hash functions one-to-one onto maps between the sets and E·-computable in polynomial time.  ...  computable minimal perfect hash functions: If E = Ef, then all infinite NP sets have polynomial-time computable minimal perfect hash functions.  ...  Acknowledgements We would like to thank Richard Karp for the discussions that originally inspired this work.  ... 
doi:10.1007/3-540-54967-6_70 fatcat:zcvrcx62fvcozje2wdazst5rte
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