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Does the Polynomial Hierarchy Collapse if Onto Functions are Invertible?

2008
*
Theory of Computing Systems
*

Do we have evidence that such

doi:10.1007/s00224-008-9160-8
fatcat:pxvzqb3xnvemjcipo54baee7ky
*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. ...##
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Inverting onto functions

2003
*
Information and Computation
*

We look at

doi:10.1016/s0890-5401(03)00119-6
fatcat:vlvq2munznb3npgbtw2da5iivi
*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. ...##
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Efficient approximate unitary t-designs from partially invertible universal sets and their application to quantum speedup
[article]

2020
*
arXiv
*
pre-print

Phys. (2016).) required choosing gates from ensembles which contained inverses for all elements, and that

arXiv:1905.01504v4
fatcat:k477biqocbgbtgdjmrtgf2c44m
*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. ...##
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Non-uniform proof systems: A new framework to describe non-uniform and probabilistic complexity classes
[chapter]

1988
*
Lecture Notes in Computer Science
*

Relatedly, we prove that

doi:10.1007/3-540-50517-2_81
fatcat:xrsn7c5lyzfs7cszkernbfwxom
*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. ...##
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Tally NP Sets and Easy Census Functions
[article]

1998
*
arXiv
*
pre-print

We prove that every #P_1^PH

arXiv:cs/9809002v1
fatcat:i5htntseirblroresznnlwz65m
*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. ...##
###
Inverting onto functions

*
Proceedings of Computational Complexity (Formerly Structure in Complexity Theory)
*

We look at

doi:10.1109/ccc.1996.507683
dblp:conf/coco/FennerFNR96
fatcat:kijgls2gcncp3ib36p6ixorqv4
*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. ...##
###
Human balancing of an inverted pendulum: is sway size controlled by ankle impedance?

2001
*
Journal of Physiology
*

Increasing ankle impedance, stiffness or viscosity

doi:10.1111/j.1469-7793.2001.0879e.x
pmid:11313453
fatcat:xvrjdv4g5fgaver2kjkrxgkq6u
*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. ...##
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A hierarchy based on output multiplicity

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. ...

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

2014
*
arXiv
*
pre-print

This article is an appreciation, on

arXiv:1406.4106v2
fatcat:jdychuy42fdg7gtpturzlmtt3a
*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*. ...##
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Tally NP Sets and Easy Census Functions

2000
*
Information and Computation
*

We prove that every *P 1 PH

doi:10.1006/inco.1999.2810
fatcat:6pdom6ck2javbhwsuito7kyhlq
*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. ...##
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Tally NP sets and easy census functions
[chapter]

1998
*
Lecture Notes in Computer Science
*

We prove that every *P 1 PH

doi:10.1007/bfb0055798
fatcat:q4723pfln5bb5drp57hlw7ecmy
*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. ...##
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On sets polynomially enumerable by iteration

1991
*
Theoretical Computer Science
*

In

doi:10.1016/0304-3975(91)90388-i
fatcat:rd2srjzg4zb5vnhgm2vnq4237a
*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. ...##
###
NP might not be as easy as detecting unique solutions

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. ...

##
###
The complexity of perfect zero-knowledge

1987
*
Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87
*

This result implies that there

doi:10.1145/28395.28418
dblp:conf/stoc/Fortnow87
fatcat:aom3t3xkdben5m3fipabnf6fhi
*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. ...##
###
On the structure and complexity of infinite sets with minimal perfect hash functions
[chapter]

1991
*
Lecture Notes in Computer Science
*

This paper studies

doi:10.1007/3-540-54967-6_70
fatcat:zcvrcx62fvcozje2wdazst5rte
*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. ...
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