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With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy

1989
*
Journal of computer and system sciences (Print)
*

As

doi:10.1016/0022-0000(89)90033-0
fatcat:mkqhwf2x2vfp3cmpdqprjn3abm
*a*consequence, we show that*a**random**oracle*set*A**separates**PSPACE**from**the*entire*polynomial*-*time**hierarchy**with**probability**one*. 0 1989 Academic PRESS, IUC. ... We show that*with*an exponential bound of*the*form exp(n")*on**the*size of*the*circuits, they make*a*50% error*on*all possible inputs, asymptotically and uniformly. ...*The*author also thanks Professors R. Book, N. Immerman, D. Joseph, S. Mahaney, Y. Moschovakis, R. Shore, and*A*. Yao and my fellow students L. Hemachandra, J. Johnstone, and S. ...##
###
The relativized relationship between probabilistically checkable debate systems, IP and PSPACE

1995
*
Information Processing Letters
*

F or

doi:10.1016/0020-0190(94)00185-2
fatcat:dad2d7c43bcdbgc2dzkdz43osy
*the*relationships between RPCDS rn;*a*n and IP and RPCDS rn;*a*n and*PSPACE*we determine*a*natural boundary in terms of*the*parameters rn and an*separating*direct-simulability and inequality*with*... Recently,*PSPACE*was again recharacterized, this*time*in terms of*Random*Probabilistically Checkable Debate Systems 7, 8 . In particular, it was shown that*PSPACE*= PCDS log n; 1 = RPCDS logn; 1 . ... We w ould also like to thank*the*anonymous referees, who gave an improved proof of Theorem 2.42 and improved*the*general presentation. ...##
###
Polynomial-Time Random Oracles and Separating Complexity Classes
[article]

2018
*
arXiv
*
pre-print

Bennett and Gill (1981) showed that P^

arXiv:1801.07317v1
fatcat:r5hkdcdc4rg63c525vkba4y65a
*A*!= NP^*A*!= coNP^*A*for*a**random**oracle**A*,*with**probability*1. We investigate whether this result extends to individual*polynomial*-*time**random**oracles*. ... Rossman, Servedio, and Tan (2015) showed that*the**polynomial*-*time**hierarchy*is infinite relative to*a**random**oracle*, solving*a*longstanding open problem. ... We show that it cannot be improved to p-*random**oracles*or improved to*separate**the**polynomial*-*time**hierarchy*without*separating*BPP or NP*from*EXP, respectively. ...##
###
The random oracle hypothesis is false

1994
*
Journal of computer and system sciences (Print)
*

For example, for

doi:10.1016/s0022-0000(05)80084-4
fatcat:adqfb3vehbbarbh5zufdalwfde
*a*long*time*it was not known if*the**polynomial*-*time**hierarchy*(PH) can be*separated*by*oracles**from**PSPACE*. ... IP*A*#*PSPACE**A**with**Probability*1 Before we begin*the*construction of*the*counterexamples to*the**Random**Oracle*Hypothesis, we need to establish some conventions.. ...##
###
On the intellectual terrain around NP
[chapter]

1994
*
Lecture Notes in Computer Science
*

of complete problems for NP and

doi:10.1007/3-540-57811-0_1
fatcat:nin6driz7nektlvo5afle2l76e
*PSPACE*, through*the*results of structural complexity to*the*recent insights about interactive proofs. ...*The*paper shortly reviews*the*history of*the*developments*from*Göde1's 1956 letter asking for*the*computational complexity of finding proofs of theorems, through computational complexity,*the*exploration ... This startling result also gives*a*natural eounterexelJIlple to refute*the**random**oracle*hypothesis. It is shown in [HCRR90b) that*with**probability*1, for*a**random**oracle**A*, IPA #PSPACEA. ...##
###
Relativization: a Revisionistic Retrospective
[chapter]

1993
*
Current Trends in Theoretical Computer Science
*

We begin

doi:10.1142/9789812794499_0040
dblp:series/wsscs/HartmanisCCRR93
fatcat:xi32jv2ukjg4lbvrk62gorwjyu
*with**the*twice-told tale of*the*relativization principle and ponder upon its possible demise. Then, we discuss whether usual assumptions are historically accurate. ... In this column we examine*the*role of relativization in complexity theory in light of recent non-relativizing results involving interactive protocols. ... Subsequently, it was shown that*with**probability*1, IP*A*=*PSPACE**A*for*a**random**oracle**A*, and thus*the*IP =*PSPACE*result also convincingly refutes*the**random**oracle*hypothesis [HCRR90, CGH90] . ...##
###
Page 6317 of Mathematical Reviews Vol. , Issue 93k
[page]

1993
*
Mathematical Reviews
*

Since

*a*randomly selected*oracle*is*pspace*-*random**with**probability**one*, (i) and (ii) immediately imply*the*corresponding*random**oracle**separations*, thus improving*a*result of Bennett and Gill (1981) and ...*The*following*separations*are shown to hold relative to every*pspace*-*random**oracle**A*, and relative to almost 68Q Theory of computing 93k:68039 every*oracle**A*€ ESPACE. ...##
###
Complexity Barriers as Independence
[chapter]

2017
*
The Incomputable
*

*The*fact that certain classes of proof techniques,

*ones*that have specific properties, are eliminated gives us

*a*direction to search for new techniques. ... Yet, we are as far

*from*resolving it as ever. Much work has been done to unravel

*the*intricate structure in

*the*complexity world,

*the*"complexity zoo" contains hosts of inhabitants. ... Acknowledgements I am very grateful to

*a*number of people for suggestions and comments for this survey. ...

##
###
Beyond NP

2005
*
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05
*

*One*graduate student at

*the*Massachusetts Institute of Technology started to look beyond NP, asking what problems have

*a*higher complexity and how do we classify them. ... Stockmeyer passed away

*on*July 31, 2004 at

*the*age of 55 and in this paper we review some of his research and

*the*legacy he has left

*on*

*the*community. ... Yao first proved

*a*strong enough lower bound in

*a*paper [71] entitled "

*Separating*

*the*

*Polynomial*-

*Time*

*Hierarchy*by

*Oracles*." ...

##
###
Circuit size relative to pseudorandom oracles

1993
*
Theoretical Computer Science
*

For example,

doi:10.1016/0304-3975(93)90256-s
fatcat:gkkyocptl5ewvlidjyicjk7tqe
*the**separation*PA $ REC holds for*a*randomly selected*oracle**with**probability*1, but fails for every decidable*pspace*-*random**oracle*.) ... Thus,*separations*that hold relative to every*pspace*-*random**oracle*also hold relative to*a*randomly selected*oracle**with**probability*1. ...##
###
Complexity limitations on quantum computation
[article]

1998
*
arXiv
*
pre-print

There exists

arXiv:cs/9811023v1
fatcat:mxiruxhcyneehe3qa6pey6wcoq
*a*relativized world where P=BQP and*the**polynomial*-*time**hierarchy*is infinite. 3. There exists*a*relativized world where BQP does not have complete sets. 4. ... We use*the*powerful tools of counting complexity and generic*oracles*to help understand*the*limitations of*the*complexity of quantum computation. ... Conclusions We give results in this paper indicating severe restrictions*on**the*complexity of quantum computing. ...##
###
Complexity Limitations on Quantum Computation

1999
*
Journal of computer and system sciences (Print)
*

We show several results for

doi:10.1006/jcss.1999.1651
fatcat:2pkcasoih5enzg3lvd7e2b7ayi
*the*probabilistic quantum class BQP: BQP is low for PP, i.e., PP BQP =PP; There exists*a*relativized, world, where P=BQP and*the**polynomial*-*time**hierarchy*is infinite; There ... We use*the*powerful tools of counting complexity and generic*oracles*to help understand*the*limitations of*the*complexity of quantum computation. ... ACKNOWLEDGMENTS We thank Andre Berthiaume, Harry Buhrman, Richard Cleve, Ronald de Wolf, Wim van Dam, and John Watrous for*a*number of illuminating conversations*on*quantum computation. ...##
###
The Minimum Oracle Circuit Size Problem

2016
*
Computational Complexity
*

When

doi:10.1007/s00037-016-0124-0
fatcat:jbrp3snmjreknjbl24w2maqqbi
*the**oracle*is QBF,*the*resulting problem MCSP QBF is known to be complete for*PSPACE*under ZPP reductions. ... We consider variants of*the*Minimum Circuit Size Problem MCSP, where*the*goal is to minimize*the*size of*oracle*circuits computing*a*given function. ... Some of this work was carried out at*the*2014 Dagstuhl Workshop*on*Algebra in Computational Complexity (Dagstuhl Seminar 14391). ...##
###
Constructing Oracles by Lower Bound Techniques for Circuits
[chapter]

1989
*
Mathematics and Its Applications
*

In this paper, we survey recent

doi:10.1007/978-94-009-2411-6_2
fatcat:3txbzmcjlzdqrfyxoeoc377lqe
*separating*and collapsing results*on*several complexity*hierarchies*, including*the**polynomial**hierarchy*,*the*probabilistic*polynomial**hierarchy*,*the*bounded Arthur-Merlin ... In particular, some*separating*results*on**the*relativized*polynomial**hierarchy*have been found using*the*lower bound results*on*constant-depth circuits [Yao, 1985; . ... Then, there exists an*oracle**A*such that*the*classes Σ P fi(n) (*A*) form*a*proper infinite*hierarchy*between*polynomial**hierarchy*P H(*A*) and*the*class*PSPACE*(*A*). ...##
###
PH = PSPACE
[article]

2021
*
arXiv
*
pre-print

In this paper we show that

arXiv:1411.0628v16
fatcat:wk2uc6rxtnbivo6zbb5n3sidje
*PSPACE*is equal to 4th level in*the**polynomial**hierarchy*. We also deduce*a*lot of important consequences. ... BQP (bounded-error quantum*polynomial**time*) is*the*class of decision problems solvable by*a*quantum computer in*polynomial**time*,*with*an error*probability*of at most 1/3 for all instances, see [1][6]. ...*with*an error*probability*bounded away*from*1/3 for all instances, see [1] . ...
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