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

Jin-Yi Cai
1989 Journal of computer and system sciences (Print)  
As 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.  ... 
doi:10.1016/0022-0000(89)90033-0 fatcat:mkqhwf2x2vfp3cmpdqprjn3abm

The relativized relationship between probabilistically checkable debate systems, IP and PSPACE

Alexander Russell, Ravi Sundaram
1995 Information Processing Letters  
F or 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.  ... 
doi:10.1016/0020-0190(94)00185-2 fatcat:dad2d7c43bcdbgc2dzkdz43osy

Polynomial-Time Random Oracles and Separating Complexity Classes [article]

John M. Hitchcock, Adewale Sekoni, Hadi Shafei
2018 arXiv   pre-print
Bennett and Gill (1981) showed that P^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.  ... 
arXiv:1801.07317v1 fatcat:r5hkdcdc4rg63c525vkba4y65a

The random oracle hypothesis is false

Richard Chang, Benny Chor, Oded Goldreich, Juris Hartmanis, Johan Håstad, Desh Ranjan, Pankaj Rohatgi
1994 Journal of computer and system sciences (Print)  
For example, for 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..  ... 
doi:10.1016/s0022-0000(05)80084-4 fatcat:adqfb3vehbbarbh5zufdalwfde

On the intellectual terrain around NP [chapter]

Juris Hartmanis, Suresh Chari
1994 Lecture Notes in Computer Science  
of complete problems for NP and 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.  ... 
doi:10.1007/3-540-57811-0_1 fatcat:nin6driz7nektlvo5afle2l76e

Relativization: a Revisionistic Retrospective [chapter]

Juris Hartmanis, Richard Chang, Suresh Chari, Desh Ranjan, Pankaj Rohatgi
1993 Current Trends in Theoretical Computer Science  
We begin 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] .  ... 
doi:10.1142/9789812794499_0040 dblp:series/wsscs/HartmanisCCRR93 fatcat:xi32jv2ukjg4lbvrk62gorwjyu

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]

Antonina Kolokolova
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.  ... 
doi:10.1007/978-3-319-43669-2_10 fatcat:i3d4hczuajb5fkwgsw3q4g3h7a

Beyond NP

Lance Fortnow
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."  ... 
doi:10.1145/1060590.1060609 dblp:conf/stoc/Fortnow05 fatcat:5ikpf2sn4bhkbpodkx3lzh4cdq

Circuit size relative to pseudorandom oracles

Jack H. Lutz, William J. Schmidt
1993 Theoretical Computer Science  
For example, 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.  ... 
doi:10.1016/0304-3975(93)90256-s fatcat:gkkyocptl5ewvlidjyicjk7tqe

Complexity limitations on quantum computation [article]

Lance Fortnow, John D. Rogers
1998 arXiv   pre-print
There exists 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.  ... 
arXiv:cs/9811023v1 fatcat:mxiruxhcyneehe3qa6pey6wcoq

Complexity Limitations on Quantum Computation

Lance Fortnow, John Rogers
1999 Journal of computer and system sciences (Print)  
We show several results for 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.  ... 
doi:10.1006/jcss.1999.1651 fatcat:2pkcasoih5enzg3lvd7e2b7ayi

The Minimum Oracle Circuit Size Problem

Eric Allender, Dhiraj Holden, Valentine Kabanets
2016 Computational Complexity  
When 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).  ... 
doi:10.1007/s00037-016-0124-0 fatcat:jbrp3snmjreknjbl24w2maqqbi

Constructing Oracles by Lower Bound Techniques for Circuits [chapter]

Ker-I Ko
1989 Mathematics and Its Applications  
In this paper, we survey recent 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).  ... 
doi:10.1007/978-94-009-2411-6_2 fatcat:3txbzmcjlzdqrfyxoeoc377lqe

PH = PSPACE [article]

Valerii Sopin
2021 arXiv   pre-print
In this paper we show that 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] .  ... 
arXiv:1411.0628v16 fatcat:wk2uc6rxtnbivo6zbb5n3sidje
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