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Derandomizing ArthurMerlin Games under Uniform Assumptions
 Computational Complexity
, 2000
"... We study how the nondeterminism versus determinism problem and the time versus space problem are related to the problem of derandomization. In particular, we show two ways of derandomizing the complexity class AM under uniform assumptions, which was only known previously under nonuniform assumption ..."
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Cited by 15 (0 self)
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We study how the nondeterminism versus determinism problem and the time versus space problem are related to the problem of derandomization. In particular, we show two ways of derandomizing the complexity class AM under uniform assumptions, which was only known previously under nonuniform assumptions [13, 14]. First, we prove that either AM = NP or it appears to any nondeterministic polynomial time adversary that NP is contained in deterministic subexponential time infinitely often. This implies that to any nondeterministic polynomial time adversary, the graph nonisomorphism problem appears to have subexponentialsize proofs infinitely often, the first nontrivial derandomization of this problem without any assumption. Next, we show that either all BPP = P, AM = NP, and PH P hold, or for any t(n) = 2 n) , DTIME(t(n)) DSPACE(t (n)) infinitely often for any constant > 0. Similar tradeoffs also hold for a whole range of parameters. This improves previous results [17, 5] ...
Improved derandomization of BPP using a hitting set generator
 Proceedings of Random99, LNCS 1671
, 1999
"... A hittingset generator is a deterministic algorithm which generates a set of strings that intersects every dense set recognizable by a small circuit. A polynomial time hittingset generator readily implies RP = P . Andreev et. al. (ICALP'96, and JACM 1998) showed that if polynomialtime hitting ..."
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Cited by 13 (2 self)
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A hittingset generator is a deterministic algorithm which generates a set of strings that intersects every dense set recognizable by a small circuit. A polynomial time hittingset generator readily implies RP = P . Andreev et. al. (ICALP'96, and JACM 1998) showed that if polynomialtime hittingset generator in fact implies the much stronger conclusion BPP = P .
Streaming Computation of Combinatorial Objects
 In Proceedings of the Seventeenth Annual IEEE Conference on Computational Complexity
, 2002
"... We prove (mostly tight) space lower bounds for "streaming " (or "online") computations of four fundamental combinatorial objects: errorcorrecting codes, universal hash functions, extractors, and dispersers. Streaming computations for these objects are motivated algorithmically by massive data set ..."
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Cited by 7 (2 self)
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We prove (mostly tight) space lower bounds for "streaming " (or "online") computations of four fundamental combinatorial objects: errorcorrecting codes, universal hash functions, extractors, and dispersers. Streaming computations for these objects are motivated algorithmically by massive data set applications and complexitytheoretically by pseudorandomness and derandomization for spacebounded probabilistic algorithms.
Dimension characterizations of complexity classes
 In Proceedings of the Thirtieth International Symposium on Mathematical Foundations of Computer Science
, 2006
"... We use derandomization to show that sequences of positive pspacedimension – in fact, even positive ∆ p kdimension for suitable k – have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose ∆ p 3dimension is positive, then BPP ⊆ PS and, ..."
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Cited by 4 (0 self)
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We use derandomization to show that sequences of positive pspacedimension – in fact, even positive ∆ p kdimension for suitable k – have, for many purposes, the full power of random oracles. For example, we show that, if S is any binary sequence whose ∆ p 3dimension is positive, then BPP ⊆ PS and, moreover, every BPP promise problem is PSseparable. We prove analogous results at higher levels of the polynomialtime hierarchy. The dimensionalmostclass of a complexity class C, denoted by dimalmostC, is the class consisting of all problems A such that A ∈ CS for all but a Hausdorff dimension 0 set of oracles S. Our results yield several characterizations of complexity classes, such as BPP = dimalmostP, PromiseBPP = dimalmostPSep, and AM = dimalmostNP, that refine previously known results on almostclasses. 1
NLprintable sets and Nondeterministic Kolmogorov Complexity
, 2003
"... This paper introduces nondeterministic spacebounded Kolmogorov complexity, and we show that it has some nice properties not shared by some other resourcebounded notions of Kcomplexity. ..."
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Cited by 2 (0 self)
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This paper introduces nondeterministic spacebounded Kolmogorov complexity, and we show that it has some nice properties not shared by some other resourcebounded notions of Kcomplexity.
Simplified derandomization of BPP using a hitting set generator
 Proceedings of Random99, LNCS 1671
, 2000
"... A hittingset generator is a deterministic algorithm which generates a set of strings that intersects every dense set recognizable by a small circuit. A polynomial time hittingset generator readily implies RP = P . Andreev et. al. (ICALP'96, and JACM 1998) showed that if polynomialtime hitting ..."
Abstract
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A hittingset generator is a deterministic algorithm which generates a set of strings that intersects every dense set recognizable by a small circuit. A polynomial time hittingset generator readily implies RP = P . Andreev et. al. (ICALP'96, and JACM 1998) showed that if polynomialtime hittingset generator in fact implies the much stronger conclusion BPP = P . We simplify and improve their (and later) constructions. Keywords: Derandomization, RP, BPP , onesided error versus twosided error A preliminary version of this work has appeared in the proceedings of Random99. y Department of Computer Science, Weizmann Institute of Science, Rehovot, Israel. oded@wisdom.weizmann.ac.il. z MIT Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139. salil@theory.lcs.mit.edu. Supported by an NSF Mathematical Sciences Postdoctoral Research Fellowship. x Institute of Computer Science, The Hebrew University of Jerusalem, GivatRam, Jerusalem, Israel. avi@cs.huji.ac....
Comparing Notions of Full Derandomization
 In Proceedings of the Sixteenth Annual IEEE Conference on Computational Complexity
, 2001
"... Most of the hypotheses of full derandomization fall into two sets of equivalent statements: Those equivalent to the existence of ecient pseudorandom generators and those equivalent to approximating the accepting probability of a circuit. We give the rst relativized world where these sets of equival ..."
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Most of the hypotheses of full derandomization fall into two sets of equivalent statements: Those equivalent to the existence of ecient pseudorandom generators and those equivalent to approximating the accepting probability of a circuit. We give the rst relativized world where these sets of equivalent statements are not equivalent to each other.