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38
Pseudorandomness and averagecase complexity via uniform reductions
 In Proceedings of the 17th Annual IEEE Conference on Computational Complexity
, 2002
"... Abstract. Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP � = BPP). Unlike results in the nonuniform setting, their result does not provide a continuous tradeoff between worstcase hardness an ..."
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Cited by 55 (8 self)
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Abstract. Impagliazzo and Wigderson (36th FOCS, 1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP � = BPP). Unlike results in the nonuniform setting, their result does not provide a continuous tradeoff between worstcase hardness and pseudorandomness, nor does it explicitly establish an averagecase hardness result. In this paper: ◦ We obtain an optimal worstcase to averagecase connection for EXP: if EXP � ⊆ BPTIME(t(n)), then EXP has problems that cannot be solved on a fraction 1/2 + 1/t ′ (n) of the inputs by BPTIME(t ′ (n)) algorithms, for t ′ = t Ω(1). ◦ We exhibit a PSPACEcomplete selfcorrectible and downward selfreducible problem. This slightly simplifies and strengthens the proof of Impagliazzo and Wigderson, which used a #Pcomplete problem with these properties. ◦ We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via “blackbox ” uniform reductions.
What can be efficiently reduced to the Kolmogorovrandom strings?
 ANNALS OF PURE AND APPLIED LOGIC
, 2004
"... We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorovrandom strings RC. This question arises because PSPACE ⊆ P RC and NEXP ⊆ NP RC, and no larger complexity classes are known to be reducib ..."
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Cited by 20 (5 self)
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We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorovrandom strings RC. This question arises because PSPACE ⊆ P RC and NEXP ⊆ NP RC, and no larger complexity classes are known to be reducible to RC in this way. We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. What follows is a list of some of our main results. • Although Kummer showed that, for every universal machine U there is a time bound t such that the halting problem is disjunctive truthtable reducible to RCU in time t, there is no such time bound t that suffices for every universal machine U. We also show that, for some machines U, the disjunctive reduction can be computed in as little as doublyexponential time. • Although for every universal machine U, there are very complex sets that are ≤P dttreducible to RCU, it is nonetheless true that P = REC ∩ ⋂ {A: U A ≤P dtt RCU}. (A similar statement holds for paritytruthtable reductions.)
Derandomization and Distinguishing Complexity
, 2003
"... We continue an investigation of resourcebounded Kolmogorov complexity and derandomization techniques begun in [2, 3]. ..."
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Cited by 9 (5 self)
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We continue an investigation of resourcebounded Kolmogorov complexity and derandomization techniques begun in [2, 3].
A note on dimensions of polynomial size circuits
 Electronic Colloquium on Computational Complexity
, 2004
"... In this paper, we use resourcebounded dimension theory to investigate polynomial size circuits. We show that for every i ≥ 0, P/poly has ith order scaled p 3strong dimension 0. We also show that P/poly i.o. has p ..."
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Cited by 7 (0 self)
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In this paper, we use resourcebounded dimension theory to investigate polynomial size circuits. We show that for every i ≥ 0, P/poly has ith order scaled p 3strong dimension 0. We also show that P/poly i.o. has p
Limits on the Computational Power of Random Strings
"... Let C(x) andK(x) denote plain and prefix Kolmogorov complexity, respectively, and let RC and RK denote the sets of strings that are “random ” according to these measures; both RK and RC are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both RK and RC, and that every ..."
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Cited by 6 (3 self)
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Let C(x) andK(x) denote plain and prefix Kolmogorov complexity, respectively, and let RC and RK denote the sets of strings that are “random ” according to these measures; both RK and RC are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both RK and RC, and that every set in BPP is polynomialtime truthtable reducible to both RK and RC [ABK06a, BFKL10]. (All of these inclusions hold, no matter which “universal ” Turing machine one uses in the definitions of C(x) andK(x).) Since each machine U gives rise to a slightly different measure CU or KU, these inclusions can be stated as: • BPP ⊆ DEC ∩ ⋂ U
What can be efficiently reduced to the Krandom strings?, in
 of Lecture Notes in Computer Science
"... STACS Topic Classification: Computational and structural complexity We investigate the question of whether one can characterize complexity classes in terms of efficient reducibility to the set of Kolmogorovrandom strings RK. We show that this question is dependent on the choice of universal machine ..."
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Cited by 6 (3 self)
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STACS Topic Classification: Computational and structural complexity We investigate the question of whether one can characterize complexity classes in terms of efficient reducibility to the set of Kolmogorovrandom strings RK. We show that this question is dependent on the choice of universal machine in the definition of Kolmogorov complexity. We show for a broad class of reductions that the sets reducible to RK have very low computational complexity. Further, we exhibit some other properties of RK that depend on the choice of universal machine. 1
The Pervasive Reach of ResourceBounded Kolmogorov Complexity in Computational Complexity Theory
"... We continue an investigation into resourcebounded Kolmogorov complexity [ABK + 06], which highlights the close connections between circuit complexity and Levin’s timebounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to ..."
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Cited by 6 (1 self)
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We continue an investigation into resourcebounded Kolmogorov complexity [ABK + 06], which highlights the close connections between circuit complexity and Levin’s timebounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introduced have many advantages over other approaches to defining resourcebounded Kolmogorov complexity (such as much greater independence from the underlying choice of universal machine that is used to define the measure) [ABK + 06]. Here, we study the properties of other measures that arise naturally in this framework. The motivation for introducing yet more notions of resourcebounded Kolmogorov complexity are twofold: • to demonstrate that other complexity measures such as branchingprogram size and formula size can also be discussed in terms of Kolmogorov complexity, and • to demonstrate that notions such as nondeterministic Kolmogorov complexity and distinguishing complexity [BFL02] also fit well into this framework. The main theorems that we provide using this new approach to resourcebounded Kolmogorov complexity are: • A complete set (RKNt) for NEXP/poly defined in terms of strings of high Kolmogorov complexity.
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 4 (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.
Kolmogorov complexity and computational complexity
 Complexity of Computations and Proofs. Quaderni di Matematica
, 2004
"... We describe the properties of various notions of timebounded Kolmogorov complexity and other connections between Kolmogorov complexity and computational complexity. 1 ..."
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Cited by 4 (0 self)
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We describe the properties of various notions of timebounded Kolmogorov complexity and other connections between Kolmogorov complexity and computational complexity. 1
Derandomizing from Random Strings
"... In this paper we show that BPP is truthtable reducible to the set of Kolmogorov random strings RK. It was previously known that PSPACE, and hence BPP is Turingreducible to RK. The earlier proof relied on the adaptivity of the Turingreduction to find a Kolmogorovrandom string of polynomial length ..."
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Cited by 4 (1 self)
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In this paper we show that BPP is truthtable reducible to the set of Kolmogorov random strings RK. It was previously known that PSPACE, and hence BPP is Turingreducible to RK. The earlier proof relied on the adaptivity of the Turingreduction to find a Kolmogorovrandom string of polynomial length using the set RK as oracle. Our new nonadaptive result relies on a new fundamental fact about the set RK, namely each initial segment of the characteristic sequence of RK is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorovcomplexity when used as advice are not much more useful than randomly chosen strings. 1