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28
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 54 (9 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 reducibl ..."
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Cited by 15 (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.) This is an extended version of a paper that appeared in Proceedings of the 21 st Symposium on
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 10 (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 10 (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
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.
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 5 (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
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
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
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 3 (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
Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic
"... Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorovrandom strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of prob ..."
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Cited by 2 (2 self)
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Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorovrandom strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomialtime truthtable reducibility to RK (the set of Kolmogorovrandom strings) that lies between BPP and PSPACE [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆C⊆PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.