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20
The quantitative structure of exponential time
 Complexity Theory Retrospective II
, 1997
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ResourceBounded Measure and Randomness
"... We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than ..."
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Cited by 42 (6 self)
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We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes and . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than timebounded randomness but which suffice for many of the applications in complexity theory.
Power from Random Strings
 IN PROCEEDINGS OF THE 43RD IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2002
"... We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions. These sets are provably not complete under the usual manyone reductions. Let ..."
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Cited by 41 (17 self)
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We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions. These sets are provably not complete under the usual manyone reductions. Let
When Worlds Collide: Derandomization, Lower Bounds, and Kolmogorov Complexity
 OF REDUCTIONS,IN“PROC.29THACM SYMPOSIUM ON THEORY OF COMPUTING
, 1997
"... This paper has the following goals:  To survey some of the recent developments in the field of derandomization.  To introduce a new notion of timebounded Kolmogorov complexity (KT), and show that it provides a useful tool for understanding advances in derandomization, and for putting vario ..."
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Cited by 17 (5 self)
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This paper has the following goals:  To survey some of the recent developments in the field of derandomization.  To introduce a new notion of timebounded Kolmogorov complexity (KT), and show that it provides a useful tool for understanding advances in derandomization, and for putting various results in context.  To illustrate the usefulness of KT, by answering a question that has been posed in the literature, and  To pose some promising directions for future research.
On the complexity of random strings (Extended Abstract)
 IN STACS 96
, 1996
"... We show that the set R of Kolmogorov random strings is truthtable complete. This improves the previously known Turing completeness of R and shows how the halting problem can be encoded into the distribution of random strings rather than using the time complexity of nonrandom strings. As an applic ..."
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Cited by 14 (1 self)
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We show that the set R of Kolmogorov random strings is truthtable complete. This improves the previously known Turing completeness of R and shows how the halting problem can be encoded into the distribution of random strings rather than using the time complexity of nonrandom strings. As an application we obtain that Post's simple set is truthtable complete in every Kolmogorov numbering. We also show that the truthtable completeness of R cannot be generalized to sizecomplexity with respect to arbitrary acceptable numberings. In addition we note that R is not frequency computable.
Complete Sets and Structure in Subrecursive Classes
 In Proceedings of Logic Colloquium '96
, 1998
"... In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completene ..."
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Cited by 13 (1 self)
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In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...
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 9 (2 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
Randomness is Hard
 SIAM Journal on Computing
, 2000
"... We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomi ..."
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Cited by 6 (3 self)
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We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are: polynomial time CD complexity dened by Sipser, the nondeterministic variant due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity, CS introduced by Hartmanis. For all of these measures we dene the set of random strings R CD t , R CND t , and R CS s as the set of strings x such that CD t (x), CND t (x), and CS s (x) is greater than or equal to the length of x, for s and t polynomials. We show the following: MA NP R CD t , where MA is the class of MerlinArthur games dened by Babai. AM NP R CND t , where AM is the class of ArthurMerlin games. PSPACE NP cR CS s . In the last item cR CS s is the set of pairs <x; y> so that x is random given y. These results show that the set of random strings for various resource bounds is hard for...
Truthtable closure and Turing closure of average polynomial time have different measures in EXP
 In Proceedings of the Eleventh Annual IEEE Conference on Computational Complexity
, 1996
"... Let PPcomp denote the sets that are solvable in polynomial time on average under every polynomialtime computable distribution on the instances. In this paper we show that the truthtable closure of PPcomp has measure 0 in EXP. Since, as we show, EXP is Turing reducible to PPcomp , the Turing clo ..."
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Cited by 5 (2 self)
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Let PPcomp denote the sets that are solvable in polynomial time on average under every polynomialtime computable distribution on the instances. In this paper we show that the truthtable closure of PPcomp has measure 0 in EXP. Since, as we show, EXP is Turing reducible to PPcomp , the Turing closure has measure 1 in EXP and thus, PPcomp is an example of a subclass of E such that the closure under truthtable reduction and the closure under Turing reduction have different measures in EXP. Furthermore, it is shown that there exists a set A in PPcomp such that for every k, the class of sets L such that A is ktruthtable reducible to L has measure 0 in EXP. 1 Introduction A randomized problem (or distributional problem) is a pair consisting of a decision problem and a density function. A randomized decision problem (A; ¯) is solvable in average polynomial time ((A; ¯) is in AP) if there exists a deterministic Turing machine M such that A = L(M ) and TimeM , the running time of M ...