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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 36 (15 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
Reducing the Complexity of Reductions
 Computational Complexity
, 1997
"... We prove that the BermanHartmanis isomorphism conjecture is true under AC 0 reductions. More generally, we show three theorems that hold for any complexity class C closed under (uniform) TC 0 computable manyone reductions. Isomorphism: The sets complete for C under AC 0 reductions are all i ..."
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Cited by 27 (13 self)
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We prove that the BermanHartmanis isomorphism conjecture is true under AC 0 reductions. More generally, we show three theorems that hold for any complexity class C closed under (uniform) TC 0 computable manyone reductions. Isomorphism: The sets complete for C under AC 0 reductions are all isomorphic under isomorphisms computable and invertible by AC 0 circuits of depth three. Gap: The sets that are complete for C under AC 0 and NC 0 reducibility coincide. Stop Gap: The sets that are complete for C under AC 0 [mod 2] and AC 0 reducibility do not coincide. (These theorems hold both in the nonuniform and Puniform settings.) To prove the second theorem for Puniform settings, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.) 1 Introduction The notion of complete sets in complexity classes provides one of ...
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 18 (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.
The Complexity of Satisfiability Problems: Refining Schaefer’s Theorem
 J. Comput. Sys. Sci
"... problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomialtime isomorphism (and these isomorphism types are distinct if and onl ..."
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Cited by 17 (7 self)
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problem for a given constraint language is either in P or is NPcomplete, and identified all tractable cases. Schaefer’s dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomialtime isomorphism (and these isomorphism types are distinct if and only if P ̸ = NP). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.
Arithmetic circuits and counting complexity classes
 In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica
"... Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in ..."
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Cited by 17 (3 self)
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Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in
Amplifying lower bounds by means of selfreducibility
 In IEEE Conference on Computational Complexity
, 2008
"... We observe that many important computational problems in NC 1 share a simple selfreducibility property. We then show that, for any problem A having this selfreducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ɛ for every ɛ>0 (counting the numb ..."
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Cited by 13 (4 self)
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We observe that many important computational problems in NC 1 share a simple selfreducibility property. We then show that, for any problem A having this selfreducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ɛ for every ɛ>0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC 1 and has the selfreducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC 0 circuits of size n 1+ɛd. If one were able to improve this lower bound to show that there is some constant ɛ>0 such that every TC 0 circuit family recognizing BFE has size n 1+ɛ, then it would follow that TC 0 ̸ = NC 1. We show that proving lower bounds of the form n 1+ɛ is not ruled out by the Natural Proof framework of Razborov and Rudich and hence there is currently no known barrier for separating classes such as ACC 0,TC 0 and NC 1 via existing “natural ” approaches to proving circuit lower bounds. We also show that problems with small uniform constantdepth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known timespace tradeoff lower bounds to show that SAT requires uniform depth d TC 0 and AC 0 [6] circuits of size n 1+c for some constant c depending on d. 1
A Status Report on the P versus NP Question
"... We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation. ..."
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Cited by 1 (1 self)
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We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation.
Strong Reductions and Isomorphism of Complete Sets
"... We study the structure of the polynomialtime complete sets for NP and PSPACE under strong nondeterministic polynomialtime reductions (SNPreductions). We show the following results. • If NP contains a prandom language, then all polynomialtime complete sets for PSPACE are SNPisomorphic. • If NP ..."
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We study the structure of the polynomialtime complete sets for NP and PSPACE under strong nondeterministic polynomialtime reductions (SNPreductions). We show the following results. • If NP contains a prandom language, then all polynomialtime complete sets for PSPACE are SNPisomorphic. • If NP ∩ coNP contains a prandom language, then all polynomialtime complete sets for NP are SNPisomorphic.