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43
Robust PCPs of Proximity, Shorter PCPs and Applications to Coding
 in Proc. 36th ACM Symp. on Theory of Computing
, 2004
"... We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) ..."
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Cited by 80 (25 self)
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We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) Boolean queries.
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
TimeSpace Tradeoffs for Satisfiability
 Journal of Computer and System Sciences
, 1997
"... We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved ..."
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Cited by 29 (1 self)
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We give the first nontrivial modelindependent timespace tradeoffs for satisfiability. Namely, we show that SAT cannot be solved simultaneously in n 1+o(1) time and n 1\Gammaffl space for any ffl ? 0 on general randomaccess nondeterministic Turing machines. In particular, SAT cannot be solved deterministically by a Turing machine using quasilinear time and p n space. We also give lower bounds for logspace uniform NC 1 circuits and branching programs. Our proof uses two basic ideas. First we show that if SAT can be solved nondeterministically with a small amount of time then we can collapse a nonconstant number of levels of the polynomialtime hierarchy. We combine this work with a result of Nepomnjascii that shows that a nondeterministic computation of super linear time and sublinear space can be simulated in alternating linear time. A simple diagonalization yields our main result. We discuss how these bounds lead to a new approach to separating the complexity classes NL a...
TimeSpace Tradeoffs for Nondeterministic Computation
 In Proceedings of the 15th IEEE Conference on Computational Complexity
, 2000
"... We show new tradeoffs for satisfiability and nondeterministic linear time. Satisfiability cannot be solved on general purpose randomaccess Turing machines in time n 1.618 and space n o(1) . This improves recent results of Fortnow and of Lipton and Viglas. In general, for any constant a less tha ..."
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Cited by 24 (2 self)
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We show new tradeoffs for satisfiability and nondeterministic linear time. Satisfiability cannot be solved on general purpose randomaccess Turing machines in time n 1.618 and space n o(1) . This improves recent results of Fortnow and of Lipton and Viglas. In general, for any constant a less than the golden ratio, we prove that satisfiability cannot be solved in time n a and space n b for some positive constant b. Our techniques allow us to establish this result for b < 1 2 ( a+2 a 2  a). We can do better for a close to the golden ratio, for example, satisfiability cannot be solved by a randomaccess Turing machine using n 1.46 time and n .11 space. We also show tradeoffs for nondeterministic linear time computations using sublinear space. For example, there exists a language computable in nondeterministic linear time and n .619 space that cannot be computed in deterministic n 1.618 time and n o(1) space. Higher up the polynomialtime hierarchy we can get be...
Nondeterministic Polynomial Time versus Nondeterministic Logarithmic Space
 In Proceedings, Twelfth Annual IEEE Conference on Computational Complexity
, 1996
"... We discuss the possibility of using the relatively old technique of diagonalization to separate complexity classes, in particular NL from NP. We show several results in this direction. ffl Any nonconstant level of the polynomialtime hierarchy strictly contains NL. ffl SAT is not simultaneously in ..."
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Cited by 22 (1 self)
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We discuss the possibility of using the relatively old technique of diagonalization to separate complexity classes, in particular NL from NP. We show several results in this direction. ffl Any nonconstant level of the polynomialtime hierarchy strictly contains NL. ffl SAT is not simultaneously in NL and deterministic n log j n time for any j. ffl On the negative side, we present a relativized world where P = NP but any nonconstant level of the polynomialtime hierarchy differs from P. 1 Introduction Separating complexity classes remains the most important and difficult of problems in theoretical computer science. Circuit complexity and other techniques on finite functions have seen some exciting early successes (see the survey of Boppana and Sipser [BS90]) but have yet to achieve their promise of separating complexity classes above logarithmic space. Other techniques based on logic and geometry also have given us separations only on very restricted models. We should turn back to...
A Tight Lower Bound for Online Monotonic List Labeling
 In SWAT
, 1994
"... . Maintaining a monotonic labeling of an ordered list during the insertion of n items requires\Omega (n log n) individual relabelings, in the worst case, if the number of usable labels is only polynomial in n. This follows from a lower bound for a new problem, prefix bucketing. 1. Introduction The ..."
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Cited by 21 (0 self)
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. Maintaining a monotonic labeling of an ordered list during the insertion of n items requires\Omega (n log n) individual relabelings, in the worst case, if the number of usable labels is only polynomial in n. This follows from a lower bound for a new problem, prefix bucketing. 1. Introduction The online listlabeling problem can be viewed as one of linear density control. A sequence of n distinct items from some dense, linearly ordered set, such as the real numbers, is received one at a time, in no predictable order. Using "labels" from some discrete linearly ordered set of adequate but limited cardinality, the problem is to maintain an assignment of labels to the items received so far, so that the labels are ordered in the same way as the items they label. In order to make room for the next item received, it might be necessary to change the labels assigned to some of the items previously received. The cost is the total number of labelings and relabelings performed. There are practi...
ComplexityTheoretic Aspects of Interactive Proof Systems
, 1989
"... In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zeroknowledge. Interactive proof systems also have a ..."
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Cited by 19 (3 self)
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In 1985, Goldwasser, Micali and Rackoff formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zeroknowledge. Interactive proof systems also have an important part in complexity theory merging the well established concepts of probabilistic and nondeterministic computation. This thesis will study the complexity of various models of interactive proof systems. A perfect zeroknowledge interactive protocol convinces a verifier that a string is in a language without revealing any additional knowledge in an information theoretic sense. This thesis will show that for any language that has a perfect zeroknowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zeroknowledge protocols for NPcomplete languages unless the polynomialtime hierarchy collapses. Thus knowledge comp...
SHORT PCPS WITH POLYLOG QUERY COMPLEXITY
 SIAM J. COMPUT. VOL. 38, NO. 2, PP. 551–607
, 2008
"... We give constructions of probabilistically checkable proofs (PCPs) of length n · polylog n proving satisfiability of circuits of size n that can be verified by querying polylog n bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping n information bits to n ..."
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Cited by 18 (4 self)
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We give constructions of probabilistically checkable proofs (PCPs) of length n · polylog n proving satisfiability of circuits of size n that can be verified by querying polylog n bits of the proof. We also give analogous constructions of locally testable codes (LTCs) mapping n information bits to n · polylog n bit long codewords that are testable with polylog n queries. Our constructions rely on new techniques revolving around properties of codes based on relatively highdegree polynomials in one variable, i.e., Reed–Solomon codes. In contrast, previous constructions of
TimeSpace Lower Bounds for the PolynomialTime Hierarchy on Randomized Machines
 SIAM Journal on Computing
, 2006
"... We establish the first polynomialstrength timespace lower bounds for problems in the lineartime hierarchy on randomized machines with twosided error. We show that for any integer ℓ> 1 and constant c < ℓ, there exists a positive constant d such that QSAT ℓ cannot be computed by such machines in ti ..."
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Cited by 15 (5 self)
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We establish the first polynomialstrength timespace lower bounds for problems in the lineartime hierarchy on randomized machines with twosided error. We show that for any integer ℓ> 1 and constant c < ℓ, there exists a positive constant d such that QSAT ℓ cannot be computed by such machines in time n c and space n d, where QSAT ℓ denotes the problem of deciding the validity of a quantified Boolean formula with at most ℓ − 1 quantifier alternations. Moreover, d approaches 1/2 from below as c approaches 1 from above for ℓ = 2, and d approaches 1 from below as c approaches 1 from above for ℓ ≥ 3. In fact, we establish the stronger result that for any constants a ≤ 1 and c < 1+(ℓ −1)a, there exists a positive constant d such that lineartime alternating machines using space n a and ℓ − 1 alternations cannot be simulated by randomized machines with twosided error running in time n c and space n d, where d approaches a/2 from below as c approaches 1 from above for ℓ = 2 and d approaches a from below as c approaches 1 from above for ℓ ≥ 3. Corresponding to ℓ = 1, we prove that there exists a positive constant d such that the set of Boolean tautologies cannot be decided by a randomized machine with onesided error in time n 1.759 and space n d. As a corollary, this gives the same lower bound for satisfiability on deterministic machines, improving on the previously best known such result. 1
Weakly Hard Problems
, 1994
"... A weak completeness phenomenon is investigated in the complexity class E = DTIME(2 linear ). According to standard terminology, a language H is P m hard for E if the set Pm (H), consisting of all languages A P m H , contains the entire class E. A language C is P m complete for E if it ..."
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Cited by 14 (6 self)
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A weak completeness phenomenon is investigated in the complexity class E = DTIME(2 linear ). According to standard terminology, a language H is P m hard for E if the set Pm (H), consisting of all languages A P m H , contains the entire class E. A language C is P m complete for E if it is P m hard for E and is also an element of E. Generalizing this, a language H is weakly P m hard for E if the set Pm (H) does not have measure 0 in E. A language C is weakly P m complete for E if it is weakly P m hard for E and is also an element of E. The main result of this paper is the construction of a language that is weakly P m complete, but not P m complete, for E. The existence of such languages implies that previously known strong lower bounds on the complexity of weakly P m hard problems for E (given by work of Lutz, Mayordomo, and Juedes) are indeed more general than the corresponding bounds for P m hard problems for E. The proof of this result in...