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Partial Biimmunity, Scaled Dimension, and NPCompleteness
"... The Turing and manyone completeness notions for NP have been previously separated under measure, genericity, and biimmunity hypotheses on NP. The proofs of all these results rely on the existence of a language in NP with almost everywhere hardness. In this ..."
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Cited by 14 (6 self)
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The Turing and manyone completeness notions for NP have been previously separated under measure, genericity, and biimmunity hypotheses on NP. The proofs of all these results rely on the existence of a language in NP with almost everywhere hardness. In this
Comparing reductions to NPcomplete sets
 Electronic Colloquium on Computational Complexity
, 2006
"... Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. (2) Strong nondetermin ..."
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Cited by 13 (4 self)
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Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. (2) Strong nondeterministic reductions are more powerful than deterministic reductions: there is a problem that is SNPcomplete for NP but not Turingcomplete. (3) Every problem that is manyone complete for NP is complete under lengthincreasing reductions that are computed by polynomialsize circuits. The first item solves one of Lutz and Mayordomo’s “Twelve Problems in ResourceBounded Measure ” (1999). We also show that every manyone complete problem for NE is complete under onetoone, lengthincreasing reductions that are computed by polynomialsize circuits. 1
Properties of NPcomplete sets
 In Proceedings of the 19th IEEE Conference on Computational Complexity
, 2004
"... We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreo ..."
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Cited by 11 (7 self)
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We study several properties of sets that are complete for NP. We prove that if L is an NPcomplete set and S � ⊇ L is a pselective sparse set, then L − S is ≤p mhard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p mhard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a sparse S ∈ EXP such that L − S is not ≤ p mhard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure oneway permutations to derive consequences for longstanding open questions about whether NPcomplete sets are immune. For example, assuming that pseudorandom generators and secure oneway permutations exist, it follows easily that NPcomplete sets are not pimmune. Assuming only that secure oneway permutations exist, we prove that no NPcomplete set is DTIME(2nɛ)immune. Also, using these hypotheses we show that no NPcomplete set is quasipolynomialclose to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turingcomplete sets for NP are not closed under union. Our hypothesis asserts existence of a UPmachine M that accepts 0 ∗ such that for some 0 < ɛ < 1, no 2nɛ timebounded machine can correctly compute infinitely many accepting computations of M. We show that if UP∩coUP contains DTIME(2nɛ)biimmune sets, then this hypothesis is true.
Scaled dimension and the Kolmogorov complexity of Turinghard sets
 In Proceedings of the 29th International Symposium on Mathematical Foundations of Computer Science
, 2004
"... We study constructive and resourcebounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension ..."
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Cited by 6 (2 self)
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We study constructive and resourcebounded scaled dimension as an information content measure and obtain several results that parallel previous work on unscaled dimension. Scaled dimension for finite strings is developed and shown to be closely related to Kolmogorov complexity. The scaled dimension of an infinite sequence is characterized by the scaled dimensions of its prefixes. We obtain an exact Kolmogorov complexity characterization of scaled dimension. Juedes and Lutz (1996) established a small span theorem for P/polyTuring reductions which asserts that for any problem A in ESPACE, either the class of problems reducible to A (the lower span) or the class of problems to which A is reducible (the upper span) has measure 0 in ESPACE. We apply our Kolmogorov complexity characterization to improve this to (−3) rdorder scaled dimension 0 in ESPACE. As a consequence we obtain a new upper bound on the Kolmogorov complexity of Turinghard sets for ESPACE. 1
Autoreducibility, mitoticity and immunity
 Mathematical Foundations of Computer Science: Thirtieth International Symposium, MFCS 2005
, 2005
"... We show the following results regarding complete sets. • NPcomplete sets and PSPACEcomplete sets are manyone autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are manyone autoreducible. • EXPcomplete sets are manyone mitotic. • NEXPcomplete sets are we ..."
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Cited by 6 (4 self)
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We show the following results regarding complete sets. • NPcomplete sets and PSPACEcomplete sets are manyone autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are manyone autoreducible. • EXPcomplete sets are manyone mitotic. • NEXPcomplete sets are weakly manyone mitotic. • PSPACEcomplete sets are weakly Turingmitotic. • If oneway permutations and quick pseudorandom generators exist, then NPcomplete languages are mmitotic. • If there is a tally language in NP ∩ coNP − P, then, for every ɛ> 0, NPcomplete sets are not 2 n(1+ɛ)immune. These results solve several of the open questions raised by Buhrman and Torenvliet in their 1994 survey paper on the structure of complete sets. 1
Upward Separations and Weaker Hypotheses in ResourceBounded Measure
"... We consider resourcebounded measure in doubleexponentialtime complexity classes. In contrast to complexity class separation translating downwards, we show that measure separation translates upwards. For example, µp(NP) ̸ = 0 ⇒ µe(NE) ̸ = 0 ⇒ µexp(NEXP) ̸ = 0. We also show that if NE does not have ..."
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Cited by 3 (2 self)
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We consider resourcebounded measure in doubleexponentialtime complexity classes. In contrast to complexity class separation translating downwards, we show that measure separation translates upwards. For example, µp(NP) ̸ = 0 ⇒ µe(NE) ̸ = 0 ⇒ µexp(NEXP) ̸ = 0. We also show that if NE does not have emeasure 0, then the NPmachine hypothesis holds. We give oracles relative to which the converses of these statements do not hold. Therefore the hypothesis on the emeasure of NE is relativizably weaker than the ofteninvestigated pmeasure hypothesis on NP, but it has many of the same consequences.
Pushdown dimension
 Theoretical Computer Science
, 2007
"... Abstract Resourcebounded dimension is a notion of computational information density of infinite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finitestate dimension.The pushdown dimension of any sequence ..."
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Cited by 2 (0 self)
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Abstract Resourcebounded dimension is a notion of computational information density of infinite sequences based on computationally bounded gamblers. This paper develops the theory of pushdown dimension and explores its relationship with finitestate dimension.The pushdown dimension of any sequence is trivially bounded above by its finitestate dimension, since a pushdown gambler can simulate any finitestate gambler. We show thatfor every rational 0 < d < 1, there exists a sequence with finitestate dimension d whosepushdown dimension is at most d/2. This provides a stronger quantitative analogue of thewellknown fact that pushdown automata decide strictly more languages than finitestate
A Post's Program For Complexity Theory
"... Some of the most important and chalenging problems in complexity theory deal with the separation of complexity classes. Apart from the results derived from the hierarchy theorems and some lower bounds for restricted models of computation, there are only few results in the eld implying the separation ..."
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Some of the most important and chalenging problems in complexity theory deal with the separation of complexity classes. Apart from the results derived from the hierarchy theorems and some lower bounds for restricted models of computation, there are only few results in the eld implying the separation of two classes. Harry Buhrmann and Leen Torenvliet present in this column a di erent approach to obtain complexity separations. The method is not new, in fact it is based on old ideas from Emil Post in the area of recursion theory. It has gained new attention recently, since it has some advantages with respect to other separation techniques. For example the proposed method does not relativize. The following survey provides a very good introduction to this line of research.
The Measure Hypothesis and Efficiency of Polynomial Time Approximation Schemes
, 2008
"... A polyomial time approximation scheme for an optimization problem X is an algorithm A such that for each instance x of X and each ffl> 0, A computes a (1 + ffl)approximate solution to instance x of Xin time is O(xf(1/ffl)) for some function f. If the running time of A isinstead bounded by g(1/ff ..."
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A polyomial time approximation scheme for an optimization problem X is an algorithm A such that for each instance x of X and each ffl> 0, A computes a (1 + ffl)approximate solution to instance x of Xin time is O(xf(1/ffl)) for some function f. If the running time of A isinstead bounded by g(1/ffl) * xO(1) for some function g, A is called anefficient polynomial time approximation scheme.