Results 1  10
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20
The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 29 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Pseudorandom generators and structure of complete degrees
 In 17th Annual IEEE Conference on Computational Complexity
, 2002
"... It is shown that if there exist sets in E that requiresized circuits then sets that are hard for class P, and above, under 11 reductions are also hard under 11, sizeincreasing reductions. Under the assumption of the hardness of solving RSA or Discrete Log problem, it is shown that sets that are h ..."
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Cited by 23 (2 self)
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It is shown that if there exist sets in E that requiresized circuits then sets that are hard for class P, and above, under 11 reductions are also hard under 11, sizeincreasing reductions. Under the assumption of the hardness of solving RSA or Discrete Log problem, it is shown that sets that are hard for class NP, and above, under manyone reductions are also hard under (nonuniform) 11, and sizeincreasing reductions. 1
Splittings, Robustness and Structure of Complete Sets
, 1993
"... We investigate the structure of EXP and NEXP complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. To address this question for a given hard set A, we construct a sparse set S, ..."
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Cited by 15 (4 self)
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We investigate the structure of EXP and NEXP complete and hard sets under various kinds of reductions. In particular, we are interested in the way in which information that makes the set complete is stored in the set. To address this question for a given hard set A, we construct a sparse set S, and ask whether A \Gamma S is still hard. It turns out that for most of the reductions considered and for an arbitrary given sparseness condition, there is a single subexponential time computable set S that meets this condition, such that A \Gamma S is not hard for any A. Not only is this set S subexponential time computable, but a slight modification of the construction can make the complexity of S meet any reasonable superpolynomial function. On the other hand we show that for any polynomialtime computable sparse set S, the set A \Gamma S remains hard. There are other properties than time complexity that make a set `almost' polynomialtime computable. For sparse pselective sets...
With Quasilinear Queries EXP is not Polynomial Time Turing Reducible to Sparse Sets
, 1993
"... . We investigate the lower bounds of queries required by the polynomial time Turing reductions from exponential time classes to the sets of small density. For complexity classes E= DTIME(2 O(n) ) and EXP=DTIME(2 n O(1) ), the following results are shown in this paper. (1) For any a ! 1, every E ..."
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Cited by 12 (1 self)
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. We investigate the lower bounds of queries required by the polynomial time Turing reductions from exponential time classes to the sets of small density. For complexity classes E= DTIME(2 O(n) ) and EXP=DTIME(2 n O(1) ), the following results are shown in this paper. (1) For any a ! 1, every EXP P n a \GammaT hard set is exponentially dense. This yields EXP6` Pn a \GammaT (SPARSE) for all a ! 1. (2)For any a ! 1 2 , every E P n a \GammaT hard set is exponentially dense. (3)E6`P o( n log n )\GammaT (TALLY). Our result substantially improve Watanabe's earlier theorem: E6`P log n\Gammatt (SPARSE) [Wa87,HOW92]. 1.Introduction The study of the density of hard sets for complexity classes has long history. Berman and Hartmanis[BH77] conjectured that all NP P m complete sets are isomorphic. Since all of the known NP P m complete sets are exponentially dense, they proposed a weaker conjecture that all NP P m complete sets are not sparse. Culminating consider...
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 12 (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
The Isomorphism Conjecture Holds and Oneway Functions Exist Relative to an Oracle
 Journal of Computer and System Sciences
, 1994
"... In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which t ..."
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Cited by 9 (2 self)
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In this paper we demonstrate an oracle relative to which there are oneway functions but every paddable 1lidegree collapses to an isomorphism type, thus yielding a relativized failure of the JosephYoung Conjecture (JYC) [JY85]. We then use this result to construct an oracle relative to which the Isomorphism Conjecture (IC) is true but oneway functions exist, which answers an open question of Fenner, Fortnow, and Kurtz [FFK92]. Thus, there are now relativizations realizing every one of the four possible states of affairs between the IC and the existence of oneway functions. 1 Introduction Berman and Hartmanis [BH76, BH77] showed that if two languages A and B are equivalent to one another under polynomialtime manytoone reductions and if they are both paddable then they are polynomialtime isomorphic. After surveying all of the thenknown NPcomplete languages and discovering that each was indeed paddable, they posed: The Isomorphism Conjecture (IC) Every NPcomplete lan...
Completeness for Nondeterministic Complexity Classes
, 1991
"... We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomialtime bounded (even logarithmic space bounded) reducibilities turn out to be different for any class ..."
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Cited by 9 (0 self)
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We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomialtime bounded (even logarithmic space bounded) reducibilities turn out to be different for any class containing NE . For space classes the completeness notions under logspace reducibilities can be separated for any class properly containing LOGSPACE . Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes. 1 Introduction Efficient reducibilities and completeness are two of the central concepts of complexity theory. Since the first use of polynomial time bounded Turing reductions by Cook [4] and the introduction of polynomial time bounded manyone reductions by Karp[9], considerable effort has been put in the investigation of properties and the relative strengt...
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
On Pimmunity of nondeterministic complete sets
 In Proceedings of the 10th Annual Conference on Structure in Complexity Theory '95
, 1995
"... We show that every mcomplete set for NEXP as well as ats complement have an infinite subset in P. Thw nnswers an open question first raised in [Ber76]. 1 ..."
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Cited by 4 (0 self)
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We show that every mcomplete set for NEXP as well as ats complement have an infinite subset in P. Thw nnswers an open question first raised in [Ber76]. 1
DSPACE(n) ? = NSPACE(n): A degree theoretic characterization
 in Proc. 10th Structure in Complexity Theory Conference
, 1995
"... It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1 ..."
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Cited by 3 (3 self)
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It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1