Results 1  10
of
30
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
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Cited by 47 (10 self)
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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
Inverting Onto Functions
, 1996
"... We look at the hypothesis that all honest onto polynomialtime computable functions have a polynomialtime computable inverse. We show this hypothesis equivalent to several other complexity conjectures including ffl In polynomial time, one can find accepting paths of nondeterministic polynomialtim ..."
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Cited by 35 (5 self)
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We look at the hypothesis that all honest onto polynomialtime computable functions have a polynomialtime computable inverse. We show this hypothesis equivalent to several other complexity conjectures including ffl In polynomial time, one can find accepting paths of nondeterministic polynomialtime Turing machines that accept \Sigma . ffl Every total multivalued nondeterministic function has a polynomialtime computable refinement. ffl In polynomial time, one can compute satisfying assignments for any polynomialtime computable set of satisfiable formulae. ffl In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments. We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse. 1 Introduction Understanding the power of nondeterminism has been one of the pri...
Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem
 Journal of Computer and System Sciences
"... We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions. ..."
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Cited by 30 (12 self)
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We show that all sets that arecomplete for NP under nonuniform AC are isomorphic under nonuniform AC computable isomorphisms. Furthermore, these sets remain NPcomplete even under nonuniform NC reductions.
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
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 ...
Separability and Oneway Functions
, 2000
"... We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are Pseparable. ..."
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Cited by 25 (13 self)
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We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are Pseparable.
A FirstOrder Isomorphism Theorem
 SIAM JOURNAL ON COMPUTING
, 1993
"... We show that for most complexity classes of interest, all sets complete under firstorder projections are isomorphic under firstorder isomorphisms. That is, a very restricted version of the BermanHartmanis Conjecture holds. ..."
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Cited by 24 (5 self)
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We show that for most complexity classes of interest, all sets complete under firstorder projections are isomorphic under firstorder isomorphisms. That is, a very restricted version of the BermanHartmanis Conjecture holds.
The random oracle hypothesis is false
, 1990
"... The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which holdforalmost all relativized worlds must also hold in the unrelativized case. Although this paper is not the rst to provideacounterexample to the Random Oracle Hy ..."
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Cited by 24 (2 self)
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The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which holdforalmost all relativized worlds must also hold in the unrelativized case. Although this paper is not the rst to provideacounterexample to the Random Oracle Hypothesis, it does provide a most compelling counterexample by showing that for almost all oracles A, IP A 6=PSPACE A. If the Random Oracle Hypothesis were true, it would contradict Shamir's result that IP = PSPACE. In fact, it is shown that for almost all oracles A, coNP A 6 IP A. These results extend to the multiprover proof systems of BenOr, Goldwasser, Kilian and Wigderson. In addition, this paper shows that the Random Oracle Hypothesis is sensitive to small changes in the de nition. A class IPP, similar to IP, is de ned. Surprisingly, the IPP = PSPACE result holds for all oracle worlds. Warning: Essentially this paper has been published in Information and Computation and is hence subject to copyright restrictions. It is for personal use only. 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
Applications of TimeBounded Kolmogorov Complexity in Complexity Theory
 Kolmogorov complexity and computational complexity
, 1992
"... This paper presents one method of using timebounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functi ..."
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Cited by 18 (4 self)
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This paper presents one method of using timebounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functions, pseudorandom generators, and hierarchy theorems in circuit complexity.