Results 1  10
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39
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.
Degrees of random sets
, 1991
"... An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrando ..."
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Cited by 46 (4 self)
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An explicit recursiontheoretic definition of a random sequence or random set of natural numbers was given by MartinLöf in 1966. Other approaches leading to the notions of nrandomness and weak nrandomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of nrandom and weakly nrandom sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of nrandomness and weak nrandomness including a new definition in terms of a forcing relation analogous to the characterization of ngeneric sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than nrandomness. Chapter III is concerned with intrinsic properties of nrandom sequences. The main results are that an (n + 1)random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that nrandom sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is nrandom then A is nrandom relative to B. It follows that any countable distributive lattice can be embedded
Large cardinals and definable counterexamples to the continuum hypothesis
 Ann. Pure Appl. Logic
, 1995
"... In this paper we consider whether L(R) has “enough information ” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexa ..."
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Cited by 20 (4 self)
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In this paper we consider whether L(R) has “enough information ” to contain a counterexample to the continuum hypothesis. We believe this question provides deep insight into the difficulties surrounding the continuum hypothesis. We show sufficient conditions for L(R) not to contain such a counterexample. Along the way we establish many results about nonstationary towers, nonreflecting stationary sets, generalizations of proper and semiproper forcing and Chang’s conjecture. §0. In this paper we present some work related to the continuum problem, which can be rephrased as the problem of the existence of a surjective (possibly partial) function f: R → ω2 (ω2 is the least ordinal of cardinality the second uncountable cardinal.) The existence of such a function is equivalent (in ZermeloFraenkel set theory with the axiom of choice, ZFC) with the failure of the continuum hypothesis. Godel [G], in 1933, showed that the continuum hypothesis is consistent with ZFC and Cohen [C], in 1963, showed that the negation of the continuum hypothesis
Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called f ..."
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Cited by 11 (2 self)
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this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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Cited by 8 (2 self)
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
The npcompleteness column: Finding needles in haystacks
 ACM Transactions on Algorithms
, 2007
"... Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, h ..."
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Cited by 7 (0 self)
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Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, hereinafter referred to as “[G&J]. ” Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g., “Column 1 [1981]. ” Full bibliographic details on the previous columns, as well as downloadable unofficial versions of them, can be found at
On the Independence of P Versus NP
, 1991
"... We investigate the possibility that the current failure to resolve basic complexity theoretic questions stems from the independence of these questions with respect to the formal theories underlying our mathematical reasoning. We show that, any question in the field of computational complexity tha ..."
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Cited by 6 (1 self)
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We investigate the possibility that the current failure to resolve basic complexity theoretic questions stems from the independence of these questions with respect to the formal theories underlying our mathematical reasoning. We show that, any question in the field of computational complexity that is independent of a certain extension of the axioms of Peano Arithmetic, using currntly available techniques, is `practically insignificant'. This implies that if P 6= NP can be shown to be independent of Peano Arithmetic, using any currently known mathematical paradigm, then NP has extremelyclosetopolynomial deterministic time upper bounds. In particular, in such a case, there is a DTIME(n log (n) ) algorithm that computes SAT correctly on infinitely many huge intervals of input lengths. We provide a complete characterization of the worst case behavior of languages whose location in the complexity hierarchy is independent (with respect to sufficiently strong proof systems, including Peano Arithmetic). Such languages are, on one hand easily computable for long stretches of inputs, and, on the other hand, they are complex infinitely often. (We also construct an explicit example of such a language). Our results hold for both the Turing Machine and the NonUniform Circuit complexity models. email: shai@cs.technion.ac.il 0 1
A Tight Relationship between Generic Oracles and Type2 Complexity Theory
, 1997
"... We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct. ..."
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Cited by 6 (1 self)
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We show that any two complexity classes satisfying some general conditions are distinct relative to a generic oracle iff the corresponding type2 classes are distinct.