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## Hausdorff dimension and oracle constructions (2004)

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Venue: | Theoretical Computer Science |

Citations: | 4 - 1 self |

### Citations

1027 |
Fractal Geometry: Mathematical Foundations and Applications
- Falconer
- 1990
(Show Context)
Citation Context ... 0, then dimH(X) = 1. For each α ∈ [0, 1] there exist sets X with dimH(X) = α. Hausdorff dimension therefore makes quantitative distinctions among the measure 0 sets. We refer to the book by Falconer =-=[4]-=- for more information about Hausdorff dimension. We now recall an equivalent definition of Hausdorff dimension involving log-loss prediction [10]. Definition. A predictor is a function that satisfies ... |

336 | Algebraic methods for interactive proof systems
- Lund, Fortnow, et al.
- 1992
(Show Context)
Citation Context ...t xy is not reserved for Ac and add xy to A. As argued in [8], we can always choose such a y. This completes stage m. The most famous counterexample to the random oracle hypothesis [1] is IP = PSPACE =-=[12, 14, 3]-=-. While IP = PSPACE holds unrelativized, the set O [IP=PSPACE] has measure 0. Since NP A ⊆ IP A ⊆ PSPACE A ⊆ EXP A relative to every oracle A, we have the following corollary of Theorem 4.3. Corollary... |

212 |
Separating the polynomial–time hierarchy by oracles
- Yao
- 1985
(Show Context)
Citation Context ...n show this oracle construction is paddable and relativizable to establish the following. Theorem 4.6. O [BPP=NEXP] has Hausdorff dimension 1. Corollary 4.7. O [P�=BPP] has Hausdorff dimension 1. Yao =-=[15]-=- (see also H˚astad [6]) constructed an oracle relative to which the polynomial-time hierarchy has infinitely many distinct levels. Whether this holds relative to a random oracle is an open problem. We... |

200 |
Computational Limitations for Small Depth Circuits
- Hastad
- 1986
(Show Context)
Citation Context ...truction is paddable and relativizable to establish the following. Theorem 4.6. O [BPP=NEXP] has Hausdorff dimension 1. Corollary 4.7. O [P�=BPP] has Hausdorff dimension 1. Yao [15] (see also H˚astad =-=[6]-=-) constructed an oracle relative to which the polynomial-time hierarchy has infinitely many distinct levels. Whether this holds relative to a random oracle is an open problem. We now use Theorem 3.2 a... |

101 | 2003): Dimension in Complexity Classes
- Lutz
(Show Context)
Citation Context ...dimension admits an equivalent definition as log-loss unpredictability. Let Π be the set of all predictors. The proof of the following theorem used Lutz’s gale characterization of Hausdorff dimension =-=[13]-=-. Theorem 2.1. (Hitchcock [10]) For every X ⊆ C, dimH(X) = inf π∈Π Llog (π, X). The following lemma can be derived from [13] and [10]; a direct proof is included here for completeness. Intuitively, if... |

47 |
Dimension und äußeres Maß
- Hausdorff
- 1919
(Show Context)
Citation Context ...acle A is chosen uniformly at random, then P A �= NP A with probability 1. More precisely, they proved that the set of oracles O [P=NP] = {A | P A = NP A } has Lebesgue measure 0. Hausdorff dimension =-=[7]-=-, the most commonly used fractal dimension, provides a quantitative distinction among the measure 0 sets. Every set O of oracles has a Hausdorff dimension dimH(O), a real number in [0, 1]. If O does n... |

46 |
Relative to a random oracle P A �= NP A �= coNP A with probability 1
- Bennett, Gill
- 1981
(Show Context)
Citation Context ...ynomial-time hierarchy is infinite relative to a Hausdorff dimension 1 set of oracles and that P A �= NP A ∩ coNP A relative to a Hausdorff dimension 1 set of oracles. 1 Introduction Bennett and Gill =-=[1]-=- initiated the study of random oracles in computational complexity theory. They showed that if an oracle A is chosen uniformly at random, then P A �= NP A with probability 1. More precisely, they prov... |

36 |
On relativized exponential and probabilistic complexity classes
- Heller
- 1986
(Show Context)
Citation Context ... Corollary 4.5. O [P=NP∩coNP] and O [P�=NP∩coNP] both have Hausdorff dimension 1. Bennett and Gill also showed that P A = BPP A relative to a random oracle A, or that O [P�=BPP] has measure 0. Heller =-=[9]-=- constructed an oracle A with BPP A = NEXP A . We can show this oracle construction is paddable and relativizable to establish the following. Theorem 4.6. O [BPP=NEXP] has Hausdorff dimension 1. Corol... |

31 | Fractal dimension and logarithmic loss unpredictability
- Hitchcock
(Show Context)
Citation Context ...ted in part by National Science Foundation grant 0515313. 1shas Hausdorff dimension 1. The proof of this theorem is facilitated by the equivalence of Hausdorff dimension and log-loss unpredictability =-=[10]-=-. In Section 4 we give several applications of the general theorem, including (1.1) and that some other measure 0 oracle sets including O [NP=EXP] and O [P�=BPP] also have Hausdorff dimension 1. It is... |

27 |
Relative to a random oracle A, PA 6= NPA 6= co-NPA with probability 1
- Bennett, Gill
- 1981
(Show Context)
Citation Context ...polynomial-time hierarchy isinfinite relative to a Hausdorff dimension 1 set of oracles and that P A 6= NPA " coNPA relative to a Hausdorff dimension 1 set of oracles. 1 Introduction Bennett and Gill =-=[1]-=- initiated the study of random oracles in computational complexity theory. They showed that if an oracle A is chosen uniformly at random, then PA 6= NPA with probability 1. More precisely, they proved... |

26 | The random oracle hypothesis is false
- Chor, Goldreich, et al.
- 1990
(Show Context)
Citation Context ...t xy is not reserved for Ac and add xy to A. As argued in [8], we can always choose such a y. This completes stage m. The most famous counterexample to the random oracle hypothesis [1] is IP = PSPACE =-=[12, 14, 3]-=-. While IP = PSPACE holds unrelativized, the set O [IP=PSPACE] has measure 0. Since NP A ⊆ IP A ⊆ PSPACE A ⊆ EXP A relative to every oracle A, we have the following corollary of Theorem 4.3. Corollary... |

20 | Scaled dimension and nonuniform complexity
- Hitchcock, Lutz, et al.
(Show Context)
Citation Context ..., C) ≥ 1 since m is a constant and n(αn + βn) = o(2 n − 1). Since C ∈ O [Φ], we have L log (π, O [Φ]) ≥ 1. We remark that the proof of Theorem 3.1 can be extended to yield a stronger scaled dimension =-=[11]-=- result. It can be shown that the set of oracles has −2 nd -order dimension 1. We conclude this section with a variation of Theorem 3.1 involving random oracles that will be useful in an application. ... |

7 |
On collapsing the polynomial-time hierarchy
- Book
- 1994
(Show Context)
Citation Context ...to which the polynomial-time hierarchy has infinitely many distinct levels. Whether this holds relative to a random oracle is an open problem. We now use Theorem 3.2 and a relativized theorem of Book =-=[2, 5]-=- to show that it holds relative to a dimension 1 set of oracles. Theorem 4.8. O [(∀i)Σ p i �=Σp i+1 ] has Hausdorff dimension 1. Proof. Let A be an oracle such that Σ p,A i for a random oracle R, Σ p,... |

7 |
Relativized worlds with an infinite hierarchy
- Fortnow
- 1999
(Show Context)
Citation Context ...to which the polynomial-time hierarchy has infinitely many distinct levels. Whether this holds relative to a random oracle is an open problem. We now use Theorem 3.2 and a relativized theorem of Book =-=[2, 5]-=- to show that it holds relative to a dimension 1 set of oracles. Theorem 4.8. O [(∀i)Σ p i �=Σp i+1 ] has Hausdorff dimension 1. Proof. Let A be an oracle such that Σ p,A i for a random oracle R, Σ p,... |

4 |
On relativized polynomial and exponential computations
- Heller
- 1984
(Show Context)
Citation Context ...f dimension 1. Since Bennett and Gill [1] proved that NP A �= coNP A relative to a random oracle A, we know that O [NP=EXP] has measure 0. Using Heller’s construction of an oracle A with NP A = EXP A =-=[8]-=-, we have a contrasting dimension result. Theorem 4.3. O [NP=EXP] has Hausdorff dimension 1. Proof. We will show that Heller’s oracle construction is paddable and relativizable. Let k ≥ 1 and let B ∈ ... |

3 | Relativized worlds with an in hierarchy - Fortnow - 1999 |

2 | Dimension und aueres Ma - Hausdor - 1919 |

1 |
Separating the polynomial-time hierarchy by oracles
- PSPACE
- 1992
(Show Context)
Citation Context ...t xy is not reserved for Ac and add xy to A. As argued in [8], we can always choose such a y. This completes stage m. The most famous counterexample to the random oracle hypothesis [1] is IP = PSPACE =-=[12, 14, 3]-=-. While IP = PSPACE holds unrelativized, the set O[IP=PSPACE] has measure 0. Since NPA ` IPA ` PSPACEA ` EXPA relative to every oracle A, we have the following corollary of Theorem 4.3. Corollary 4.4.... |