## Extracting Kolmogorov complexity with applications to dimension zero-one laws (2006)

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Venue: | IN PROCEEDINGS OF THE 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING |

Citations: | 20 - 4 self |

### BibTeX

@INPROCEEDINGS{Fortnow06extractingkolmogorov,

author = {Lance Fortnow and John M. Hitchcock and A. Pavan and N. V. Vinodchandran and Fengming Wang},

title = {Extracting Kolmogorov complexity with applications to dimension zero-one laws},

booktitle = {IN PROCEEDINGS OF THE 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING},

year = {2006},

pages = {335--345},

publisher = {Springer-Verlag}

}

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### Abstract

We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y) ? (1 \Gamma ffl)jyj. This result holds for both classical and space-bounded Kolmogorov complexity. We use the extraction procedure for space-bounded complexity to establish zero-one laws for polynomial-space strong dimension. Our results include: (i) If Dimpspace(E) ? 0, then Dimpspace(E=O(1)) = 1. (ii) Dim(E=O(1) j ESPACE) is either 0 or 1. (iii) Dim(E=poly j ESPACE) is either 0 or 1. In other words,

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Citation Context ...M(π) prints x using at most f(|x|) space}. There is a universal machine U such that for every machine M, there is some constant c such that for all x, KU(x) ≤ KM(x)+c and KS cf+c U (x) ≤ KS f M (x)+c =-=[9]-=-. We fix such a machine U and drop the subscript, writing K(x) and KS f (x), which are called the (plain) Kolmogorov complexity of x and f-bounded (plain) Kolmogorov complexity of x. While we use plai... |

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Citation Context ...epends on x. This computation needs only polynomial time, and yet it extracts unbounded Kolmogorov complexity. Our proofs use a construction of a multi-source extractor. Traditional extractor results =-=[17, 28, 24, 16, 26, 20, 21, 25, 10, 23, 22, 5]-=- show how to take a distribution with high min-entropy and some truly random bits to create a close to uniform distribution. A multi-source extractor takes several independent distributions with high ... |

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Citation Context ...epends on x. This computation needs only polynomial time, and yet it extracts unbounded Kolmogorov complexity. Our proofs use a construction of a multi-source extractor. Traditional extractor results =-=[17, 28, 24, 16, 26, 20, 21, 25, 10, 23, 22, 5]-=- show how to take a distribution with high min-entropy and some truly random bits to create a close to uniform distribution. A multi-source extractor takes several independent distributions with high ... |

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Citation Context ... resource-bounded versions of Hausdorff dimension and packing dimension, respectively, the two most important fractal dimensions. Polynomial-space dimension and strong dimension refine PSPACE-measure =-=[10]-=- and have been shown to be duals of each other in many ways [1]. Additionally, polynomial-space strong dimension is closely related to PSPACE-category [7]. In this paper we focus on polynomial-space s... |

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Citation Context ...epends on x. This computation needs only polynomial time, and yet it extracts unbounded Kolmogorov complexity. Our proofs use a construction of a multi-source extractor. Traditional extractor results =-=[17, 28, 24, 16, 26, 20, 21, 25, 10, 23, 22, 5]-=- show how to take a distribution with high min-entropy and some truly random bits to create a close to uniform distribution. A multi-source extractor takes several independent distributions with high ... |

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Citation Context ...olds for polynomial-space bounded Kolmogorov complexity. We apply this to obtain zero-one laws for the strong dimensions of certain complexity classes. Resource-bounded dimension and strong dimension =-=[11, 1]-=- were developed as extensions of the classical Hausdorff and packing fractal dimensions to study the structure of complexity classes. Dimension and strong dimension both refine resource-bounded measur... |

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Citation Context ..., we show that the strong dimension Dim(E | ESPACE) is either 0 or 1. The zero-one law also holds for various other complexity classes. Our techniques also apply in the constructive dimension setting =-=[12]-=-. Miller and Nies [14] asked if it is possible to compute a set of higher constructive dimension from an arbitrary set of positive constructive dimension. We answer the strong dimension variant of thi... |

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Citation Context ...olds for polynomial-space bounded Kolmogorov complexity. We apply this to obtain zero-one laws for the strong dimensions of certain complexity classes. Resource-bounded dimension and strong dimension =-=[11, 1]-=- were developed as extensions of the classical Hausdorff and packing fractal dimensions to study the structure of complexity classes. Dimension and strong dimension both refine resource-bounded measur... |

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Citation Context ...gorov complexity. Definition. Given a language L and a polynomial g the g-rate of L is strong g-rate of L is rate g (L) = lim inf n→∞ rateg (L↾n). Rate g (L) = lim sup rate n→∞ g (L↾n). Theorem 2.1. (=-=[13, 6]-=-) Let poly denote all polynomials. For every class X of languages, and dimpspace(X) = inf g∈poly Dimpspace(X) = inf g∈poly sup L∈X sup L∈X 3 Extracting Kolmogorov Complexity rate g (L). Rate g (L). Ba... |

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Citation Context ...o uniform distribution. Thus multi-source extractors eliminate the need for a truly random source. Substantial progress has been made recently in the construction of efficient multi-source extractors =-=[2, 3, 19, 18]-=-. In this paper we use the construction due to Barak, Impagliazzo, and Wigderson [2] for our main result on extracting Kolmogorov complexity. To make the connection, consider the uniform distribution ... |

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Citation Context ...o uniform distribution. Thus multi-source extractors eliminate the need for a truly random source. Substantial progress has been made recently in the construction of efficient multi-source extractors =-=[2, 3, 19, 18]-=-. In this paper we use the construction due to Barak, Impagliazzo, and Wigderson [2] for our main result on extracting Kolmogorov complexity. To make the connection, consider the uniform distribution ... |

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Citation Context ...ereshchagin and Vyugin showed that this is not possible in general [27], i.e., they showed that there is no algorithm that can extract Kolmogorov complexity. Buhrman, Fortnow, Newman and Vereshchagin =-=[4]-=- showed that if one allows a small amount of extra information then Kolmogorov extraction is indeed possible. More specifically, they showed there is an efficient procedure A such ∗ This research was ... |

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Citation Context ...o uniform distribution. Thus multi-source extractors eliminate the need for a truly random source. Substantial progress has been made recently in the construction of efficient multi-source extractors =-=[2, 3, 19, 18]-=-. In this paper we use the construction due to Barak, Impagliazzo, and Wigderson [2] for our main result on extracting Kolmogorov complexity. To make the connection, consider the uniform distribution ... |

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Citation Context ...ong dimension Dim(E | ESPACE) is either 0 or 1. The zero-one law also holds for various other complexity classes. Our techniques also apply in the constructive dimension setting [12]. Miller and Nies =-=[14]-=- asked if it is possible to compute a set of higher constructive dimension from an arbitrary set of positive constructive dimension. We answer the strong dimension variant of this question. 2 Prelimin... |

27 | Effective Fractal Dimension: Foundations and Applications
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Citation Context ...gorov complexity. Definition. Given a language L and a polynomial g the g-rate of L is strong g-rate of L is rate g (L) = lim inf n→∞ rateg (L↾n). Rate g (L) = lim sup rate n→∞ g (L↾n). Theorem 2.1. (=-=[13, 6]-=-) Let poly denote all polynomials. For every class X of languages, and dimpspace(X) = inf g∈poly Dimpspace(X) = inf g∈poly sup L∈X sup L∈X 3 Extracting Kolmogorov Complexity rate g (L). Rate g (L). Ba... |

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Citation Context ...ad of Theorem 3.7 we use Theorem 3.5. The reduction we obtain is actually an exponential-time truth-table reduction, so in particular it is a weak truth-table reduction. In contrast, Nies and Reimann =-=[15]-=- showed that this is sometimes impossible for constructive dimension: there exists S with dim(S) > 0 such that every set which weak truth-table reduces to S has dim(R) ≤ dim(S). Acknowledgments We tha... |

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Citation Context ...Pr[X = x] . 2.2 Polynomial-Space Dimension We now review the definitions of polynomial-space dimension [11] and strong dimension [1]. For more background we refer to these papers and the survey paper =-=[7]-=-. Let s > 0. An s-gale is a function d : {0, 1} ∗ → [0, ∞) satisfying 2 s d(w) = d(w0) + d(w1) for all w ∈ {0, 1} ∗ . For a language A, we write A↾n for the first n bits of A’s characteristic sequence... |

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Citation Context ...he Kolmogorov randomness from a string? That is, given x, is it possible to compute a string of length m that is Kolmogorov-random? Vereshchagin and Vyugin showed that this is not possible in general =-=[27]-=-, i.e., they showed that there is no algorithm that can extract Kolmogorov complexity. Buhrman, Fortnow, Newman and Vereshchagin [4] showed that if one allows a small amount of extra information then ... |

13 | Constructive dimension and weak truth-table degrees
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Citation Context ...eduction, so in particular it is a truth-table reduction. Therefore we also have a 0-1 law for the truth-table degrees. Subsequent to the conference version of this paper, Bienvenu, Doty, and Stephan =-=[4]-=- obtained a different proof of Theorem 5.1 and other related results using quite different techniques. In contrast, Miller [15] recently showed that there is no analogous 0-1 law for constructive dime... |

8 |
Extracting information is hard
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Citation Context ...uent to the conference version of this paper, Bienvenu, Doty, and Stephan [4] obtained a different proof of Theorem 5.1 and other related results using quite different techniques. In contrast, Miller =-=[15]-=- recently showed that there is no analogous 0-1 law for constructive dimension: there exists S with dim(S) = 1/2 such that every sequence R ≤T S has dim(R) ≤ 1/2. Acknowledgments We thank Xiaoyang Gu ... |

5 | Resource-bounded strong dimension versus resource-bounded category
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Citation Context ...ture of complexity classes. Dimension and strong dimension both refine resource-bounded measure and are duals of each other in many ways. Strong dimension is also related to resource-bounded category =-=[8]-=-. In this paper we focus on strong dimension. The strong dimension of each complexity class is a real number between zero and one inclusive. While there are examples of nonstandard complexity classes ... |