## Scaled dimension and nonuniform complexity (2004)

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Venue: | Journal of Computer and System Sciences |

Citations: | 22 - 9 self |

### BibTeX

@ARTICLE{Hitchcock04scaleddimension,

author = {John M. Hitchcock and Jack H. Lutz and Elvira Mayordomo},

title = {Scaled dimension and nonuniform complexity},

journal = {Journal of Computer and System Sciences},

year = {2004},

volume = {69},

pages = {2004}

}

### Years of Citing Articles

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### Abstract

Resource-bounded dimension is a complexity-theoretic extension of classical Hausdorff dimension introduced by Lutz (2000) in order to investigate the fractal structure of sets that have resource-bounded measure 0. For example, while it has long been known that the Boolean circuit-size complexity class SIZE � α 2n n has measure 0 in ESPACE for all 0 ≤ α ≤ 1, we now know that SIZE � α 2n n has dimension α in ESPACE for all 0 ≤ α ≤ 1. The present paper furthers this program by developing a natural hierarchy of “rescaled” resource-bounded dimensions. For each integer i and each set X of decision problems, we define the ith-order dimension of X in suitable complexity classes. The 0th-order dimension is precisely the dimension of Hausdorff (1919) and Lutz (2000). Higher and lower orders are useful for various sets X. For example, we prove the following for 0 ≤ α ≤ 1 and any polynomial q(n) ≥ n2. 1. The class SIZE(2 αn) and the time- and space-bounded Kolmogorov complexity classes KT q (2 αn) and KS q (2 αn) have 1 st-order dimension α in ESPACE. 2. The classes SIZE(2nα), KT q (2nα), and KS q (2nα) have 2nd-order dimension α in ESPACE.

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Citation Context ...nian sample path in the plane has Hausdor dimension 2, but a more careful analysis with a rescaled version of Hausdor dimension shows that the dimension is actually \logarithmically smaller" than=-= 2 [Fal90]-=-. In this paper we extend the resource-bounded dimension of [Lut00a] by introducing the general notion of a scale according to which dimension may be measured. The choice of which scale to use for a p... |

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Citation Context ...tative structures that are toosne to be elucidated by resource-bounded measure. For example, it has long been known that the Boolean circuit-size complexity class SIZE 2 n n has measure 0 in ESPACE [=-=Lut92-=-], so resourcebounded measure cannot make quantitative distinctions among subclasses of SIZE 2 n n . In early 2000, Lutz [Lut00a] developed resource-bounded dimension in order to remedy this situatio... |

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Citation Context ...s a 1-gale. Remarks. 1. Martingales were introduced by Lévy [11] and named by Ville [22], who used them in early investigations of random sequences. Martingales were later used extensively by Schnorr =-=[18, 19, 20, 21]-=- in his investigations of random sequences and by Lutz [13, 16] in the development of resource-bounded measure. Gales were introduced by Lutz [14, 15] in the development of resource-bounded and constr... |

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Citation Context ...esource-bounded dimension of Lutz [14]. This sort of difficulty has already been encountered in the classical theory of Hausdorff dimension and dealt with by rescaling the dimension. The 1970 classic =-=[17]-=- by C.A. Rogers describes the resulting theory of generalized dimension, in which Hausdorff dimension may be rescaled by any element of a very large class of extended real-valued functions. (In fact, ... |

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Citation Context ...ean circuit-size complexity class SIZE 2 n n has measure 0 in ESPACE [Lut92], so resourcebounded measure cannot make quantitative distinctions among subclasses of SIZE 2 n n . In early 2000, Lutz [L=-=ut00a]-=- developed resource-bounded dimension in order to remedy this situation. Just as resource-bounded measure is a complexity-theoretic generalization of classical Lebesgue measure, resource-bounded dimen... |

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Citation Context ...es were later used extensively by Schnorr [18, 19, 20, 21] in his investigations of random sequences and by Lutz [13, 16] in the development of resource-bounded measure. Gales were introduced by Lutz =-=[14, 15]-=- in the development of resource-bounded and constructive dimension. Scaled gales are introduced here in order to formulate scaled dimension. 2. Although the martingale condition is usually stated in t... |

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Citation Context ...s a 1-gale. Remarks. 1. Martingales were introduced by Lévy [11] and named by Ville [22], who used them in early investigations of random sequences. Martingales were later used extensively by Schnorr =-=[18, 19, 20, 21]-=- in his investigations of random sequences and by Lutz [13, 16] in the development of resource-bounded measure. Gales were introduced by Lutz [14, 15] in the development of resource-bounded and constr... |

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Citation Context ...s with limiting frequency α has dimension H(α) in E, where H is the binary entropy function of Shannon information theory. (This is a complexity-theoretic extension of a classical result of Eggleston =-=[4]-=-.) These preliminary results suggest new relationships between information and complexity and open the way for investigating the fractal structure of complexity classes. More recent work has already u... |

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36 |
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Citation Context ...med by Ville [22], who used them in early investigations of random sequences. Martingales were later used extensively by Schnorr [18, 19, 20, 21] in his investigations of random sequences and by Lutz =-=[13, 16]-=- in the development of resource-bounded measure. Gales were introduced by Lutz [14, 15] in the development of resource-bounded and constructive dimension. Scaled gales are introduced here in order to ... |

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Citation Context ...besgue measure 0, resource-bounded dimension enables us to quantify the structures of some sets that have measure 0 in complexity classes. For example, Lutz [14] showed that for every real number α ∈ =-=[0, 1]-=-, the class SIZE � α 2n � n has dimension α in ESPACE. He also showed that for every p-computable α ∈ [0, 1], the class of languages with limiting frequency α has dimension H(α) in E, where H is the b... |

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Citation Context ... open the way for investigating the fractal structure of complexity classes. More recent work has already used resource-bounded dimension to illuminate a variety of topics in computational complexity =-=[1, 2, 6, 8, 9, 3]-=-. However, there is a conspicuous obstacle to further progress along these lines. Many classes that occur naturally in computational complexity are parametrized in such a way as to remain out of reach... |

26 | MAX3SAT is exponentially hard to approximate if NP has positive dimension. Theoretical Computer Science
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(Show Context)
Citation Context ... open the way for investigating the fractal structure of complexity classes. More recent work has already used resource-bounded dimension to illuminate a variety of topics in computational complexity =-=[1, 2, 6, 8, 9, 3]-=-. However, there is a conspicuous obstacle to further progress along these lines. Many classes that occur naturally in computational complexity are parametrized in such a way as to remain out of reach... |

24 |
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(Show Context)
Citation Context |

14 |
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Citation Context ...eneralized dimension, in which Hausdorff dimension may be rescaled by any element of a very large class of extended real-valued functions. (In fact, this idea was introduced in Hausdorff’s 1919 paper =-=[7]-=-.) Choosing the right such function for a particular set often yields more precise information about that set’s dimension. For example, it is known that with probability 1 a Brownian sample path in th... |

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10 |
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Citation Context ...n ; ) gates. Proof. Let m(n) = dlog g i (2 n ; )e. For n large enough, m(n)sn. Then there are 2 2 m(n) 2 g i (2 n ;) dierent sets C f0; 1g m(n) . Fix " > 0. For all suciently large n, Lupanov [=-=Lup58] has shown -=-that each of these sets is decided by a circuit of at most 2 m(n) m(n) (1 + ") gates. Now for suciently large n, 2 m(n) m(n) (1 + ") 2g i (2 n ; ) log(g i (2 n ; )) (1 + ")si (2 n ; ):... |

9 | Completeness and weak completeness under polynomial-size circuits
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Citation Context ...bounded dimension of [Lut00a] cannot provide the sort of quantitative classication that is needed. Similarly, in their investigations of the information content of complete problems, Juedes and Lutz [=-=JL9-=-6] established tight bounds on space-bounded Kolmogorov complexity of the forms 2 n and 2 n+1 2 n ; in the investigation of completeness in E one is typically interested in dense languages, which ha... |

8 |
The Fractional Dimension of a Set De by Decimal Properties
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(Show Context)
Citation Context ...s with limiting frequency has dimension H() in E, where H is the binary entropy function of Shannon information theory. (This is a complexity-theoretic extension of a classical result of Eggleston [E=-=gg49]-=-.) These preliminary results are hopeful because they suggest new relationships between information and complexity and open the way for investigating the fractal structure of complexity classes. Howev... |

6 |
Dimension und ausseres
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(Show Context)
Citation Context ...cales, one for each integer, and use these to dene the i th -order dimension of arbitrary sets X in suitable complexity classes. The 0 th -order dimension is precisely the dimension used by Hausdor [H=-=au1-=-9] and Lutz [Lut00a]. We propose that higher- and lower-order dimensions will be useful for many investigations in computational complexity. In support of this proposal we prove the following for 0 ... |

6 | Infinitely-often autoreducible sets
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Citation Context ... open the way for investigating the fractal structure of complexity classes. More recent work has already used resource-bounded dimension to illuminate a variety of topics in computational complexity =-=[1, 2, 6, 8, 9, 3]-=-. However, there is a conspicuous obstacle to further progress along these lines. Many classes that occur naturally in computational complexity are parametrized in such a way as to remain out of reach... |

2 | Dimension und "ausseres - Hausdorff - 1919 |