## Gales suffice for constructive dimension (2003)

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Venue: | Information Processing Letters |

Citations: | 19 - 4 self |

### BibTeX

@ARTICLE{Hitchcock03galessuffice,

author = {John M. Hitchcock},

title = {Gales suffice for constructive dimension},

journal = {Information Processing Letters},

year = {2003},

volume = {86},

pages = {2003}

}

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### Abstract

Supergales, generalizations of supermartingales, have been used by Lutz (2002) to define the constructive dimensions of individual binary sequences. Here it is shown that gales, the corresponding generalizations of martingales, can be equivalently used to define constructive dimension. 1

### Citations

332 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ...ntroduction Effective martingales have been very useful objects in theoretical computer science. Schnorr [7, 8] used constructive martingales to give an equivalent definition of Martin-Löf randomness =-=[6]-=-. Martingales computable within resource bounds have been used by Lutz [3] to define various resource-bounded measures that have been successful in complexity theory. In all these cases, it is known t... |

184 | The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms
- Zvonkin, Levin
- 1970
(Show Context)
Citation Context ... α > 0 such that d ∗ (w) ≥ αd(w) for all w ∈ {0, 1} ∗ . 2. We say that d ∗ is successively optimal for D if for every d ∈ D, S ∞ [d] ⊆ S ∞ [d ∗ ]. Lutz used Levin’s universal constructive semimeasure =-=[9]-=- to show that there exist multiplicatively optimal supergales. 2 � . (3.1)sTheorem 3.1. (Lutz [2])For any computable s ∈ [0, ∞) there is a constructive s-supergale d (s) that is multiplicatively optim... |

172 | Almost everywhere high nonuniform complexity
- Lutz
- 1992
(Show Context)
Citation Context ...ical computer science. Schnorr [7, 8] used constructive martingales to give an equivalent definition of Martin-Löf randomness [6]. Martingales computable within resource bounds have been used by Lutz =-=[3]-=- to define various resource-bounded measures that have been successful in complexity theory. In all these cases, it is known that replacing the constructive or resource-bounded martingales with constr... |

149 |
Zufälligkeit und Wahrscheinlichkeit
- Schnorr
- 1971
(Show Context)
Citation Context ... generalizations of martingales, can be equivalently used to define constructive dimension. 1 Introduction Effective martingales have been very useful objects in theoretical computer science. Schnorr =-=[7, 8]-=- used constructive martingales to give an equivalent definition of Martin-Löf randomness [6]. Martingales computable within resource bounds have been used by Lutz [3] to define various resource-bounde... |

114 | Dimension in complexity classes
- Lutz
- 2003
(Show Context)
Citation Context ...heory. In all these cases, it is known that replacing the constructive or resource-bounded martingales with constructive or resource-bounded supermartingales results in an equivalent definition. Lutz =-=[4]-=- recently introduced supergales and gales as natural generalizations of supermartingales and martingales, respectively. He showed that gales can be used to characterize classical Hausdorff dimension. ... |

94 | The dimensions of individual strings and sequences
- Lutz
- 2003
(Show Context)
Citation Context ...urce-bounded dimensions that work inside of complexity classes. He also showed that supergales may be used in place of gales to give equivalent definitions of these dimensions. Constructive dimension =-=[5]-=- refines the theory of Martin-Löf randomness by assigning each individual binary sequence a dimension. Lutz used constructive supergales to define constructive dimension. Supergales were used rather t... |

88 |
A unified approach to the definition of random sequences
- Schnorr
- 1971
(Show Context)
Citation Context ... generalizations of martingales, can be equivalently used to define constructive dimension. 1 Introduction Effective martingales have been very useful objects in theoretical computer science. Schnorr =-=[7, 8]-=- used constructive martingales to give an equivalent definition of Martin-Löf randomness [6]. Martingales computable within resource bounds have been used by Lutz [3] to define various resource-bounde... |

43 |
Gales and the constructive dimension of individual sequences
- Lutz
- 2000
(Show Context)
Citation Context ...ass of constructive s-supergales. Theorem 3.1 was used to prove the following cornerstone of constructive dimension theory. Theorem 3.2. (Lutz [2])For any X ⊆ C, cdim(X) = sup cdim(S). S∈X Remark. In =-=[5]-=-, a conference paper preceding [2], Lutz defined constructive dimension using constructive gales. There Lutz used a incorrect assertion about martingales to argue that for each computable s there exis... |

20 | Correspondence principles for effective dimensions
- Hitchcock
(Show Context)
Citation Context ...ere it is shown that constructive gales give an equivalent definition of constructive dimension. The proof is a simple and direct construction that uses some ideas from an earlier paper by the author =-=[1]-=-. As a corollary we obtain a form of optimal constructive gales. 2 Preliminaries The set of natural numbers is N = {0, 1, 2, . . .}. The set of binary strings of length n ∈ N is {0, 1} n . The set of ... |

12 | A uni approach to the de of random sequences - Schnorr - 1971 |

11 |
The dimensions of individual strings and sequences, Inform. and Comput. 187 (2003), 49–79 (preliminary version: Gales and the constructive dimension of individual sequences
- Lutz
- 2000
(Show Context)
Citation Context ...urce-bounded dimensions that work inside of complexity classes. He also showed that supergales may be used in place of gales to give equivalent definitions of these dimensions. Constructive dimension =-=[2]-=- refines the theory of Martin-Löf randomness by assigning each individual binary sequence a dimension. Lutz used constructive supergales to define constructive dimension. Supergales were used rather t... |

11 | The complexity of objects and the development of the concepts of information and randomness by means of the theory of algorithms - Zvonkin, Levin - 1970 |

4 | Correspondence principles for eective dimensions - Hitchcock - 2002 |