## Dimension Extractors and Optimal Decompression

### Cached

### Download Links

- [www.cs.iastate.edu]
- [www.cs.iastate.edu]
- [arxiv.org]
- [www.cs.iastate.edu]
- DBLP

### Other Repositories/Bibliography

Citations: | 9 - 4 self |

### BibTeX

@MISC{Doty_dimensionextractors,

author = {David Doty},

title = {Dimension Extractors and Optimal Decompression},

year = {}

}

### OpenURL

### Abstract

### Citations

6050 | E.: A mathematical theory of communication - SHANNON - 1948 |

2345 | Computational Complexity - Papadimitriou - 1994 |

1682 | An Introduction to Kolmogorov Complexity and its Applications - Li, Vitányi - 1997 |

730 | Compression of individual sequences via variable-rate coding - Ziv, Lempel - 1978 |

473 | Recursively Enumerable Sets and Degrees - Soare - 1987 |

332 | The definition of random sequences - Martin-Löf - 1966 |

330 | A theory of program size formally identical to information theory - Chaitin |

222 | Computational complexity of probabilistic Turing machines - Gill - 1977 |

209 | Various techniques used in connection with random digits - Neumann - 1951 |

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 |

172 | Almost everywhere high nonuniform complexity - Lutz - 1992 |

144 | Recent developments in explicit constructions of extractors - Shaltiel |

114 | Dimension in complexity classes - Lutz - 2003 |

94 | The dimensions of individual strings and sequences - Lutz - 2003 |

88 | A unified approach to the definition of random sequences - Schnorr - 1971 |

85 | Process complexity and effective random tests - Schnorr - 1973 |

81 | Effective strong dimension, algorithmic information, and computational complexity
- Athreya, Hitchcock, et al.
(Show Context)
Citation Context ... developed lead directly to new characterizations of some effective dimensions in terms of optimal decompression by Turing reductions. 1 Introduction Effective dimension [33, 34] and strong dimension =-=[2]-=- are effectivizations of classical Hausdorff [21] and packing [49, 50] dimension, which can each be characterized in terms of betting strategies called martingales. By placing resource bounds on the m... |

73 | A Kolmogorov complexity characterization of constructive Hausdorff dimension - Mayordomo |

60 | Two definitions of fractional dimension - Tricot - 1982 |

51 | Counting complexity - Fortnow - 1997 |

50 |
Relative to a random oracle A, P A = NP A = coNP A with probability 1
- Bennett, Gill
- 1981
(Show Context)
Citation Context ... with probability of correctness at least 2/3, is generally regarded as the set of decision problems feasibly decidable by a randomized algorithm. Bennett [4] (refining measure-theoretic arguments of =-=[5]-=- and [1]) has demonstrated that, given access to any oracle sequence that is algorithmically random in the sense of Martin-Löf, every language in BPP can be decided deterministically in polynomial tim... |

50 | Coding of combinatorial sources and hausdorff dimension - Ryabko - 1984 |

48 | Logical Depth and Physical Complexity
- Bennett
- 1988
(Show Context)
Citation Context ...ble by a randomized polynomial time algorithm with probability of correctness at least 2/3, is generally regarded as the set of decision problems feasibly decidable by a randomized algorithm. Bennett =-=[4]-=- (refining measure-theoretic arguments of [5] and [1]) has demonstrated that, given access to any oracle sequence that is algorithmically random in the sense of Martin-Löf, every language in BPP can b... |

42 | Every sequence is reducible to a random one - Gács - 1986 |

37 | Fractal dimension and logarithmic loss unpredictability - Hitchcock |

32 | Randomness and computability: Open questions - Nies, Miller - 2006 |

28 | classes and complete extensions of - Kucera, Measure - 1985 |

27 | Effective Fractal Dimension: Foundations and Applications - Hitchcock - 2003 |

24 | Switching and Finite Automata Theory (Second Edition - Kohavi - 1978 |

24 | A lower cone in the wtt degrees of non-integral effective dimension - Nies, Reimann - 2006 |

21 | Independent minimum length programs to translate between given strings. Theor. Comput. Sci - Vereshchagin, Vyugin - 2002 |

20 | An observation on probability versus randomness with applications to complexity classes; Mathematical Systems Theory 27 - BOOK, LUTZ, et al. - 1994 |

20 | Classics on Fractals - Edgar - 2004 |

20 | Extracting Kolmogorov complexity with applications to dimension zero-one laws - Fortnow, Hitchcock, et al. - 2006 |

19 | Gales suffice for constructive dimension - Hitchcock - 2002 |

17 | Finite-state dimension. Theoretical Computer Science - Dai, Lathrop, et al. - 2004 |

16 | On the use of diagonally nonrecursive functions - Kučera - 1987 |

16 | Dimension is compression - López-Valdés, Mayordomo |

15 | Canonical forms for information-lossless finite-state logical machines - Huffman - 1959 |

13 | Constructive dimension and weak truth-table degrees
- Bienvenu, Doty, et al.
- 2007
(Show Context)
Citation Context ...ble reducible to S satisfies dim(P ) ≤ α. Since the Turing reduction in our proof is also a weak truth-table reduction, it cannot always be the case that dim(P ) > dim(S). Bienvenu, Doty, and Stephan =-=[7]-=- have shown that the stronger results of the next subsection hold for constructive dimension as well, using weak truth-table reductions, thereby improving the present paper’s Corollary 4.3 to be as st... |

13 | Relative to a random oracle A, PA 6= NPA 6= co-NPA with probability 1 - Bennett, Gill - 1981 |

13 | A pseudorandom oracle characterization of BPP - Lutz - 1993 |

10 | Entropy rates and finite-state dimension. Theoretical Computer Science - Bourke, Hitchcock, et al. - 2005 |

10 | Every sequence is decompressible from a random one - Doty - 2006 |

10 | Noiseless coding of combinatorial sources. Problems of Information Transmission - Ryabko - 1986 |

8 |
Randomness, relativizations, and polynomial reducibilities
- Ambos-Spies
- 1986
(Show Context)
Citation Context ...obability of correctness at least 2/3, is generally regarded as the set of decision problems feasibly decidable by a randomized algorithm. Bennett [4] (refining measure-theoretic arguments of [5] and =-=[1]-=-) has demonstrated that, given access to any oracle sequence that is algorithmically random in the sense of Martin-Löf, every language in BPP can be decided deterministically in polynomial time. Book,... |

7 | On the construction of effective random sets - Merkle, Mihailovic |

6 | Dimension und äusseres Mass. Mathematische Annalen, 79:157– 179 - Hausdorff - 1919 |

4 |
Barzdin ′ . Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set
- M
- 1968
(Show Context)
Citation Context ... more compact description. Consider the following example. It is well known that K, the characteristic sequence of the halting language, has constructive dimension and constructive strong dimension 0 =-=[3]-=-. The binary representation of Chaitin’s halting probability Ω = � M halts 2−|M| (where M ranges over all halting programs and |M| is M’s binary description length) is an algorithmically random sequen... |

4 | Finite-state dimension and lossy decompressors - Doty, Moser - 2006 |