## Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences (2007)

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Citations: | 18 - 6 self |

### BibTeX

@TECHREPORT{Zimand07twosources,

author = {Marius Zimand},

title = {Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences},

institution = {},

year = {2007}

}

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### Citations

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

51 | Counting complexity
- Fortnow
- 1997
(Show Context)
Citation Context ...observed earlier, there is no effective transformation that increases the absolute amount of Kolmogorov complexity. However, some positive results do exist. Buhrman, Fortnow, Newman, and Vereshchagin =-=[BFNV05]-=- show that, for any nonrandom string of length n, one can flip O( √ n) of its bits and obtain a string with higher Kolmogorov complexity. Fortnow, Hitchcock, Pavan, Vinodchandran, and Wang [FHP + 06] ... |

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

48 | On partial randomness - Calude, Staiger, et al. - 2006 |

33 | A generalization of Chaitin’s halting probability ω and halting self-similar sets - Tadaki |

32 |
Randomness and computability: Open questions
- Nies, Miller
- 2006
(Show Context)
Citation Context ...r randomness rate. This is a natural question, first raised by Reimann [Rei04], which has received significant attention recently (it is Question 10.1 in the list of open questions of Miller and Nies =-=[MN06]-=-). So far, there exist a series of partial results, mostly negative, obtained by restricting the type of transformation. Reimann and Terwijn [Rei04, Th 3.10] have shown that for every constant c < 1, ... |

24 | A lower cone in the wtt degrees of non-integral effective dimension
- Nies, Reimann
- 2006
(Show Context)
Citation Context ...for every constant c < 1, there exists a sequence x such that if f is a many-one reduction, then the randomness rate of f(x) cannot be larger than c. This result has been improved by Nies and Reimann =-=[NR06]-=- ∗ The author is supported by NSF grant CCF 0634830. Part of this work was done while visiting University of Auckland, New Zealand, partially supported by a CDMTCS grant. 1 The randomness rate of x is... |

21 |
Computability and fractal dimension
- Reimann
- 2004
(Show Context)
Citation Context ... sparsely distributed) and thus it cannot be ruled out that there may be a way to concentrate it and obtain a sequence with higher randomness rate. This is a natural question, first raised by Reimann =-=[Rei04]-=-, which has received significant attention recently (it is Question 10.1 in the list of open questions of Miller and Nies [MN06]). So far, there exist a series of partial results, mostly negative, obt... |

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

19 |
Algorithmic Information Theory and Kolmogorov Complexity, lecture notes of a course taught at Uppsala University, available as a technical report at
- Shen
(Show Context)
Citation Context ...te some basic algorithmic-information theoretical properties of the blocks x1, x2, . . . . and y1, y2, . . .. We first recall the following basic fact (for example, see Alexander Shen’s lecture notes =-=[She00]-=-). Theorem 4.3 For all finite binary strings u and v, (i) K(vu) ≤ K(u) + K(v | u) + O(log K(u) + log K(v)). (ii) K(vu) ≥ K(u) + K(v | u) − O(log K(u) + log K(v)). The hidden constants depend only on t... |

18 | Constructive dimension equals Kolmogorov complexity - Staiger - 2005 |

13 | Constructive dimension and weak truth-table degrees
- Bienvenu, Doty, et al.
- 2007
(Show Context)
Citation Context ...precisely, they showed that for all rational c ∈ (0, 1), there exists a sequence x with randomness rate c such that for all wtt-reductions f, f(x) has randomness rate ≤ c. Bienvenu, Doty, and Stephan =-=[BDS07]-=- have obtained an impossibility result for the general case of Turing reductions, which, however, is valid only for uniform reductions. Relying on the result of Nies and Reimann, they show that for ev... |

9 | Dimension extractors and optimal decompression. Theory of Computing Systems
- Doty
(Show Context)
Citation Context ...effective transformation that on input σ ∈ (0, 1], ɛ > 0, and x, a sequence with randomness rate σ, produces a string y with randomness rate σ + ɛ ?” has a negative answer. On the positive side, Doty =-=[Dot07]-=- has shown that for every constant c there exists a uniform effective transformation f able to transform any x with randomness rate c ∈ (0, 1] into a sequence f(x) that, for infinitely many n, has the... |