## EXTRACTING INFORMATION IS HARD: A TURING DEGREE OF NON-INTEGRAL EFFECTIVE HAUSDORFF DIMENSION

Citations: | 8 - 0 self |

### BibTeX

@MISC{Miller_extractinginformation,

author = {Joseph S. Miller},

title = {EXTRACTING INFORMATION IS HARD: A TURING DEGREE OF NON-INTEGRAL EFFECTIVE HAUSDORFF DIMENSION},

year = {}

}

### OpenURL

### Abstract

Abstract. We construct a ∆0 2 infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A ∈ 2ω is the lim inf of the ratio between the information content and length of initial segments of A. Thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density. 1.

### Citations

1682 | An Introduction to Kolmogorov Complexity and its Applications
- Li, Vitányi
- 1997
(Show Context)
Citation Context ...o algorithmic randomness can be found in the upcoming monographs of Downey and Hirschfeldt [4] and Nies [19] or the excellent survey paper of Downey, Hirschfeldt, Nies and Terwijn [6]. Li and Vitányi =-=[13]-=- is another useful source, although it does not cover effective dimension. One common approach to measuring the information content of binary strings is prefix-free complexity, as introduced by Levin ... |

332 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ... Note that this was not the definition given by Martin-Löf; it was proposed independently by Levin [12] and Chaitin [3] and proved by Schnorr to be equivalent to Martin-Löf’s definition of randomness =-=[16]-=-. Another notion characterized in terms of initial segment complexity is effective dimension. A thorough survey of effective dimension is given by Lutz [15]. As stated in the introduction, the effecti... |

330 | A theory of program size formally identical to information theory
- Chaitin
(Show Context)
Citation Context ...l source, although it does not cover effective dimension. One common approach to measuring the information content of binary strings is prefix-free complexity, as introduced by Levin [12] and Chaitin =-=[3]-=-. Call S ⊆ 2 <ω prefix-free if no element of S is a proper prefix of anther element. A prefix-free machine M : 2 <ω → 2 <ω is a partial computable function whose domain is prefixfree. We say that U is... |

209 |
Various techniques used in connection with random digits
- Neumann
- 1951
(Show Context)
Citation Context ...he right choice of bias—in particular, if heads comes up about 89% of the time—will produce a sequence with effective dimension 1/2. As it turns out, using a simple technique described by von Neumann =-=[29]-=-, randomness can also be extracted from these sequences. Consider pairs of coin flips; output a 1 if you see HT and a 0 if you see TH. Produce no output for pairs of the form HH or TT. The resulting s... |

156 |
Algorithmic randomness and complexity
- Downey, Hirschfeldt
- 2004
(Show Context)
Citation Context ...tor space determined by setting µ([σ]) = 2−|σ| , for each σ ∈ 2 <ω. Algorithmic randomness. An introduction to algorithmic randomness can be found in the upcoming monographs of Downey and Hirschfeldt =-=[4]-=- and Nies [19] or the excellent survey paper of Downey, Hirschfeldt, Nies and Terwijn [6]. Li and Vitányi [13] is another useful source, although it does not cover effective dimension. One common appr... |

99 |
Generating quasi-random sequences from semi-random sources
- Santha, Vazirani
- 1986
(Show Context)
Citation Context ...on, in such a4 JOSEPH S. MILLER way that the induced distribution is nearly uniform. In other words, composing the distribution with the map essentially produces a random source. Sántha and Vazirani =-=[25]-=- observed that this ideal goal cannot be met with a single source, but proved that extraction can be done with several independent sources. The fact that two independent sources were sufficient was pr... |

95 |
of information conservation (nongrowth) and aspects of the foundation of probability theory, Probl
- Levin, Laws
(Show Context)
Citation Context ... is another useful source, although it does not cover effective dimension. One common approach to measuring the information content of binary strings is prefix-free complexity, as introduced by Levin =-=[12]-=- and Chaitin [3]. Call S ⊆ 2 <ω prefix-free if no element of S is a proper prefix of anther element. A prefix-free machine M : 2 <ω → 2 <ω is a partial computable function whose domain is prefixfree. ... |

93 | Computability and Randomness
- Nies
- 2009
(Show Context)
Citation Context ...ermined by setting µ([σ]) = 2−|σ| , for each σ ∈ 2 <ω. Algorithmic randomness. An introduction to algorithmic randomness can be found in the upcoming monographs of Downey and Hirschfeldt [4] and Nies =-=[19]-=- or the excellent survey paper of Downey, Hirschfeldt, Nies and Terwijn [6]. Li and Vitányi [13] is another useful source, although it does not cover effective dimension. One common approach to measur... |

81 | Effective strong dimension, algorithmic information, and computational complexity
- Athreya, Hitchcock, et al.
(Show Context)
Citation Context ... fairly regularly. To formalize this observation, define the effective strong dimension of A ∈ 2ω to be Dim(A) = lim supn→∞ K(A ↾ n)/n. Clearly Dim(A) ≥ dim(A). Athreya, Hitchcock, Lutz and Mayordomo =-=[1]-=- proved that effective strong dimension is the effective analogue of packing dimension, another classical fractal dimension, in the same way that effective dimension is the analogue of Hausdorff dimen... |

73 |
A Kolmogorov complexity characterization of constructive Hausdorff dimension
- Mayordomo
(Show Context)
Citation Context ...ness. 1 is2 JOSEPH S. MILLER the classical Hausdorff dimension of a singleton set is zero, the effective Hausdorff dimension may not be. The equivalence of these two definitions, proved by Mayordomo =-=[17]-=- (but essentially implicit in Ryabko [24]), is evidence that effective dimension is a robust notion. Another indication is that the use of prefix-free complexity in the definition above is unnecessary... |

59 | Calibrating randomness
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...s. An introduction to algorithmic randomness can be found in the upcoming monographs of Downey and Hirschfeldt [4] and Nies [19] or the excellent survey paper of Downey, Hirschfeldt, Nies and Terwijn =-=[6]-=-. Li and Vitányi [13] is another useful source, although it does not cover effective dimension. One common approach to measuring the information content of binary strings is prefix-free complexity, as... |

54 |
Strong communication complexity or generating quasirandom sequences from two communicating semi-random sources
- Vazirani
- 1987
(Show Context)
Citation Context ...al cannot be met with a single source, but proved that extraction can be done with several independent sources. The fact that two independent sources were sufficient was proved soon after by Vazirani =-=[27]-=-. The analogy with infinite sequences is clear and, as we have pointed out, the study of randomness extractors has been applied to variants of Question 1.1 [7, 30]. Outline. In the next section we giv... |

50 |
Coding of combinatorial sources and hausdorff dimension
- Ryabko
- 1984
(Show Context)
Citation Context ...al Hausdorff dimension of a singleton set is zero, the effective Hausdorff dimension may not be. The equivalence of these two definitions, proved by Mayordomo [17] (but essentially implicit in Ryabko =-=[24]-=-), is evidence that effective dimension is a robust notion. Another indication is that the use of prefix-free complexity in the definition above is unnecessary; replacing it with plain Kolmogorov comp... |

46 | Kolmogorov complexity and the recursion theorem
- Kjos-Hanssen, Merkle, et al.
(Show Context)
Citation Context ...tudy of randomness extractors, a subject that is discussed in more detail below. Attempts to answer Question 1.1 in the negative have also led to interesting results. Kjos-Hanssen, Merkle and Stephan =-=[11]-=- call A complex if there is an unbounded, nondecreasing computable function f such that (∀n) K(A ↾ n) ≥ f(n). Although a complex sequence can have very low information density, we can effectively find... |

43 |
Gales and the constructive dimension of individual sequences
- Lutz
- 2000
(Show Context)
Citation Context ... 1/2 is guaranteed to have nearly n/2 bits of information in the first n bits, although it can have more for some n. This is not the original definition of effective dimension. That was given by Lutz =-=[14]-=-, who effectivized a martingale characterization of Hausdorff dimension and defined dim(A) to be the effective Hausdorff dimension of {A}. Note that although Date: December 29, 2008. 2000 Mathematics ... |

24 | A lower cone in the wtt degrees of non-integral effective dimension
- Nies, Reimann
- 2006
(Show Context)
Citation Context ...) who constructed a ∆ 0 2 sequence A with effective dimension 1/2 such that if B is many-one reducible to A, then dim(B) ≤ 1/2. Nies and Reimann later generalized this to weak truthtable reducibility =-=[20]-=-. Many-one and weak truth-table reduction are strong forms of computation; without going into the definitions, the point is that each of these results showed that a certain restricted family of algori... |

21 |
and André Nies. Randomness and computability: open questions
- Miller
(Show Context)
Citation Context ...sequence with effective dimension 1 and minimal Turing degree. This gives a negative answer to a variant of Question 1.1 that appeared, for example, in the open questions paper of Nies and the author =-=[18]-=-: if dim(A) = 1, does A compute a Martin-Löf random sequence? Another line of attack that yielded partial negative solutions was to place a limit on the type of algorithms used to extract information ... |

21 |
Recent developments in Extractors
- Shaltiel
- 2002
(Show Context)
Citation Context ...ing that is simple relative to x is unconditionally simple. Moving beyond algorithmic information theory, there is a large body of work on randomness extractors, much of which is surveyed by Shaltiel =-=[26]-=-. The usual assumption is that you are given a distribution on 2 n with a certain guaranteed min-entropy, which simply means that no element of 2 n is too likely. Intuitively, the min-entropy is a low... |

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

20 | Extracting Kolmogorov complexity with applications to dimension zero-one laws
- Fortnow, Hitchcock, et al.
- 2006
(Show Context)
Citation Context ...s of nearly random behavior unpredictably followed by periods of relative order. We mention two other positive results on the problem of extracting information from infinite sequences. Fortnow et al. =-=[7]-=- proved that if Dim(A) > 0, then AEXTRACTING INFORMATION IS HARD 3 computes sequences with effective strong dimension arbitrarily close to 1 (see also Bienvenu et al. [2]). Secondly, Zimand [30] show... |

18 | Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences
- Zimand
(Show Context)
Citation Context ...t al. [7] proved that if Dim(A) > 0, then AEXTRACTING INFORMATION IS HARD 3 computes sequences with effective strong dimension arbitrarily close to 1 (see also Bienvenu et al. [2]). Secondly, Zimand =-=[30]-=- showed that if A, B ∈ 2ω are sufficiently independent and both have positive effective dimension, then together they compute (in fact, truth-table compute, uniformly in a lower bound on the dimension... |

18 |
Computability and Randomness, volume 51 of Oxford Logic Guides
- Nies
- 2009
(Show Context)
Citation Context ...ermined by setting µ([σ]) = 2−|σ| , for each σ ∈ 2 <ω. Algorithmic randomness. An introduction to algorithmic randomness can be found in the upcoming monographs of Downey and Hirschfeldt [4] and Nies =-=[17]-=- or the excellent survey paper of Downey, Hirschfeldt, Nies and Terwijn [6]. Li and Vitányi [11] is another useful source, although it does not cover effective dimension. One common approach to measur... |

15 | Effective fractal dimensions
- Lutz
(Show Context)
Citation Context ...nt to Martin-Löf’s definition of randomness [16]. Another notion characterized in terms of initial segment complexity is effective dimension. A thorough survey of effective dimension is given by Lutz =-=[15]-=-. As stated in the introduction, the effective (Hausdorff) dimension of A ∈ 2ω is dim(A) = lim infn→∞ K(A ↾ n)/n and the effective strong dimension is Dim(A) = lim supn→∞ K(A ↾ n)/n. Note that 0 ≤ dim... |

11 |
Computability and fractal dimension. Doctoral dissertation, Universität
- Reimann
- 2004
(Show Context)
Citation Context ...n f such that (∀n) K(A ↾ n) ≥ f(n). Although a complex sequence can have very low information density, we can effectively find initial segments with as much information as we want. Reimann and Slaman =-=[21, 23]-=-, and independently Kjos-Hanssen, et al. [11, Cor. 7], proved that complex sets need not compute Martin-Löf random sequences. Along similar lines, Downey and Greenberg [5] proved that there is an A ∈ ... |

8 |
Members of random closed sets
- Diamondstone, Kjos-Hanssen
- 2009
(Show Context)
Citation Context ...sults of Kjos-Hanssen and Reimann. (Although we have specialized to the case s = 1/2, the proof works generally.) Kjos-Hanssen showed that scapacitability implies vehement s-randomness [22] (see also =-=[9, 10]-=-), which together with Reimann’s result that strong s-randomness implies s-capacitability [22], proves that all three notions are the same. See [22, 10] for the relevant definitions. The forcing condi... |

7 | Infinite subsets of random sets of integers
- Kjos-Hanssen
- 2009
(Show Context)
Citation Context ...sults of Kjos-Hanssen and Reimann. (Although we have specialized to the case s = 1/2, the proof works generally.) Kjos-Hanssen showed that scapacitability implies vehement s-randomness [22] (see also =-=[9, 10]-=-), which together with Reimann’s result that strong s-randomness implies s-capacitability [22], proves that all three notions are the same. See [22, 10] for the relevant definitions. The forcing condi... |

7 | Effectively closed sets of measures and randomness
- Reimann
(Show Context)
Citation Context ...follows from results of Kjos-Hanssen and Reimann. (Although we have specialized to the case s = 1/2, the proof works generally.) Kjos-Hanssen showed that scapacitability implies vehement s-randomness =-=[22]-=- (see also [9, 10]), which together with Reimann’s result that strong s-randomness implies s-capacitability [22], proves that all three notions are the same. See [22, 10] for the relevant definitions.... |

6 | Constructive dimension and Turing degrees
- Bienvenu, Doty, et al.
(Show Context)
Citation Context ...Hausdorff dimension. If A ∈ 2ω is a sequence of effective dimension 1/2 obtained either through dilution or from a biased coin, as described above, then Dim(A) is also 1/2. Bienvenu, Doty and Stephan =-=[2]-=- showed that this is enough to guarantee that A computes sequences of higher effective dimension. Specifically, they proved that if ε > 0 and Dim(A) > 0, then A computes a set B such that dim(B) ≥ dim... |

6 | Diagonally non-recursive functions and effective Hausdorff dimension
- Greenberg, Miller
(Show Context)
Citation Context ... sequences with effective dimension arbitrarily close to 1. (Note that it is open whether such an A must always compute a sequence with effective dimension 1, but it follows from Greenberg and Miller =-=[8]-=- that A need not compute a Martin-Löf random sequence.) The result of Bienvenu et al. demonstrates that any sequence refuting Question 1.1 must be irregular, having periods of nearly random behavior u... |

3 | Turing degrees of reals of positive effective packing dimension
- Downey, Greenberg
(Show Context)
Citation Context ...ant. Reimann and Slaman [21, 23], and independently Kjos-Hanssen, et al. [11, Cor. 7], proved that complex sets need not compute Martin-Löf random sequences. Along similar lines, Downey and Greenberg =-=[5]-=- proved that there is an A ∈ 2ω such that Dim(A) = 1 and A has minimal (Turing) degree, meaning that any noncomputable set computed from A must compute A. This property implies that A does not compute... |

1 |
Constructive dimension and Turing degrees. Theory Comput
- Bienvenu, Doty, et al.
(Show Context)
Citation Context ...Hausdorff dimension. If A ∈ 2ω is a sequence of effective dimension 1/2 obtained either through dilution or from a biased coin, as described above, then Dim(A) is also 1/2. Bienvenu, Doty and Stephan =-=[2]-=- showed that this is enough to guarantee that A computes sequences of higher effective dimension. Specifically, they proved that if ε > 0 and Dim(A) > 0, then A computes a set B such that dim(B) ≥ dim... |