## Randomness, computability, and density (2002)

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Venue: | SIAM Journal of Computation |

Citations: | 13 - 6 self |

### BibTeX

@INPROCEEDINGS{Downey02randomness,computability,,

author = {Rod G. Downey and Denis R. Hirschfeldt},

title = {Randomness, computability, and density},

booktitle = {SIAM Journal of Computation},

year = {2002},

pages = {1169--1183},

publisher = {Springer-Verlag}

}

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### Abstract

1 Introduction In this paper we are concerned with effectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying n + ff and ff for any n 2! and ff 2 (0; 1], and we do this below without further comment.

### Citations

1687 | Introduction to Kolmogorov Complexity and Its Applications. Springer-Verlag, 2nd edition edition
- Li, Vitanyi
- 1997
(Show Context)
Citation Context ...tandard Kolmogorov complexity. (Most of the statements below also hold with K(o/ ) in place of H(o/ ). For the definitions and basic properties of H(o/ ) and K(o/ ), see Calude [3] and Li and Vitanyi =-=[22]-=-. Among the many works dealing with these and related topics, and in addition to those mentioned elsewhere in this paper, we may cite Solomonoff [28, 29], Kolmogorov [18], Levin [20, 21], Schnorr [25]... |

523 |
Three approaches to the quantitative definition of information
- Kolmogorov
- 1965
(Show Context)
Citation Context ...Calude [3] and Li and Vitanyi [22]. Among the many works dealing with these and related topics, and in addition to those mentioned elsewhere in this paper, we may cite Solomonoff [28, 29], Kolmogorov =-=[18]-=-, Levin [20, 21], Schnorr [25], and the expository article Calude and Chaitin [4].) We identify a real ff 2 (0; 1] with the infinite binary string oe such that ff = 0:oe. (The fact that certain reals ... |

403 |
A Formal Theory of Inductive Inference
- Solomonoff
- 1964
(Show Context)
Citation Context ...o/ ) and K(o/ ), see Calude [3] and Li and Vitanyi [22]. Among the many works dealing with these and related topics, and in addition to those mentioned elsewhere in this paper, we may cite Solomonoff =-=[28, 29]-=-, Kolmogorov [18], Levin [20, 21], Schnorr [25], and the expository article Calude and Chaitin [4].) We identify a real ff 2 (0; 1] with the infinite binary string oe such that ff = 0:oe. (The fact th... |

332 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ...oles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte [14]. A real is random if its dyadic expansion forms a random infinite binary sequence (in the sense of, for instance, Martin-L"of =-=[23]-=-). Chaitin's number \Omega , the halting probability of a universal self-delimiting computer, is a standard random c.e. real. (We will define these concepts more formally below.) Many authors have stu... |

330 | A theory of program size formally identical to information theory
- Chaitin
- 1975
(Show Context)
Citation Context ... mentioned elsewhere in this paper, we may cite Solomonoff [30, 31], Kolmogorov [19], Levin [22, 23], Schnorr [27], and the expository article Calude and Chaitin [4]. As shown by Schnorr (see Chaitin =-=[9]-=-), a real ff is random if and only if there is a constant c such that H(ff _ n) ? n \Gammasc for all n 2 !. (We identify a real ff 2 (0; 1] with the infinite binary string oe such that ff = 0:oe. The ... |

321 | Algorithmic Information Theory
- Chaitin
(Show Context)
Citation Context ...lity of a universal self-delimiting computer, is a standard random c.e. real. (We will define these concepts more formally below.) Many authors have studied \Omegasand its properties, notably Chaitin =-=[9, 10, 11]-=- and Martin-L"of [23]. In the very long and widely circulated manuscript [30] (a fragment of which appeared in [31]), Solovay carefully investigated relationships between Martin-L"ofChaitin prefix-fre... |

72 |
Soare, Recursively Enumerable Sets and Degrees
- I
- 1987
(Show Context)
Citation Context ...n ff? In the process of understanding a degree structure, the question of density has always played a key role, and been one of the first to be addressed. For instance, the Sacks Density Theorem (see =-=[29]-=-) was one of the earliest and most important results in the study of the c.e. Turing degrees. In this paper, we show that the Solovay degrees of c.e. reals are dense. To do this we divide the proof in... |

66 | Randomness and recursive enumerability
- Kučera, Slaman
(Show Context)
Citation Context ...specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [32] and studied by Calude, Hertling, Khoussainov, and Wang [6], Calude [2], Ku^cera and Slaman =-=[20]-=-, and Downey, Hirschfeldt, and LaForte [15], among others. This measure is called domination or Solovay reducibility, and is defined by saying that ff dominates fi if there are a constant c and a part... |

53 |
of a paper (or series of papers) on Chaitin’s work ... done for the most part during the period of Sept
- Solovay, Draft
- 1974
(Show Context)
Citation Context ... University of Chicago August 20, 2000 Abstract We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay =-=[30]-=- and studied by Calude, Hertling, Khoussainov, and Wang [6], Calude [2], Slaman [26], and Coles, Downey, and LaForte [12], among others. This measure is called domination or Solovay reducibility, and ... |

32 | Incompleteness theorems for random reals
- Chaitin
- 1987
(Show Context)
Citation Context ...lity of a universal self-delimiting computer, is a standard random c.e. real. (We will define these concepts more formally below.) Many authors have studied \Omegasand its properties, notably Chaitin =-=[9, 10, 11]-=- and Martin-L"of [23]. In the very long and widely circulated manuscript [30] (a fragment of which appeared in [31]), Solovay carefully investigated relationships between Martin-L"ofChaitin prefix-fre... |

26 | Randomness and reducibility
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...enumerable reals) under a measure of relative randomness introduced by Solovay [30] and studied by Calude, Hertling, Khoussainov, and Wang [6], Calude [2], Slaman [26], and Coles, Downey, and LaForte =-=[12]-=-, among others. This measure is called domination or Solovay reducibility, and is defined by saying that ff dominates fi if there are a constant c and a partial computable function ' such that for all... |

25 |
Various measures of complexity for finite objects (axiomatic description
- Levin
- 1976
(Show Context)
Citation Context ...nd Li and Vitanyi [22]. Among the many works dealing with these and related topics, and in addition to those mentioned elsewhere in this paper, we may cite Solomonoff [28, 29], Kolmogorov [18], Levin =-=[20, 21]-=-, Schnorr [25], and the expository article Calude and Chaitin [4].) We identify a real ff 2 (0; 1] with the infinite binary string oe such that ff = 0:oe. (The fact that certain reals have two differe... |

20 |
On the definitions of some complexity classes of real numbers
- Ko
- 1983
(Show Context)
Citation Context ... addition to the papers and books mentioned elsewhere in this introduction, we may cite, among others, early work of Rice [24], Lachlan [19], Soare [27], and Ce*itin [8], and more recent papers by Ko =-=[16, 17]-=-, Calude, Coles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte [14]. A real is random if its dyadic expansion forms a random infinite binary sequence (in the sense of, for instance, M... |

17 |
Calude, Information and Randomness: An Algorithmic Perspective, 2nd Edition, Revised and Extended
- S
- 2002
(Show Context)
Citation Context ...complexity of and K() its standard Kolmogorov complexity. (Most of the statements below also hold with K() in place of H( ). For the denitions and basic properties of H() and K( ), see Calude [3] and Li and Vitanyi [22]. Among the many works dealing with these and related topics, and in addition to those mentioned elsewhere in this paper, we may cite Solomono [28, 29], Kolmogorov [18], Levin... |

17 |
On random r.e. sets
- Solovay
- 1977
(Show Context)
Citation Context ... below.) Many authors have studied \Omegasand its properties, notably Chaitin [9, 10, 11] and Martin-L"of [23]. In the very long and widely circulated manuscript [30] (a fragment of which appeared in =-=[31]-=-), Solovay carefully investigated relationships between Martin-L"ofChaitin prefix-free complexity, Kolmogorov complexity, and properties of random languages and reals. See Chaitin [9] for an account o... |

16 |
Weakly computable real numbers
- Ambos-Spies, Weihrauch, et al.
- 2000
(Show Context)
Citation Context ...ble increasing sequences of rationals. We call such reals computably enumerable (c.e.), though they have also been called recursively enumerable, left computable (by Ambos-Spies, Weihrauch, and Zheng =-=[1]-=-), and left semicomputable. If, in addition to the existence of a computable increasing sequence q0; q1; : : : of rationals with limit ff, there is a total computable function f such that ff \Gammasqf... |

13 | Degree-theoretic aspects of computably enumerable reals
- Calude, Coles, et al.
- 1999
(Show Context)
Citation Context ...re in this introduction, we may cite, among others, early work of Rice [24], Lachlan [19], Soare [27], and Cetin [8], and more recent papers by Ko [16, 17], Calude, Coles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte [14]. A real is random if its dyadic expansion forms a random innite binary sequence (in the sense of, for instance, Martin-Lof [23]). Chaitin's numbersthe halting ... |

10 | Khoussainov Recursively enumerable reals and Chaitin Omega numbers
- Calude, Hertlund, et al.
- 2001
(Show Context)
Citation Context ...ectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [30] and studied by Calude, Hertling, Khoussainov, and Wang =-=[6]-=-, Calude [2], Slaman [26], and Coles, Downey, and LaForte [12], among others. This measure is called domination or Solovay reducibility, and is defined by saying that ff dominates fi if there are a co... |

10 | Presentations of computably enumerable reals
- Downey, LaForte
(Show Context)
Citation Context ... among others, early work of Rice [24], Lachlan [19], Soare [27], and Ce*itin [8], and more recent papers by Ko [16, 17], Calude, Coles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte =-=[14]-=-. A real is random if its dyadic expansion forms a random infinite binary sequence (in the sense of, for instance, Martin-L"of [23]). Chaitin's number \Omega , the halting probability of a universal s... |

10 |
Cohesive sets and recursively enumerable dedekind cuts
- Soare
- 1969
(Show Context)
Citation Context ...se and related concepts have been widely studied. In addition to the papers and books mentioned elsewhere in this introduction, we may cite, among others, early work of Rice [26], Lachlan [21], Soare =-=[28]-=-, and Ce*itin [8], and more recent papers by Ko [17, 18], Calude, Coles, Hertling, and Khoussainov [5], Ho [16], and Downey and LaForte [14]. A computer M is self-delimiting if, for each binary string... |

6 |
K.: Relatively recursive reals and real functions. Theoretical Computer Science 210
- Ho
- 1999
(Show Context)
Citation Context ...is introduction, we may cite, among others, early work of Rice [24], Lachlan [19], Soare [27], and Ce*itin [8], and more recent papers by Ko [16, 17], Calude, Coles, Hertling, and Khoussainov [5], Ho =-=[15]-=-, and Downey and LaForte [14]. A real is random if its dyadic expansion forms a random infinite binary sequence (in the sense of, for instance, Martin-L"of [23]). Chaitin's number \Omega , the halting... |

5 |
Chaitin \Omega numbers and strong reducibilities
- Calude, Nies
- 1997
(Show Context)
Citation Context ...e, the degree of \Omega . For proofs of these facts and more on c.e. reals and Solovay reducibility, see for instance Chaitin [9, 10, 11], Calude, Hertling, Khoussainov, and Wang [6], Calude and Nies =-=[7]-=-, Calude [2], Slaman [26], and Coles, Downey, and LaForte [12]. Despite the many attractive features of the Solovay degrees, their structure is largely unknown. Coles, Downey, and LaForte [12] have sh... |

4 |
Randomness everywhere, Nature 400
- Calude, Chaitin
- 1999
(Show Context)
Citation Context ...elated topics, and in addition to those mentioned elsewhere in this paper, we may cite Solomono [28, 29], Kolmogorov [18], Levin [20, 21], Schnorr [25], and the expository article Calude and Chaitin [4].) We identify a real 2 (0; 1] with the innite binary string such that = 0:. (The fact that certain reals have two dierent dyadic expansions need not concern us here, since all such reals are... |

4 |
On the continued fraction representation of computable real numbers, Theor
- Ko
- 1986
(Show Context)
Citation Context ... addition to the papers and books mentioned elsewhere in this introduction, we may cite, among others, early work of Rice [24], Lachlan [19], Soare [27], and Ce*itin [8], and more recent papers by Ko =-=[16, 17]-=-, Calude, Coles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte [14]. A real is random if its dyadic expansion forms a random infinite binary sequence (in the sense of, for instance, M... |

3 |
Computability, definability, and algebraic structures, to appear
- Downey
(Show Context)
Citation Context ...nonrandom, but ff + fi = \Omegasis random. Before turning to the details of the paper, we point out that there are other reducibilities one can study in this context. Downey, Hirschfeldt, and LaForte =-=[15, 13]-=- introduced two such reducibilities, sw-reducibility and rH-reducibility, and showed, among other things, that the results of this paper also hold for rH-reducibility. The proofs are essentially the s... |

1 |
Calude, A characterization of c.e. random reals
- S
(Show Context)
Citation Context ...ven positive reals (more specically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [30] and studied by Calude, Hertling, Khoussainov, and Wang [6], Calude [=-=-=-2], Slaman [26], and Coles, Downey, and LaForte [12], among others. This measure is called domination or Solovay reducibility, and is dened by saying that dominatessif there are a constant c and a pa... |

1 |
A pseudofundamental sequence that is not equivalent to a monotone one
- Ceitin
- 1971
(Show Context)
Citation Context ...cepts have been widely studied. In addition to the papers and books mentioned elsewhere in this introduction, we may cite, among others, early work of Rice [24], Lachlan [19], Soare [27], and Ce*itin =-=[8]-=-, and more recent papers by Ko [16, 17], Calude, Coles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte [14]. A real is random if its dyadic expansion forms a random infinite binary seq... |

1 |
Symbolic Logic 28
- Lachlan, numbers, et al.
- 1963
(Show Context)
Citation Context ...putable. These and related concepts have been widely studied. In addition to the papers and books mentioned elsewhere in this introduction, we may cite, among others, early work of Rice [24], Lachlan =-=[19]-=-, Soare [27], and Ce*itin [8], and more recent papers by Ko [16, 17], Calude, Coles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte [14]. A real is random if its dyadic expansion forms... |

1 |
Process complexity and effective random tests
- Soc
- 1954
(Show Context)
Citation Context ... is called computable. These and related concepts have been widely studied. In addition to the papers and books mentioned elsewhere in this introduction, we may cite, among others, early work of Rice =-=[24]-=-, Lachlan [19], Soare [27], and Ce*itin [8], and more recent papers by Ko [16, 17], Calude, Coles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte [14]. A real is random if its dyadic e... |

1 |
Randomness and recursive enumerability, to appear. [27] R. Soare, Cohesive sets and recursively enumerable Dedekind cuts
- Slaman
- 1969
(Show Context)
Citation Context ...reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [30] and studied by Calude, Hertling, Khoussainov, and Wang [6], Calude [2], Slaman =-=[26]-=-, and Coles, Downey, and LaForte [12], among others. This measure is called domination or Solovay reducibility, and is defined by saying that ff dominates fi if there are a constant c and a partial co... |