## Randomness, relativization, and Turing degrees (2005)

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Venue: | J. Symbolic Logic |

Citations: | 38 - 17 self |

### BibTeX

@ARTICLE{Nies05randomness,relativization,,

author = {Andre Nies and Frank Stephan and Sebastiaan A. Terwijn},

title = {Randomness, relativization, and Turing degrees},

journal = {J. Symbolic Logic},

year = {2005},

volume = {70},

pages = {2005}

}

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### Abstract

We compare various notions of algorithmic randomness. First we consider relativized randomness. A set is n-random if it is Martin-Lof random relative to . We show that a set is 2-random if and only if there is a constant c such that infinitely many initial segments x of the set are c-incompressible: C(x) c. The `only if' direction was obtained independently by Joseph Miller. This characterization can be extended to the case of time-bounded C-complexity.

### Citations

1681 | An Introduction to Kolmogorov Complexity and its Applications
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- 1993
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Citation Context ...tion of randomness of infinite strings based on constructive measure theory. Especially the strong connections with the theory of randomness for finite objects made this notion very popular, see e.g. =-=[17], -=-to name only one of the many references that the reader can consult for this. Another landmark in the theory of randomness is Schnorr’s book [26], containing a thorough discussion (and criticism) of... |

472 |
Recursively Enumerable Sets and Degrees
- Soare
- 1987
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Citation Context ...xity. Usually, we use V to denote a universal plain machine (for the definition of C) and U to denote a universal prefix-free machine (for K). Our recursion theoretic notation is standard and follows =-=[25, 28]. As usual, subse-=-ts A ⊆ N can be identified with infinite binary sequences and sometimes we interpret an A ⊆ N as the real number � n∈A 2−n−1 . A↾n is the initial segment of A of length n and σ ≺ A de... |

332 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ...une-free and PA-complete degrees. Mathematical Subject Classification: 68Q30, 03D15, 03D28, 03D80, 28E15. 1. Introduction The study of algorithmic randomness received a strong impulse when Martin-Löf=-= [19]-=- defined his notion of randomness of infinite strings based on constructive measure theory. Especially the strong connections with the theory of randomness for finite objects made this notion very pop... |

329 | A theory of program size formally identical to information theory
- Chaitin
- 1975
(Show Context)
Citation Context ...many sets that are low for the Schnorr random sets. These all have hyperimmune-free degree, hence cannot be in ∆ 0 2. In this section we study lowness for an individual random set, namely Chaitin’=-=s Ω [3]. Following a-=- tradition of Chaitin, we denote by the symbol Ω not only the sets10 ANDRÉ NIES, FRANK STEPHAN AND SEBASTIAAN A. TERWIJN but also the real number � n∈Ω 2−n−1 represented by the set. Fixin... |

149 |
Zufälligkeit und Wahrscheinlichkeit
- Schnorr
- 1971
(Show Context)
Citation Context ...nite objects made this notion very popular, see e.g. [17], to name only one of the many references that the reader can consult for this. Another landmark in the theory of randomness is Schnorr’s boo=-=k [26]-=-, containing a thorough discussion (and criticism) of several of the randomness notions used in this paper, in particular A. Nies: Department of Computer Science, University of Auckland, 38 Princes St... |

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

78 | Lowness properties and randomness
- Nies
(Show Context)
Citation Context ...grees of sets that are low for classes of random sets. • (Kučera and Terwijn [15]) There is a nonrecursive r.e. set that is low for the Martin-Löf random sets. Every such set must be in ∆ 0 2 by=-= Nies [24]. ��-=-� (Nies [24]) A set is low for the recursively random sets if and only if it is recursive. • (Terwijn and Zambella [33]) There are uncountably many sets that are low for the Schnorr random sets. The... |

66 | Randomness and recursive enumerability
- Kučera, Slaman
(Show Context)
Citation Context ...1s extends a program p such that U(p) halts. The main reason for being interested in Ω is that Ω is a natural example for a left-r.e. random set and in a certain sense the only one: Kučera and Sl=-=aman [14] s-=-howed that all random left-r.e. sets are Ω-numbers, that is, represent the halting probability of some universal prefix-free machine.sRANDOMNESS, RELATIVIZATION, AND TURING DEGREES 3 At the beginnin... |

56 | Trivial reals
- Downey, Hirschfeldt, et al.
- 2003
(Show Context)
Citation Context ...cΩ(x + n) > ΨA (x), where cΩ(z) is the least s such that Ωs↾z = Ω↾z. So we have that x ∈ A ′ if and only if x is enumerated into A ′ within cΩ(x + n) many steps, hence A ′ ≤T =-=A ⊕ Ω. � Definition 3.3 ([9]). A is K-trivial if -=-K(X↾n) ≤ K(n) + O(1) for every n. Definition 3.4. An r.e. set W ⊆ N × {0, 1} ∗ is a Kraft-Chaitin set (KC set) if � 2 −r ≤ 1. 〈r,y〉∈W The pairs enumerated into W are called axioms... |

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 ...elativized Martin-Löf random sequences. We start off with some observations about the complexity of finite strings. The method used to prove the following inequality goes back to Solovay’s manuscri=-=pt [29], -=-and was further used in [7]. Proposition 2.1. For all strings x and y, C(xy) ≤ K(x) + C(y) + O(1). Proof. Recall that V is the universal machine for C and U is the universal prefixfree machine for K... |

47 |
Π 0 1 classes, and complete extensions of PA
- Measure
- 1984
(Show Context)
Citation Context .... 2, Proposition XI.2.10]. In particular, our “natural examples” for sets which are low for Ω do not build a minimal pair with every 1-generic set. • Above every set there is a 1-random set by=-= Kučera [12].-=- In particular, no 2-generic set builds a minimal pair with every 1-random set. 4. Separating randomness notions in Turing degrees In this section we show that the notions of Martin-Löf randomness, r... |

47 | Classical Recursion Theory (North-Holland - Odifreddi - 1989 |

46 | Degrees of Random Sets - Kautz - 1991 |

41 | The Unknowable - Chaitin - 1999 |

39 |
Algebras of sets binumerable in complete extensions of arithmetic, in Recursive Function Theory
- Scott
- 1962
(Show Context)
Citation Context ...or Ω. Since any 2-random set A is 1-random the equivalence follows. � Every PA-complete set A bounds a 1-random set B. (This is because there is a Π 0 1class of 1-random sets and by a result of Scott =-=[27]-=- PA-complete sets can compute an element in every Π 0 1-class.) If the PA-complete set has hyperimmune-free or ∆ 0 2 Turing degree, then B is not 2-random and thus not low for Ω. It follows that in th... |

30 |
Randomness and Complexity
- Wang
(Show Context)
Citation Context ...tion 4 we discuss the separation of the notions of Martin-Löf randomness, recursive randomness, and Schnorr randomness. It was known that all of these notions are different (see Schnorr [26] and Wang=-= [34]-=-). Here we indicate precisely what computational resources are needed to separate them: we show that the three notions can be separated in every high degree, and conversely that if a set separates any... |

23 |
Complexity oscillations in infinite binary sequences, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
- Martin-Löf
(Show Context)
Citation Context ...b)(∃ ∞ n) � C(X↾n) ≥ n − b � , where C is the plain Kolmogorov complexity. This notion was studied earlier in several equivalent forms by Loveland, Schnorr, Daley and others, see section=-= 2. MartinLöf [20] proved that th-=-ere are no sets X such that (∃b)(∀n) � C(X↾n) ≥ n − b � and he also showed that Kolmogorov randomness implies Martin-Löf randomness. We give a simple proof of this last fact in Proposit... |

23 |
Probabilities over Rich Languages
- Gaifman, Snir
- 1982
(Show Context)
Citation Context ...imple proof of this last fact in Proposition 2.4. We then compare Kolmogorov randomness with relativized Martin-Löf randomness. The first authors to study relativized randomness were Gaifman and Snir =-=[8]-=-. A set is n-random if it is Martin-Löf random relative to ∅ (n−1) . So it is 1-random if it is Martin-Löf random, 2-random if it is Martin-Löf random relative to ∅ ′ , etc. Ding, Downey and Yu [5] pr... |

22 | The axiomatization of randomness
- Lambalgen
- 1990
(Show Context)
Citation Context ...order to reach this conclusion directly. We next give a further characterization of 2-randomness. Theorem 3.10. A set A is 2-random if and only if A is 1-random and low for Ω. Proof. M. van Lambalge=-=n [16] show-=-ed that for any two sets A and B, A ⊕ B is Martin-Löf random if and only if B is Martin-Löf random and A is Martin-Löf random relative to B. Thus, for any 1-random set A it holds that A is 2-rand... |

17 | Recursively enumerable sets modulo iterated jumps and extensions of Arslanov’s completeness criterion - Jockusch, Lerman, et al. - 1989 |

13 | Computability and Measure
- Terwijn
- 1998
(Show Context)
Citation Context ...te that every Kolmogorov random set is Kolmogorov random with time bound g, for every recursive g. As noted above, a set A is Kolmogorov random if and only if (∃b)(∃ ∞ n) � C(X↾n|n) ≥ n−=-=b � . Terwijn [31, 32]-=- showed that a similar equivalence holds for time-bounded Kolmogorov complexity. The next theorem shows that 2-randomness is characterized by Kolmogorov randomness, as well as by its time-bounded vers... |

13 | Every 2-random real is Kolmogorov random
- Miller
- 2004
(Show Context)
Citation Context ...lmogorov randomness coincides with 2-randomness (Theorem 2.8). This had been conjectured by C. Calude. That 2-randomness implies Kolmogorov randomness was proven independently (and earlier) by Miller =-=[21]-=-. Note that Martin-Löf randomness was characterized by Schnorr in terms of the prefix-free Kolmogorov complexity K, whereas the above characterization is in terms of the plain Kolmogorov complexity C.... |

11 | Complexity and randomness
- Terwijn
- 2003
(Show Context)
Citation Context ...tforward characterization is that {q ∈ Q : q < A} is an r.e. set. We will now list very briefly some preliminaries from effective measure theory. More discussion on these notions can be found e.g. i=-=n [1, 32]. We -=-also refer there for complete references and suppress these in the following. A martingale is a functions4 ANDRÉ NIES, FRANK STEPHAN AND SEBASTIAAN A. TERWIJN M : {0, 1} ∗ → R + that satisfies fo... |

10 |
Ambos-Spies and Antonín Kučera. Randomness in computability theory. In Computability theory and its applications
- Klaus
- 1999
(Show Context)
Citation Context ...n a clear interest in the interplay of the various randomness notions and relative computability, or Turing reducibility. The reader can e.g. consult the recent survey paper by Ambos-Spies and Kučera=-= [1]-=-. In the present paper we prove some new results on randomness relating to both Turing reducibility and Kolmogorov complexity. The outline of the paper is as follows. In Section 2 we consider relativi... |

7 |
Terwijn and Domenico Zambella. Computational randomness and lowness
- Sebastiaan
(Show Context)
Citation Context ...is low for the Martin-Löf random sets. Every such set must be in ∆ 0 2 by Nies [24]. • (Nies [24]) A set is low for the recursively random sets if and only if it is recursive. • (Terwijn and Za=-=mbella [33]) -=-There are uncountably many sets that are low for the Schnorr random sets. These all have hyperimmune-free degree, hence cannot be in ∆ 0 2. In this section we study lowness for an individual random ... |

5 | The Kolmogorov complexity of random reals
- Ding, Downey, et al.
- 2004
(Show Context)
Citation Context ...relating to both Turing reducibility and Kolmogorov complexity. The outline of the paper is as follows. In Section 2 we consider relativized randomness and Kolmogorov complexity. Ding, Downey, and Yu =-=[7] call a set X Ko-=-lmogorov random if (∃b)(∃ ∞ n) � C(X↾n) ≥ n − b � , where C is the plain Kolmogorov complexity. This notion was studied earlier in several equivalent forms by Loveland, Schnorr, Daley ... |

5 |
private communication
- Griffiths
- 1959
(Show Context)
Citation Context ...by left-r.e. sets. if the high degree happens to be an r.e. degree. That Schnorr randomness and recursive randomness can be separated by left-r.e. set was independently proven by Downey and Griffiths =-=[8]. Recall t-=-hat a set A is high if and only if A ′ ≥T ∅ ′′ . Martin [28, Theorem XI.1.3] showed that a set A is high if and only if there is an A-recursive function which dominates every recursive funct... |

5 | and Nenad Mihailović, On the construction of effective random sets - Merkle |

4 |
On relative randomness, Annals of Pure and Applied Logic 63
- Kucera
- 1993
(Show Context)
Citation Context ...Löf random sets by [24]. However, this is not true for sets in general, since all 2-random sets are low for Ω, so this class has in fact measure 1! The following proof is similar to the one of Kuč=-=era [13] that all sets-=- which are low for Martin-Löf randomness are in the class GL1. Theorem 3.2. Let A be low for Ω. Then A is generalized low: A ′ ≤T A ⊕ ∅ ′ . Proof. Let ψA be an A-recursive function with ... |

3 |
On minimal program complexity measures
- Loveland
- 1969
(Show Context)
Citation Context ...|z| + c − d. � Definition 2.3. (Ding, Downey, and Yu [7]) A set X is Kolmogorov random if (∃b)(∃ ∞ n) � C(X↾n) ≥ n − b � . This notion was studied earlier in several forms, see Sch=-=norr [27], Loveland [18], Daley [5]. E.g. -=-Daley [5] proved that a set A is Kolmogorov random if and only if (∃b)(∃ ∞ n) � C(X↾n|n) ≥ n − b � , where C(σ|n) is the complexity of σ given n. We now give a simple proof of [17, T... |

2 | K.: Randomness Relative to Cantor Expansions
- Calude, Staiger, et al.
(Show Context)
Citation Context ...IAAN A. TERWIJN Schnorr [26, 27] proved that the converse direction of Proposition 2.4 does not hold. An argument similar to the one in Proposition 2.1 can be used to answer a question of Calude e.a. =-=[2] -=-for each infinite recursive R, if Z has high prefix complexity on all initial segments whose length is in R, then Z is Martin-Löf random. This was independently proved by Lance Fortnow. Proposition 2... |

2 |
and Antonín Kučera, Remarks on 1-genericity, semigenericity and related concepts, Commentationes Mathematicae Universitatis Carolinae 28
- Demuth
- 1987
(Show Context)
Citation Context ...ence has hyperimmune degree. The question remains whether Corollary 2.16 can be strengthened, namely, Question 3.13. Does every set that is low for Ω have hyperimmune Turing degree? Demuth and Kuče=-=ra [6]-=- proved that no 1-random set is below a 1-generic set, which implies that no 2-random set is below a 2-generic set. The next theorem shows that the conversely no 2-generic set is below a 2-random set.... |

2 |
Kolmogorov random reals are 2-random
- Miller
(Show Context)
Citation Context ...). This had been conjectured by C. Calude (personal communication to André Nies, Auckland, June 2003). That 2-randomness implies Kolmogorov randomness was proved independently (and earlier) by Miller=-= [22].-=- Note that Martin-Löf randomness was characterized by Schnorr in terms of the prefix-free Kolmogorov complexity K, whereas the above characterization is in terms of the plain Kolmogorov complexity C.... |

1 |
Complexity and randomness
- Daley
- 1971
(Show Context)
Citation Context ... � Definition 2.3. (Ding, Downey, and Yu [7]) A set X is Kolmogorov random if (∃b)(∃ ∞ n) � C(X↾n) ≥ n − b � . This notion was studied earlier in several forms, see Schnorr [27], Lov=-=eland [18], Daley [5]. E.g. Daley [5] p-=-roved that a set A is Kolmogorov random if and only if (∃b)(∃ ∞ n) � C(X↾n|n) ≥ n − b � , where C(σ|n) is the complexity of σ given n. We now give a simple proof of [17, Theorem 2.14... |

1 | Each Low(CR) set is computable - Nies - 2003 |

1 |
Martin-Löf random and PA-complete sets, Forschungsberichte
- Stephan
- 2002
(Show Context)
Citation Context ... A degree contains a set which is Kurtz-random but not Schnorr random if and only if the degree is hyperimmune. On the hyperimmune-free degrees, all considered notions of randomness coincide. Stephan =-=[30] inves-=-tigated the connection between PA-completeness and Martin-Löf randomness. He showed that no PA-complete set A �≥T K is in the Turing degrees20 ANDRÉ NIES, FRANK STEPHAN AND SEBASTIAAN A. TERWIJN... |

1 |
André Nies and Frank Stephan, Trivial reals
- Downey, Hirschfeldt
- 2003
(Show Context)
Citation Context ...cΩ(x + n) > ΨA (x), where cΩ(z) is the least s such that Ωs↾z = Ω↾z. So we have that x ∈ A ′ if and only if x is enumerated into A ′ within cΩ(x + n) many steps, hence A ′ ≤T A ⊕ Ω. � Definition 3.3 (=-=[7]-=-). A is K-trivial if K(X↾n) ≤ K(n) + O(1) for every n. Definition 3.4. An r.e. set W ⊆ N × {0, 1} ∗ is a Kraft-Chaitin set (KC set) if � 2 −r ≤ 1. 〈r,y〉∈WsRANDOMNESS, RELATIVIZATION AND TURING DEGREES... |