## On initial segment complexity and degrees of randomness

Venue: | Trans. Amer. Math. Soc |

Citations: | 32 - 6 self |

### BibTeX

@ARTICLE{Miller_oninitial,

author = {Joseph S. Miller and Liang Yu},

title = {On initial segment complexity and degrees of randomness},

journal = {Trans. Amer. Math. Soc},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define X ≤K Y to mean that (∀n) K(X ↾ n) ≤ K(Y ↾ n) +O(1). The equivalence classes under this relation are the K-degrees. We prove that if X ⊕ Y is 1-random, then X and Y have no upper bound in the K-degrees (hence, no join). We also prove that n-randomness is closed upward in the K-degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the vL-degrees. Unlike the K-degrees, many basic properties of the vL-degrees are easy to prove. We show that X ≤K Y implies X ≤vL Y, so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for ≤C, the analogue of ≤K for plain Kolmogorov complexity. Two other interesting results are included. First, we prove that for any Z ∈ 2ω, a 1-random real computable from a 1-Z-random real is automatically 1-Z-random. Second, we give a plain Kolmogorov complexity characterization of 1-randomness. This characterization is related to our proof that X ≤C Y implies X ≤vL Y. 1.

### Citations

1680 | An introduction to Kolmogorov complexity and its applications
- Li, Vitányi
- 1997
(Show Context)
Citation Context ... the 1-random reals remains entirely open. 2. Preliminaries We begin with a brief review of effective randomness and Kolmogorov complexity. A more complete introduction can be found in Li and Vitányi =-=[18]-=- or the upcoming monograph of Downey and Hirschfeldt [5]. We assume that the reader is familiar with the basics of computability theory and measure theory. Soare [27] and Oxtoby [25] are good resource... |

521 |
Three approaches to the quantitive definition of information
- Kolmogorov
- 1965
(Show Context)
Citation Context ...ump operator: Theorem 2.1 (Kurtz [14]). For n ∈ ω and Z ∈ 2 ω , n-Z-randomness is equivalent to 1-Z (n−1) -randomness. We write n-random for n-∅-random, or equivalently, 1-∅ (n−1) -random. Kolmogorov =-=[11]-=- and Solomonoff [28] defined the complexity of a finite string to be the length of its shortest description. Formally, we use a partial computables4 JOSEPH S. MILLER AND LIANG YU function M :2 <ω → 2 ... |

472 |
Recursively Enumerable Sets and Degrees
- Soare
- 1987
(Show Context)
Citation Context ...on can be found in Li and Vitányi [18] or the upcoming monograph of Downey and Hirschfeldt [5]. We assume that the reader is familiar with the basics of computability theory and measure theory. Soare =-=[27]-=- and Oxtoby [25] are good resources for these subjects. Martin-Löf [19] introduced the most successful notion of effective randomness for real numbers. A Martin-Löf test is a uniform sequence {Gn}n∈ω ... |

404 |
A formal theory of inductive inference
- Solomonoff
- 1964
(Show Context)
Citation Context ...m 2.1 (Kurtz [14]). For n ∈ ω and Z ∈ 2 ω , n-Z-randomness is equivalent to 1-Z (n−1) -randomness. We write n-random for n-∅-random, or equivalently, 1-∅ (n−1) -random. Kolmogorov [11] and Solomonoff =-=[28]-=- defined the complexity of a finite string to be the length of its shortest description. Formally, we use a partial computables4 JOSEPH S. MILLER AND LIANG YU function M :2 <ω → 2 <ω to “decode” descr... |

332 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ...plexity of initial segments of 1-random (i.e., Martin-Löf random) reals, with respect to either plain or prefix-free Kolmogorov complexity (denoted by C and K, respectively). These include Martin-Löf =-=[19, 20]-=-, Chaitin [1, 3], Solovay [29] and van Lambalgen [30]. Our approach is different. While previous work focuses on describing the behavior of the initial segment complexity of a real number, we instead ... |

226 | On the length of the programs for computing finite binary sequences: Statistical considerations
- Chaitin
- 1969
(Show Context)
Citation Context ... segments of 1-random (i.e., Martin-Löf random) reals, with respect to either plain or prefix-free Kolmogorov complexity (denoted by C and K, respectively). These include Martin-Löf [19, 20], Chaitin =-=[1, 3]-=-, Solovay [29] and van Lambalgen [30]. Our approach is different. While previous work focuses on describing the behavior of the initial segment complexity of a real number, we instead focus on interpr... |

155 |
Algorithmic randomness and complexity
- Downey, Hirschfeldt
- 2010
(Show Context)
Citation Context ...es We begin with a brief review of effective randomness and Kolmogorov complexity. A more complete introduction can be found in Li and Vitányi [18] or the upcoming monograph of Downey and Hirschfeldt =-=[5]-=-. We assume that the reader is familiar with the basics of computability theory and measure theory. Soare [27] and Oxtoby [25] are good resources for these subjects. Martin-Löf [19] introduced the mos... |

95 |
of information conservation (nongrowth) and aspects of the foundation of probability theory, Probl
- Levin, Laws
(Show Context)
Citation Context ... :2 <ω → 2 <ω is any other partial computable function, then (∀σ ∈ 2 <ω ) CV (σ) ≤ CM(σ)+O(1). We call V the universal machine and call C(σ)=CV (σ)theplain (Kolmogorov) complexity of σ ∈ 2 <ω . Levin =-=[16]-=- and Chaitin [2] introduced a modified form of Kolmogorov complexity that has natural connections to the Martin-Löf definition of randomness. For finite binary strings σ, τ ∈ 2 <ω ,wewriteσ≺τto mean t... |

73 |
Degrees of Unsolvability
- Lerman
- 1983
(Show Context)
Citation Context ...k/∈ F(j) andsoYj⊕Xkis 1-random. But Yk ⊕ Xk is not 1-random since k ∈ F(k). So Yj �vL Yk. � Note that part (v) of the corollary implies that the Σ 0 1 theory of (2 ω , ≤vL) is decidable, as in Lerman =-=[15]-=-. We finish the section by considering the vL-degrees of specific reals. Chaitin [2] proposed the halting probability Ω of the universal prefix-free machine U as a natural example of a 1-random real. ... |

62 |
Degrees of Unsolvability
- Sacks
- 1963
(Show Context)
Citation Context ...A e = Z}. Because Z is a ∆0 2 set, it is the limit of a computable sequence {Zs}s∈ω of finite sets. Thus G = {A ∈ 2 ω :(∀n)(∀t)(∃s ≥ t) ϕ A e,s ↾ n = Zs ↾ n}, so G is a Π 0 2 class. A result of Sacks =-=[26]-=- states that µ{A ∈ 2ω : A ≥T Z} =0becauseZ is not computable. Hence, µG = 0. Kurtz [14] observed that no 2-random real is contained in a measure zero Π 0 2 class, so ϕ � Z e �= Z. But the choice of e ... |

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 ...andom (i.e., Martin-Löf random) reals, with respect to either plain or prefix-free Kolmogorov complexity (denoted by C and K, respectively). These include Martin-Löf [19, 20], Chaitin [1, 3], Solovay =-=[29]-=- and van Lambalgen [30]. Our approach is different. While previous work focuses on describing the behavior of the initial segment complexity of a real number, we instead focus on interpreting that beh... |

50 |
Randomness and genericity in the degrees of unsolvability
- Kurtz
- 1981
(Show Context)
Citation Context ...∈ ω and oracle Z ∈ 2 ω , define n-Z-randomness by replacing the Σ 0 1 classes with Σ 0 n[Z] classes in Martin-Löf’s definition. The two parameters are related by the jump operator: Theorem 2.1 (Kurtz =-=[14]-=-). For n ∈ ω and Z ∈ 2 ω , n-Z-randomness is equivalent to 1-Z (n−1) -randomness. We write n-random for n-∅-random, or equivalently, 1-∅ (n−1) -random. Kolmogorov [11] and Solomonoff [28] defined the ... |

38 | relativization and Turing degrees
- Nies, Stephan, et al.
- 2005
(Show Context)
Citation Context ...ng the Computational Aspects of Infinity program in 2005. 1 c○2008 American Mathematical Society Reverts to public domain 28 years from publications2 JOSEPH S. MILLER AND LIANG YU Stephan and Terwijn =-=[24]-=-). These results raise obvious questions: can 1-randomness be characterized in terms of initial segment C-complexity—a long elusive goal— or 2-randomness in terms of initial segment K-complexity? We w... |

22 | The axiomatization of randomness
- Lambalgen
- 1990
(Show Context)
Citation Context ... propose a different approach—one based on the global behavior of real numbers, rather than their local structure. Our definition will be motivated by the following result. Theorem 3.1 (van Lambalgen =-=[31]-=-). If X,Y ∈ 2 ω ,thenX ⊕ Y is 1-random iff X is 1-random and Y is 1-X-random. Nies [23] defined X ≥LR Y to mean (∀Z ∈ 2 ω )[Z is 1-X-random =⇒ Z is 1-Y -random]. By Theorem 3.1, if X and Y are both 1-... |

21 | Relativizing Chaitin’s halting probability
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...rst, it is worth asking if the vL-degree of ΩZ is independent of the choice of U Z . It is not. ThevL-degree of Ω is well defined, but this is not even true for Ω∅′ . It can be proved (using ideas in =-=[8]-=-) that if Z ∈ 2ω has 1-random degree, then different choices of U Z can give values of ΩZ that are 1-random relative to each other. Therefore, by Theorem 3.4 (iii), different versions of ΩZ have diffe... |

18 |
Π 0 1 -classes and complete extensions of PA, in: Recursion Theory Week (Oberwolfach
- Kučera, Measure
- 1984
(Show Context)
Citation Context ...ion to Theorem 3.1, we will use the following two facts. The first is closely related to a result proved independently by Gács [10]: every real is computable from a 1-random real. Theorem 3.2 (Kučera =-=[12]-=-). There is a 1-random real in every Turing degree ≥ 0 ′ . Theorem 3.3 (Kučera and Terwijn [13]). For every X ∈ 2 ω ,thereisaW �T X so that every 1-X-random real is 1-X ⊕ W-random. Theorem 3.4 (Basic ... |

16 |
Lambalgen, Random Sequences
- van
- 1987
(Show Context)
Citation Context ... random) reals, with respect to either plain or prefix-free Kolmogorov complexity (denoted by C and K, respectively). These include Martin-Löf [19, 20], Chaitin [1, 3], Solovay [29] and van Lambalgen =-=[30]-=-. Our approach is different. While previous work focuses on describing the behavior of the initial segment complexity of a real number, we instead focus on interpreting that behavior. We argue that th... |

14 |
Exact expressions for some randomness tests
- Gács
- 1980
(Show Context)
Citation Context ... n − G(n) − O(1). This theorem has precursors in the literature. Results closely related to (i) implies (iii) were proved by Martin-Löf [20]. Condition (ii) is similar to a characterization that Gács =-=[9]-=- gave of 1-randomness in terms of length conditional Kolmogorov complexity. He proved that X ∈ 2 ω is 1-random iff (∀n) C(X ↾ n | n) ≥ n − K(n) − O(1), where C(X ↾ n | n) denotes the Kolmogorov comple... |

13 | Every 2-random real is Kolmogorov random
- Miller
- 2004
(Show Context)
Citation Context ...out X. An obvious example is Schnorr’s theorem that X ∈ 2ω is 1-random iff (∀n) K(X ↾ n) ≥ n − O(1). A more recent example is the fact that X ∈ 2ω is 2-random iff (∃∞n) C(X ↾n) ≥ n − O(1) (see Miller =-=[21]-=-; Nies, Received by the editors May 30, 2006. 2000 Mathematics Subject Classification. Primary 68Q30, 03D30, 03D28. The first author was partially supported by the Marsden Fund of New Zealand and by a... |

10 |
Oscillation in the initial segment complexity of random reals
- Miller, Yu
- 2010
(Show Context)
Citation Context ...r which µ{Y ∈ 2ω : X ≤K Y } =0. Several results in this paper produce incomparable 1-random K-degrees, but none prove the existence of comparable 1-random K-degrees. That is done in a companion paper =-=[22]-=-, where we prove that for any 1-random Y ∈ 2ω ,thereisa 1-random X ∈ 2ω such that X<K Y (in fact, limn→∞ K(Y ↾n) − K(X ↾n)=∞). Another problem that is not addressed in this paper is whether the C-degr... |

5 | The Kolmogorov complexity of random reals
- Ding, Downey, et al.
- 2004
(Show Context)
Citation Context ...s. It follows from work of Solovay [29] that Chaitin’s halting probability Ω has a different K-degree than any arithmetically random real. Hence, there are at least two K-degrees. Yu, Ding and Downey =-=[33]-=- proved that µ{X ∈ 2ω : X ≤K Y } = 0, for any Y ∈ 2ω .Fromthis, they conclude that there are uncountably many 1-random K-degrees (an explicit construction of an antichain of size 2ℵ0 is given in [32])... |

1 |
theorems for random reals, Adv
- Incompleteness
- 1987
(Show Context)
Citation Context ... segments of 1-random (i.e., Martin-Löf random) reals, with respect to either plain or prefix-free Kolmogorov complexity (denoted by C and K, respectively). These include Martin-Löf [19, 20], Chaitin =-=[1, 3]-=-, Solovay [29] and van Lambalgen [30]. Our approach is different. While previous work focuses on describing the behavior of the initial segment complexity of a real number, we instead focus on interpr... |

1 | Every 1-generic computes a properly 1-generic
- Csima, Downey, et al.
(Show Context)
Citation Context ...generic and Y ≤T X is 2-generic, then Y is n-generic. 2 Thefactthatitisnotsufficientto assume that Y is 1-generic is the subject of a recent paper of Csima, Downey, Greenberg, Hirschfeldt, and Miller =-=[4]-=-. Note that Nies, Stephan and Terwijn [24, Theorem 3.10] showed that a set A is 2-random iff A is 1-random and low for Ω (i.e., Ω is 1-A-random). From this result—a consequence of van Lambalgen’s theo... |

1 |
Randomness and reducibility (extended abstract), Mathematical foundations of computer science
- Downey, Hirschfeldt, et al.
- 2001
(Show Context)
Citation Context ... of initial segment K-complexity? We will give positive answers to both questions. Many of our results will be stated in terms of the K-degrees, whichwereintroduced by Downey, Hirschfeldt and LaForte =-=[6, 7]-=-. Write X ≤K Y if Y has higher initial segment prefix-free complexity than X (up to a constant). Formally, (∀n) K(X ↾ n) ≤ K(Y ↾ n)+O(1). The induced partial order is called the Kdegrees. Define the C... |

1 |
See [6] for an extended abstract
- Randomness, reducibility
(Show Context)
Citation Context ... of initial segment K-complexity? We will give positive answers to both questions. Many of our results will be stated in terms of the K-degrees, whichwereintroduced by Downey, Hirschfeldt and LaForte =-=[6, 7]-=-. Write X ≤K Y if Y has higher initial segment prefix-free complexity than X (up to a constant). Formally, (∀n) K(X ↾ n) ≤ K(Y ↾ n)+O(1). The induced partial order is called the Kdegrees. Define the C... |

1 |
sequence is reducible to a random one
- Every
- 1986
(Show Context)
Citation Context ...operties of the vL-degrees are easy to prove from known results. In addition to Theorem 3.1, we will use the following two facts. The first is closely related to a result proved independently by Gács =-=[10]-=-: every real is computable from a 1-random real. Theorem 3.2 (Kučera [12]). There is a 1-random real in every Turing degree ≥ 0 ′ . Theorem 3.3 (Kučera and Terwijn [13]). For every X ∈ 2 ω ,thereisaW ... |

1 |
Decheng Ding, There are 2 ℵ0 many H-degrees in the random reals
- Yu
(Show Context)
Citation Context ... [33] proved that µ{X ∈ 2ω : X ≤K Y } = 0, for any Y ∈ 2ω .Fromthis, they conclude that there are uncountably many 1-random K-degrees (an explicit construction of an antichain of size 2ℵ0 is given in =-=[32]-=-). An early goal of the present research was to calculate the measure of {Y ∈ 2ω : X ≤K Y }. Itmustbenoted that this measure depends on the choice of X ∈ 2ω .IfXis computable, then it is K-below every... |

1 |
André Nies,Relativizing Chaitin’s halting probability
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...st, it is worth asking if the vL-degree of ΩZ is independent of the choice of U Z . It is not. The vL-degree of Ω is well defined, but this is not even true for Ω∅′ . It can be proved (using ideas in =-=[8]-=-) that if Z ∈ 2ω has 1-random degree, then different choices of U Z can give values of ΩZ that are 1-random relative to each other. Therefore, by Theorem 3.4 (iii), different versions of ΩZ have diffe... |