## Degrees of random sets (1991)

Citations: | 46 - 4 self |

### BibTeX

@TECHREPORT{Kautz91degreesof,

author = {Steven M. Kautz},

title = {Degrees of random sets},

institution = {},

year = {1991}

}

### Years of Citing Articles

### OpenURL

### Abstract

An explicit recursion-theoretic definition of a random sequence or random set of natural numbers was given by Martin-Löf in 1966. Other approaches leading to the notions of n-randomness and weak n-randomness have been presented by Solovay, Chaitin, and Kurtz. We investigate the properties of n-random and weakly n-random sequences with an emphasis on the structure of their Turing degrees. After an introduction and summary, in Chapter II we present several equivalent definitions of n-randomness and weak n-randomness including a new definition in terms of a forcing relation analogous to the characterization of n-generic sequences in terms of Cohen forcing. We also prove that, as conjectured by Kurtz, weak nrandomness is indeed strictly weaker than n-randomness. Chapter III is concerned with intrinsic properties of n-random sequences. The main results are that an (n + 1)-random sequence A satisfies the condition A (n) ≡T A⊕0 (n) (strengthening a result due originally to Sacks) and that n-random sequences satisfy a number of strong independence properties, e.g., if A ⊕ B is n-random then A is n-random relative to B. It follows that any countable distributive lattice can be embedded

### Citations

2054 | Probability theory and its applications (Vol - Feller - 1950 |

838 |
Theory of Recursive Functions and Effective Computability
- ROGERS
- 1967
(Show Context)
Citation Context ...or Π 0 n -classes. Since notions of computation can be expressed in a simple way in L ∗ we can, for example, represent a Σ 0 n -class in the form if n is odd, and in the form if n is even. See Rogers =-=[30]-=- for details. {A : (∃x1)(∀x2) . . . (∃xn)[ϕ A e (x1, . . . , xn)↓]} {A : (∃x1)(∀x2) . . . (∀xn)[ϕ A e (x1, . . . , xn)↑]} The definitions of arithmetical classes can all be relativized; e.g., a ΣC 1 -... |

530 |
Three approaches to the quantitative definition of information
- Kolmogorov
- 1965
(Show Context)
Citation Context ...previous section. Kolmogorov and Chaitin Complexity What we have just described is, in effect, a complexity measure based on “program size” originating with the following definition due to Kolmogorov =-=[14]-=-. We call a 19 = p = p,spartial recursive function ψ : 2 <ω −→ 2 <ω universal if it interprets its input as a pair 〈e, σ〉 in some canonical way and simulates ϕe(σ). Definition II.2.2 Let ψ : 2 <ω → 2 ... |

477 |
Recursively Enumerable Sets and Degrees
- Soare
- 1987
(Show Context)
Citation Context ...egrees may not be exactly the same for all atomic computable measures. I.3 Notation and Conventions Most of our notation is standard; any undefined terminology can be found in Odifreddi [26] or Soare =-=[32]-=-. Lowercase ω denotes the natural numbers; the central objects of our attention are elements of the continuum 2 ω or {0, 1} ω . We refer to an element of 2 ω both as an infinite binary sequence and as... |

335 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ...C 1 -approximation; moreover, given a Σ C 1 -approximation {Ui}i∈ω with index e, we have Ui = Ext(W C h(g(e,i),i) ), so every ΣC 1 -approximation does appear on the list. ✷ Theorem II.1.7 (Martin-Löf =-=[22]-=-) For any C ∈ 2ω and any n ≥ 1 there exists a universal ΣC n -approximation. That is, there is a recursive sequence of ΣC(n−1) 1 -classes {Ui}i∈ω, with µ(Ui) ≤ 2−i , such that every ΣC n -approximable... |

323 | Algorithmic Information Theory
- Chaitin
- 1987
(Show Context)
Citation Context ...A is n-random with respect to µ if for every recursive sequence of Σ 0(n−1) 1 -classes {Si}i∈ω with µ(Si) ≤ 2 −i , A �∈ � i Si. Each measure µ induces a natural correspondence between real numbers in =-=[0, 1]-=- and sequences in 2 ω . For a ∈ [0, 1] the representation of a with respect to µ is denoted seq µ(a); thus seq λ(a) is the usual binary expansion of a. Loosely speaking, the correspondence between [0,... |

312 |
Classical Recursion Theory I
- Odifreddi
- 1989
(Show Context)
Citation Context ...the 1-random degrees may not be exactly the same for all atomic computable measures. I.3 Notation and Conventions Most of our notation is standard; any undefined terminology can be found in Odifreddi =-=[26]-=- or Soare [32]. Lowercase ω denotes the natural numbers; the central objects of our attention are elements of the continuum 2 ω or {0, 1} ω . We refer to an element of 2 ω both as an infinite binary s... |

173 | Almost everywhere high nonuniform complexity
- Lutz
- 1992
(Show Context)
Citation Context ...ithm”, and one useful feature of the recursion-theoretic definitions of Martin-Löf and Kolmogorov is the ease with which they specialize to resource-bounded computations (Hartmanis [8], Ko [13], Lutz =-=[20]-=-). In the setting of computer science, computational definitions of randomness seem to capture all of its relevant properties (see Chaitin [2], for example) and are usually much stronger than necessar... |

127 |
Π0 1-classes an degrees of theories
- Jockusch, Soare
- 1972
(Show Context)
Citation Context ...ning a 1-random set also contains a fixed-point-free function; see Ku˘cera [17, 18]. There are also 1-random degrees strictly below 0 ′ , as the next theorem shows. Theorem IV.1.2 (Jockusch and Soare =-=[12]-=-) Every nonempty Π 0 1 -class contains a member of low degree, i.e., an A >T 0 such that A ′ ≡T 0 ′ . The following definition will be needed several times. 52sDefinition IV.1.3 A set A is hyperimmune... |

93 |
A model of set-theory in which every set of reals is Lebesgue measurable
- Solovay
(Show Context)
Citation Context .... It is known that for both set theory and for arithmetic there are kinds of requirements which can’t be forced using only finite information, e.g., producing a set of minimal degree. In 1970 Solovay =-=[33]-=- introduced a method of forcing with closed sets of positive measure to produce a model of set theory in which all sets of reals are Lebesgue measurable; it is essentially this method, specialized to ... |

79 |
Measure and Category
- Oxtoby
- 1980
(Show Context)
Citation Context ... one. For any B, the section SB is a Σ B n -class, so the class S ∗ = {B : SB has measure one} = {B : (∀ɛ > 0)[µ(SB) > 1 − ɛ]} is a Π 0 n+1 -class, and has measure one by Fubini’s theorem (see Oxtoby =-=[28]-=-). Therefore S ∗ is an intersection of Σ 0 n -classes, all of which have measure one and therefore contain every weakly n-random set. If S is a Π 0 n+1 -class, then the class S ∗ defined as above is s... |

74 | Degrees of unsolvability - Lerman - 1983 |

62 |
The independence of the continuum hypothesis
- Cohen
- 1963
(Show Context)
Citation Context ... weakly (n + 1)-generic ⇒ n-generic ⇒ weakly n-generic, which justifies the use of the word “generic” in Definition II.4.2(i). The method of forcing with finite initial segments was invented by Cohen =-=[4]-=- in 1963 to construct a model of set theory in which the continuum hypothesis is false. The method was applied to arithmetic in 1965 by Feferman [5], and the original constructions of Kleene and Post ... |

62 |
Degrees of Unsolvability
- Sacks
- 1966
(Show Context)
Citation Context ...III.3.14 that for any 2-random A ⊕ B, A and B form a minimal pair. Most of the known facts are summarized in Theorem IV.2.4: Theorem IV.2.4 (i) The class {A : A is not minimal} has measure one (Sacks =-=[31]-=-), and includes every 1-random set. (ii) The class {A⊕B : A, B form a minimal pair} has measure one (Stillwell [34]); it includes every 2-random set but not every 1-random set. (iii) For each n, the c... |

60 | Random Sequences
- Lambalgen
- 1987
(Show Context)
Citation Context ...r example), which we closely follow in spirit, but can also be traced back to early measure-theoretic results of Spector and Sacks (see Theorem II.5.2). We will also cite Kurtz [15] and van Lambalgen =-=[36]-=- frequently. Whether the nature of randomness can actually be characterized in terms of computation is at best a contentious question. Computation certainly has a valid descriptive role in the study o... |

54 |
On the concept of a random sequence
- Church
- 1940
(Show Context)
Citation Context ...we invoke Church’s thesis to conclude that intuitively what we mean by a “betting strategy” is an algorithm ϕ : 2 <ω → {0, 1}. This yeilds precisely the definition of randomness proposed by Church in =-=[3]-=-. (We emphasize, however, that invoking Church’s thesis is by no means the only way, or necessarily the correct way, to resolve the controversy. See van Lambalgen’s [36] for a much deeper analysis of ... |

51 |
Randomness and genericity in the degrees of unsolvability
- Kurtz
- 1981
(Show Context)
Citation Context ...andomness (see [17], for example), which we closely follow in spirit, but can also be traced back to early measure-theoretic results of Spector and Sacks (see Theorem II.5.2). We will also cite Kurtz =-=[15]-=- and van Lambalgen [36] frequently. Whether the nature of randomness can actually be characterized in terms of computation is at best a contentious question. Computation certainly has a valid descript... |

47 |
Π 0 1 classes, and complete extensions of PA
- Measure
- 1984
(Show Context)
Citation Context ...seful tool, analogous to the well-established notion of genericity, for understanding the structure of the Turing degrees. The same point of view can be seen in the work of Ku˘cera on randomness (see =-=[17]-=-, for example), which we closely follow in spirit, but can also be traced back to early measure-theoretic results of Spector and Sacks (see Theorem II.5.2). We will also cite Kurtz [15] and van Lambal... |

45 |
Generalized Kolmogorov complexity and the structure of feasible computations
- Hartmanis
- 1983
(Show Context)
Citation Context ...by a feasible algorithm”, and one useful feature of the recursion-theoretic definitions of Martin-Löf and Kolmogorov is the ease with which they specialize to resource-bounded computations (Hartmanis =-=[8]-=-, Ko [13], Lutz [20]). In the setting of computer science, computational definitions of randomness seem to capture all of its relevant properties (see Chaitin [2], for example) and are usually much st... |

26 |
Degrees of generic sets
- Jockusch
- 1979
(Show Context)
Citation Context ...eting all the dense requirements φ. Such sets are called generic; a generic set is “typical” in that it has every property that can be produced by a finite extension argument. It is shown in Jockusch =-=[9]-=- that if the sentence φ is Σn, the class Ext({σ : σ ||− φ}) is a Σ 0 n -class; this suggests the following definition, due to Kurtz: Definition II.4.2 Let A ∈ 2 ω . (i) A is generic if A is a member o... |

25 |
On the notion of infinite pseudorandom sequences, Theoretical Computer Science 48
- Ko
- 1986
(Show Context)
Citation Context ...sible algorithm”, and one useful feature of the recursion-theoretic definitions of Martin-Löf and Kolmogorov is the ease with which they specialize to resource-bounded computations (Hartmanis [8], Ko =-=[13]-=-, Lutz [20]). In the setting of computer science, computational definitions of randomness seem to capture all of its relevant properties (see Chaitin [2], for example) and are usually much stronger th... |

22 | The axiomatization of randomness - Lambalgen - 1990 |

18 |
An alternative, priority-free solution to Post’s Problem, in: Lecture Notes in Computer Science 233, 493–500
- Kučera
- 1986
(Show Context)
Citation Context ...andom set is never minimal, since by Theorem III.1.4 the columns are recursively independent. (ii) By Corollary III.3.12, every 2-random set is the join of a minimal pair. On the other hand, Ku˘cera (=-=[16]-=-) has shown that if A and B are 1-random sets <T 0 ′ , then there is a nonzero r.e. set C below A and B. By the Low Basis Theorem (Theorem IV.1.2) there is a 1-random set A ⊕ B <T 0 ′ , so A and B are... |

16 |
The degree of hyperimmune sets
- Miller, Martin
- 1968
(Show Context)
Citation Context ...contains a hyperimmune set. A degree is hyperimmunefree if it contains no hyperimmune sets. We will also need the characterization provided by the following theorem. Theorem IV.1.4 (Miller and Martin =-=[24]-=-) A degree a contains a hyperimmune set iff there is a function f recursive in a which is not dominated by any recursive function. The next result implies, then, that there are 1-random sets of hyperi... |

15 |
On the notion of randomness
- Martin-Löf
- 1968
(Show Context)
Citation Context ...niformly find, for each i and j, a ΠC(n−2) 1 -class Vi,j ⊆ Ti with µ(Ti) − µ(Vi,j) ≤ 2−j . Let V = � Vi,j. Clearly V ⊆ S and µ(V) = µ(S). The proof for (ii) is similar. ✷ i,j 1 Martin-Löf suggests in =-=[23]-=- that the “right” choice is to consider only the hyperarithmetical classes of measure one. 21sTheorem II.3.4 Let A, C ∈ 2ω and n ≥ 2. A is C-weakly n-random ⇐⇒ A is a member of every ΣC(n−2) 2 -class ... |

11 |
Some applications of the notions of forcing and generic sets
- Feferman
- 1965
(Show Context)
Citation Context ... with finite initial segments was invented by Cohen [4] in 1963 to construct a model of set theory in which the continuum hypothesis is false. The method was applied to arithmetic in 1965 by Feferman =-=[5]-=-, and the original constructions of Kleene and Post were recast as forcing arguments. It is known that for both set theory and for arithmetic there are kinds of requirements which can’t be forced usin... |

11 |
Degrees of members of Π0 1 classes
- Soare
- 1972
(Show Context)
Citation Context ..., to conclude that there are 1-random sets with property P . (The class of sets with property P is called a basis for Π 0 1 -classes.) Some known facts include the following. Theorem IV.1.1 (Jockusch =-=[11]-=-) Every nonempty Π 0 1 -class contains a member of r.e. degree. Theorem IV.1.1 can be relativized to show that there are n-random sets of Σ 0 n -degree, and hence recursive in 0 (n) (but not, however,... |

8 | A note on monte carlo primality tests and algorithmic information theory - Chaitin, Schwartz - 1978 |

8 | Automorphism bases for degrees of unsolvability - Jockusch, Posner - 1981 |

8 |
Forcing and reducibilities, The
- Odifreddi
- 1983
(Show Context)
Citation Context ...f genericity with respect to this forcing relation which exactly coincides with weak n-randomness. The presentation follows the uniform treatment of Cohen forcing and Sacks forcing given in Odifreddi =-=[27]-=-. Let IPn denote the collection of all Π 0 n -classes of positive measure, ordered by inclusion. Definition II.4.4 (i) Let T ⊆ 2 ω . We say T ||n− φ if T ∈ IPn and for all A ∈ T , A |= φ. (ii) For A ∈... |

6 |
Decidability of the “almost all” theory of degrees
- Stillwell
- 1972
(Show Context)
Citation Context ...der to talk about structural properties of random degrees in relation to one another. It is known that the first-order theory of D is as complicated as second order arithmetic. By contrast, Stillwell =-=[34]-=- has shown that the “a.e. theory” of D, that is, the theory of D in a language containing only quantifiers of the form “for all except a set of measure 0”, is decidable. This result is tantalizing in ... |

5 |
Measure, category, and degrees of unsolvability. Unpublished manuscript
- Martin
- 1960
(Show Context)
Citation Context ...: A (n−1) ≡T A ⊕ 0 (n−1) } has measure one (Sacks, Stillwell [34]); it includes every n-random set but not every (n − 1)-random set. (iv) The class {A : deg(A) is hyperimmune} has measure one (Martin =-=[21]-=-); it 7sincludes every 2-random set but not every 1-random set. (v) The class {A : A has a 1-generic predecessor} has measure one (Kurtz [15]); it includes every 2-random set but not every 1-random se... |

3 |
Probabilities over rich languages, randomness and testing
- Gaifman, Snir
- 1982
(Show Context)
Citation Context ...ince Pe is a Π 0 1 -class we can take Pe(s) to be (the extension of) a finite set of strings. 2 The result was evidently known in some form to Gaifman and Snir; a version appears (without a proof) in =-=[7]-=-. 28sLet Se(s) denote the approximation of Se at stage s, and let Se(s) denote the corresponding approximation of the r.e. set of strings Se such that Se = Ext(Se). Let Γ(s) denote a “moveable marker”... |

3 |
Randomness and generalizations of fixed point free functions
- Kucera
- 1990
(Show Context)
Citation Context ...e of the application of this approach is Ku˘cera’s use of results on random sets and fixed-point-free degrees to settle an open question about the generalized Arslanov completeness criterion (Ku˘cera =-=[18]-=-). Another reason that an explicit definition of randomness is necessary is in order to talk about structural properties of random degrees in relation to one another. It is known that the first-order ... |

3 |
Category methods in recursion theory
- Myhill
- 1961
(Show Context)
Citation Context ...with an illustration of the relationship between finite extension constructions, forcing, and category. The connection between finite extension constructions and category was first observed by Myhill =-=[25]-=-. 22sLet L ∗ be the usual language of first order arithmetic with an additional set constant X and a membership symbol ∈. For φ a sentence of L ∗ and A ∈ 2 ω , we write A |= φ if φ is true in the stan... |

3 |
1977], Measure and minimal degrees
- Paris
- 1977
(Show Context)
Citation Context ...there are 1-random degrees which are not relatively r.e. ✷ We mention two other significant measure-theoretic results which have not yet yeilded to the type of analysis used in Theorem IV.2.4: Paris (=-=[29]-=-) showed that the class {A : A has no minimal predecessor} has measure one, and Kurtz ([15]) used a similar argument to show that the class has measure one. {A : The 1-generic degrees are downward den... |