## TOWARD THE DEFINABILITY OF THE ARRAY NONCOMPUTABLE DEGREES (1999)

### BibTeX

@MISC{Cholak99towardthe,

author = {Peter A. Cholak and Stephen M. Walk},

title = {TOWARD THE DEFINABILITY OF THE ARRAY NONCOMPUTABLE DEGREES},

year = {1999}

}

### OpenURL

### Abstract

by

### Citations

472 |
Recursively Enumerable Sets and Degrees
- Soare
- 1987
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Citation Context ...m 4.10, page 118. A function f ≤t K is c n+1 if and only if f ≤n·rct K. All of the results presented in this thesis are finite-injury or infinite-injury constructions. The reader is directed to Soare =-=[39]-=- for background on these types of construction. 1.3 Conventions and Notation Our notation is mostly standard as in [39] and other references. Some conventions we will use freely throughout this work a... |

47 |
Two recursively enumerable sets of incomparable degrees of unsolvability
- Friedberg
- 1957
(Show Context)
Citation Context ... computable and computably enumerable rather than recursive and recursively enumerable.) This structure has revealed itself to be rich, complex, and in many ways unpredictable. For example, Friedberg =-=[22]-=- and also Muchnik [33] showed that there are c.e. degrees besides 0 (the degree of the computable sets) and 0 ′ (the degree of the Halting Problem), and that the ordering ≤ is not linear; Sacks [34] s... |

42 | Lower bounds for pairs of recursively enumerable degrees
- Lachlan
- 1966
(Show Context)
Citation Context ...e. degrees besides 0 (the degree of the computable sets) and 0 ′ (the degree of the Halting Problem), and that the ordering ≤ is not linear; Sacks [34] showed that the c.e. degrees are dense; Lachlan =-=[28]-=- and Yates [47] showed that some pairs of degrees fail to have a greatest lower bound, so that (R, ≤) is not a lattice, but some pairs of degrees do have greatest lower bounds, so that some lattices c... |

36 | Soare, An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees - Ambos-Spies, Jockusch, et al. |

35 |
The recursively enumerable degrees are dense
- Sacks
- 1964
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Citation Context ...g [22] and also Muchnik [33] showed that there are c.e. degrees besides 0 (the degree of the computable sets) and 0 ′ (the degree of the Halting Problem), and that the ordering ≤ is not linear; Sacks =-=[34]-=- showed that the c.e. degrees are dense; Lachlan [28] and Yates [47] showed that some pairs of degrees fail to have a greatest lower bound, so that (R, ≤) is not a lattice, but some pairs of degrees d... |

32 |
On the unsolvability of the problem of reducibility in the theory of algorithms
- Muchnik
- 1956
(Show Context)
Citation Context ...ably enumerable rather than recursive and recursively enumerable.) This structure has revealed itself to be rich, complex, and in many ways unpredictable. For example, Friedberg [22] and also Muchnik =-=[33]-=- showed that there are c.e. degrees besides 0 (the degree of the computable sets) and 0 ′ (the degree of the Halting Problem), and that the ordering ≤ is not linear; Sacks [34] showed that the c.e. de... |

30 |
Array nonrecursive sets and multiple permitting arguments
- Downey, Jockusch, et al.
- 1990
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Citation Context ... however, each requirement Re may need as many as f(e) permissions to succeed, where f is some computable function; in this case we have a multiple-permitting construction. Downey, Jockusch, and Stob =-=[14]-=- provide examples of multiple-permitting constructions that arise naturally in computability theory, such as the construction of non-f-c.e. sets: Let f be any computable function. A set D ≤ 0 ′ is sai... |

28 |
Embedding nondistributive lattices in the recursively enumerable degrees
- Lachlan
- 1970
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Citation Context ... who showed that every countable distributive lattice can be embedded into any nontrivial interval of R. Thus the question became, “Which nondistributive lattices can be embedded into R?” Lachlan, in =-=[29]-=-, showed that both the 5-element modular lattice and the 5-element nonmodular lattice, denoted as M5 and N5 respectively, can be embedded into R; but an 8-element lattice was soon found that cannot be... |

24 | Array nonrecursive degrees and genericity
- Downey, Jockusch, et al.
- 1996
(Show Context)
Citation Context ...≤t. α The reader is no doubt wondering how all of this relates to the second of Questions 4.7. Our attempts to settle the question have involved the generalized definition of array computability from =-=[15]-=-: that a degree a is array computable if and only if there is some function f ≤wtt K that dominates every g ≤t a. Thatis,we havestartedwithac.e.setA of array computable (c 1 -c) degree, taken an f ≤wt... |

23 | Kolmogorov complexity and instance complexity of recursively enumerable sets
- Kummer
- 1996
(Show Context)
Citation Context ...pect to complexity theory. The Kolmogorov complexity of any c.e. set A is bounded by 2 log n (that is, for some c, for all n the complexity of the first n bits of A is less than 2 log n−c; see Kummer =-=[27]-=-). Kummer [27, Theorem 3.2] showed that a degree d contains a set that attains this bound if and only if d is anc. In fact there is quite a gap between anc degrees and array computable degrees [27, al... |

19 | Double jumps of minimal degrees - Jockusch, Posner - 1978 |

18 |
The weak truth-table degrees of the recursively enumerable sets
- Ladner, Sasso
- 1975
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Citation Context ...iven two sets A and B, A is weak truth-table reducible to B, A ≤wtt B, if there are a functional � Φanda computable function f such that A = � Φ(B)andforanyx, the use u( � Φ,B,x) ≤ f(x). Ladner–Sasso =-=[31]-=-: A computably enumerable degree a is contiguous if for every c.e. A, B ∈ a we have A ≡wtt B. An array noncomputable degree cannot be contiguous; see Downey [13, Theorem 2.8] or Downey–Stob [19, Theor... |

18 |
A minimal pair of recursively enumerable degrees
- Yates
(Show Context)
Citation Context ...des 0 (the degree of the computable sets) and 0 ′ (the degree of the Halting Problem), and that the ordering ≤ is not linear; Sacks [34] showed that the c.e. degrees are dense; Lachlan [28] and Yates =-=[47]-=- showed that some pairs of degrees fail to have a greatest lower bound, so that (R, ≤) is not a lattice, but some pairs of degrees do have greatest lower bounds, so that some lattices can be embedded ... |

17 |
Splitting theorems in recursion theory
- Downey, Stob
- 1993
(Show Context)
Citation Context ...that is, σ ⊆ fs) and l N (e, s) > max{l N (e, t):t<sand σ ⊆ ft}. 155sWe meet the Ne’s via a Sacks preservation strategy. Our strategy for the Re’s is similar to strategies in Downey [13], Downey-Stob =-=[19]-=-, and Schweiter [36, Theorem 1]. We pick a witness x and wait until � Φe,s(D1 s ; x) ↓= 0. We put x into B and restrain D, in order to restrain D 1 below �ϕe(x) and preserve the disagreement �Φe,s(D 1... |

16 | Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees
- Cholak, Coles, et al.
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Citation Context ...he past few years. One fact that has come to light recently involves Π0 1-classes, whereasetXis a Π01 -class if X is the set of paths through some computable tree. Cholak, Coles, Downey, and Herrmann =-=[6]-=- have studied the lattice EΠ of Π 0 1-classes under set inclusion and in particular the automorphism group of EΠ. They show [6, Theorem 7.10] that the array noncomputable c.e. degrees form an invarian... |

15 |
Lattice nonembeddings and initial segments of the recursively enumerable degrees
- Downey
- 1990
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Citation Context ...ment Boolean algebra can be embedded, and by Slaman’s result [37] it followed that N5 can be embedded into all nontrivial intervals. The modular lattice M5 (Figure 2.1) is not so well-behaved. Downey =-=[12]-=- and independently Weinstein [46] showed that it cannot be embedded into every initial segment of R; that is, there is a noncomputable degree a such that M5 cannot be embedded into R below a. Cholak a... |

15 | Reducibility and completeness for sets of integers - Friedberg, Rogers - 1959 |

13 | Degree theoretic splitting properties of recursively enumerable sets - Ambos-Spies, Fejer - 1988 |

13 |
0 2 degrees and transfer theorems
- Downey
- 1987
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Citation Context ...y requires it only for the c.e. sets in d. Clearly, then, if a < d are c.e. degrees and d is contiguous over a, thend is a-contiguous. α We will also need the following definition: Definition. Downey =-=[10]-=-: A c.e. degree is strongly contiguous if for all sets C1,C2 ∈ c, C1 ≡wtt C2. On the face of it, this definition is stronger than the definition of contiguous, since in the definition of contiguity th... |

13 |
Not every finite lattice is embeddable in the recursively enumerable degrees
- Lachlan, Soare
- 1980
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Citation Context ... modular lattice and the 5-element nonmodular lattice, denoted as M5 and N5 respectively, can be embedded into R; but an 8-element lattice was soon found that cannot be so embedded (Lachlan and Soare =-=[30]-=-). 10sFor M5 and N5 there remained the question, “Where within R can these lattices be embedded?” Ambos-Spies and Fejer [3, Proposition 2.5] showed that the nonmodular lattice N5 can be enbedded into ... |

13 | Splitting properties and jump classes - Maass, Shore, et al. - 1981 |

12 |
jump classes and strong reducibilities
- Downey, Jockusch
- 1987
(Show Context)
Citation Context ...ve been unable to prove our conjecture by this sort of effort. We note that, of all the published constructions of contiguous degrees that we are aware of (in [43], for example, and in [1], [11], and =-=[16]-=-, among others), the positive requirements are no more complicated than “Φe �= A”; specifically, the requirements work independently, each requirement needs only one permission to succeed, and if one ... |

12 |
The density of infima in the recursively enumerable degrees
- Slaman
- 1991
(Show Context)
Citation Context ... to all finite distributive lattices by Thomason [45] and by Lerman (unpublished—see Soare [39, p. 161]) and then to the countable atomless Boolean algebra by Lachlan and by Lerman (see [39]). Slaman =-=[37]-=- showed that the 4-element lattice could be embedded into any nontrivial interval (where an interval [a,b] is simply the set of degrees {d : a ≤ d ≤ b}), and Ambos-Spies [1, Corollary 4.3 and Remark 4... |

11 | On pairs of recursively enumerable degrees - Ambos-Spies - 1984 |

9 |
The density of the nonbranching degrees
- Fejer
- 1983
(Show Context)
Citation Context ...e Theorem 3.29 until we had moved on to an investigation of tardiness. In hindsight, this result would have been enough to prove the existence theorems in the Subsections 3.2.2 and 3.2.3, since Fejer =-=[20]-=- has 92sshown that the nonbranching c.e. t-degrees are dense. Thus, given a c.e. degree d, there is a nonbranching a ∈ [0,d]; so in particular there is no c.e. degree b such that d ∩ b = a, and thus d... |

9 |
Relative recursive enumerability
- Soare, Stob
- 1982
(Show Context)
Citation Context ...a. It is certainly not true that if A is c.e. and D is c.e.a. in A,thenD must be c.e. itself, or must even have c.e. degree. This is obvious if A is not low, but it is true in general. Soare and Stob =-=[41]-=- demonstrated that for every noncomputable A, there is some c.e.a.-in-A degree that is not c.e. (It is worth noting that by relativizing the proof of this result, Soare and Stob refuted a conjecture o... |

8 | Lattice embeddings below a non-low2 recursively enumerable degree
- Downey, Shore
- 1996
(Show Context)
Citation Context ...h by showing that Theorem 2.6, page 26. There is an array noncomputable degree below which the lattice M5 cannot be embedded into the computably enumerable degrees. Given a result of Downey and Shore =-=[18]-=-, this shows that there is a definable difference between the anc degrees and the nonlow2 degrees, since every nonlow2 degree bounds a copy of M5 and “x bounds a copy of M5” is easily rendered as a fo... |

6 |
Sublattices of the recursively enumerable degrees
- Thomason
- 1971
(Show Context)
Citation Context ...edded into,” R—and where. In the papers [28] and [47] it was proved that the 4-element Boolean algebra can be embedded into R. This result was extended to all finite distributive lattices by Thomason =-=[45]-=- and by Lerman (unpublished—see Soare [39, p. 161]) and then to the countable atomless Boolean algebra by Lachlan and by Lerman (see [39]). Slaman [37] showed that the 4-element lattice could be embed... |

5 | Lattice nonembeddings and intervals in the recursively enumerable degrees - Cholak, Downey - 1993 |

5 | Embedding lattices with top preserved below non-GL 2 degrees - Fejer - 1989 |

4 | Intervals without critical triples
- Cholak, Downey, et al.
- 1998
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Citation Context ...a first-order formula in the language {≤}—it shows that the class of anc degrees is not just different from the class of nonlow2 degrees, but is definably so. 2.2 The Cholak-Downey-Shore Module As in =-=[8]-=-, we construct a degree e that does not bound a weak critical triple. The following definitions appear in that paper: Definition. Let a, b0, and b1 be elements of an uppersemilattice L. Then a, b0, an... |

4 |
Soare, Dynamic properties of computably enumerable sets, In: Computability, Enumerability, Unsolvability: Directions in Recursion Theory
- Harrington, I
- 1996
(Show Context)
Citation Context ...y if there is no a < d such that d is prompt over a,” and see what characteristics might be shared by “quite 80stardy” degrees. (The name quite tardy would have been chosen since Harrington and Soare =-=[24]-=- have already used the term very tardy for a different concept.) However, Theorem 3.17 shows that no “quite tardy” degrees exist; all degrees are “somewhat prompt.” The next best thing would have been... |

3 |
Interpolating d.r.e. and REA degrees between r.e. degrees
- Arslanov, Lempp, et al.
- 1994
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Citation Context ... by relativizing the proof of this result, Soare and Stob refuted a conjecture of Cooper [9]—that any degree that is c.e.a. in 0 ′ is c.e.a. in any high degree.) Arslanov, Lempp, and Shore have shown =-=[5]-=- that there is a noncomputable c.e. set A such that every D that is c.e.a. in A and below 0 ′ has c.e. degree; note that, by the Soare-Stob result, this A cannot be low. Stob [44] has asked what happe... |

3 |
Array nonrecursive degrees and lattice embeddings of the diamond
- Downey
- 1993
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Citation Context ... and thus unusable) computable function. We will always read the expression “ � Φe,s(D 1 s; x)↓”tomean“Φi,s(D 1 s; x)↓, ϕj,s(x)↓, andu(Φi,D 1 ,x,s) ≤ ϕj(x).” The set D 1 is a “kick set” (as in Downey =-=[13]-=-) of D that we construct along with D. Since it is a kick set, D 1 ≡t D (see below), so that if B ≤t D, and if all Re are met we will have deg(D) noncontiguous. The Pe’s make A noncomputable, the Ne’s... |

3 |
wtt-degrees and T-degrees of r.e. sets
- Stob
- 1983
(Show Context)
Citation Context ... functions for Ce. So far, however, we have been unable to prove our conjecture by this sort of effort. We note that, of all the published constructions of contiguous degrees that we are aware of (in =-=[43]-=-, for example, and in [1], [11], and [16], among others), the positive requirements are no more complicated than “Φe �= A”; specifically, the requirements work independently, each requirement needs on... |

2 |
A contiguous nonbranching degree
- Downey
- 1989
(Show Context)
Citation Context ...ver, we have been unable to prove our conjecture by this sort of effort. We note that, of all the published constructions of contiguous degrees that we are aware of (in [43], for example, and in [1], =-=[11]-=-, and [16], among others), the positive requirements are no more complicated than “Φe �= A”; specifically, the requirements work independently, each requirement needs only one permission to succeed, a... |

2 |
Contiguity and distributivity in the enumerable degrees
- Downey, Lempp
- 1997
(Show Context)
Citation Context ... contiguous degree d that is not noncontiguous over any a < d. In fact this is true for any contiguous degree. The last two propositions are easy immediate consequences of results by Downey and Lempp =-=[17]-=- and Downey and Stob [19]. Since we have conjectured that array noncomputability is definable, we investigate a relativization of it also and show that Theorem 3.45, page 103. If d is an array computa... |

2 |
Abstract complexity theory and the degrees of unsolvability
- Schaeffer
- 1998
(Show Context)
Citation Context ...ationship between the ≤n·rct (recursivelycontrolled Turing) reducibilities developed by Yuefei [48] and the finite levels of the composition hierarchy {c α } of ∆ 0 2 functions developed by Schaeffer =-=[35]-=-: Theorem 4.10, page 118. A function f ≤t K is c n+1 if and only if f ≤n·rct K. All of the results presented in this thesis are finite-injury or infinite-injury constructions. The reader is directed t... |

1 |
Sets recursively enumerable in high degrees
- Cooper
- 1972
(Show Context)
Citation Context ...ated that for every noncomputable A, there is some c.e.a.-in-A degree that is not c.e. (It is worth noting that by relativizing the proof of this result, Soare and Stob refuted a conjecture of Cooper =-=[9]-=-—that any degree that is c.e.a. in 0 ′ is c.e.a. in any high degree.) Arslanov, Lempp, and Shore have shown [5] that there is a noncomputable c.e. set A such that every D that is c.e.a. in A and below... |

1 |
The R.E. WTT-Structure of Certain Turing Degrees
- Schweiter
- 1987
(Show Context)
Citation Context ...e, there is a low anc degree b ≥ a. We discovered another result independently during our pursuit of the conjecture, but later learned that it follows from work in the Ph.D. thesis of G. A. Schweiter =-=[36]-=-: Corollary 4.2, page 110. There are c.e. sets A and D such that A is noncomputable, deg(D) is noncontiguous, and for all � D ∈ deg(D), A ≤wtt � D. 7sThe final result exhibits a nice relationship betw... |

1 | Correction to promptly simple theorem - Soare - 1998 |

1 |
Embedding M5. Unpublished handwritten notes
- Stob
- 1980
(Show Context)
Citation Context ... explicitly low2 anc degree that bounds a copy of M5, and this result accomplishes that as well. The construction is very easy, however, given the paper of Downey and Shore [18] and the notes of Stob =-=[42]-=-, so we relegate it to Appendix A. 47sThus it is possible for every degree in a copy of M5 to be array noncomputable. It is certainly possible for the bottom of an M5 to be array computable, sincethe ... |

1 |
On Embedding of the Lattice 1-3-1 into the Recursively Enumerable Degrees
- Weinstein
- 1988
(Show Context)
Citation Context ...ded, and by Slaman’s result [37] it followed that N5 can be embedded into all nontrivial intervals. The modular lattice M5 (Figure 2.1) is not so well-behaved. Downey [12] and independently Weinstein =-=[46]-=- showed that it cannot be embedded into every initial segment of R; that is, there is a noncomputable degree a such that M5 cannot be embedded into R below a. Cholak and Downey [7, Theorem 1] extended... |

1 |
A new reducibility between Turing- and wtt-reducibility
- Yuefei
- 1994
(Show Context)
Citation Context ...le, deg(D) is noncontiguous, and for all � D ∈ deg(D), A ≤wtt � D. 7sThe final result exhibits a nice relationship between the ≤n·rct (recursivelycontrolled Turing) reducibilities developed by Yuefei =-=[48]-=- and the finite levels of the composition hierarchy {c α } of ∆ 0 2 functions developed by Schaeffer [35]: Theorem 4.10, page 118. A function f ≤t K is c n+1 if and only if f ≤n·rct K. All of the resu... |