## Totally ω-computably enumerable degrees and bounding critical triples, preprint

Citations: | 8 - 4 self |

### BibTeX

@MISC{Downey_totallyω-computably,

author = {Rod Downey and Noam Greenberg and Rebecca Weber},

title = {Totally ω-computably enumerable degrees and bounding critical triples, preprint},

year = {}

}

### OpenURL

### Abstract

Abstract. We characterize the class of c.e. degrees that bound a critical triple (equivalently, a weak critical triple) as those degrees that compute a function that has no ω-c.e. approximation. 1.

### Citations

477 |
Recursively enumerable sets and degrees
- Soare
- 1987
(Show Context)
Citation Context ...ided by the theorem are not natural definitions of objects in computability theory (as outlined in Shore [Sho00].) Here we are thinking of results such as the following. (We refer the reader to Soare =-=[Soa87]-=- for unexplained definitions in the sequel, since they are used to provide background for the results of the current paper.) Theorem 1.2 (Ambos-Spies, Jockusch, Shore, and Soare [ASJSS84]). A c.e. deg... |

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.
(Show Context)
Citation Context ...eader to Soare [Soa87] for unexplained definitions in the sequel, since they are used to provide background for the results of the current paper.) Theorem 1.2 (Ambos-Spies, Jockusch, Shore, and Soare =-=[ASJSS84]-=-). A c.e. degree a is promptly simple iff it is not cappable. The authors’ research was supported by the Marsden Fund of New Zealand, via postdoctoral fellowships. 1s2 ROD DOWNEY, NOAM GREENBERG, AND ... |

34 | Interpretability and definability in the Recursively Enumerable Degrees
- Nies, Shore, et al.
- 1998
(Show Context)
Citation Context ...finability issues within the computably enumerable Turing degrees. In terms of abstract results on definability, there has been significant success in recent years, culminating in Nies, Shore, Slaman =-=[NSS98]-=-, where the following is proven. Theorem 1.1 (Nies, Shore, Slaman [NSS98]). Any relation on the c.e. degrees invariant under the double jump is definable in the c.e. degrees iff it is definable in fir... |

30 |
Array nonrecursive sets and multiple permitting arguments
- Downey, Jockusch, et al.
- 1990
(Show Context)
Citation Context ...n, the array computable degrees also capture the combinatorics of a wide class of constructions. These include constructions of perfect thin Π 0 1 classes ([CCDH01]), incomparable separating classes (=-=[DJS90]-=-), computably enumerable degrees containing sets of infinitely often maximal Kolmogorov complexity (Kummer [Kum]), and those computably enumerable degrees with strong minimal covers (Ishmukhametov [Is... |

28 |
Embedding nondistributive lattices in the recursively enumerable degrees
- Lachlan
- 1970
(Show Context)
Citation Context ... � � � � 1 ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ a b0 0 Figure 1. The lattice M5 capture the need for “continuous tracing” which is used in an embedding of the lattice M5 into the c.e. degrees (first embedded by Lachlan =-=[Lac72]-=-). Indeed the first nonembeddability result was by Lachlan and Soare [LS80] who demonstrated that an “infimum into an M5” could not be embedded in the c.e. degrees by showing that the lattice S8 below... |

24 | Array nonrecursive degrees and genericity
- Downey, Jockusch, et al.
- 1996
(Show Context)
Citation Context ..., s + 1)}| < h(x). We remark that the key difference between being totally ω-c.e. and being array computable is that for the latter, the function h(x) = x can always be chosen (Downey, Jockusch, Stob =-=[DJS96]-=-). A central notion for lattice embeddings into the c.e. degrees is the notion of a weak critical triple. The reader should recall from Downey [Dow90] and Weinstein [Wei88] that three incomparable ele... |

23 | Kolmogorov complexity and instance complexity of recursively enumerable sets
- Kummer
- 1996
(Show Context)
Citation Context ...structions of perfect thin Π 0 1 classes ([CCDH01]), incomparable separating classes ([DJS90]), computably enumerable degrees containing sets of infinitely often maximal Kolmogorov complexity (Kummer =-=[Kum]-=-), and those computably enumerable degrees with strong minimal covers (Ishmukhametov [Is]), to name but a few.sTOTALLY ω-C.E. DEGREES AND BOUNDING CRITICAL TRIPLES 5 Given the nature of the totally ω-... |

16 | Automorphisms of the lattice of Π 0 1 classes: perfect thin classes and anc degrees
- Cholak, Coles, et al.
(Show Context)
Citation Context ...ructions such as the maximal set construction, the array computable degrees also capture the combinatorics of a wide class of constructions. These include constructions of perfect thin Π 0 1 classes (=-=[CCDH01]-=-), incomparable separating classes ([DJS90]), computably enumerable degrees containing sets of infinitely often maximal Kolmogorov complexity (Kummer [Kum]), and those computably enumerable degrees wi... |

15 |
Lattice nonembeddings and initial segments of the recursively enumerable degrees
- Downey
- 1990
(Show Context)
Citation Context ... = x can always be chosen (Downey, Jockusch, Stob [DJS96]). A central notion for lattice embeddings into the c.e. degrees is the notion of a weak critical triple. The reader should recall from Downey =-=[Dow90]-=- and Weinstein [Wei88] that three incomparable elements a0, a1 and b in an upper semilattice form a weak critical triple if a0 ∪ b = a1 ∪ b and there is no c � a0, a1 with a0 � b ∪ c. We say that inco... |

13 |
Not every finite lattice is embeddable in the recursively enumerable degrees
- Lachlan, Soare
- 1980
(Show Context)
Citation Context ...d for “continuous tracing” which is used in an embedding of the lattice M5 into the c.e. degrees (first embedded by Lachlan [Lac72]). Indeed the first nonembeddability result was by Lachlan and Soare =-=[LS80]-=- who demonstrated that an “infimum into an M5” could not be embedded in the c.e. degrees by showing that the lattice S8 below could not be embedded (as suggested by Lerman.) The necessity of the “cont... |

11 |
Weak recursive degrees and a problem of Spector, in Recursion Theory and Complexity
- Ishmukhametov
- 1999
(Show Context)
Citation Context ...90]), computably enumerable degrees containing sets of infinitely often maximal Kolmogorov complexity (Kummer [Kum]), and those computably enumerable degrees with strong minimal covers (Ishmukhametov =-=[Is]-=-), to name but a few.sTOTALLY ω-C.E. DEGREES AND BOUNDING CRITICAL TRIPLES 5 Given the nature of the totally ω-c.e. degrees, we would therefore hope that this class will also encode the combinatorics ... |

11 |
A finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees
- Lempp, Lerman
- 1997
(Show Context)
Citation Context ...lity results for the c.e. degrees. Moreover, these definability results will be related to the central topic of lattice embeddings into the c.e. degrees as analysed by, for instance, Lempp and Lerman =-=[LL97]-=-, Lempp, Lerman and Solomon [LLS], and Lerman [Ler85]. Additionally our new definability results will allow us to tie a number of natural constructions together in new degree classes in the same way a... |

8 | Lattice embeddings below a non-low2 recursively enumerable degree
- Downey, Shore
- 1996
(Show Context)
Citation Context ... Downey [Dow90] that the embedding of critical triples seemed to be tied up with multiple permitting in a way that was similar to non-low2-ness. Indeed this intuition was verified by Downey and Shore =-=[DS96]-=- where it is shown that if a is non-low2 then a bounds a copy of M5. The notion of non-low2-ness seemed too strong to capture the class of degrees which bound M5’s but it was felt that something like ... |

6 |
Contiguity and distributivity in the enumerable Turing degrees
- Downey, Lempp
- 1997
(Show Context)
Citation Context ... is not cappable. The authors’ research was supported by the Marsden Fund of New Zealand, via postdoctoral fellowships. 1s2 ROD DOWNEY, NOAM GREENBERG, AND REBECCA WEBER Theorem 1.3 (Downey and Lempp =-=[DL97]-=-). A c.e. degree a is contiguous iff it is locally distributive, meaning that holds in the c.e. degrees. ∀a1, a2, b(a1 ∪ a2 = a ∧ b � a → ∃b1, b2(b1 ∪ b2 = b ∧ b1 � a1 ∧ b2 � a2)) Theorem 1.4 (Ambos-S... |

6 |
Dynamic notions of genericity and array noncomputability, Annals of Pure and Applied Logic 95
- Schaeffer
- 1998
(Show Context)
Citation Context ...nswers a question of André Nies. Recall that a set A is called superlow iff A ′ ≡tt ∅ ′ . Corollary 1.11. The low degrees and the superlow degrees are not elementarily equivalent. Proof. As Schaeffer =-=[Sch98]-=- and Walk [Wal99] observe, all superlow degrees are array computable, and hence totally ω-c.e. Thus we cannot put a copy of M5 below one. On the other hand there are indeed low copies of M5. � One of ... |

4 | Embeddings of N5 and the contiguous degrees
- Ambos-Spies, Fejer
- 2001
(Show Context)
Citation Context ...e a is contiguous iff it is locally distributive, meaning that holds in the c.e. degrees. ∀a1, a2, b(a1 ∪ a2 = a ∧ b � a → ∃b1, b2(b1 ∪ b2 = b ∧ b1 � a1 ∧ b2 � a2)) Theorem 1.4 (Ambos-Spies and Fejer =-=[ASF01]-=-). A c.e. degree a is contiguous iff it is not the top of the non-modular 5 element lattice N5 (the pentagon) in the c.e. degrees. Theorem 1.5 (Downey and Shore [DS95]). A c.e. truth table degree is l... |

4 | Intervals without critical triples
- Cholak, Downey, et al.
- 1998
(Show Context)
Citation Context ...comparable, and such that Ai �T B ⊕ A1−i (i < 2). To show that they do not form a weak critical triple we need to construct a set E �T A0, A1 such that A0 �T B ⊕ E. The strategy is similar to that of =-=[CDS98]-=-. Recall that a fundamental notion is that of a layer, which we get by repeatedly applying the use function of the given reductions of A0 and A1 to the top of the triple. To preserve the correctness o... |

4 | Embedding finite lattices into the computably enumerable degrees - a status survey
- Lempp, Lerman, et al.
- 2002
(Show Context)
Citation Context ... Moreover, these definability results will be related to the central topic of lattice embeddings into the c.e. degrees as analysed by, for instance, Lempp and Lerman [LL97], Lempp, Lerman and Solomon =-=[LLS]-=-, and Lerman [Ler85]. Additionally our new definability results will allow us to tie a number of natural constructions together in new degree classes in the same way as the array noncomputable degrees... |

3 |
2002], Natural definability in degree structure, Computability theory and its applications: Current trends and open problems
- Shore
- 1999
(Show Context)
Citation Context ...em; these are the kinds of sets we investigate here. Another issue is that the definitions provided by the theorem are not natural definitions of objects in computability theory (as outlined in Shore =-=[Sho00]-=-.) Here we are thinking of results such as the following. (We refer the reader to Soare [Soa87] for unexplained definitions in the sequel, since they are used to provide background for the results of ... |

3 |
On embeddings of the 1-3-1 lattice into the recursively enumerable degrees
- Weinstein
- 1988
(Show Context)
Citation Context ...sen (Downey, Jockusch, Stob [DJS96]). A central notion for lattice embeddings into the c.e. degrees is the notion of a weak critical triple. The reader should recall from Downey [Dow90] and Weinstein =-=[Wei88]-=- that three incomparable elements a0, a1 and b in an upper semilattice form a weak critical triple if a0 ∪ b = a1 ∪ b and there is no c � a0, a1 with a0 � b ∪ c. We say that incomparable a0, a1 and b ... |

2 |
1986], Structural interactions of the recursively enumerable T- and w-degrees
- Downey, Stob
- 1984
(Show Context)
Citation Context ...fore hope that this class will also encode the combinatorics of other constructions aside from the critical triple one. In this paper we make a modest contribution to this program. In Downey and Stob =-=[DS86]-=-, the authors observed that there seemed to be a deep connection between the structure of the c.e. weak truth table degrees within a c.e. Turing degrees and lattice embeddings. To wit, Downey and Stob... |

2 |
The embedding problem for the recursively enumerable degrees
- Lerman
- 1982
(Show Context)
Citation Context ...efinability results will be related to the central topic of lattice embeddings into the c.e. degrees as analysed by, for instance, Lempp and Lerman [LL97], Lempp, Lerman and Solomon [LLS], and Lerman =-=[Ler85]-=-. Additionally our new definability results will allow us to tie a number of natural constructions together in new degree classes in the same way as the array noncomputable degrees did in Downey, Jock... |

2 |
1986], Structural interactions of the recursively enumerable Tand w-degrees
- Downey, Stob
- 1984
(Show Context)
Citation Context ...fore hope that this class will also encode the combinatorics of other constructions aside from the critical triple one. In this paper we make a modest contribution to this program. In Downey and Stob =-=[DS86]-=-, the authors observed that there seemed to be a deep connection between the structure of the c.e. weak truth table degrees within a c.e. Turing degrees and lattice embeddings. To wit, Downey and Stob... |

1 |
Working with strong reducibilities above array computable and totally ω-c.e. degrees, submitted
- Barmpalias, Downey, et al.
(Show Context)
Citation Context ...not totally ω-c.e. iff there exists a left c.e. real α �T a and a c.e. set B <T α such that if A presents α, then A �T B. In yet another sequel to the present paper, Barmpalias, Downey and Greenberg (=-=[BDG]-=-) demonstrate that the class of totally ω-c.e. degrees captures a number of constructions. For example, it is shown that a is totally ω-c.e. iff every set in a is weak truth table reducible to a ranke... |

1 |
The degrees which bound M5
- Downey, Greenberg
(Show Context)
Citation Context ...f it has no minimal cover in the c.e. truth table degrees. At the present time, as articulated in Shore [Sho00], there are very few such natural definability results. In this paper, and in the sequel =-=[DGa]-=-, we will give some new natural definability results for the c.e. degrees. Moreover, these definability results will be related to the central topic of lattice embeddings into the c.e. degrees as anal... |

1 |
Totally ω-c.e. degrees and presentations of left c.e. reals
- Downey, Greenberg
(Show Context)
Citation Context ...lled left c.e. if it is the limit of a computable nondecreasing sequence of rationals, and that a c.e., prefix free set of strings A presents α if α = � σ∈A 2−|σ| . Theorem 1.14 (Downey and Greenberg =-=[DGb]-=-). A degree a is not totally ω-c.e. iff there exists a left c.e. real α �T a and a c.e. set B <T α such that if A presents α, then A �T B. In yet another sequel to the present paper, Barmpalias, Downe... |

1 |
Degree-theoretic definitions of the low2 recursively enumerable sets
- Downey, Shore
- 1995
(Show Context)
Citation Context ...orem 1.4 (Ambos-Spies and Fejer [ASF01]). A c.e. degree a is contiguous iff it is not the top of the non-modular 5 element lattice N5 (the pentagon) in the c.e. degrees. Theorem 1.5 (Downey and Shore =-=[DS95]-=-). A c.e. truth table degree is low2 iff it has no minimal cover in the c.e. truth table degrees. At the present time, as articulated in Shore [Sho00], there are very few such natural definability res... |

1 |
Towards a definitioon of the array computable degrees
- Walk
- 1999
(Show Context)
Citation Context ... a bounds a copy of M5. The notion of non-low2-ness seemed too strong to capture the class of degrees which bound M5’s but it was felt that something like that should suffice. On the other hand, Walk =-=[Wal99]-=- constructed an array noncomputable c.e. degree bounding no weak critical triple, and hence it was already known that array non-computability was not enough for such embeddings. We prove the following... |