## THE COMPLEXITY OF ORBITS OF COMPUTABLY ENUMERABLE SETS (2007)

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### BibTeX

@MISC{Cholak07thecomplexity,

author = {Peter A. Cholak and Rodney Downey and Leo A. Harrington},

title = {THE COMPLEXITY OF ORBITS OF COMPUTABLY ENUMERABLE SETS },

year = {2007}

}

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### Abstract

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, E, such that the question of membership in this orbit is Σ1 1-complete. This result and proof have a number of nice corollaries: the Scott rank of E is ωCK 1 + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E; for all finite α ≥ 9, there is a properly ∆0 α orbit (from the proof).

### Citations

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Systems of Logic based on Ordinals
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- 1939
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Citation Context ...some effective process. The basic example is the set of consequences of a computably enumerable set of axioms for a formal system. Of course, the other key concept in computability was that of Turing =-=[36]-=- who introduced the notion of reducibility. Reducibilities are pre-orderings used to measure relative computational complexity. The interplay of these two basic objects, (Turing) reducibility and effe... |

117 |
Recursively enumerable sets of positive integers and their decision problems
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- 1944
(Show Context)
Citation Context ...l orbits are elementarily definable; there is no arithmetic description of all orbits of E; for all finite α ≥ 9, there is a properly ∆0 α orbit (from the proof). 1. Introduction In the classic paper =-=[32]-=-, Post suggested that the study of the lattice of computably (recursively) enumerable (c.e.) sets was fundamental in computability theory. Post observed that, at the time, all known undecidability pro... |

56 |
Computable Structures and the Hyperarithmetical Hierarchy, 1st edn
- Ash, Knight
- 2000
(Show Context)
Citation Context ...hat there is an orbit such that membership in that orbit is Σ1 1-complete. There are other equivalent definitions of a structure having Scott Rank ωCK 1 + 1 and we refer the readers to Ash and Knight =-=[1]-=-. � A consequence of the method of the proof (and some further effort to preserve quantifiers) is the following. Theorem 2.5. For all finite α > 8 there is a properly ∆ 0 α orbit. Hitherto this paper ... |

53 |
Classes of recursively enumerable sets and degrees of unsolvability
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- 1966
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Citation Context ...long we focus on infinite and co-infinite sets). Early work by Lachlan and others showed that both of the automorphism groups Aut(E) and Aut(E ∗ ) were large since each had 2 ℵ0 automorphisms. Martin =-=[30]-=- used a priority construction to show that a certain construction of Post (hypersimplicity) was not invariant under automorphisms of E. Post’s original programme was to look at thinness properties of ... |

37 |
The upper semi-lattice of degrees of recursive unsolvability
- Kleene, Post
- 1954
(Show Context)
Citation Context ...arantee Turing incompleteness. While Post’s problem was eventually solved by the development of the priority method independently by Friedberg [18] and Muchnik [31] out of the work of Kleene and Post =-=[24]-=-, whether Post’s Programme 1 Indeed the reader should recall that, more generally, a set A is lown iff A (n) = ∅ (n) iff ∆ 0 n+1 = ∆ A n+1, and A is highn iff A (n) = ∅ (n+1) iff ∆ 0 n+2 = ∆ A n+1.sOR... |

32 | Automorphisms of the Lattice of Recursively Enumerable Sets
- Cholak
- 1995
(Show Context)
Citation Context ...are [34], who showed that maximal sets form an orbit in Aut(E). In particular, no “extra” property together with maximality could guarantee incompleteness. Moreover, the paper Cholak, Downey and Stob =-=[3]-=-, showed that no property of L∗ (A) alone could guarantee Turing incompleteness for a computably enumerable set A. That is, Cholak, Downey and Stob proved that if for any computably enumerable set A t... |

32 |
On the unsolvability of the problem of reducibility in the theory of algorithms
- Muchnik
- 1956
(Show Context)
Citation Context ...erty of a c.e. set in E ∗ which would guarantee Turing incompleteness. While Post’s problem was eventually solved by the development of the priority method independently by Friedberg [18] and Muchnik =-=[31]-=- out of the work of Kleene and Post [24], whether Post’s Programme 1 Indeed the reader should recall that, more generally, a set A is lown iff A (n) = ∅ (n) iff ∆ 0 n+1 = ∆ A n+1, and A is highn iff A... |

27 |
A criterion for completeness of degrees of unsolvability
- Friedberg
- 1957
(Show Context)
Citation Context ...ny definable property of a c.e. set in E ∗ which would guarantee Turing incompleteness. While Post’s problem was eventually solved by the development of the priority method independently by Friedberg =-=[18]-=- and Muchnik [31] out of the work of Kleene and Post [24], whether Post’s Programme 1 Indeed the reader should recall that, more generally, a set A is lown iff A (n) = ∅ (n) iff ∆ 0 n+1 = ∆ A n+1, and... |

22 |
Automorphisms of the lattice of recursively enumerable sets, Part I: Maximal sets
- Soare
- 1974
(Show Context)
Citation Context ...will be important for our subsequent discussion. Definition 1.1. A ≈ Â iff there is a map, Φ, from the c.e. sets to the c.e. sets preserving inclusion, ⊆, (so Φ ∈ Aut(E)) such that Φ(A) = Â. By Soare =-=[34]-=-, E can be replaced with E ∗ , since Soare showed that every automorphism of E ∗ is equivalent to one on E and conversely (as long we focus on infinite and co-infinite sets). Early work by Lachlan and... |

21 | Post’s Program and incomplete recursively enumerable sets
- Harrington, Soare
- 1991
(Show Context)
Citation Context ...e of investigation, where failures of the automorphism machinery could be exploited to provide definability results in E ∗ . A classic example of this is the following theorem of Harrington and Soare =-=[21]-=- [23] who showed that a more general form of Post’s Programme indeed has a positive solution. Theorem 1.3 (Harrington and Soare [21]). There is a definable property Q(A), such that, if a c.e. set A sa... |

21 |
On the lattice of recursively enumerable sets
- Lachlan
- 1968
(Show Context)
Citation Context ...th a parameter for A. The following result says that the full complexity of the isomorphism problem for Boolean algebras of Theorem 2.6 is present in the supersets of a c.e. set. Theorem 2.7 (Lachlan =-=[25]-=-). Effectively in i there is a c.e. set Hi such that L ∗ (Hi) ∼ = Bi. Corollary 2.8. The set {〈i, j〉 : L ∗ (Hi) ∼ = L ∗ (Hj)} is Σ 1 1-complete. 2 We think it is well known that the isomorphism proble... |

13 | 1996b]. The # 3 -automorphism method and noninvariant classes of degrees
- Harrington, Soare
(Show Context)
Citation Context ...ism ∆0 3, and A ≈∆0 Â. While later papers presented Soare’s 3 automorphism machinery argument as a more thematic and flexible tree argument (beginning with Cholak [5] and [6] and Harrington and Soare =-=[22]-=-) most of the key underlying ideas for constructing automorphisms of (E) are in Soare’s original paper. The principal tool used is called the (or, in view of recent work, an) Extension Lemma. Roughly ... |

12 |
d-Simple sets, small sets, and degree classes
- Lerman, Soare
- 1980
(Show Context)
Citation Context ...the Harrington-Soare result. Harrington used the idea of exploiting the failure of the machinery to get a definition of being a halting problem in the lattice of c.e. sets. Similarly Lerman and Soare =-=[26]-=- showed that there are low simple sets that are elementarily inequivalent, in that one has a property called d-simplicity and one has not, where d-simplicity is an elementary property implying certain... |

10 | Codable sets and orbits of computably enumerable sets
- Harrington, Soare
- 1998
(Show Context)
Citation Context ...ton and Soare [21], we know that not every c.e. set is automorphic to a complete set, and partial classifications of precisely which sets can be found in Downey and Stob [16] and Harrington and Soare =-=[22, 20]-=-. Question 3.2 (Cone Avoidance). Given an incomplete c.e. degree d and an incomplete c.e. set A, is there an Â automorphic to A such that d �≤T Â? Question 3.3 (Can single jumps be coded into E?). Let... |

10 |
On the orbit of hyperhypersimple sets
- Maass
(Show Context)
Citation Context ...ce of c.e. sets. Almost all of it either uses Soare’s original Extension Lemma as a black box, or modified it, to prove various results on the lattice of c.e. sets. Examples include the work of Maass =-=[27]-=-, Maass and Stob [28], and Downey and Stob [16]. Early on, the methods seemed so powerful that anything seemed possible. Perhaps all sets were automorphic to complete sets, as suggested by Soare [35].... |

9 | On the definability of the double jump in the computably enumerable sets
- Cholak, Harrington
(Show Context)
Citation Context ...ims, we direct the reader to Section 5.3. 4. Past Work and Other Connections The paper [9] is a fourth paper in a series of loosely connected papers, the previous three being by Harrington and Cholak =-=[13]-=-, [7], and [8]. We have seen above that results from [8] determine the direction one must take to prove Theorem 2.2. The above results from [8] depend heavily on the main result in [7] whose proof dep... |

8 | 1996a]. Definability, automorphisms, and dynamic properties of computably enumerable sets, Bull. Symbolic Logic 2(2
- Harrington, Soare
(Show Context)
Citation Context ...investigation, where failures of the automorphism machinery could be exploited to provide definability results in E ∗ . A classic example of this is the following theorem of Harrington and Soare [21] =-=[23]-=- who showed that a more general form of Post’s Programme indeed has a positive solution. Theorem 1.3 (Harrington and Soare [21]). There is a definable property Q(A), such that, if a c.e. set A satisfi... |

6 | Some orbits for
- Cholak, Downey, et al.
- 2001
(Show Context)
Citation Context ...therefore Λ maps R to a computable subset of Â. This observation of Herrmann was never published and is one of the key facts he used in showing that the Herrmann sets form an orbit; see Cholak et al. =-=[10]-=-. Theorem 4.9 (Soare [34]). The maximal sets form an orbit. Proof. Assume that A and Â are maximal. Then C(A) = E. If W ⊆ A then let Ψ(W ) = Λ(W ). If W ∪ A = ∗ ω there is a computable set RW such tha... |

6 | There is no fat orbit
- Downey, Harrington
- 1996
(Show Context)
Citation Context ...s a property called d-simplicity and one has not, where d-simplicity is an elementary property implying certain facts about entry states. Another example of this can be found in Downey and Harrington =-=[17]-=- where the “no fat orbit” theorem is proven. The simplest form of the Downey-Harrington result below says that no c.e. set has an orbit hitting all nonzero degrees. Theorem 1.4 (Downey and Harrington ... |

6 |
The intervals of the lattice of recursively enumerable sets determined by major subsets
- Maass, Stob
- 1983
(Show Context)
Citation Context ...st all of it either uses Soare’s original Extension Lemma as a black box, or modified it, to prove various results on the lattice of c.e. sets. Examples include the work of Maass [27], Maass and Stob =-=[28]-=-, and Downey and Stob [16]. Early on, the methods seemed so powerful that anything seemed possible. Perhaps all sets were automorphic to complete sets, as suggested by Soare [35]. Certainly Harrington... |

6 | Characterizations for Computable Structures
- White
- 2000
(Show Context)
Citation Context ...r of places where something very close to what we want appears; for example, see the example at the end of Section 5 of Goncharov et al. [19] and surely there are earlier examples (for example, White =-=[39]-=-). All of these constructions work by coding the Harrison ordering. In the full paper we give self-contained proofs of the folklore theorems we use. 3 See Section 5 of the full paper [9] for more info... |

6 | Π 1 1 relations and paths through O
- Goncharov, Harizanov, et al.
(Show Context)
Citation Context ...ely that these theorems were known to Kleene. There are a number of places where something very close to what we want appears; for example, see the example at the end of Section 5 of Goncharov et al. =-=[19]-=- and surely there are earlier examples (for example, White [39]). All of these constructions work by coding the Harrison ordering. In the full paper we give self-contained proofs of the folklore theor... |

5 | Definable encodings in the computably enumerable sets
- Cholak, Harrington
- 2000
(Show Context)
Citation Context ...he following conjecture of Harrington. Conjecture 1.7 (Harrington). For all A and degrees d if A ′ ≤T d ′ is there Â ∈ d such that L∗ (A) ∼ = L ∗ ( Â). For more of these results one can see the paper =-=[12]-=-. 2. New Results The present work is motivated by basic questions about the automorphism group of E ∗ . How complicated is it? If A ≈ Â is A ≈ Â witnessed by an arithmetical automorphism? How complica... |

4 | Isomorphisms of splits of computably enumerable sets
- Cholak, Harrington
(Show Context)
Citation Context ...e direct the reader to Section 5.3. 4. Past Work and Other Connections The paper [9] is a fourth paper in a series of loosely connected papers, the previous three being by Harrington and Cholak [13], =-=[7]-=-, and [8]. We have seen above that results from [8] determine the direction one must take to prove Theorem 2.2. The above results from [8] depend heavily on the main result in [7] whose proof depends ... |

4 | Extension theorems, orbits, and automorphisms of the computably enumerable sets
- Cholak, Harrington
(Show Context)
Citation Context ...s the following. Theorem 2.5. For all finite α > 8 there is a properly ∆ 0 α orbit. Hitherto this paper [9] all known orbits were ∆ 0 3 with the single exception of the orbit of Cholak and Harrington =-=[8]-=- which constructed a pair of sets ∆ 0 5 automorphic but not ∆ 0 3. Before we turn to the proof of Theorem 2.2, we will discuss the background to the Slaman-Woodin Conjecture. Certainly the set {〈i, j〉... |

4 |
A class of incomplete sets
- Marchenkov
- 1976
(Show Context)
Citation Context ...hortly this original programme cannot be solved. But there several solutions to various modified versions of this programme. The earliest solution to a modified Post’s Programme was due to Marchenkov =-=[29]-=- who showed that a certain type of maximal set in a related quotient structure gave a solution. Specifically, if you change the game and replace the integers by computably enumerable equivalence class... |

3 |
The translation theorem
- Cholak
- 1994
(Show Context)
Citation Context ... similar) results, rely on dynamic extension lemmas, of one type or another. In the paper Cholak and Harrington [8], Theorem 4.3 is proven using a modification of Cholak’s Translation Theorem, Cholak =-=[2]-=- Whilst it is not directly pertinent to the present paper, we point out how Cholak and Harrington applied theorems like Theorem 4.3 using the idea of supports. This notion is related to the relationsh... |

3 | On the orbits of computably enumerable sets
- Cholak, Downey, et al.
(Show Context)
Citation Context ...oxes. These four theorems provide a clean interface between the two papers. If one wants to understand the proofs of these four theorems one must go to Cholak and Harrington [8]; otherwise, the paper =-=[9]-=- is completely independent from its three predecessors. In the next section we will explore the statements of Theorems 5.10 and 6.3 of Cholak and Harrington [8] in more detail. 4.1. An Algebraic Frame... |

3 |
Invariance and noninvariance in the lattice of Π 0 1 classes
- Cholak, Downey
(Show Context)
Citation Context ...S 7 We remark that this “failure” methodology has yielded similar definability results in other structures such as the lattice of Π 0 1 classes, as witnessed by Weber [37] and [38], Cholak and Downey =-=[11]-=-, and Downey and Montalbán [14]. Perhaps the best example of the methodology is the following proof of the definability of the double jump classes, the proof using “patterns” which are more or less di... |

3 |
Jumps of hemimaximal sets
- Downey, Stob
- 1991
(Show Context)
Citation Context ...ile it is known by Downey and Harrington [17] that there is no orbit containing sets of all nonzero degrees, the orbit of hemimaximal sets contain representatives of all jump classes (Downey and Stob =-=[15]-=-). We are able to also show that we can construct our orbits to contain at least a fixed hemimaximal degree (possibly along others) or contain all hemimaximal degrees (again possibly along others). Ho... |

2 | Slender classes
- Downey, Montalbán
- 2006
(Show Context)
Citation Context ...e” methodology has yielded similar definability results in other structures such as the lattice of Π 0 1 classes, as witnessed by Weber [37] and [38], Cholak and Downey [11], and Downey and Montalbán =-=[14]-=-. Perhaps the best example of the methodology is the following proof of the definability of the double jump classes, the proof using “patterns” which are more or less direct reflections of blockages t... |

2 |
A definable relation between c.e. sets and ideals
- Weber
- 2004
(Show Context)
Citation Context ...if d1 ≤T D ≤T d2 then S(D).sORBITS 7 We remark that this “failure” methodology has yielded similar definability results in other structures such as the lattice of Π 0 1 classes, as witnessed by Weber =-=[37]-=- and [38], Cholak and Downey [11], and Downey and Montalbán [14]. Perhaps the best example of the methodology is the following proof of the definability of the double jump classes, the proof using “pa... |

2 | Invariance in E ∗ and EΠ
- Weber
(Show Context)
Citation Context ...D ≤T d2 then S(D).sORBITS 7 We remark that this “failure” methodology has yielded similar definability results in other structures such as the lattice of Π 0 1 classes, as witnessed by Weber [37] and =-=[38]-=-, Cholak and Downey [11], and Downey and Montalbán [14]. Perhaps the best example of the methodology is the following proof of the definability of the double jump classes, the proof using “patterns” w... |

1 |
The Computably Enumerable Sets: the Past, the Present and the Future. Theory and Applications of Models of Computation, 2006, Beijing China, Slides can be found at http://www.nd.edu/ ∼cholak
- Cholak
- 2006
(Show Context)
Citation Context ... of these questions. Furthermore, it not clear how this current work impacts possible approaches to these questions. At this point we will just direct the reader to slides of a presentation of Cholak =-=[4]-=-; perhaps a paper reflecting on these issues will appear later. One of the issues that will impact all of these questions are which degrees can be realized in the orbits that we construct in Theorem 2... |

1 |
relations and paths through O
- Goncharov, Harizanov, et al.
(Show Context)
Citation Context ...ely that these theorems were known to Kleene. There are a number of places where something very close to what we want appears; for example, see the example at the end of Section 5 of Goncharov et al. =-=[19]-=- and surely there are earlier examples (for example, White [39]). All of these constructions work by coding the Harrison ordering. In the full paper we give self-contained proofs of the folklore theor... |