## Effectively Closed Sets and Enumerations (2007)

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### BibTeX

@MISC{Brodhead07effectivelyclosed,

author = {Paul Brodhead and Douglas Cenzer},

title = {Effectively Closed Sets and Enumerations},

year = {2007}

}

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### Abstract

An effectively closed set, or Π 0 1 class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of Π 0 1 classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of Π 0 1 classes and for the subclasses of decidable and of homogeneous Π 0 1 classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends.

### Citations

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Citation Context ...al computable functionals take natural number (m) and real (x) inputs and are indexed as Φe; we will write Φ x e(m) for the result of applying Φe to m and x. We generally follow the notation of Soare =-=[25]-=- for these functions. For example, φe,s denotes that portion φe defined by stage s, and φe(x)↓ means that φe is defined on x (and ↑ means undefined). Let 〈•, •〉 : ω 2 → ω be a computable bijection suc... |

324 | Classical recursion theory - Odifreddi - 1989 |

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(Show Context)
Citation Context ...n an effective numbering. To see that this is a computable numbering, let σ ∈ Re ⇐⇒ Φ σ e ↑ . so that ψ3(e) = [Re] and the trees Re are uniformly primitive recursive. Numbering 4: The Halting Problem =-=[17]-=- Consider the mapping given by ψ4(e) = {x : Φ x e(e)↑}]. This is a computable numbering, since ψ4(e) = [Te], where σ ∈ Te ⇐⇒ Φ σ e (e) ↑ . For any computable tree T , choose a so that Φ σ a(n) converg... |

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(Show Context)
Citation Context ...ays exists. Then let g(σ⌢i) = τ ⌢i for i ∈ {0, 1}. Corollary 6.3. There is no computable numbering of all thin or of all perfect thin Π 0 1 classes. Proof. All thin classes have Lebesgue measure zero =-=[24]-=-. Therefore if e ↦→ Pe were a numbering of (perfect) thin classes then Theorem 6.2 would provide a (perfect) thin class P such that P �= Pe for all e, a contradiction. 7 Small Classes Binns defined in... |

29 |
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(Show Context)
Citation Context ...ew c.e. extensions, (2) essentially undecidable, and (3) well-generated. Some authors have chosen to only impose (1) [4], while others (1) and (2) [5], [8], and finally others (1), (2), and (3) [12], =-=[11]-=-, [3]. The complete consistent extensions of these theories correspond to thin, perfect thin (or equivalently, special thin [5]), and homogenous thin classes, respectively. This section is devoted tow... |

25 |
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(Show Context)
Citation Context ...2 ω , by modifying Friedberg’s original presentation. 6sTheorem 3.1. There is a 1-1 computable numbering of all Π 0 1 classes in 2 ω . Proof. The proof is a modification of the Friedburg construction =-=[14]-=- of an injective numbering for the c.e. sets of natural numbers. Let {We}e∈ω be the computable enumeration of the c.e. subsets of 2 <ω \{∅}. We will construct a computable numbering {Ye : e ∈ ω} in st... |

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Citation Context .... extensions, (2) essentially undecidable, and (3) well-generated. Some authors have chosen to only impose (1) [4], while others (1) and (2) [5], [8], and finally others (1), (2), and (3) [12], [11], =-=[3]-=-. The complete consistent extensions of these theories correspond to thin, perfect thin (or equivalently, special thin [5]), and homogenous thin classes, respectively. This section is devoted towards ... |

13 |
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(Show Context)
Citation Context ... ↑ means undefined). Let 〈•, •〉 : ω 2 → ω be a computable bijection such that 〈0, 0〉 = 0. A and P(A) denote the complement and power set of A, respectively. We generally follow the notation of Cenzer =-=[5]-=- for Π 0 1 classes. In particular, for any σ ∈ {0, 1} { < ω}, I(σ) is the interval of all infinite sequences extending σ. Now a c.e. open set is defined to be the complement of a Π 0 1 class. That is,... |

13 | Theory of Recursive Functions and Effective - Rogers |

9 | Random closed sets
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(Show Context)
Citation Context ...ed. For a given enumeration ψ(e) = Pe of the Π 0 1 classes and a property R of sets, {e : R(Pe)} is said to be an index set. For example, {e : Pe has a computable member} is a Σ 0 3 complete set. See =-=[6]-=- for many more results on index sets. There are several equivalent definitions of Π 0 1 classes; in particular P is a Π 0 1 class if and only if P = [T ] for some primitive recursive tree T and also i... |

8 |
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(Show Context)
Citation Context ...ecidable Π 0 1 class must contain a computable member. P is said to be special if it does not contain a computable member. Enumerations of Π 0 1 classes were given by Lempp [18] and Cenzer and Remmel =-=[7, 8]-=-, where index sets for various families of Π 0 1 classes were analyzed. For a given enumeration ψ(e) = Pe of the Π 0 1 classes and a property R of sets, {e : R(Pe)} is said to be an index set. For exa... |

6 | Countable thin Π01 classes
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(Show Context)
Citation Context ...varied depending upon the context and motivation of the authors, but include: (1) few c.e. extensions, (2) essentially undecidable, and (3) well-generated. Some authors have chosen to only impose (1) =-=[4]-=-, while others (1) and (2) [5], [8], and finally others (1), (2), and (3) [12], [11], [3]. The complete consistent extensions of these theories correspond to thin, perfect thin (or equivalently, speci... |

5 |
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(Show Context)
Citation Context ... that includes all computable trees without dead ends. 1 Introduction The general theory numbers was initiated in the mid-1950s by Kolmogorov, and continued under the direction of Mal’tsev and Ershov =-=[13]-=-. A numbering, or enumeration, of a collection C of objects is a surjective map F : ω → C. In one of the earliest results, Friedberg constucted an injective computable numbering ψ of the Σ 0 1 or comp... |

5 |
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(Show Context)
Citation Context ...putable member, whereas a decidable Π 0 1 class must contain a computable member. P is said to be special if it does not contain a computable member. Enumerations of Π 0 1 classes were given by Lempp =-=[18]-=- and Cenzer and Remmel [7, 8], where index sets for various families of Π 0 1 classes were analyzed. For a given enumeration ψ(e) = Pe of the Π 0 1 classes and a property R of sets, {e : R(Pe)} is sai... |

3 |
Π 0 1 classes
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(Show Context)
Citation Context ...lasses, called string verifiable families in Section 4. In Sections 5-6, numberings for decidable, homogeneous, and thin classes are considered. Finally, in Section 7, motivated by some work by Binns =-=[1]-=-, numberings for small classes are considered. 2sThe partial computable {0, 1}–valued functions are indexed as {φe}e∈ω and primitive recursive functions as {πe}e∈ω. Partial computable functionals take... |

2 | Index Sets for Computable Real Functions - Cenzer, Remmel - 2003 |

2 |
Friedburg Numberings of Families of n-Computably Enumerable Sets
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(Show Context)
Citation Context ...e) = ω ω \ O({n : χe(n) = 1}) is an alternative effective numbering based on total computable functions. It is known, for fixed n > 0, that there is a effective injective numbering of the n-c.e. sets =-=[15]-=-. We conjecture that, for each n, there is a numbering e ↦→ Ne of n-c.e. sets such that there is an injective computable numbering e ↦→ ω ω \ O(Ne) of all closed sets of this form. For n = 1 the resul... |

2 | Relative Randomness via RK-Reducibility
- Raichev
- 2006
(Show Context)
Citation Context ...ve Numberings In this section, we construct a computable injective numbering of the Π 0 1 classes in 2 ω , by modifying Friedberg’s original presentation. An alternative proof was sketched by Raichev =-=[21]-=-. 6Theorem 3.1. There is a 1-1 computable numbering of all Π 0 1 classes in 2 ω . Proof. The proof is a modification of the Friedburg construction [14] of an injective numbering for the c.e. sets of ... |

2 |
Enumerations of recursive sets
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(Show Context)
Citation Context ...nt only as long as it is larger than any previously enumerated element. Applying Friedberg’s argument to this class of c.e. sets yields an effective injective numbering e ↦→ Ce of the computable sets =-=[27]-=-. Furthermore each Ce still enumerates its elements in increasing order. Now suppose {χe}e∈ω is a corresponding set of characteristic functions. One characterization of a Π 0 1 class P is that P = ω ω... |

2 | Computable Aspects of Closed Sets - Brodhead - 2008 |

2 | Thin classes of separating sets - Solomon - 2007 |

1 |
Enumerations of Π 0 1 Classes: Acceptability and Decidable
- Brodhead
- 2006
(Show Context)
Citation Context ... effective numberings, then µ is said to be acceptable. The ordering A preliminary version of this paper appeared in the Proceedings of the 2006 Conference on Computability and Complexity in Analysis =-=[2]-=-. This is part of a Ph.D. dissertation. Research partially supported by National Science Foundation grants DMS 0075899, 0532644 and 0554841. Keywords: Computability, Numberings, Π 0 1 Classes 1s≤ give... |

1 |
Index Sets for Π 0 1
- Cenzer, Remmel
- 1998
(Show Context)
Citation Context ...ecidable Π 0 1 class must contain a computable member. P is said to be special if it does not contain a computable member. Enumerations of Π 0 1 classes were given by Lempp [18] and Cenzer and Remmel =-=[7, 8]-=-, where index sets for various families of Π 0 1 classes were analyzed. For a given enumeration ψ(e) = Pe of the Π 0 1 classes and a property R of sets, {e : R(Pe)} is said to be an index set. For exa... |

1 | Recursively enumerable classes and their application to recursive sequences of formal theories - El, M, et al. - 1965 |

1 |
Thin classes of separating sets, “Proceedings: North Texas Logic Conference
- Solomon
(Show Context)
Citation Context ...tion is devoted towards demonstrating the nonexistence of computable numberings of the first two cases by modifying the classical Martin-Pour El construction of a perfect thin class. Recently Solomon =-=[26]-=- also modified this theorem to construct a homogeneous thin class and therefore we conjecture that no computable numberings exist for these classes. A perfect class may be defined by a function g : 2 ... |

1 | Enumerations of Π 0 1 classes: acceptability and decidable classes, Elect - Brodhead - 2007 |

1 | Index sets for computable real functions, Theor - Cenzer, Remmel - 1999 |