## Dynamic Properties of Computably Enumerable Sets (1995)

Venue: | In Computability, Enumerability, Unsolvability, volume 224 of London Math. Soc. Lecture Note Ser |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Harrington95dynamicproperties,

author = {Leo Harrington and Robert I. Soare},

title = {Dynamic Properties of Computably Enumerable Sets},

booktitle = {In Computability, Enumerability, Unsolvability, volume 224 of London Math. Soc. Lecture Note Ser},

year = {1995},

pages = {105--121},

publisher = {Cambridge Univ. Press}

}

### OpenURL

### Abstract

A set A ` ! is computably enumerable (c.e.), also called recursively enumerable, (r.e.), or simply enumerable, if there is a computable algorithm to list its members. Let E denote the structure of the c.e. sets under inclusion. Starting with Post [1944] there has been much interest in relating the denable (especially E-denable) properties of a c.e. set A to its iinformation contentj, namely its Turing degree, deg(A), under T , the usual Turing reducibility. [Turing 1939]. Recently, Harrington and Soare answered a question arising from Post's program by constructing a nonemptly E-denable property Q(A) which guarantees that A is incomplete (A !T K). The property Q(A) is of the form (9C)[A ae m C & Q \Gamma (A; C)], where A ae m C abbreviates that iA is a major subset of Cj, and Q \Gamma (A; C) contains the main ingredient for incompleteness. A dynamic property P (A), such as prompt simplicity, is one which is dened by considering how fast elements elements enter A relat...

### Citations

120 |
Recursively enumerable sets of positive integers and their decision problems
- Post
- 1944
(Show Context)
Citation Context ...undation Grant DMS 91-06714 and DMS 94-00825. 1 Dynamic Properties of Computably Enumerable Sets 2 1 Introduction Warning. From now on all sets and degrees will be c.e. unless specied otherwise. Post =-=[16]-=- initiated the study of the relationship between denable properties of a c.e. set A and its iinformation contentj as measured by its Turing degree, deg(A), under the usual Turing reducibility T . By t... |

54 |
Classes of recursively enumerable sets and degrees of unsolvability. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik
- Martin
- 1966
(Show Context)
Citation Context ...000 -method) and used to prove a number of theorems on c.e. sets and degrees. Sacks used the second method to construct an incomplete maximal set, Yates constructed a complete maximal set, and Martin =-=[15]-=- brought these results together and extended them in his beautiful theorem that the degrees of maximal sets are exactly H 1 , the high degrees. Then Lachlan [8] and Shoeneld [17] proved that the degre... |

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 ...me dramatic advances in the subject. Maass [12] proved that any two promptly simple low sets are automorphic and discovered other properties of these sets [13]. AmbosSpies, Jockusch, Shore, and Soare =-=[1]-=- used prompt degrees to unify and extend results about r.e. degrees, and promptness has been very inAEuential ever since. (See [19, Chap. XIII].) 2.4 Almost Prompt Sets and Degrees The material from t... |

32 | Automorphisms of the Lattice of Recursively Enumerable Sets - Cholak - 1995 |

24 | Soare, Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets - I - 1987 |

21 | Post’s Program and incomplete recursively enumerable sets
- Harrington, Soare
- 1991
(Show Context)
Citation Context ...by showing that every nonrecursive set A has a complete set in its orbit. However, Harrington and Soare gave a negative answer to this question by proving the following. Theorem 1.1 (Harrington-Soare =-=[3]-=-) There is a nonempty E-denable property Q(A) such that every c.e. set A satisfying Q(A) is noncomputable and Turing incomplete. The property, which we shall describe fully in #4, is in two parts, Q(A... |

19 | Turing, “Systems of logic based on ordinals - M - 1939 |

17 | On some games which are relevant to the theory of recursively enumerable sets - Lachlan - 1970 |

15 |
Recursively enumerable generic sets
- Maass
- 1982
(Show Context)
Citation Context ...aining insight into the completeness phenomenon and the rst part of the question. 2.3 Promptly Simple Sets The next signicant advance came with the following denition of promptly simple sets by Maass =-=[12]-=-. Dynamic Properties of Computably Enumerable Sets 5 Denition 2.3 (i) A coinnite r.e. set A is promptly simple if there is a computable function p and a computable enumeration f A s g s2! of A such th... |

14 |
Degrees of recursively enumerable sets which have no maximal supersets
- Lachlan
(Show Context)
Citation Context ...ted a complete maximal set, and Martin [15] brought these results together and extended them in his beautiful theorem that the degrees of maximal sets are exactly H 1 , the high degrees. Then Lachlan =-=[8]-=- and Shoeneld [17] proved that the degrees of the atomless sets (those with no maximal supersets) are L 2 , the complement of the low 2 degrees. Both properties of being maximal or atomless are E-dena... |

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

12 |
d-Simple sets, small sets, and degree classes
- Lerman, Soare
- 1980
(Show Context)
Citation Context ...ays of r.e. sets, fU n g n2! and f b V g n2! , and it was important to measure for an element x which U n sets it entered before entering certain b Vm sets. 2.2 d-simple sets In 1980 Lerman and Soare =-=[11]-=- attempted to capture part of the dynamic property of the Extension Theorem with an E-denable property which is called d-simple, but they succeeded in capturing only a small part. Denition 2.1 A coinn... |

10 | Codable sets and orbits of computably enumerable sets
- Harrington, Soare
- 1998
(Show Context)
Citation Context ... ivery tardy.j Note that A is 0-tardy ioe A = !, and A is 1-tardy ioe A is recursive. The 2-tardy sets play a special role in our work and have additional characterizations as follows, as we prove in =-=[5]-=-. Proposition 2.8 (Harrington-Soare [5]) For an r.e. set A the following are equivalent: (i) A is 2-tardy; (ii) For every nondecreasing recursive function p(s), (9W i ' A)(9W e = A)(8y)(8s)[y 2 W i;s ... |

7 |
The elementary theory of recursively enumerable sets
- Lachlan
- 1968
(Show Context)
Citation Context ...e following are equivalent: (i) A is 2-tardy; (ii) For every nondecreasing recursive function p(s), (9W i ' A)(9W e = A)(8y)(8s)[y 2 W i;s \Gamma W e;s =) y 62 A p(s) ]]: (14) 3 Small Subsets Lachlan =-=[9]-=- introduced small sets in his program to construct canonical examples of certain diagrams and then rule out possible extensions so as to give a decision procedure for the \Gamma! 8 \Gamma! 9 -theory o... |

6 |
Variations on promptly simple sets
- Maass
- 1985
(Show Context)
Citation Context ...mptly simply sets and degrees helped bring some dramatic advances in the subject. Maass [12] proved that any two promptly simple low sets are automorphic and discovered other properties of these sets =-=[13]-=-. AmbosSpies, Jockusch, Shore, and Soare [1] used prompt degrees to unify and extend results about r.e. degrees, and promptness has been very inAEuential ever since. (See [19, Chap. XIII].) 2.4 Almost... |

6 |
Soare, Automorphisms of the recursively enumerable sets, Part I: Maximal sets
- I
- 1974
(Show Context)
Citation Context ...he degrees of the atomless sets (those with no maximal supersets) are L 2 , the complement of the low 2 degrees. Both properties of being maximal or atomless are E-denable properties. Meanwhile Soare =-=[18]-=- developed a new method for generating automorphisms of E, and used it to show that maximal sets form an orbit. (The orbit of A 2 E is the set of all sets B which are automorphic to A, written A ' B.)... |

5 | Soare, The \Delta 3 -Automorphism Method and Noninvariant Classes of Degrees - Harrington, I - 1996 |

3 | Soare, Denable properties of the computably enumerable sets, in preparation - Harrington, I - 1996 |

2 | Soare, Noninvariance of the nonlow computably enumerable degrees, in preparation - Harrington, I |

2 |
Degrees of classes of r.e
- Shoeneld
- 1976
(Show Context)
Citation Context ...ximal set, and Martin [15] brought these results together and extended them in his beautiful theorem that the degrees of maximal sets are exactly H 1 , the high degrees. Then Lachlan [8] and Shoeneld =-=[17]-=- proved that the degrees of the atomless sets (those with no maximal supersets) are L 2 , the complement of the low 2 degrees. Both properties of being maximal or atomless are E-denable properties. Me... |

2 |
The Structure and Elementary Theory of the Recursive Enumerable Sets
- Stob
- 1979
(Show Context)
Citation Context ...C, written A ae s C, in connection with his decision procedure for part of the elementary theory of E as described in #3. This notion proved useful and other facts about small sets were added by Stob =-=[20]-=- (see [19, pp. 193195]), and others. The property b Q(A) = (9C)[A ae s C] comes tantalizingly close to being a property like Q(A) which guarantees A incomplete, but not quite. We note that b Q(A) impl... |