AsetAof nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E =({We}e∈ω,⊆). Most previously known automorphisms Φ of the structure E of

A set X of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let E denote the structure of the computably enumerable sets under inclusion, E = (fW e g e2! ; `). We previously exhibited a first order E-definable property Q(X) such that Q(X) guarantees that X is not Turing complete (i.e., does not code complete information about c.e. sets). Here we show first that Q(X) implies that X has a certain "slowness " property whereby the elements must enter X slowly (under a certain precise complexity measure of speed of computation) even though X may have high information content. Second we prove that every X with this slowness property is computable in some member of any nontrivial orbit, namely for any noncomputable A 2 E there exists B in the orbit of A such that X T B under relative Turing computability ( T ). We produce B using the \Delta 0 3 -automorphism method we introduced earli...

Abstract. We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: Let C = {a: a is the Turing degree of a � 0 3 set J ≥T 0 ′ ′}. Let D ⊆ C such that D is upward closed in C. Then there is an L(A) property ϕD(A) such that F ′ ′ ∈ D iff there is an A where A ≡T F and ϕD(A). A corollary of this is that, for all n ≥ 2, the highn (lown) computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin’s Invariance Conjecture. 1.

1. Introduction. The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤T B, where arbitrary (Turing) machines, ϕe, can be used; access to

The purpose of this communication is to announce some recent results on the computably enumerable sets. There are two disjoint sets of results; the first involves invariant classes and the second involves automorphisms of the computably enumerable sets. What these results have in common is

We show the existence of a high r.e. degree bounding only joins of minimal pairs and of a high 2 nonbounding r.e. degree. 0 Introduction An important topic in the study of recursively enumerable sets and degrees has been the interaction between the jump operator and the order theoretic properties of an r. e. set A (in the lattice E of all r. e. sets) and of its degree a in R, the upper semilattice of the r. e. degrees. An early theme in this area was the idea that sets with "low" jumps should behave like the recursive sets while those with "high" jumps should exhibit properties like the complete sets. For example, in the lattice E of r. e. sets modulo finite sets, we know from Soare[23] that if A is low, i. e. A 0 j T ; 0 , then L (A), the lattice of r. e. supersets of A, is isomorphic to E . In R there are many instances of the low Partially supported by IGC of Vicoria University, Wellington and ARO through MSI, Cornell University, DAAL-03-C-0027. y Partailly suppo...

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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...

The purpose of this article is to summarize some of the results on the algebraic structure of the computably enumerable (c.e.) sets since 1987 when the subject was covered in Soare 1987 , particularly Chapters X, XI, and XV. We study the c.e. sets as a partial ordering under inclusion, (E; `). We do not study the partial ordering of the c.e. degrees under Turing reducibility, although a number of the results here relate the algebraic structure of a c.e. set A to its (Turing) degree in the sense of the information content of A. We consider here various properties of E: (1) deønable properties; (2) automorphisms; (3) invariant properties; (4) decidability and undecidability results; miscellaneous results. This is not intended to be a comprehensive survey of all results in the subject since 1987, but we give a number of references in the bibliography to other results. 1 A Brief History of C.E. Sets G#del 1934 introduced the deønition of a (general) recursive function, and Church 1936 p...

Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, h-simple, and hh-simple, with the intent of finding a property guaranteeing incompleteness of A. From observations of Post 1943 and Myhill 1956, attention focused by the 1950's on properties definable in the inclusion ordering of c.e. subsets of!, namely E = (fWngn2! ; ae). In the 1950's and 1960's Tennenbaum, Martin, Yates, Sacks, Lachlan, Shoenfield and others produced a number of elegant results relating E-definable properties of A, like maximal, hh-simple, atomless, to the information content (usually the