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Codable Sets and Orbits of Computably Enumerable Sets
 J. Symbolic Logic
, 1995
"... 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 ..."
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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 Edefinable 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...
On the definability of the double jump in the computably enumerable sets
 J. MATH. LOG
, 2002
"... 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 ..."
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Cited by 9 (5 self)
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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.
The complexity of orbits of computably enumerable sets
 BULLETIN OF SYMBOLIC LOGIC
, 2008
"... 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 1complete. 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; ..."
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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 1complete. 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).
An Overview of the Computably Enumerable Sets
"... 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 ..."
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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.
Graph Theory. 1. Fragmentation of Structural Graphs
 Journal of Practices and Technologies
, 2002
"... The investigation of structural graphs has many fields of applications in engineering, especially in applied sciences like as applied chemistry and physics, computer sciences and automation, electronics and telecommunication. The main subject of the paper is to express fragmentation criteria in grap ..."
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The investigation of structural graphs has many fields of applications in engineering, especially in applied sciences like as applied chemistry and physics, computer sciences and automation, electronics and telecommunication. The main subject of the paper is to express fragmentation criteria in graph using a new method of investigation: terminal paths. Using terminal paths are defined most of the fragmentation criteria that are in use in molecular topology, but the fields of applications are more generally than that, as I mentioned before. Graphical examples of fragmentation are given for every fragmentation criteria. Note that all fragmentation is made with a computer program that implements a routine for every criterion.[1] A web routine for tracing all terminal paths in graph can be found at the address:
Dynamic Properties of Computably Enumerable Sets
 In Computability, Enumerability, Unsolvability, volume 224 of London Math. Soc. Lecture Note Ser
, 1995
"... 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 t ..."
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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 Edenable) 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 Edenable 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...
Definable properties of the computably enumerable sets
 Proceedings of the Oberwolfach Conference on Computability Theory
, 1996
"... Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, hsimple, and hhsimple, 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 ..."
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Post 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, hsimple, and hhsimple, 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 Edefinable properties of A, like maximal, hhsimple, atomless, to the information content (usually the
Review of Peter Cholak, “Automorphisms of the Lattice of Recursively Enumerable Sets ” Memoirs of the American Math. Soc. (1995) viii+151 pp and
"... Historical Origins. Computability (or recursion) theory grew from our efforts to understand the algorithmic content of mathematics. One of the great achievements of the 20th century is the development of a precise formulation of the notion of a computable function via the ChurchTuring Thesis. In an ..."
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Historical Origins. Computability (or recursion) theory grew from our efforts to understand the algorithmic content of mathematics. One of the great achievements of the 20th century is the development of a precise formulation of the notion of a computable function via the ChurchTuring Thesis. In an very influential paper [Po44], Post articulated some of the fundamental notions at the heart of most undecidability proofs. He observed that these proofs worked by coding some “noncomputability ” into the theory at hand thereby arguing that the relevant structures could emulate computation. One key concept was that of effective enumeration which leads to the notion of a computably (recursively) enumerable set. A computably enumerable set is a subset of N which is the range of a computable (total) function. The intuitive idea is that if f is computable, then I can “effectively list ” (not necessarily in order) the range of f as {f(0), f(1),...}. Think of consequences of a computably enumerable set of axioms for a formal system. The other key concept discovered by Turing [1939] and further developed in Post’s paper