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Definability, Automorphisms, And Dynamic Properties Of Computably Enumerable Sets
 this Bulletin
, 1996
"... . We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly ..."
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. We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility). 1. Introduction. All functions are on the nonnegative integers, # = {0, 1, 2, . . . }, and all sets will be subsets of #. Turing and G odel informally called a function computable if it can be calculated by a mechanical procedure, and regarded this as being synonymous with being specified by an "algorithm" or a "finite combinatorial procedure." They each formalized it as follows. 1 A function is Turing computable if it is definable by a Turing machine, as defined by Turing ...
On the Orbits of Computable Enumerable Sets
, 2007
"... The goal of this paper is to show 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; th ..."
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The goal of this paper is to show 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).
Extending and Interpreting Post’s Programme
, 2008
"... Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic in ..."
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Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals which produced richness of local computabilitytheoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Posts programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Posts programme, and look at two interesting developments of Posts approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computabilitytheoretic infrastructure. We will conclude by viewing Posts programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world.
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.
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).
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...
Extensions, Automorphisms, and Definability
 CONTEMPORARY MATHEMATICS
"... This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to ..."
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This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here
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 ..."
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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