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Applications of TimeBounded Kolmogorov Complexity in Complexity Theory
 Kolmogorov complexity and computational complexity
, 1992
"... This paper presents one method of using timebounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functi ..."
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Cited by 18 (4 self)
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This paper presents one method of using timebounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functions, pseudorandom generators, and hierarchy theorems in circuit complexity.
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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Cited by 10 (5 self)
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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...
1995], Degree theoretic definitions of the low 2 recursively enumerable sets
 J. Symbolic Logic
, 1995
"... 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) mac ..."
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Cited by 7 (5 self)
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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 recursively enumerable degrees
 in Handbook of Computability Theory, Studies in Logic and the Foundations of Mathematics 140
, 1996
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Natural Definability in Degree Structures
"... . A major focus of research in computability theory in recent years has involved denability issues in degree structures. There has been much success in getting general results by coding methods that translate rst or second order arithmetic into the structures. In this paper we concentrate on the ..."
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Cited by 5 (1 self)
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. A major focus of research in computability theory in recent years has involved denability issues in degree structures. There has been much success in getting general results by coding methods that translate rst or second order arithmetic into the structures. In this paper we concentrate on the issues of getting denitions of interesting, apparently external, relations on degrees that are ordertheoretically natural in the structures D and R of all the Turing degrees and of the r.e. Turing degrees, respectively. Of course, we have no formal denition of natural but we oer some guidelines, examples and suggestions for further research. 1. Introduction A major focus of research in computability theory in recent years has involved denability issues in degree structures. The basic question is, which interesting apparently external relations on degrees can actually be dened in the structures themselves, that is, in the rst order language with the single fundamental relation...
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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Cited by 2 (0 self)
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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.
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