The Upper Semi-Lattice of Degrees of Recursive Unsolvability

Kleene, S. C.; Post, Emil L.

doi:10.2307/1969708

Cited by 243 publications

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“…However, Kleene and Post [76] showed that this was not the case by exhibiting a pair of degrees a, b (≤ 0 ) which were incomparable. (i.e.…”

Section: The Global Degreesmentioning

confidence: 99%

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Computability theory, algorithmic randomness and Turing's anticipation

Downey

2014

Turing's Legacy

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“…However, Kleene and Post [76] showed that this was not the case by exhibiting a pair of degrees a, b (≤ 0 ) which were incomparable. (i.e.…”

Section: The Global Degreesmentioning

confidence: 99%

“…Due to the work of Kleene and Post [76], as we discuss below, we know that it is not one to one on the degrees. For example, there are sets X ≡ T ∅ with X ≡ T ∅ .…”

mentioning

confidence: 99%

Computability theory, algorithmic randomness and Turing's anticipation

Downey

2014

Turing's Legacy

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“…More generally, Post's theorem [23] (that X eA^+ l^> X < T A {n) ) relates the quantifier forms of many naturally occurring sets of numbers to the ascending sequence 0 < 0' < 0" < < 0 (n+1) = (0 (n) )' < (0 being the degree of the recursive sets). Because of this connection, this jumpderived sequence plays an essential role in calculating bounds on complexity as in bounds for embedding problems in Kleene-Post [14] onwards.…”

Section: S Barry Coopermentioning

confidence: 99%

“…These degrees form a partial ordering 3 under the induced reducibility relation < . The structural analysis of the partial ordering 3 has been a major area of research in recursion theory since the pioneering paper of Kleene and Post [14].Kleene and Post proved a number of results on the structure of 3 including the embeddability of arbitrary countable partial orders into 3, and obtained partial results on extendability of a given embedding to a larger domain. This line of investigation was pursued by many people over the next twenty-five years, culminating in essentially complete solutions of these problems, and a characterization of the possible ideals of the structure 3 (see Lachlan and Lebeuf [16] and Lerman [17], [18]).…”

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The jump is definable in the structure of the degrees of unsolvability

Cooper¹

1990

Bull. Amer. Math. Soc.

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Recursion theory deals with computability on the natural numbers. A function ƒ from N to N is computable (or recursive) if it can be calculated by some program on a Turing machine, or equivalently on any other general purpose computer. A major topic of interest, introduced in Post [23], is the notion of relative difficulty of computation. A function ƒ is computable relative to a function g if after equipping the machine with a black box subroutine that provides the values of g, there is a program (which now may call g via the subroutine) which computes ƒ. In this case we write ƒ < T g. Two functions are Turing equivalent if each is computable relative to the other; the equivalence classes are called Turing degrees. These degrees form a partial ordering 3 under the induced reducibility relation < . The structural analysis of the partial ordering 3 has been a major area of research in recursion theory since the pioneering paper of Kleene and Post [14].Kleene and Post proved a number of results on the structure of 3 including the embeddability of arbitrary countable partial orders into 3, and obtained partial results on extendability of a given embedding to a larger domain. This line of investigation was pursued by many people over the next twenty-five years, culminating in essentially complete solutions of these problems, and a characterization of the possible ideals of the structure 3 (see Lachlan and Lebeuf [16] and Lerman [17], [18]).Kleene and Post also considered the enriched structure 3 1 equipped with the "jump operator", denoted ', which is a canonical operation on degrees which takes each degree d to a strictly

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On a Conjecture of Kleene and Post

Cooper¹

2001

MLQ - Math. Log. Quart.

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A proof is given that 0 (the largest Turing degree containing a computably enumerable set) is definable in the structure of the degrees of unsolvability. This answers a long-standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.Mathematics Subject Classification: 03D25, 03D30, 03D35.Following Gödel [8], Church [1] and Turing's [33] discovery that most familiar mathematical theories are undecidable, the existence of a noncomputable universe intimately connected with the relatively small world of everyday mathematics has led to the development (see Kleene and Post [14]) of the degrees of unsolvability D as an appropriate theoretical framework, or fine structure theory. While more mundane considerations dictated a direct interest in the details of the computable universe, the early work of Kleene and Post [14] presaged an ever deepening understanding of the nature of the noncomputable universe, a development both of philosophical significance and of wider technical application.The most important noncomputable degree 0 is that containing the (coded) undecidable axiomatic theories of Gödel, as well as many other natural mathematical objects. When relativised to a set A of degree a, it gives rise to a jump operator taking a to a strictly higher degree a , defined as the largest degree containing sets which are computably enumerable (written c. e.) using oracle A. Post's Theorem [23] showed a close relationship between the quantifier forms of most naturally occuring sets of numbers and the ascending sequence 0 < 0 < 0 < · · · < 0 (n+1) = (0 (n) ) < · · · (0 being the degree of the computable sets). 0 and the jump operator are basic

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The Upper Semi-Lattice of Degrees of Recursive Unsolvability

Cited by 243 publications

References 10 publications

Computability theory, algorithmic randomness and Turing's anticipation

Computability theory, algorithmic randomness and Turing's anticipation

The jump is definable in the structure of the degrees of unsolvability

On a Conjecture of Kleene and Post

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