Relative recursive enumerability of generic degrees

1991 ◽  
Vol 56 (3) ◽  
pp. 1075-1084 ◽  
Author(s):  
Masahiro Kumabe
Keyword(s):  
Characteristic Function ◽  
Initial Segment ◽  
Minimal Degree ◽  
Turing Degree ◽  
Natural Numbers ◽  
Recursive Enumerability ◽  
Degree Bounds ◽  
Generic Set ◽  
The Given ◽  
Dense Set

Let ω be the set of natural numbers, i.e. {0, 1, 2, 3, …}. A string is a mapping from an initial segment of ω into {0, 1}. We identify a set A ≤ ω with its characteristic function. A set A ≤ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every set of strings S, there is a σ < A such that σ ∈ S or (∀ν ≥ σ)(ν ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, D(≤ a) denotes the set of degrees recursive in a.The relation between generic degrees and minimal degrees has been widely studied. Spector [9] proved the existence of minimal degrees. Shoenfield [8] simplified the proof by using trees. In the construction of a minimal degree, given σ we extend σ to ν so that ν is in the (splitting or nonsplitting) subtree of a given tree. But in the construction of a generic set, given σ we extend σ to ν to meet the given dense set. So these two constructions are quite different. Jockusch [5] showed that any 2-generic degree bounds no minimal degree. Chong and Jockusch [3] showed that any 1-generic degree below 0′ bounds no minimal degree.

A 1-generic degree which bounds a minimal degree

1990 ◽  
Vol 55 (2) ◽  
pp. 733-743 ◽  
Author(s):  
Masahiro Kumabe
Keyword(s):  
Affirmative Answer ◽  
Minimal Degree ◽  
Turing Degree ◽  
Natural Numbers ◽  
Degree Bounds

Let ω be the set of natural numbers, i.e. {0,1,2,…}. A set A (≤ω) is called n-generic if it is Cohen-generic for n-quantifier arithmetic. As characterized by Jockusch [4], this is equivalent to saying that for every set of strings S, there is a σ < A such that σ ∈ S or ∀ν ≥ σ(ν ∉ S). When we say degree, we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. A nonrecursive degree a is called minimal if there is no nonrecursive degree b with b < a. Jockusch [4] exhibited various properties of generic degrees, and he showed that any 2-generic degree bounds no minimal degree. Chong and Jockusch [1] showed that any 1-generic degree below 0′ bounds no minimal degree. Haught [3] refuted one of the conjectures in [1] and showed that if a is a 1-generic degree and 0 < b < a < 0′ then b is also 1-generic. We show here that there is a 1-generic degree which bounds a minimal degree. This gives an affirmative answer to questions in [1] and [4], As any 1-generic degree below 0′ bounds no minimal degree, we see that our 1-generic degree which bounds a minimal degree is not below 0′, but can be constructed recursively in 0″. Furthermore we see that the initial segments below 1-generic degrees are not order isomorphic.

Every n-generic degree is a minimal cover of an n-generic degree

1993 ◽  
Vol 58 (1) ◽  
pp. 219-231 ◽  
Author(s):  
Masahiro Kumabe
Keyword(s):  
Characteristic Function ◽  
Set Theory ◽  
Initial Segment ◽  
Continuum Hypothesis ◽  
Recursion Theory ◽  
Turing Degree ◽  
Minimal Cover ◽  
The Axiom Of Choice ◽  
Generic Set ◽  
The Continuum

The notions of forcing and generic set were introduced by Cohen in 1963 to prove the independence of the Axiom of Choice and the Continuum Hypothesis in set theory. Let ω be the set of natural numbers, i.e., {0,1,2,3,…}. A string is a mapping from an initial segment of ω into {0,1}. We identify a set A ⊆ ω to with its characteristic function.We now consider a set generic over the arithmetic sets. A set A ⊆ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every -set of strings S, there is a σ ⊂ A such that σ ∈ S or (∀v ≥ σ)(v ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤a) denote the set of degrees which are recursive in a.Before Cohen's work, there was a precursor of the notion of forcing in recursion theory. Friedberg showed that for every degree b above the complete degree 0', i.e., the degree of a complete r.e. set, there is a degree a such that a′ = a ⋃ 0′ = b. He actually proved this result by using the notion of forcing for statements.

A note on degrees of subsets1

1969 ◽  
Vol 34 (2) ◽  
pp. 256-256 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Minimal Degree ◽  
Turing Degree ◽  
Lower Degree ◽  
Natural Numbers ◽  
Infinite Set

In [2] we constructed an infinite set of natural numbers containing no subset of higher (Turing) degree. Since it is well known that there are nonrecursive sets (e.g. sets of minimal degree) containing no nonrecursive subset of lower degree, it is natural to suppose that these arguments may be combined, but this is false. We prove that every infinite set must contain a nonrecursive subset of either higher or lower degree.

Diagonalization in degree constructions

1978 ◽  
Vol 43 (2) ◽  
pp. 280-283 ◽  
Author(s):  
D. Posner ◽  
R. Epstein
Keyword(s):  
Characteristic Function ◽  
Initial Segment ◽  
Partial Function ◽  
Natural Numbers ◽  
Degree Constructions ◽  
Degrees Of Unsolvability

We present two theorems whose applications are to eliminate diagonalization arguments from a variety of constructions of degrees of unsolvability.All definitions and notations come from [1, Chapter 1]. We give a brief resumé of them here.We identify a set with its characteristic function. (A(x)= 1 if x ∈ A and A (x) = 0 if x ∉ A.) A string σ is the restriction of a characteristic function to a finite initial segment of natural numbers, lh(σ) = length of σ = n + 1 if σ = A[n] for some set A. (A [n] is the restriction of A to {m: m ≤ n}.) If i = 0 or 1, σ * i is defined as the string of length lh(σ) + 1 such that σ * i ⊇ σ and σ * i(lh(σ)) = i. We write σ ∣ τ if σ ⊉ τ and τ ⊉ σ.{Φn} is a listing of the partial recursive functionals. We write “A ≤TB” (“A is Turing reducible to B”) if ∃n∀xΦn(B)(x) = A(x).A partial function, T, from strings to strings is a tree if T is order preserving and for all strings, σ, if one of T(σ * 0), T(σ * 1) is defined then T(σ), T(σ * 0), T(σ * 1) are all defined and T(σ * 0)∣(σ * 1).

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

2020 ◽  
Vol 26 (3-4) ◽  
pp. 268-286
Author(s):  
YONG CHENG
Keyword(s):  
Model Theory ◽  
Recursion Theory ◽  
Minimal Degree ◽  
Incompleteness Theorem ◽  
Turing Degree ◽  
Turing Reducibility ◽  
Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

Degrees of classes of RE sets

1976 ◽  
Vol 41 (3) ◽  
pp. 695-696 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Order Theory ◽  
Turing Degree ◽  
Natural Numbers ◽  
First Order ◽  
First Order Theory ◽  
Simple Sets ◽  
Natural Classes

In [3], Martin computed the degrees of certain classes of RE sets. To state the results succinctly, some notation is useful.If A is a set (of natural numbers), dg(A) is the (Turing) degree of A. If A is a class of sets, dg(A) = {dg(A): A ∈ A). Let M be the class of maximal sets, HHS the class of hyperhypersimple sets, and DS the class of dense simple sets. Martin showed that dg(M), dg(HHS), and dg(DS) are all equal to the set H of RE degrees a such that a′ = 0″.Let M* be the class of coinfinite RE sets having no superset in M; and define HHS* and DS* similarly. Martin showed that dg(DS*) = H. In [2], Lachlan showed (among other things) that dg(M*)⊆K, where K is the set of RE degrees a such that a″ > 0″. We will show that K ⊆ dg (HHS*). Since maximal sets are hyperhypersimple, this gives dg(M*) = dg (HHS*) = K.These results suggest a problem. In each case in which dg(A) has been calculated, the set of nonzero degrees in dg(A) is either H or K or the empty set or the set of all nonzero RE degrees. Is this always the case for natural classes A? Natural here might mean that A is invariant under all automorphisms of the lattice of RE sets; or that A is definable in the first-order theory of that lattice; or anything else which seems reasonable.

A Note About Linear Systems on Curves

1984 ◽  
Vol 27 (3) ◽  
pp. 371-374
Author(s):  
Allen Tannenbaum
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Minimal Degree ◽  
Maximal Dimension ◽  
Degree Relative ◽  
Irreducible Curve ◽  
The Given

AbstractInverting the Castelnuovo bound in two ways, we show that for given integers p ≥ 0, d > 1, n > 1, we can find a smooth irreducible curve of genus p which contains a linear system of degree d and of maximal dimension relative to the given data p and d, and a smooth irreducible curve of genus p which contains a linear system of dimension n and of minimal degree relative to the data p and n.

1-genericity in the enumeration degrees

1988 ◽  
Vol 53 (3) ◽  
pp. 878-887 ◽  
Author(s):  
Kate Copestake
Keyword(s):  
Partial Function ◽  
Partial Ordering ◽  
Turing Degree ◽  
Turing Degrees ◽  
Original Formulation ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Generic Set

The structure of the Turing degrees of generic and n-generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]).In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree.Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree ae which is minimal-like; that is, all functions in degrees below ae have partial recursive extensions.

Functional Pearls: The Minout problem

1991 ◽  
Vol 1 (1) ◽  
pp. 121-124
Author(s):  
Richard S. Bird
Keyword(s):  
Natural Number ◽  
Linear Time ◽  
Constant Time ◽  
Natural Numbers ◽  
Free Object ◽  
Functional Program ◽  
Practical Applications ◽  
The World ◽  
The Given ◽  
Easy Solution

The problem of computing the smallest natural number not contained in a given set of natural numbers has a number of practical applications. Typically, the given set represents the indices of a class of objects ‘in use’ and it is required to find a ‘free’ object with smallest index. Our purpose in this article is to derive a linear-time functional program for the problem. There is an easy solution if arrays capable of being accessed and updated in constant time are available, but we aim for an algorithm that employs only standard lists. Noteworthy is the fact that, although an algorithm using lists is the result, the derivation is carried out almost entirely in the world of sets.

scholarly journals On Study of Vertex Labeling of Graph Operations

2011 ◽  
Vol 3 (2) ◽  
pp. 291-301
Author(s):  
M. A. Rajan ◽  
N. M. Kembhavimath ◽  
V. Lokesha
Keyword(s):  
Direct Product ◽  
Graph Labeling ◽  
Natural Numbers ◽  
Graph Vertex ◽  
Direct Sum ◽  
Graph Operations ◽  
Subdivision Graph ◽  
Maximal Vertex ◽  
The Given

Vertices of the graphs are labeled from the set of natural numbers from 1 to the order of the given graph. Vertex adjacency label set (AVLS) is the set of ordered pair of vertices and its corresponding label of the graph. A notion of vertex adjacency label number (VALN) is introduced in this paper. For each VLS, VLN of graph is the sum of labels of all the adjacent pairs of the vertices of the graph. is the maximum number among all the VALNs of the  different labeling of the graph and the corresponding VALS is defined as maximal vertex  adjacency label set . In this paper  for different graph operations are discussed. Keywords: Subdivision; Graph labeling; Direct sum; Direct product.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i26222                  J. Sci. Res. 3 (2), 291-301 (2011) 

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