Wtt-degrees and T-degrees of r.e. sets

1983 ◽  
Vol 48 (4) ◽  
pp. 921-930 ◽  
Author(s):  
Michael Stob
Keyword(s):  
Minimal Pair ◽  
Turing Degree ◽  
Turing Reducibility

AbstractWe use some simple facts about the wtt-degrees of r.e. sets together with a construction to answer some questions concerning the join and meet operators in the r.e. degrees. The construction is that of an r.e. Turing degree a with just one wtt-degree in a such that a is the join of a minimal pair of r.e. degrees. We hope to illustrate the usefulness of studying the stronger reducibility orderings of r.e. sets for providing information about Turing reducibility.

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.

m-topped degrees

2020 ◽  
pp. 134-148
Author(s):  
Rod Downey ◽  
Noam Greenberg
Keyword(s):  
Turing Degree ◽  
General Study ◽  
Turing Reducibility ◽  
The Many ◽  
Original Construction ◽  
Computable Presentations

This chapter assesses m-topped degrees. The notion of m-topped degrees comes from a general study of the interaction between Turing reducibility and stronger reducibilities among c.e. sets. For example, this study includes the contiguous degrees. A c.e. Turing degree d is m-topped if it contains a greatest degree among the many one degrees of c.e. sets in d. Such degrees were constructed in Downey and Jockusch. The dynamics of the cascading phenomenon occurring in the construction of m-topped degrees strongly resemble the dynamics of the embedding of the 1–3–1 lattice in the c.e. degrees. Similar dynamics occurred in the original construction of a noncomputable left–c.e. real with only computable presentations, which was discussed in the previous chapter.

Principal filters definable by parameters in 𝓔bT

2009 ◽  
Vol 19 (1) ◽  
pp. 153-167
Author(s):  
ANGSHENG LI ◽  
WEILIN LI ◽  
YICHENG PAN ◽  
LINQING TANG
Keyword(s):  
Turing Degree ◽  
Turing Degrees ◽  
Turing Reducibility

We show that there exist c.e. bounded Turing degrees a, b such that 0 < a < 0′, and that for any c.e. bounded Turing degree x, we have b ∨ x = 0′ if and only if x ≥ a. The result gives an unexpected definability theorem in the structure of bounded Turing reducibility.

Transfinite extensions of Friedberg's completeness criterion

1977 ◽  
Vol 42 (1) ◽  
pp. 1-10 ◽  
Author(s):  
John M. Macintyre
Keyword(s):  
Turing Degree ◽  
Completeness Criterion ◽  
Turing Reducibility ◽  
Answering Questions ◽  
Degrees Of Unsolvability

In [3] Friedberg showed that every Turing degree ≥ 0′ is the jump of some degree. Using the relativized version of this theorem it can be shown by finite induction that if a ≥ 0(n)0(n) then there is a b such that b(n) = a (our notation is defined in §1). It is natural to ask whether these results can be extended into the transfinite. Is it true for example, that whenever a ≥ 0(ω) there is a b such that = b(ω)= a? In §2 we use forcing to prove this result. (The usefulness of forcing in answering questions involving the jump operation on degrees of unsolvability has been previously demonstrated in Selman [5] where, for example, forcing was used to construct degrees a and b such that a(ω) = b(ω) = a ∪ b = 0(ω).) In §3 we generalize the methods of §2 to show that if α is a recursive ordinal and a ≥ 0(α) then there is a bsuch that b(α) = a, i.e. the Friedberg result can be extended to all recursive ordinal levels.Thomason [6] used a forcing argument to show: If (the Kleene set of notations for the recursive ordinals) then there is a B such that (the set of notations for ordinals recursive in B). In §4 we show this result holds when hyperarithmetic reducibility is replaced by Turing reducibility: If then there is a B such that .

scholarly journals The importance of Π10 classes in effective randomness

2010 ◽  
Vol 75 (1) ◽  
pp. 387-400 ◽  
Author(s):  
George Barmpalias ◽  
Andrew E.M. Lewis ◽  
Keng Meng Ng
Keyword(s):  
Essential Role ◽  
Minimal Pair ◽  
Turing Degree ◽  
Turing Degrees ◽  
Halting Problem

AbstractWe prove a number of results in effective randomness, using methods in which Π10 classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.

A minimal pair joining to a plus cupping Turing degree

2003 ◽  
Vol 49 (6) ◽  
pp. 553-566 ◽  
Author(s):  
Dengfeng Li ◽  
Angsheng Li
Keyword(s):  
Minimal Pair ◽  
Turing Degree

The universal splitting property. II

1984 ◽  
Vol 49 (1) ◽  
pp. 137-150 ◽  
Author(s):  
M. Lerman ◽  
J. B. Remmel
Keyword(s):  
Turing Degree ◽  
Splitting Theorem ◽  
Splitting Property ◽  
Exhaustive Study

We say that a pair of r.e. sets B and C split an r.e. set A if B ∩ C = ∅ and B ∪ C = A. Friedberg [F] was the first to study the degrees of splittings of r.e. sets. He showed that every nonrecursive r.e. set A has a splitting into nonrecursive sets. Generalizations and strengthenings of Friedberg's result were obtained by Sacks [Sa2], Owings [O], and Morley and Soare [MS].The question which motivated both [LR] and this paper is the determination of possible degrees of splittings of A. It is easy to see that if B and C split A, then both B and C are Turing reducible to A (written B ≤TA and C ≤TA). The Sacks splitting theorem [Sa2] is a result in this direction, as are results by Lachlan and Ladner on mitotic and nonmitotic sets. Call an r.e. set A mitotic if there is a splitting B and C of A such that both B and C have the same Turing degree as A; A is nonmitotic otherwise. Lachlan [Lac] showed that nonmitotic sets exist, and Ladner [Lad1], [Lad2] carried out an exhaustive study of the degrees of mitotic sets.The Sacks splitting theorem [Sa2] shows that if A is r.e. and nonrecursive, then there are r.e. sets B and C splitting A such that B <TA and C <TA. Since B is r.e. and nonrecursive, we can now split B and continue in this manner to produce infinitely many r.e. degrees below the degree of A which are degrees of sets forming part of a splitting of A. We say that an r.e. set A has the universal splitting property (USP) if for any r.e. set D ≤T A, there is a splitting B and C of A such that B and D are Turing equivalent (written B ≡TD).

Durational contrast in gemination and informativity

2018 ◽  
Vol 4 (s2) ◽  
Author(s):  
Shin-Ichiro Sano
Keyword(s):  
Shannon’S Entropy ◽  
Minimal Pair ◽  
Shannon's Entropy ◽  
Linguistic Behavior ◽  
Manner Of Articulation

AbstractRecent studies in Message Oriented Phonology (MOP) have provided increasing evidence that informativity plays a non-trivial role in linguistic behavior. This paper provides a case study of MOP focusing on the durational contrast of singleton and geminate consonants in spoken Japanese. In modern Japanese, short consonants (singletons) and long consonants (geminates) are lexically contrastive, and the durational properties of these consonants are affected by a variety of factors. This provides a useful test of the assumptions of MOP. Based on the assumption that the higher the informativity, the more robustly the contrast is phonetically implemented, this study examines the hypothesis that the durations of singletons and geminates increase or decrease according to the informativity of their durational contrast. The study confirms that (i) the distribution of singletons and geminates is affected by the manner of articulation and positional differences (morpheme-initial, medial, and final); (ii) the distributional differences follow from the informativity of contrasts as represented by Shannon’s entropy; and (iii) the durational contrast is enhanced by the presence or absence of a minimal pair.

scholarly journals Ramsey's theorem and cone avoidance

2009 ◽  
Vol 74 (2) ◽  
pp. 557-578 ◽  
Author(s):  
Damir D. Dzhafarov ◽  
Carl G. Jockusch
Keyword(s):  
Minimal Pair ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Homogeneous Set

AbstractIt was shown by Cholak, Jockusch, and Slaman that every computable 2-coloring of pairs admits an infinite low2 homogeneous set H. We answer a question of the same authors by showing that H may be chosen to satisfy in addition C ≰rH, where C is a given noncomputable set. This is shown by analyzing a new and simplified proof of Seetapun's cone avoidance theorem for Ramsey's theorem. We then extend the result to show that every computable 2-coloring of pairs admits a pair of low2 infinite homogeneous sets whose degrees form a minimal pair.

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