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

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 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.

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.

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.

scholarly journals AL-AKHTHA’ FII IKTISAB AL-TSUNAIYAT AL-SHUGHRA

2011 ◽  
Vol 5 (1) ◽  
Author(s):  
Miftahul Huda
Keyword(s):  
Saudi Arabia ◽  
Language Acquisition ◽  
Foreign Language ◽  
Error Analysis ◽  
Language Learning ◽  
Mother Tongue ◽  
Minimal Pair ◽  
Minimal Pairs ◽  
The Subject ◽  
Child Age

Language acquisition starts from the ability of listening basic letter(iktisab al-ashwat) since child age. The letter of a language is limited in number, and sometimes there is similarity of letters among languages. The similarity of letters in two languages make it easy to learn the language. On the contrary, the obstacle of language learning can be caused by different letters between two languages (mother tongue and second/foreign language). The problem may be caused by minimal pairs (tsunaiyat al-shughro). This research aims at finding out the error of minimal pair acquisition, with the subject of Indonesian students in Jami’ah Malik Saud Saudi Arabia, with the method of error analysis. The study concludes that in iktisab al-ashwat of minimal pairs, the error is around 3,3 %-58,3%. Second: the error on minimal pairs occurs on the letters shift ?? ?? ? to be ? , letter ? to be ? , letter ? to be ? , letter ? to be ? , letter ? to be ?? , and letter ? to be ?.

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.

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