TURING DEGREES AND THE ERSHOV HIERARCHY

2009 ◽  
Author(s):  
FRANK STEPHAN ◽  
YUE YANG ◽  
LIANG YU
Keyword(s):  
Turing Degrees ◽  
Ershov Hierarchy

scholarly journals On Transfinite Levels of the Ershov Hierarchy

2021 ◽  
Vol 27 (2) ◽  
pp. 220-221
Author(s):  
Cheng Peng
Keyword(s):  
Recursion Theory ◽  
Density Theorem ◽  
Turing Degrees ◽  
Elementary Substructure ◽  
Ordered Structures ◽  
Research Areas ◽  
Ershov Hierarchy ◽  
Sigma 1 ◽  
E Mail

AbstractIn this thesis, we study Turing degrees in the context of classical recursion theory. What we are interested in is the partially ordered structures $\mathcal {D}_{\alpha }$ for ordinals $\alpha <\omega ^2$ and $\mathcal {D}_{a}$ for notations $a\in \mathcal {O}$ with $|a|_{o}\geq \omega ^2$ .The dissertation is motivated by the $\Sigma _{1}$ -elementary substructure problem: Can one structure in the following structures $\mathcal {R}\subsetneqq \mathcal {D}_{2}\subsetneqq \dots \subsetneqq \mathcal {D}_{\omega }\subsetneqq \mathcal {D}_{\omega +1}\subsetneqq \dots \subsetneqq \mathcal {D(\leq \textbf {0}')}$ be a $\Sigma _{1}$ -elementary substructure of another? For finite levels of the Ershov hierarchy, Cai, Shore, and Slaman [Journal of Mathematical Logic, vol. 12 (2012), p. 1250005] showed that $\mathcal {D}_{n}\npreceq _{1}\mathcal {D}_{m}$ for any $n < m$ . We consider the problem for transfinite levels of the Ershov hierarchy and show that $\mathcal {D}_{\omega }\npreceq _{1}\mathcal {D}_{\omega +1}$ . The techniques in Chapters 2 and 3 are motivated by two remarkable theorems, Sacks Density Theorem and the d.r.e. Nondensity Theorem.In Chapter 1, we first briefly review the background of the research areas involved in this thesis, and then review some basic definitions and classical theorems. We also summarize our results in Chapter 2 to Chapter 4. In Chapter 2, we show that for any $\omega $ -r.e. set D and r.e. set B with $D<_{T}B$ , there is an $\omega +1$ -r.e. set A such that $D<_{T}A<_{T}B$ . In Chapter 3, we show that for some notation a with $|a|_{o}=\omega ^{2}$ , there is an incomplete $\omega +1$ -r.e. set A such that there are no a-r.e. sets U with $A<_{T}U<_{T}K$ . In Chapter 4, we generalize above results to higher levels (up to $\varepsilon _{0}$ ). We investigate Lachlan sets and minimal degrees on transfinite levels and show that for any notation a, there exists a $\Delta ^{0}_{2}$ -set A such that A is of minimal degree and $A\not \equiv _T U$ for all a-r.e. sets U.Abstract prepared by Cheng Peng.E-mail: [email protected]

TRIAL AND ERROR MATHEMATICS II: DIALECTICAL SETS AND QUASIDIALECTICAL SETS, THEIR DEGREES, AND THEIR DISTRIBUTION WITHIN THE CLASS OF LIMIT SETS

2016 ◽  
Vol 9 (4) ◽  
pp. 810-835 ◽  
Author(s):  
JACOPO AMIDEI ◽  
DUCCIO PIANIGIANI ◽  
LUCA SAN MAURO ◽  
ANDREA SORBI
Keyword(s):  
Mathematical Practice ◽  
Turing Degrees ◽  
Trial And Error ◽  
Close Inspection ◽  
Enumeration Degrees ◽  
Deductive Systems ◽  
Ershov Hierarchy ◽  
Image Position ◽  
Ordinal Notation ◽  
Computably Enumerable

AbstractThis paper is a continuation of Amidei, Pianigiani, San Mauro, Simi, & Sorbi (2016), where we have introduced the quasidialectical systems, which are abstract deductive systems designed to provide, in line with Lakatos’ views, a formalization of trial and error mathematics more adherent to the real mathematical practice of revision than Magari’s original dialectical systems. In this paper we prove that the two models of deductive systems (dialectical systems and quasidialectical systems) have in some sense the same information content, in that they represent two classes of sets (the dialectical sets and the quasidialectical sets, respectively), which have the same Turing degrees (namely, the computably enumerable Turing degrees), and the same enumeration degrees (namely, the ${\rm{\Pi }}_1^0$ enumeration degrees). Nonetheless, dialectical sets and quasidialectical sets do not coincide. Even restricting our attention to the so-called loopless quasidialectical sets, we show that the quasidialectical sets properly extend the dialectical sets. As both classes consist of ${\rm{\Delta }}_2^0$ sets, the extent to which the two classes differ is conveniently measured using the Ershov hierarchy: indeed, the dialectical sets are ω-computably enumerable (close inspection also shows that there are dialectical sets which do not lie in any finite level; and in every finite level n ≥ 2 of the Ershov hierarchy there is a dialectical set which does not lie in the previous level); on the other hand, the quasidialectical sets spread out throughout all classes of the hierarchy (close inspection shows that for every ordinal notation a of a nonzero computable ordinal, there is a quasidialectical set lying in ${\rm{\Sigma }}_a^{ - 1}$, but in none of the preceding levels).

Weakly precomplete equivalence relations in the Ershov hierarchy

2019 ◽  
Vol 58 (3) ◽  
pp. 297-319
Author(s):  
N. A. Bazhenov ◽  
B. S. Kalmurzaev
Keyword(s):  
Equivalence Relations ◽  
Ershov Hierarchy

Turing jumps in the Ershov hierarchy

2011 ◽  
Vol 50 (3) ◽  
pp. 279-289
Author(s):  
M. Kh. Faizrakhmanov
Keyword(s):  
Ershov Hierarchy

scholarly journals THE n-r.e. DEGREES: UNDECIDABILITY AND Σ1 SUBSTRUCTURES

2012 ◽  
Vol 12 (01) ◽  
pp. 1250005 ◽  
Author(s):  
MINGZHONG CAI ◽  
RICHARD A. SHORE ◽  
THEODORE A. SLAMAN
Keyword(s):  
Turing Degrees ◽  
First Order ◽  
Global Properties

We study the global properties of [Formula: see text], the Turing degrees of the n-r.e. sets. In Theorem 1.5, we show that the first order of [Formula: see text] is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, [Formula: see text] is not a Σ1-substructure of [Formula: see text].

CEA Operators and the Ershov Hierarchy

2021 ◽  
Vol 65 (8) ◽  
pp. 63-69
Author(s):  
M. M. Arslanov ◽  
I. I. Batyrshin ◽  
M. M. Yamaleev
Keyword(s):  
Ershov Hierarchy

scholarly journals PERMUTATIONS OF THE INTEGERS INDUCE ONLY THE TRIVIAL AUTOMORPHISM OF THE TURING DEGREES

2018 ◽  
Vol 24 (2) ◽  
pp. 165-174
Author(s):  
BJØRN KJOS-HANSSEN
Keyword(s):  
Open Problem ◽  
Main Idea ◽  
Computability Theory ◽  
Turing Degrees ◽  
Nontrivial Automorphism

AbstractIs there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial automorphism must be computable.A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more.

scholarly journals Bounding computably enumerable degrees in the Ershov hierarchy

2006 ◽  
Vol 141 (1-2) ◽  
pp. 79-88 ◽  
Author(s):  
Angsheng Li ◽  
Guohua Wu ◽  
Yue Yang
Keyword(s):  
Ershov Hierarchy ◽  
Computably Enumerable
Sign in / Sign up

Export Citation Format

Share Document