scholarly journals Classifying equivalence relations in the Ershov hierarchy

2020 ◽  
Vol 59 (7-8) ◽  
pp. 835-864
Author(s):  
Nikolay Bazhenov ◽  
Manat Mustafa ◽  
Luca San Mauro ◽  
Andrea Sorbi ◽  
Mars Yamaleev
Keyword(s):  
Equivalence Relations ◽  
Special Focus ◽  
Ershov Hierarchy ◽  
Standard Tool ◽  
Algebraic Properties ◽  
Sigma 1 ◽  
Degree Structure ◽  
Computably Enumerable

Abstract Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $$\leqslant _c$$ ⩽ c . This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the $$\Delta ^0_2$$ Δ 2 0 case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by $$\leqslant _c$$ ⩽ c on the $$\Sigma ^{-1}_{a}\smallsetminus \Pi ^{-1}_a$$ Σ a - 1 \ Π a - 1 equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.

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

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

scholarly journals Nutritional Status as a Mediator of Fatigue and Its Underlying Mechanisms in Older People

Nutrients ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 444 ◽  
Author(s):  
Domenico Azzolino ◽  
Beatrice Arosio ◽  
Emanuele Marzetti ◽  
Riccardo Calvani ◽  
Matteo Cesari
Keyword(s):  
Older People ◽  
Healthcare Professionals ◽  
Special Focus ◽  
Incomplete Knowledge ◽  
Level Of Activity ◽  
Standard Tool ◽  
Underlying Mechanisms ◽  
Inadequate Nutrition ◽  
Composition Changes

Fatigue is an often-neglected symptom but frequently complained of by older people, leading to the inability to continue functioning at a normal level of activity. Fatigue is frequently associated with disease conditions and impacts health status and quality of life. Yet, fatigue cannot generally be completely explained as a consequence of a single disease or pathogenetic mechanism. Indeed, fatigue mirrors the exhaustion of the physiological reserves of an older individual. Despite its clinical relevance, fatigue is typically underestimated by healthcare professionals, mainly because reduced stamina is considered to be an unavoidable corollary of aging. The incomplete knowledge of pathophysiological mechanisms of fatigue and the lack of a gold standard tool for its assessment contribute to the poor appreciation of fatigue in clinical practice. Inadequate nutrition is invoked as one of the mechanisms underlying fatigue. Modifications in food intake and body composition changes seem to influence the perception of fatigue, probably through the mechanisms of inflammation and/or mitochondrial dysfunction. Here, we present an overview on the mechanisms that may mediate fatigue levels in old age, with a special focus on nutrition.

scholarly journals Jumps of computably enumerable equivalence relations

2018 ◽  
Vol 169 (3) ◽  
pp. 243-259 ◽  
Author(s):  
Uri Andrews ◽  
Andrea Sorbi
Keyword(s):  
Equivalence Relations ◽  
Computably Enumerable

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]

Algebraic Properties of Rough Set on Two Universal Sets based on Multigranulation

2014 ◽  
Vol 1 (2) ◽  
pp. 49-61 ◽  
Author(s):  
Mary A. Geetha ◽  
D. P. Acharjya ◽  
N. Ch. S. N. Iyengar
Keyword(s):  
Information System ◽  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Granular Computing ◽  
Real Life ◽  
Equivalence Relations ◽  
Life Problems ◽  
Algebraic Properties ◽  
The Universe

The rough set philosophy is based on the concept that there is some information associated with each object of the universe. The set of all objects of the universe under consideration for particular discussion is considered as a universal set. So, there is a need to classify objects of the universe based on the indiscernibility relation (equivalence relation) among them. In the view of granular computing, rough set model is researched by single granulation. The granulation in general is carried out based on the equivalence relation defined over a universal set. It has been extended to multi-granular rough set model in which the set approximations are defined by using multiple equivalence relations on the universe simultaneously. But, in many real life scenarios, an information system establishes the relation with different universes. This gave the extension of multi-granulation rough set on single universal set to multi-granulation rough set on two universal sets. In this paper, we define multi-granulation rough set for two universal sets U and V. We study the algebraic properties that are interesting in the theory of multi-granular rough sets. This helps in describing and solving real life problems more accurately.

UNIVERSAL COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 60-88 ◽  
Author(s):  
URI ANDREWS ◽  
STEFFEN LEMPP ◽  
JOSEPH S. MILLER ◽  
KENG MENG NG ◽  
LUCA SAN MAURO ◽  
...  
Keyword(s):  
Computable Function ◽  
Order Theory ◽  
Equivalence Relations ◽  
Index Set ◽  
First Order ◽  
First Order Theory ◽  
Bounded Poset ◽  
Image Position ◽  
Open Question ◽  
Computably Enumerable

Abstract We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$ , where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

LINEAR ORDERS REALIZED BY C.E. EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (2) ◽  
pp. 463-482 ◽  
Author(s):  
EKATERINA FOKINA ◽  
BAKHADYR KHOUSSAINOV ◽  
PAVEL SEMUKHIN ◽  
DANIEL TURETSKY
Keyword(s):  
Ordered Set ◽  
Equivalence Relations ◽  
Ordered Sets ◽  
Linear Orders ◽  
Algebraic Properties ◽  
Image Position ◽  
Quotient Set ◽  
The Relationship ◽  
Induced Structure ◽  
Linearly Ordered Set

AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$(or equivalently, the relationE)realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure ($\omega /E$; ⊴) is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.

THE COMPLEXITY OF INDEX SETS OF CLASSES OF COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (4) ◽  
pp. 1375-1395 ◽  
Author(s):  
URI ANDREWS ◽  
ANDREA SORBI
Keyword(s):  
Open Problem ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Index Set ◽  
Index Sets ◽  
Image Position ◽  
Computably Enumerable

AbstractLet$ \le _c $be computable the reducibility on computably enumerable equivalence relations (or ceers). We show that for every ceerRwith infinitely many equivalence classes, the index sets$\left\{ {i:R_i \le _c R} \right\}$(withRnonuniversal),$\left\{ {i:R_i \ge _c R} \right\}$, and$\left\{ {i:R_i \equiv _c R} \right\}$are${\rm{\Sigma }}_3^0$complete, whereas in caseRhas only finitely many equivalence classes, we have that$\left\{ {i:R_i \le _c R} \right\}$is${\rm{\Pi }}_2^0$complete, and$\left\{ {i:R \ge _c R} \right\}$(withRhaving at least two distinct equivalence classes) is${\rm{\Sigma }}_2^0$complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is${\rm{\Pi }}_4^0$complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a${\rm{\Sigma }}_3^0$complete preordering relation, a fact that is used to show that the preordering relation$ \le _c $on ceers is a${\rm{\Sigma }}_3^0$complete preordering relation.

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