Bounded query classes and the difference hierarchy

1989 ◽  
Vol 29 (2) ◽  
pp. 69-84 ◽  
Author(s):  
Richard Beigel ◽  
William I. Gasarch ◽  
Louise Hay
Keyword(s):  
Difference Hierarchy ◽  
The Difference

Equivalence structures and isomorphisms in the difference hierarchy

2009 ◽  
Vol 74 (2) ◽  
pp. 535-556 ◽  
Author(s):  
Douglas Cenzer ◽  
Geoffrey Laforte ◽  
Jeffrey Remmel
Keyword(s):  
Equivalence Relations ◽  
Computable Categoricity ◽  
Difference Hierarchy ◽  
The Difference

AbstractWe examine the effective categoricity of equivalence structures via Ershov's difference hierarchy. We explore various kinds of categoricity available by distinguishing three different notions of isomorphism available in this hierarchy. We prove several results relating our notions of categoricity to computable equivalence relations: for example, we show that, for such relations, computable categoricity is equivalent to our notion of weak ω-c.e. categoricity, and that -categoricity is equivalent to our notion of graph-ω-c.e. categoricity.

scholarly journals Towards a descriptive theory of cb0-spaces

2016 ◽  
Vol 27 (8) ◽  
pp. 1553-1580 ◽  
Author(s):  
VICTOR SELIVANOV
Keyword(s):  
Set Theory ◽  
Descriptive Set Theory ◽  
Difference Hierarchy ◽  
Descriptive Theory ◽  
Wadge Hierarchy ◽  
The Difference ◽  
Descriptive Set

The paper tries to extend some results of the classical Descriptive Set Theory to as many countably basedT0-spaces (cb0-spaces) as possible. Along with extending some central facts about Borel, Luzin and Hausdorff hierarchies of sets we also consider the more general case ofk-partitions. In particular, we investigate the difference hierarchy ofk-partitions and the fine hierarchy closely related to the Wadge hierarchy.

Analytic determinacy and 0#

1978 ◽  
Vol 43 (4) ◽  
pp. 685-693 ◽  
Author(s):  
Leo Harrington
Keyword(s):  
Borel Sets ◽  
Difference Hierarchy ◽  
The Difference

Martin [12] has shown that the determinacy of analytic games is a consequence of the existence of sharps. Our main result is the converse of this:Theorem. If analytic games are determined, then x2 exists for all reals x.This theorem answers question 80 of Friedman [5]. We actually obtain a somewhat sharper result; see Theorem 4.1. Martin had previously deduced the existence of sharps from 3 − Π11-determinacy (where α − Π11 is the αth level of the difference hierarchy based on − Π11 see [1]). Martin has also shown that the existence of sharps implies < ω2 − Π11-determinacy.Our method also produces the following:Theorem. If all analytic, non-Borel sets of reals are Borel isomorphic, then x* exists for all reals x.The converse to this theorem had been previously proven by Steel [7], [18].We owe a debt of gratitude to Ramez Sami and John Steel, some of whose ideas form basic components in the proofs of our results.For the various notation, definitions and theorems which we will assume throughout this paper, the reader should consult [3, §§5, 17], [8], [13] and [14, Chapter 16].Throughout this paper we will concern ourselves only with methods for obtaining 0# (rather than x# for all reals x). By relativizing our arguments to each real x, one can produce x2.

Splitting and non-splitting in the difference hierarchy

2016 ◽  
Vol 28 (3) ◽  
pp. 384-391
Author(s):  
MARAT ARSLANOV
Keyword(s):  
Difference Hierarchy ◽  
Open Questions ◽  
The Difference

In this paper, we investigate splitting and non-splitting properties in the Ershov difference hierarchy, in which area major contributions have been made by Barry Cooper with his students and colleagues. In the first part of the paper, we give a brief survey of his research in this area and discuss a number of related open questions. In the second part of the paper, we consider a splitting of 0′ with some additional properties.

A noninitial segment of index sets

1974 ◽  
Vol 39 (2) ◽  
pp. 209-224 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Partial Order ◽  
Initial Segment ◽  
Finite Sets ◽  
Index Set ◽  
Difference Hierarchy ◽  
Recursively Enumerable ◽  
Index Sets ◽  
The Difference ◽  
The One ◽  
Standard Enumeration

Let {Wk}k ≥ 0 be a standard enumeration of all recursively enumerable (r.e.) sets. If A is any class of r.e. sets, let θA denote the index set of A, i.e., θA = {k ∣ Wk ∈ A}. The one-one degrees of index sets form a partial order ℐ which is a proper subordering of the partial order of all one-one degrees. Denote by ⌀ the one-one degree of the empty set, and, if b is the one-one degree of θB, denote by the one-one degree of . Let . Let {Ym}m≥0 be the sequence of index sets of nonempty finite classes of finite sets (classified in [5] and independently, in [2]) and denote by am the one-one degree of Ym. As shown in [2], these degrees are complete at each level of the difference hierarchy generated by the r.e. sets. It was proved in [3] that, for each m ≥ 0,(a) am+1 and ām+1 are incomparable immediate successors of am and ām, and(b) .For m = 0, since Y0 = θ{⌀}, it follows from (a) that(c) .Hence it follows that(d) {⌀, , ao, ā0, a1, ā1 is an initial segment of ℐ.

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