Index sets in Ershov's hierarchy

1974 ◽  
Vol 39 (1) ◽  
pp. 97-104 ◽  
Author(s):  
Jacques Grassin
Keyword(s):  
Recursion Theory ◽  
Case Note ◽  
Boolean Combination ◽  
Open Set ◽  
Recursively Enumerable ◽  
Power Set ◽  
Index Sets ◽  
Recursively Enumerable Sets ◽  
Complete Set ◽  
Recursive Isomorphism

This work is an attempt to characterize the index sets of classes of recursively enumerable sets which are expressible in terms of open sets in the Baire topology on the power set of the set N of natural numbers, usual in recursion theory. Let be a class of subsets of N and be the set of indices of recursively enumerable sets Wх belonging to .A well-known theorem of Rice and Myhill (cf. [5, p. 324, Rice-Shapiro Theorem]) states that is recursively enumerable if and only ifis a r.e. open set. In this case, note that if is not empty and does not contain all recursively enumerable sets, is a complete set. This theorem will be partially extended to classes which are boolean combinations of open sets by the following:(i) There is a canonical boolean combination which represents, namely the shortest among boolean combinations which represent.(ii) The recursive isomorphism type of depends on the length n of this canonical boolean combination (and trivial properties of ); for instance, is recursively isomorphic (in the particular case where is a boolean combination of recursive open sets) to an elementary set combination Yn or Un, constructed from {х ∣ х Wх) and depending on the length n. We can say also that is a complete set in the sense of Ershov's hierarchy [1] (in this particular case).

Index sets of finite classes of recursively enumerable sets

1969 ◽  
Vol 34 (1) ◽  
pp. 39-44 ◽  
Author(s):  
Louise Hay
Keyword(s):  
Index Set ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Index Sets ◽  
Recursively Enumerable Sets ◽  
Partial Recursive Functions ◽  
Recursive Isomorphism ◽  
Standard Enumeration

Let q0, q1,… be a standard enumeration of all partial recursive functions of one variable. For each i, let wi = range qi and for any recursively enumerable (r.e.) set α, let θα = {n | wn = α}. If A is a class of r.e. sets, let θA = the index set of A = {n | wn ∈ A}. It is the purpose of this paper to classify the possible recursive isomorphism types of index sets of finite classes of r.e. sets. The main theorem will also provide an answer to the question left open in [2] concerning the possible double isomorphism types of pairs (θα, θβ) where α ⊂ β.

scholarly journals Recursive properties of isomorphism types

1983 ◽  
Vol 34 (2) ◽  
pp. 269-286 ◽  
Author(s):  
Michael Moses
Keyword(s):  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Recursive Isomorphism

AbstractFor Γ a revursively enumerable set of formulae, a structure U on a recursive universe is said to be “Γ-recursively enumerable” if the satisfaction predicate restricted to Γ is recursively enumerable (equivalently, if the formulae of Γ uniformulae of Γ uniformly denote recursively enumerable relations on U).For recursively enumerable sets Γ1 ⊆ Γ2 of formulae we shall, under certain conditions, characterize structures U with the following properties.1) Every isomorphism form U to a Γ1-recursively enumerable structure is a recursive isomorphism.2) Every Γ1-recursively enumerable structure isomorphic to U is recursively isomorphic to U.3) Every Γ1-recursively enumerable structure isomorphic to U is Γ2-recursively enumerable.

A representation of recursively enumerable sets through Horn formulas in higher recursion theory

2016 ◽  
Vol 73 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Juan A. Nido Valencia ◽  
Julio E. Solís Daun ◽  
Luis M. Villegas Silva
Keyword(s):  
Recursion Theory ◽  
Recursively Enumerable ◽  
Horn Formulas ◽  
Recursively Enumerable Sets

Nowhere simple sets and the lattice of recursively enumerable sets

1978 ◽  
Vol 43 (2) ◽  
pp. 322-330 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Boolean Algebra ◽  
Classification Scheme ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
The Other ◽  
Finite Sets ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets ◽  
Invariant Properties

Ever since Post [4] the structure of recursively enumerable sets and their classification has been an important area in recursion theory. It is also intimately connected with the study of the lattices and of r.e. sets and r.e. sets modulo finite sets respectively. (This lattice theoretic viewpoint was introduced by Myhill [3].) Key roles in both areas have been played by the lattice of r.e. supersets, , of an r.e. set A (along with the corresponding modulo finite sets) and more recently by the group of automorphisms of and . Thus for example we have Lachlan's deep result [1] that Post's notion of A being hyperhypersimple is equivalent to (or ) being a Boolean algebra. Indeed Lachlan even tells us which Boolean algebras appear as —precisely those with Σ3 representations. There are also many other simpler but still illuminating connections between the older typology of r.e. sets and their roles in the lattice . (r-maximal sets for example are just those with completely uncomplemented.) On the other hand, work on automorphisms by Martin and by Soare [8], [9] has shown that most other Post type conditions on r.e. sets such as hypersimplicity or creativeness which are not obviously lattice theoretic are in fact not invariant properties of .In general the program of analyzing and classifying r.e. sets has been directed at the simple sets. Thus the subtypes of simple sets studied abound — between ten and fifteen are mentioned in [5] and there are others — but there seems to be much less known about the nonsimple sets. The typologies introduced for the nonsimple sets begin with Post's notion of creativeness and add on a few variations. (See [5, §8.7] and the related exercises for some examples.) Although there is a classification scheme for r.e. sets along the simple to creative line (see [5, §8.7]) it is admitted to be somewhat artificial and arbitrary. Moreover there does not seem to have been much recent work on the nonsimple sets.

A hierarchy of families of recursively enumerable degrees

1984 ◽  
Vol 49 (4) ◽  
pp. 1160-1170 ◽  
Author(s):  
Lawrence V. Welch
Keyword(s):  
Finite Sets ◽  
Recursively Enumerable ◽  
Similar Theorem ◽  
Recursively Enumerable Sets ◽  
Complete Set ◽  
Image Position

Certain investigations have been made concerning the nature of classes of recursively enumerable sets, and the relation of such classes to the recursively enumerable indices of their sets. For instance, a theorem of Rice [3, Theorem XIV(a), p. 324] states that if A is the complete set of indices for a class of recursively enumerable sets (that is, if there is a class of recursively enumerable sets such that and if A is recursive, then either A = ⌀ or A = ω. A relate theorem by Rice and Shapiro [3, Theorem XIV(b), p. 324] can be stated as follows:Let be a class of recursively enumerable sets, and let A be the complete set of indices for . Then A is r.e. if and only if there is an r.e. set D of canonical indices of finite sets Du, u ∈ D, such thatA somewhat similar theorem of Yates is the following: Let be a class of recursively enumerable sets which contains all finite sets. Let A be the complete set of indices for . Then there is a uniform recursive enumeration of the sets in if and only if A is recursively enumerable in 0(2)—that is, if and only if A is Σ3. A corollary of this is that if C is any r.e. set such that C(2)≡T⌀(2), there is a uniform recursive enumeration of all sets We such that We ≤TC [9, Theorem 9, p. 265].

The Sacks density theorem and Σ2-bounding

1996 ◽  
Vol 61 (2) ◽  
pp. 450-467 ◽  
Author(s):  
Marcia J. Groszek ◽  
Michael E. Mytilinaios ◽  
Theodore A. Slaman
Keyword(s):  
Recursion Theory ◽  
Density Theorem ◽  
Turing Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Models Of Arithmetic

AbstractThe Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P− + BΣ2. The proof has two components: a lemma that in any model of P− + BΣ2, if B is recursively enumerable and incomplete then IΣ1 holds relative to B and an adaptation of Shore's [9] blocking technique in α-recursion theory to models of arithmetic.

An invariance notion in recursion theory

1982 ◽  
Vol 47 (1) ◽  
pp. 48-66 ◽  
Author(s):  
Robert E. Byerly
Keyword(s):  
Recursion Theory ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Partial Recursive Functions

AbstractA set of gödel numbers is invariant if it is closed under automorphisms of (ω, ·), where ω is the set of all gödel numbers of partial recursive functions and · is application (i.e., n · m ≃ φn(m)). The invariant arithmetic sets are investigated, and the invariant recursively enumerable sets and partial recursive functions are partially characterized.

Sign in / Sign up

Export Citation Format

Share Document