Π01-classes and Rado's selection principle

1991 ◽  
Vol 56 (2) ◽  
pp. 684-693 ◽  
Author(s):  
C. G. Jockusch ◽  
A. Lewis ◽  
J. B. Remmel
Keyword(s):  
Recursive Function ◽  
Real Field ◽  
Partial Recursive Function ◽  
Marriage Problem ◽  
Selection Principle ◽  
Recursive Tree ◽  
Effective Version ◽  
Finite Sequences ◽  
Π01 Classes ◽  
Effective Degree

There are several areas in recursive algebra and combinatorics in which bounded or recursively bounded -classes have arisen. For our purposes we may define a -class to be a set Path(T) of all infinite paths through a recursive tree T. Here a recursive tree T is just a recursive subset of ω<ω, the set of all finite sequences of the natural numbers ω = {0,1,2,…}, which is closed under initial segments. If the tree T is finitely branching, then we say the -class Path(T) is bounded. If T is highly recursive, i.e., if there exists a partial recursive function f: T→ω such that for each node ηЄ T, f(η) equals the number of immediate successors of η, then we say that the -class Path(T) is recursively bounded (r.b.). For example, Manaster and Rosenstein in [6] studied the effective version of the marriage problem and showed that the set of proper marriages for a recursive society S was always a bounded -class and the set of proper marriages for a highly recursive society was always an r.b. -class. Indeed, Manaster and Rosenstein showed that, in the case of the symmetric marriage problem, any r.b. -class could be represented as the set of symmetric marriages of a highly recursive society S in the sense that given any r.b. Π1-class C there is a society Sc such that there is a natural, effective, degree-preserving 1:1 correspondence between the elements of C and the symmetric marriages of Sc. Jockusch conjectured that the set of marriages of a recursive society can represent any bounded -class and the set of marriages of a highly recursive society can represent any r.b. -class. These conjectures remain open. However, Metakides and Nerode [7] showed that any r.b. -class could be represented by the set of total orderings of a recursive real field and vice versa that the set of total orderings of a recursive real field is always an r.b. -class.

A universal partial recursive function

1991 ◽  
Vol 49 (2) ◽  
pp. 186-189
Author(s):  
E. A. Polyakov
Keyword(s):  
Recursive Function ◽  
Partial Recursive Function

Recursive isomorphism types of recursive Boolean algebras

1981 ◽  
Vol 46 (3) ◽  
pp. 572-594 ◽  
Author(s):  
J. B. Remmel
Keyword(s):  
Boolean Algebra ◽  
Recursive Function ◽  
Boolean Algebras ◽  
Isomorphism Type ◽  
Turing Degrees ◽  
Main Question ◽  
Partial Recursive Function ◽  
Recursive Isomorphism ◽  
Infinite Set ◽  
The Ideal

A Boolean algebra is recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Given two Boolean algebras and , we write if is isomorphic to and if is recursively isomorphic to , that is, if there is a partial recursive function f: B1 → B2 which is an isomorphism from to . will denote the set of atoms of and () will denote the ideal generated by the atoms of .One of the main questions which motivated this paper is “To what extent does the classical isomorphism type of a recursive Boolean algebra restrict the possible recursion theoretic properties of ?” For example, it is easy to see that must be co-r.e. (i.e., N − is an r.e. set), but can be immune, not immune, cohesive, etc? It follows from a result of Goncharov [4] that there exist classical isomorphism types which contain recursive Boolean algebras but do not contain any recursive Boolean algebras such that is recursive. Thus the classical isomorphism can restrict the possible Turing degrees of , but what is the extent of this restriction? Another main question is “What is the recursion theoretic relationship between and () in a recursive Boolean algebra?” In our attempt to answer these questions, we were led to a wide variety of recursive isomorphism types which are contained in the classical isomorphism type of any recursive Boolean algebra with an infinite set of atoms.

A universal embedding property of the RETs

1970 ◽  
Vol 35 (1) ◽  
pp. 51-59 ◽  
Author(s):  
Anil Nerode ◽  
Alfred B. Manaster
Keyword(s):  
Recursive Function ◽  
Relational Structure ◽  
Partial Ordering ◽  
Proper Subset ◽  
Natural Numbers ◽  
Partial Recursive Function ◽  
Cardinal Numbers ◽  
Equivalence Type ◽  
Universal Embedding

Recursive equivalence types are an effective or recursive analogue of cardinal numbers. They were introduced by Dekker in the early 1950's. The richness of various theories related to the recursive equivalence types is demonstrated in this paper by showing that the theory of any countable relational structure can be embedded in or interpreted in these theories. A more complete summary is presented in the last paragraph of this section.Let E = {0,1, 2, …} be the natural numbers. If α ⊆ E, β ⊆ E, and there is a 1-1 partial recursive function f such that the image under f of α is β, α and β are called recursively equivalent (see [3]). The recursive equivalence type or RET of α, denoted 〈α〉, is the class of all β recursively equivalent to α. Addition of RETs is defined by 〈α〉 + 〈β〉 = 〈{2x ∣ x ∈ α} ∪ 〈{2x + 1 ∣ x ∈ β}〉. The partial ordering ≤ is defined on the RETs by A ≤ B iff (EC)(A + C = B). An RET, X, is called an isol if X ≠ X + 1 or, equivalently, if no representative of X is recursively equivalent to a proper subset of itself. The isols are thus the recursive analogue of the Dedekind-finite cardinals.

Recursively enumerable sets which are uniform for finite extensions

1971 ◽  
Vol 36 (2) ◽  
pp. 271-287 ◽  
Author(s):  
Donald A. Alton
Keyword(s):  
Recursive Function ◽  
Background Information ◽  
Partial Recursive Function ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

Let W0, W1 … be one of the usual enumerations of recursively enumerable (r.e.) subsets of the set N of nonnegative integers. (Background information will be given later.) Suggestions of Anil Nerode led to the followingDefinitions. Let B be a subset of N and let ψ be a partial recursive function.

On complexity properties of recursively enumerable sets

1973 ◽  
Vol 38 (4) ◽  
pp. 579-593 ◽  
Author(s):  
M. Blum ◽  
I. Marques
Keyword(s):  
Complexity Theory ◽  
Recursive Function ◽  
Strong Sense ◽  
Important Goal ◽  
Partial Recursive Function ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Partial Recursive Functions ◽  
Do So

An important goal of complexity theory, as we see it, is to characterize those partial recursive functions and recursively enumerable sets having some given complexity properties, and to do so in terms which do not involve the notion of complexity.As a contribution to this goal, we provide characterizations of the effectively speedable, speedable and levelable [2] sets in purely recursive theoretic terms. We introduce the notion of subcreativeness and show that every program for computing a partial recursive function f can be effectively speeded up on infinitely many integers if and only if the graph of f is subcreative.In addition, in order to cast some light on the concepts of effectively speedable, speedable and levelable sets we show that all maximal sets are levelable (and hence speedable) but not effectively speedable and we exhibit a set which is not levelable in a very strong sense but yet is effectively speedable.

Splinters of recursive functions

1960 ◽  
Vol 25 (1) ◽  
pp. 33-38 ◽  
Author(s):  
J. S. Ullian
Keyword(s):  
Recursive Function ◽  
Partial Recursive Function ◽  
Recursive Functions ◽  
Basic Notation ◽  
One And Many

Basic notation in this paper is as in [3]. From [5] and [9] the following additional notation is derived, ϕi is the partial recursive function with index i, Wi its range. ∅ is the empty set. ‘≡’ denotes isomorphism between sets, ‘≡m’ many-one equivalence, ‘≡T’ Turing equivalence, ‘≦1’ and ‘≦m’ signify one-one and many-one reducibility respectively. ‘Recursive’ is used throughout for ‘general recursive’.

Constructive order types on cuts

1969 ◽  
Vol 34 (2) ◽  
pp. 285-289 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Recursive Function ◽  
Natural Numbers ◽  
Partial Recursive Function ◽  
Order Types

If A and B are subsets of natural numbers we say that A is recursively equivalent to B (denoted A ≃ B) if there is a one-one partial recursive function which maps A onto B, and that A is recursively isomorphic to B (denoted A ≅ B) if there is a one-one total recursive function which maps A onto B and Ā (the complement of A) onto B#x00AF;.

A generalisation of productive set

1966 ◽  
Vol 31 (3) ◽  
pp. 455-459 ◽  
Author(s):  
R. Mitchell
Keyword(s):  
Recursive Function ◽  
Partial Recursive Function

The purpose of the present paper is to introduce a generalisation of the concept of productive set introduced by Post (2) and studied by Dekker (3) and others. Throughout the paper we shall use small Latin letters to denote both non-negative integers (referred to as numbers) and functions (both partial and total) from numbers to numbers. Sets of numbers will be denoted by small Greek letters and classes of such sets by capital Latin letters. ωη is the range of the ηth partial recursive function.

Theoretical Pearls:Representing ‘undefined’ in lambda calculus

1992 ◽  
Vol 2 (3) ◽  
pp. 367-374 ◽  
Author(s):  
Henk Barendregt
Keyword(s):  
Normal Form ◽  
Recursive Function ◽  
Lambda Calculus ◽  
Recursion Theory ◽  
Partial Recursive Function ◽  
Normalization Theorem ◽  
Image Position

AbstractLet ψ be a partial recursive function (of one argument) with λ-defining termF∈Λ°. This meansThere are several proposals for whatF⌜n⌝ should be in case ψ(n) is undefined: (1) a term without a normal form (Church); (2) an unsolvable term (Barendregt); (3) an easy term (Visser); (4) a term of order 0 (Statman).These four possibilities will be covered by one ‘master’ result of Statman which is based on the ‘Anti Diagonal Normalization Theorem’ of Visser (1980). That ingenious theorem about precomplete numerations of Ershov is a powerful tool with applications in recursion theory, metamathematics of arithmetic and lambda calculus.

A note on frame extensions

1972 ◽  
Vol 37 (3) ◽  
pp. 543-545
Author(s):  
Louise Hay
Keyword(s):  
Recursive Function ◽  
Partial Recursive Function ◽  
Recursively Enumerable ◽  
Recursive Relations

In [2] “recursive frames” were introduced as a means of extending relations R on the nonnegative integers to relations RΛ on the isols. In [1], this extension procedure was generalized by the introduction of “partial recursive frames”; the resulting extended relation on the isols was called RΛ. It was shown in [1] that the two extension procedures agree for recursive relations R, while RΛ ⊇ RΛ if R is . The case when R is , nonrecursive was left open. We show in this note that the extension procedures in fact agree for all relations R.In the following, the notation and terminology is that of [1] and [2].Theorem. If R ⊆ XκQ is a recursively enumerable (r.e.) relation, then RΛ = RΛ.Proof. Clearly RΛ ⊆ RΛ, since every recursive frame is partial recursive. To prove RΛ ⊆ RΛ, we give a uniform effective method for expanding any partial recursiveR-frame F to a recursiveR-frame G such that F ⊆ G, so that So let F be a (nonempty) partial recursive R-frame, with CF(α) generated by . Let Rn denote the result of performing n steps in a fixed recursive enumeration of R. If g(α) is a partial recursive function, “g(α) is defined in n steps” means that in whichever coding of recursive computations is being used, a terminating computation for g with argument α has length ≤ n.

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