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 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.

An unsolved problem in the theory of constructive order types

1969 ◽  
Vol 33 (4) ◽  
pp. 565-567
Author(s):  
Alan G. Hamilton
Keyword(s):  
Recursive Function ◽  
Unsolved Problem ◽  
Order Type ◽  
Partial Answer ◽  
Linear Ordering ◽  
Partial Recursive Function ◽  
Order Types ◽  
The Difference ◽  
Definition Of

In the forthcoming monograph of Crossley [1] the question is raised whether the implication 2 + A = A ⇒ 1 + A = A is true for constructive order types. In this paper a partial answer to this question is given, in that a counterexample is constructed, using, however, not the definition of constructive order type (C.O.T.) given in [1], but an earlier one, namely that in Crossley [2]. The difference is that in [1] only orderings which can be imbedded in a standard dense r.e. ordering R by a partial recursive function are considered. The linear ordering constructed in this paper can be shown not to be such. The problem remains open in the case of orderings imbeddable in R.

A problem in the theory of constructive order types

1970 ◽  
Vol 35 (1) ◽  
pp. 119-121
Author(s):  
Robin O. Gandy ◽  
Robert I. Soare
Keyword(s):  
Recursive Function ◽  
Type A ◽  
Order Type ◽  
General Question ◽  
Partial Recursive Function ◽  
Order Types ◽  
Definition Of ◽  
Priority Argument

J. N. Crossley [1] raised the question of whether the implication 2 + A = A ⇒ 1 + A = A is true for constructive order types (C.O.T.'s). Using an earlier definition of constructive order type, A. G. Hamilton [2] presented a counterexample. Hamilton left open the general question, however, since he pointed out that Crossley considers only orderings which can be embedded in a standard dense r.e. ordering by a partial recursive function, and that his counterexample fails to meet this requirement. We resolve the question by finding a C.O.T. A which meets Crossley's requirement and such that 2 + A = A but 1 + A ≠ A. At the suggestion of A. B. Manaster and A. G. Hamilton we easily extend this construction to show that for any n ≧ 2, there is a C.O.T. A such that n + A = A but m + A ≠ A for 0 < m < n. Hence, Theorem 3 of [2] and all of its corollaries hold with the new definition of C.O.T. The construction is not difficult and requires no priority argument. The techniques are similar to those developed in [3], but no outside results are needed here.

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.

Frequency computations and the cardinality theorem

1992 ◽  
Vol 57 (2) ◽  
pp. 682-687 ◽  
Author(s):  
Valentina Harizanov ◽  
Martin Kummer ◽  
Jim Owings
Keyword(s):  
Characteristic Function ◽  
Recursive Function ◽  
Natural Numbers ◽  
New Techniques ◽  
Semirecursive Set ◽  
Image Position ◽  
Special Case ◽  
Full Proof

In 1960 G. F. Rose [R] made the following definition: A function f: ω → ω is (m, n)-computable, where 1 ≤ m ≤ n, iff there exists a recursive function R: ωn → ωn such that, for all n-tuples (x1,…, xn) of distinct natural numbers,J. Myhill (see [McN, p. 393]) asked if f had to be recursive if m was close to n; B. A. Trakhtenbrot [T] responded by showing in 1963 that f is recursive whenever 2m > n. This result is optimal, because, for example, the characteristic function of any semirecursive set is (1,2)-computable. Trakhtenbrot's work was extended by E. B. Kinber [Ki1], using similar techniques. In 1986 R. Beigel [B] made a powerful conjecture, much more general than the above results. Partial verification, falling short of a full proof, appeared in [O]. Using new techniques, M. Kummer has recently established the conjecture, which will henceforth be referred to as the cardinality theorem (CT). It is the goal of this paper to show the connections between these various theorems, to review the methods used by Trakhtenbrot, and to use them to prove a special case of CT strong enough to imply Kinber's theorem (see §3). We thus have a hierarchy of results, with CT at the top. We will also include a discussion of Kummer's methods, but not a proof of CT.

Type two partial degrees

1978 ◽  
Vol 43 (4) ◽  
pp. 623-629
Author(s):  
Ko-Wei Lih
Keyword(s):  
Recursive Function ◽  
Natural Extension ◽  
Degree Theory ◽  
Recursion Theory ◽  
Natural Numbers ◽  
Partial Functions ◽  
Reduction Procedure ◽  
Turing Reducibility

Roughly speaking partial degrees are equivalence classes of partial objects under a certain notion of relative recursiveness. To make this notion precise we have to state explicitly (1) what these partial objects are; (2) how to define a suitable reduction procedure. For example, when the type of these objects is restricted to one, we may include all possible partial functions from natural numbers to natural numbers as basic objects and the reduction procedure could be enumeration, weak Turing, or Turing reducibility as expounded in Sasso [4]. As we climb up the ladder of types, we see that the usual definitions of relative recursiveness, equivalent in the context of type-1 total objects and functions, may be extended to partial objects and functions in quite different ways. First such generalization was initiated by Kleene [2]. He considers partial functions with total objects as arguments. However his theory suffers the lack of transitivity, i.e. we may not obtain a recursive function when we substitute a recursive function into a recursive function. Although Kleene's theory provides a nice background for the study of total higher type objects, it would be unsatisfactory when partial higher type objects are being investigated. In this paper we choose the hierarchy of hereditarily consistent objects over ω as our universe of discourse so that Sasso's objects are exactly those at the type-1 level. Following Kleene's fashion we define relative recursiveness via schemes and indices. Yet in our theory, substitution will preserve recursiveness, which makes a degree theory of partial higher type objects possible. The final result will be a natural extension of Sasso's Turing reducibility. Due to the abstract nature of these objects we do not know much about their behaviour except at the very low types. Here we pay our attention mainly to type-2 objects. In §2 we formulate basic notions and give an outline of our recursion theory of partial higher type objects. In §3 we introduce the definitions of singular degrees and ω-consistent degrees which are two important classes of type-2 objects that we are most interested in.

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.

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.

scholarly journals THE POLARISED PARTITION RELATION FOR ORDER TYPES

2020 ◽  
Vol 71 (3) ◽  
pp. 823-842
Author(s):  
L D Klausner ◽  
T Weinert
Keyword(s):  
Natural Numbers ◽  
Partition Relation ◽  
Cardinal Characteristics ◽  
Order Types ◽  
The Given ◽  
The Continuum

Abstract We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition properties of these products to cardinal characteristics of the continuum. We build on work by Erd̋s, Garti, Jones, Orr, Rado, Shelah and Szemerédi. In particular, we show that a theorem of Jones extends from the natural numbers to the rational ones, but consistently extends only to three further equimorphism classes of countable orderings. This is made possible by applying a 13-year-old theorem of Orr about embedding a given order into a sum of finite orders indexed over the given order.

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