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.

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.

A universal partial recursive function

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

scholarly journals Monotone and 1–1 sets

1982 ◽  
Vol 33 (1) ◽  
pp. 62-75 ◽  
Author(s):  
D. B. Madan ◽  
R. W. Robinson
Keyword(s):  
Recursive Function ◽  
Infinite Subset ◽  
Recursively Enumerable Set ◽  
Recursively Enumerable

AbstractAn infinite subset of ω is monotone (1–1) if every recursive function is eventually monotone on it (eventually constant on it or eventually 1–1 on it). A recursively enumerable set is co-monotone (co-1–1) just if its complement is monotone (1–1). It is shown that no implications hold among the properties of being cohesive, monotone, or 1–1, though each implies r-cohesiveness and dense immunity. However it is also shown that co-monotone and co-1–1 are equivalent, that they are properly stronger than the conjunction of r-maximality and dense simplicity, and that they do not imply maximality.

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.

Variations on promptly simple sets

1985 ◽  
Vol 50 (1) ◽  
pp. 138-148 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Recursive Function ◽  
Splitting Property ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Simple Sets

In this paper we answer the question of whether all low sets with the splitting property are promptly simple. Further we try to make the role of lowness properties and prompt simplicity in the construction of automorphisms of the lattice of r.e. (recursively enumerable) sets more perspicuous. It turns out that two new properties of r.e. sets, which are dual to each other, are essential in this context: the prompt and the low shrinking property.In an earlier paper [4] we had shown (using Soare's automorphism construction [10] and [12]) that all r.e. generic sets are automorphic in the lattice ℰ of r.e. sets under inclusion. We called a set A promptly simple if Ā is infinite and there is a recursive enumeration of A and the r.e. sets (We)e∈N such that if We is infinite then there is some element (or equivalently: infinitely many elements) x of We such that x gets into A “promptly” after its appearance in We (i.e. for some fixed total recursive function f we have x ∈ Af(s), where s is the stage at which x entered We). Prompt simplicity in combination with lowness turned out to capture those properties of r.e. generic sets that were used in the mentioned automorphism result. In a following paper with Shore and Stob [7] we studied an ℰ-definable consequence of prompt simplicity: the splitting property.

Post's problem and his hypersimple set

1973 ◽  
Vol 38 (3) ◽  
pp. 446-452 ◽  
Author(s):  
Carl G. Jockusch ◽  
Robert I. Soare
Keyword(s):  
Recursive Function ◽  
Recursively Enumerable ◽  
Standard Enumeration

A standard enumeration of the recursively enumerable (r.e.) sets is an acceptable numbering {Wn}n∈N of the r.e. sets in the sense of Rogers [5, p. 41], together with a 1:1 recursive function f with range In his quest for nonrecursive incomplete r.e. sets Post [4] constructed a hypersimple set Hf, relative to a fixed but unspecified standard enumeration f. Although it was later shown that hyper-simplicity does not guarantee incompleteness, the ironic possibility remained that Post's own particular hypersimple set might be incomplete. We settle the question by proving that H, may be either complete or incomplete depending upon which standard enumeration f is used. In contrast, D. A. Martin has shown [3] that Post's simple set S [4, p. 298] is complete for any standard enumeration. Furthermore, what most modern recursion theorists would regard as the “natural” construction of a hypersimple set (which we give in §1) is also complete for any standard enumeration.There are two conclusions to be drawn from these results. First, they substantiate the often repeated remark among recursion theorists that Post's hypersimple set construction is a precursor of priority constructions because priorities play a strong role, and because there is a great deal of “restraint” which tends to keep elements out of the set. Secondly, the results warn recursion theorists that more properties than might have been supposed depend upon which standard enumeration is chosen at the beginning of the construction of some r.e. set.

Arithmetic Without the Exponential

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Recursive Function ◽  
Function Theory ◽  
Axiom System ◽  
Peano Arithmetic ◽  
Recursively Enumerable

§1. By an arithmetic term or formula, we mean a term or formula in which the exponential symbol E does not appear, and by an arithmetic relation (or set), we mean a relation (set) expressible by an arithmetic formula. By the axiom system P.A. (Peano Arithmetic), we mean the system P.E. with axiom schemes N10 and N11 deleted, and in the remaining axiom schemes, terms and formulas are understood to be arithmetic terms and formulas. The system P.A. is the more usual object of modern study (indeed, the system P.E. is rarely considered in the literature). We chose to give the incompleteness proof for P.E. first since it is the simpler. In this chapter, we will prove the incompleteness of P.A. and establish several other results that will be needed in later chapters. The incompleteness of P.A. will easily follow from the incompleteness of P.E., once we show that the relation xy = z is not only Arithmetic but arithmetic (definable from plus and times alone). We will first have to show that certain other relations are arithmetic, and as we are at it, we will show stronger results about these relations that will be needed, not for the incompleteness proof of this chapter, but for several chapters that follow—we will sooner or later need to show that certain key relations are not only arithmetic, but belong to a much narrower class of relations, the Σ1-relations, which we will shortly define. These relations are the same as those known as recursively enumerable. Before defining the Σ1-relations, we turn to a still narrower class, the Σ0-relations, that will play a key role in our later development of recursive function theory. §2. We now define the classes of Σ0-formulas and Σ0-relations and then the Σ1-formulas and relations. By an atomic Σ0-formula, we shall mean a formula of one of the four forms c1 + c2 = c3, c1 · c2 =c3, c1 = c2 or c1 ≤ c2, where each of c1, c2 or c3 is either a variable or a numeral (but some may be variables and others numerals).

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.

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