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.

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.

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.

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.

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).

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.

Minimal pairs and high recursively enumerable degrees

1974 ◽  
Vol 39 (4) ◽  
pp. 655-660 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Recursive Function ◽  
Uniform Method ◽  
Minimal Pair ◽  
Minimal Pairs ◽  
Recursively Enumerable ◽  
Finite Approximations ◽  
Degree Bounds ◽  
Recursively Enumerable Degree ◽  
Image Position ◽  
High Set

A. H. Lachlan [2] and C. E. M. Yates [4] independently showed that minimal pairs of recursively enumerable (r.e.) degrees exist. Lachlan and Richard Ladner have shown (unpublished) that there is no uniform method for producing a minimal pair of r.e. degrees below a given nonzero r.e. degree. It is not known whether every nonzero r.e. degree bounds a r.e. minimal pair, but in the present paper it is shown (uniformly) that every high r.e. degree bounds a r.e. minimal pair. (A r.e. degree is said to be high if it contains a high set in the sense of Robert W. Robinson [3].)Theorem. Let a be a recursively enumerable degree for which a′ = 0″. Then there are recursively enumerable degrees b0 and b1 such that0 < bi < a for each i ≤ 1, and b0 ⋂ b1 = 0.The proof is based on the Lachlan minimal r.e. pair construction. For notation see Lachlan [2] or S. B. Cooper [1].By Robinson [3] we can choose a r.e. representative A of the degree a, with uniformly recursive tower {As, ∣ s ≥ 0} of finite approximations to A, such that CA dominates every recursive function whereWe define, stage by stage, finite sets Bi,s, i ≤ 1, s ≥ 0, in such a way that Bi, s + 1 ⊇ Bi,s for each i, s, and {Bi,s ∣ i ≤ 1, s ≥ 0} is uniformly recursive.

On axiomatizability within a system

1953 ◽  
Vol 18 (1) ◽  
pp. 30-32 ◽  
Author(s):  
William Craig
Keyword(s):  
Recursive Function ◽  
Recursive Relation ◽  
Recursively Enumerable Set ◽  
Formal Systems ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Primitive Recursive ◽  
Recursive Set

Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q, such that Q is a symmetric subrelation of R (i.e. if Q(m, n), then Q(n, m) and R(m, n)), and such that, for each m ϵ B, Q(m, n) for infinitely many n. Then there exists a primitive recursive set A, such that C is the closure under R of A. For proof, note that , where f is a primitive recursive function which enumerates B, has the required properties. For each m ϵ B, there is an n ϵ A, such that Q(m, n) and hence Q(n, m); therefore the closure of A under Q, and hence that under R, includes B. Conversely, since Q is a subrelation of R, A is included in C. Finally, that A is primitive recursive follows from [2] p. 180.This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R(m, n) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.

Arithmetical representation of recursively enumerable sets

1956 ◽  
Vol 21 (2) ◽  
pp. 162-186 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Primitive Recursive

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

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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