Some theorems on R-maximal sets and major subsets of recursively enumerable sets

1971 ◽  
Vol 36 (2) ◽  
pp. 193-215 ◽  
Author(s):  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Elementary Equivalence ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Nonstandard Models ◽  
Relational Systems ◽  
The Relationship ◽  
Models Of Arithmetic

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

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.

1-genericity in the enumeration degrees

1988 ◽  
Vol 53 (3) ◽  
pp. 878-887 ◽  
Author(s):  
Kate Copestake
Keyword(s):  
Partial Function ◽  
Partial Ordering ◽  
Turing Degree ◽  
Turing Degrees ◽  
Original Formulation ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Generic Set

The structure of the Turing degrees of generic and n-generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]).In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree.Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree ae which is minimal-like; that is, all functions in degrees below ae have partial recursive extensions.

Omega-inconsistency without cuts and nonstandard models

2016 ◽  
Vol 13 (5) ◽  
Author(s):  
Andreas Fjellstad
Keyword(s):  
Deontic Logic ◽  
Sequent Calculus ◽  
Standard Deontic Logic ◽  
Nonstandard Models ◽  
Deontic Modality ◽  
The Relationship ◽  
Labelled Sequent Calculus ◽  
Models Of Arithmetic

This paper concerns the relationship between transitivity of entailment, omega-inconsistency and nonstandard models of arithmetic. First, it provides a cut-free sequent calculus for non-transitive logic of truth STT based on Robinson Arithmetic and shows that this logic is omega-inconsistent. It then identifies the conditions in McGee (1985) for an omega-inconsistent logic as quantified standard deontic logic, presents a cut-free labelled sequent calculus for quantified standard deontic logic based on Robinson Arithmetic where the deontic modality is treated as a predicate, proves omega-inconsistency and shows thus, pace Cobreros et al.(2013), that the result in McGee (1985) does not rely on transitivity. Finally, it also explains why the omega-inconsistent logics of truth in question do not require nonstandard models of arithmetic.

Splitting and Decomposition by Regressive Sets, II

1967 ◽  
Vol 19 ◽  
pp. 291-311 ◽  
Author(s):  
T. G. McLaughlin
Keyword(s):  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Relationship Of ◽  
The Relationship ◽  
Trivial Fact ◽  
Recursive Set

In (3), Dekker drew attention to an analogy between (a) the relationship of the recursive sets to the recursively enumerable sets, and (b) the relationship of the retraceable sets to the regressive sets. As was to be expected, this analogy limps in some respects. For example, if a number set α is split by a recursive set, then it is decomposed by a pair of recursively enumerable sets; whereas, as we showed in (6, Theorem 2), α may be split by a retraceable set and yet not decomposable (in a liberal sense of the latter term) by a pair of regressive sets. The result for recursive and recursively enumerable sets, of course, follows from the trivial fact that the complement of a recursive set is recursive.

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.

Characterization of recursively enumerable sets

1972 ◽  
Vol 37 (3) ◽  
pp. 507-511 ◽  
Author(s):  
Jesse B. Wright
Keyword(s):  
Transitive Closure ◽  
Successor Function ◽  
Function Composition ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Abstract Setting ◽  
Recursively Enumerable Sets ◽  
Bracket Operation

AbstractLet N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets P, Q, R ⊆ N2 operations of transposition, composition, and bracketing are defined as follows: P∪ = {〈x, y〉 ∣ 〈y, x〉 ∈ P}, PQ = {〈x, z〉 ∣ ∃y〈x, y〉 ∈ P & 〈y, z〉 ∈ Q}, and [P, Q, R] = ⋃n ∈ M(Pn Q Rn).Theorem. The class of recursively enumerable subsets of N2 is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing.This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and “definition by general recursion.” A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [P∪, A∪M, R].The above definitions of the transposition, composition, and bracketing operations on subsets of N2 can be generalized to subsets of X2 for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs.

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.

Partial degrees and the density problem. Part 2: The enumeration degrees of the Σ2 sets are dense

1984 ◽  
Vol 49 (2) ◽  
pp. 503-513 ◽  
Author(s):  
S. B. Cooper
Keyword(s):  
Density Theorem ◽  
Turing Degrees ◽  
Partial Functions ◽  
Enumeration Degrees ◽  
Recursively Enumerable ◽  
Recursive Sequences ◽  
Recursively Enumerable Sets ◽  
Finite Set ◽  
Image Position ◽  
Density Problem

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ withProof. We enumerate an e-operator Θ so as to satisfy the list of conditions:Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

Working below a high recursively enumerable degree

1993 ◽  
Vol 58 (3) ◽  
pp. 824-859 ◽  
Author(s):  
Richard A. Shore ◽  
Theodore A. Slaman
Keyword(s):  
Turing Degrees ◽  
Jump Operator ◽  
Recursively Enumerable Set ◽  
Enumerable Degree ◽  
Local Degree ◽  
Recursively Enumerable ◽  
Recursively Enumerable Degree ◽  
Turing Reducibility ◽  
Definition Of ◽  
The Relationship

In recent work, Cooper [3, 1990] has extended results of Jockusch and Shore [6, 1984] to show that the Turing jump is definable in the structure given by the Turing degrees and the ordering of Turing reducibility. In his definition of x′ from x, Cooper identifies an order-theoretic property shared by all of the degrees that are recursively enumerable in x and above x. He then shows that x′ is the least upper bound of all the degrees with this property. Thus, the jump of x is identified by comparing the recursively enumerable degrees with other degrees which are not recursively enumerable. Of course, once the jump operator is known to be definable, the relation of jump equivalence x′ = y′ is also known to be a definable relation on x and y. If we consider how much of the global theory of the Turing degrees is sufficient for Cooper's methods, it is immediately clear that his methods can be implemented to show that the jump operator and its weakening to the relation of jump equivalence are definable in any ideal closed under the Turing jump. However, his methods do not localize to , the degrees, or to the recursively enumerable degrees.This paper fits, as do Shore and Slaman [16, 1990] and [17, to appear], within the general project to develop an understanding of the relationship between the local degree-theoretic properties of a recursively enumerable set A and its jump class. For an analysis of the possibility of defining jump equivalence in , consult Shore [15, to appear] who shows that the relation x(3) = y(3) is definable. In this paper, we will restrict our attention to definitions expressed completely in ℛ (Note: All sets and degrees discussed for the remainder of this paper will be recursively enumerable.) Ultimately, one would like to find some degree-theoretic properties definable in terms of the ordering of Turing reducibility and quantifiers over the recursively enumerable degrees that would define the relation of jump equivalence or define one or more of the jump classes Hn = {w∣ wn = 0n+1} or Ln = {w ∣ wn = 0n}. Such a result could very likely then be used as a springboard to other general definability results for the recursively enumerable degrees. It would be especially interesting to know whether every recursively enumerable degree is definable and whether every arithmetical degree-invariant property of the recursively enumerable sets is definable in .

Sign in / Sign up

Export Citation Format

Share Document