Definability of r. e. sets in a class of recursion theoretic structures

1983 ◽  
Vol 48 (3) ◽  
pp. 662-669 ◽  
Author(s):  
Robert E. Byerly
Keyword(s):  
Recursion Theory ◽  
Universal Function ◽  
Independent Interest ◽  
Suitable Model ◽  
Natural Numbers ◽  
Construction Techniques ◽  
Partial Recursive Functions ◽  
Priority Method ◽  
Recursive Isomorphism ◽  
Direct Attack

It is known [4, Theorem 11-X(b)] that there is only one acceptable universal function up to recursive isomorphism. It follows from this that sets definable in terms of a universal function alone are specified uniquely up to recursive isomorphism. (An example is the set K, which consists of all n such that {n}(n) is defined, where λn, m{n}(m) is an acceptable universal function.) Many of the interesting sets constructed and studied by recursion theorists, however, have definitions which involve additional notions, such as a specific enumeration of the graph of a universal function. In particular, many of these definitions make use of the interplay between the purely number-theoretic properties of indices of partial recursive functions and their purely recursion-theoretic properties.This paper concerns r.e. sets that can be defined using only a universal function and some purely number-theoretic concepts. In particular, we would like to know when certain recursion-theoretic properties of r.e. sets definable in this way are independent of the particular choice of universal function (equivalently, independent of the particular way in which godel numbers are identified with natural numbers).We will first develop a suitable model-theoretic framework for discussing this question. This will enable us to classify the formulas defining r.e. sets by their logical complexity. (We use the number of alternations of quantifiers in the prenex form of a formula as a measure of logical complexity.) We will then be able to examine the question at each level.This work is an approach to the question of when the recursion-theoretic properties of an r.e. set are independent of the particular parameters used in its construction. As such, it does not apply directly to the construction techniques most commonly used at this time for defining particular r.e. sets, e.g., the priority method. A more direct attack on this question for these techniques is represented by such works as [3] and [5]. However, the present work should be of independent interest to the logician interested in recursion theory.

The hereditary partial effective functionals and recursion theory in higher types

1984 ◽  
Vol 49 (4) ◽  
pp. 1319-1332 ◽  
Author(s):  
G. Longo ◽  
E. Moggi
Keyword(s):  
Recursion Theory ◽  
Type Structure ◽  
Natural Numbers ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Higher Types ◽  
Elementary Technique

AbstractA type-structure of partial effective functionals over the natural numbers, based on a canonical enumeration of the partial recursive functions, is developed. These partial functionals, defined by a direct elementary technique, turn out to be the computable elements of the hereditary continuous partial objects; moreover, there is a commutative system of enumerations of any given type by any type below (relative numberings).By this and by results in [1] and [2], the Kleene-Kreisel countable functionals and the hereditary effective operations (HEO) are easily characterized.

TRUNCATION AND SEMI-DECIDABILITY NOTIONS IN APPLICATIVE THEORIES

2018 ◽  
Vol 83 (3) ◽  
pp. 967-990
Author(s):  
GERHARD JÄGER ◽  
TIMOTEJ ROSEBROCK ◽  
SATO KENTARO
Keyword(s):  
Recursion Theory ◽  
Natural Numbers ◽  
First Order ◽  
Recursive Functions ◽  
Partial Recursive Functions

AbstractBON+ is an applicative theory and closely related to the first order parts of the standard systems of explicit mathematics. As such it is also a natural framework for abstract computations. In this article we analyze this aspect of BON+ more closely. First a point is made for introducing a new operation τN, called truncation, to obtain a natural formalization of partial recursive functions in our applicative framework. Then we introduce the operational versions of a series of notions that are all equivalent to semi-decidability in ordinary recursion theory on the natural numbers, and study their mutual relationships over BON+ with τN.

Turing reducibility and Turing degrees

2018 ◽  
Author(s):  
Harold Hodes
Keyword(s):  
Computational Complexity ◽  
Recursion Theory ◽  
Turing Degrees ◽  
Natural Numbers ◽  
Mathematical Objects ◽  
Classical Setting ◽  
Turing Reducibility

A reducibility is a relation of comparative computational complexity (which can be made precise in various non-equivalent ways) between mathematical objects of appropriate sorts. Much of recursion theory concerns such relations, initially between sets of natural numbers (in so-called classical recursion theory), but later between sets of other sorts (in so-called generalized recursion theory). This article considers only the classical setting. Also Turing first defined such a relation, now called Turing- (or just T-) reducibility; probably most logicians regard it as the most important such relation. Turing- (or T-) degrees are the units of computational complexity when comparative complexity is taken to be T-reducibility.

SPARSE PARTITION REGULARITY

2006 ◽  
Vol 93 (3) ◽  
pp. 545-569 ◽  
Author(s):  
IMRE LEADER ◽  
PAUL A. RUSSELL
Keyword(s):  
Chromatic Number ◽  
Independent Interest ◽  
Natural Numbers ◽  
Ramsey Theorem

Our aim in this paper is to prove Deuber's conjecture on sparse partition regularity, that for every $m$, $p$ and $c$ there exists a subset of the natural numbers whose $(m,p,c)$-sets have high girth and chromatic number. More precisely, we show that for any $m$, $p$, $c$, $k$ and $g$ there is a subset $S$ of the natural numbers that is sufficiently rich in $(m,p,c)$-sets that whenever $S$ is $k$-coloured there is a monochromatic $(m,p,c)$-set, yet is so sparse that its $(m,p,c)$-sets do not form any cycles of length less than $g$.Our main tools are some extensions of Nešetřil–Rödl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.

scholarly journals The infinite injury priority method

1976 ◽  
Vol 41 (2) ◽  
pp. 513-530 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Recursion Theory ◽  
Powerful Method ◽  
Recursively Enumerable ◽  
Priority Method

One of the most important and distinctive tools in recursion theory has been the priority method whereby a recursively enumerable (r.e.) set A is constructed by stages to satisfy a sequence of conditions {Rn}n∈ω called requirements. If n < m, requirement Rn is given priority over Rm and action taken for Rm at some stage s may at a later stage t > s be undone for the sake of Rn thereby injuring Rm at stage t. The first priority method was invented by Friedberg [2] and Muchnik [11] to solve Post's problem and is characterized by the fact that each requirement is injured at most finitely often.Shoenfield [20, Lemma 1], and then independently Sacks [17] and Yates [25] invented a much more powerful method in which a requirement may be injured infinitely often, and the method was applied and refined by Sacks [15], [16], [17], [18], [19] and Yates [25], [26] to obtain many deep results on r.e. sets and their degrees. In spite of numerous simplifications and variations this infinite injury method has never been as well understood as the finite injury method because of its apparently greater complexity.The purpose of this paper is to reduce the Sacks method to two easily understood lemmas whose proofs are very similar to the finite injury case. Using these lemmas we can derive all the results of Sacks on r.e. degrees, and some by Yates and Robinson as well, in a manner accessible to the nonspecialist. The heart of the method is an ingenious observation of Lachlan [7] which is combined with a further simplification of our own.

Monotone inductive definitions over the continuum

1976 ◽  
Vol 41 (1) ◽  
pp. 188-198 ◽  
Author(s):  
Douglas Cenzer
Keyword(s):  
Mathematical Logic ◽  
Recursion Theory ◽  
Axiom System ◽  
Baire Space ◽  
Natural Numbers ◽  
Inductive Definition ◽  
Inductive Definitions ◽  
Definition Of ◽  
The Continuum

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

Metarecursively enumerable sets and their metadegrees

1968 ◽  
Vol 33 (3) ◽  
pp. 389-411 ◽  
Author(s):  
Graham C. Driscoll
Keyword(s):  
Recursion Theory ◽  
Natural Numbers

Metarecursion theory is an analogue of recursion theory which deals with sets of recursive, or constructive, ordinals rather than of natural numbers. It was originated by Kreisel and Sacks [3], who make extensive use of an equation calculus developed by Kripke. We assume that the reader is acquainted with the outline of it given in [3], and especially in [3, §3].

Sign in / Sign up

Export Citation Format

Share Document