Piergiorgio Odifreddi. Classical recursion theory. The theory of functions and sets of natural numbers. Studies in logic and the foundations of mathematics, vol. 125. North-Holland, Amsterdam etc. 1989, xvii + 668 pp.

1990 ◽  
Vol 55 (3) ◽  
pp. 1307-1308
Author(s):  
Peter G. Hinman
Keyword(s):  
Recursion Theory ◽  
Foundations Of Mathematics ◽  
Natural Numbers

J. C. Shepherdson. Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 285–308. - J. C. Shepherdson. Computational complexity of real functions. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 309–315. - A. J. Kfoury. The pebble game and logics of programs. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 317–329. - R. Statman. Equality between functionals revisited. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 331–338. - Robert E. Byerly. Mathematical aspects of recursive function theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 339–352.

1990 ◽  
Vol 55 (2) ◽  
pp. 876-878
Author(s):  
J. V. Tucker
Keyword(s):  
New York ◽  
Computational Complexity ◽  
Recursive Function ◽  
Function Theory ◽  
Recursion Theory ◽  
Foundations Of Mathematics ◽  
Real Functions ◽  
Pebble Game ◽  
Logics Of Programs

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.

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.

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].

The countably based functionals

1983 ◽  
Vol 48 (2) ◽  
pp. 458-474 ◽  
Author(s):  
John P. Hartley
Keyword(s):  
Finite Type ◽  
Recursion Theory ◽  
Type Structure ◽  
Topological Properties ◽  
Natural Numbers ◽  
Finite Amount ◽  
Amount Of Information ◽  
Natural Concept ◽  
Higher Types

In [5], Kleene extended previous notions of computations to objects of higher finite type in the maximal type-structure of functionals given by:Tp(0) = N = the natural numbers,Tp(n + 1) = NTp(n) = the set of total maps from Tp(n) to N.He gave nine schemata, S1–S9, for describing algorithms for computations from a finite list of functionals, and indices to denote these algorithms. It is generally agreed that S1-S9 give a natural concept of computations in higher types.The type-structure of countable functions, Ct(n) for n ϵ N, was first developed by Kleene [6] and Kreisel [7]. It arises from the notions of ‘constructivity’, and has been extensively studied as a domain for higher type recursion theory. Each countable functional is globally described by a countable amount of information coded in its associate, and it is determined locally by a finite amount of information about its argument. The countable functionals are well summarised in Normann [9], and treated in detail in Normann [8].The purpose of this paper is to discuss a generalisation of the countable functionals, which we shall call the countably based functions, Cb(n) for n ϵ N. It is suggested by the notions of ‘predicativity’, in which we view N as a completed totality, and build higher types on it in a constructive manner. So we shall allow quantification over N and include application of 2E in our schemata. Each functional is determined locally by a countable amount of information about its argument, and so is globally described by a continuum of information coded in its associate, which will now be a type-2 object. This generalisation, obtained via associates, was proposed by Wainer, and seems to reflect topological properties of the countable functionals.

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.

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