Some properties of invariant sets

1984 ◽  
Vol 49 (1) ◽  
pp. 9-21 ◽  
Author(s):  
Robert E. Byerly
Keyword(s):  
Recursive Function ◽  
Invariant Sets ◽  
Turing Degrees ◽  
Partial Recursive Function ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Previous Paragraph

In [1] two interesting invariance notions were introduced: the notions of a set of godel numbers being invariant to automorphisms of the structures (ω, ·) and (ω, E) respectively. Here, · and E are defined by n · m ≃ φn (m) and nEm if and only if n Є Wm, where {φn} and {Wn} are acceptable enumerations of the partial recursive functions and r.e. sets respectively. In this paper we continue the study of the invariant sets, and especially the invariant r.e. sets, of gödel numbers.We start off with an easy result which characterizes the Turing degrees containing invariant sets. We then take a closer look at r.e. sets invariant with respect to automorphisms of (ω,E). Using the characterization [1, Theorem 4.2] of such sets, we will derive a somewhat different characterization (which was stated, but not proved, in [1, Proposition 4.4]) and, using it as a tool for constructing invariant sets, prove that the r.e. sets invariant with respect to automorphisms of (ω, E) cannot be effectively enumerated.We will next discuss representations of r.e. sets invariant with respect to automorphisms of (ω, ·). Although these sets do not have as nice a characterization as the r.e. sets invariant with respect to automorphisms of (ω, E) do, the techniques of [1] can still profitably be used to investigate their structure. In particular, if f is a partial recursive function whose graph is invariant with respect to automorphisms of (ω, ·), then for every a in the domain of f, there is a term t(a) built up from a and · only such that f(a) ≃ t(a). This is an analog to [1, Corollary 4.3]. We will also prove an analog to a result mentioned in the previous paragraph: the r.e. sets invariant with respect to automorphisms of (ω, ·) cannot be effectively enumerated.

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.

Two recursion theoretic characterizations of proof speed-ups

1989 ◽  
Vol 54 (2) ◽  
pp. 522-526 ◽  
Author(s):  
James S. Royer
Keyword(s):  
Recursive Function ◽  
Formal System ◽  
Complexity Measure ◽  
Incompleteness Theorem ◽  
Similar Treatment ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Speed Up

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. (Here we equate the length of a proof with something like the number of characters making it up,notits number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa (Blum) complexity measure associated withϕ[Blu67], [DW83].

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.

Gödel numberings of partial recursive functions

1958 ◽  
Vol 23 (3) ◽  
pp. 331-341 ◽  
Author(s):  
Hartley Rogers
Keyword(s):  
Mathematical Logic ◽  
Recursive Function ◽  
Function Theory ◽  
Recursive Structure ◽  
Partial Functions ◽  
Logical Calculus ◽  
Recursive Functions ◽  
Several Variables ◽  
Partial Recursive Functions ◽  
Made In

In § 1 we present conceptual material concerning the notion of a Gödel numbering of the partial recursive functions. § 2 presents a theorem about these concepts. § 3 gives several applications. The material in § 1 and § 2 grew out of attempts by the author to find routine solutions to some of the problems discussed in § 3. The author wishes to acknowledge his debt in § 2 to the fruitful methods of Myhill in [M] and to thank the referee for an abbreviated and improved version of the proof for Lemma 3 in § 2.In the literature of mathematical logic, “Gödel numbering” usually means an effective correspondence between integers and the well-formed formulas of some logical calculus. In recursive function theory, certain such associations between the non-negative integers and instructions for computing partial recursive functions have been fundamental. In the present paper we shall be concerned only with numberings of the latter, more special, sort. By numbers and integers we shall mean non-negative integers. Our notation is, in general, that of [K]. If ϕ and ψ are two partial functions, ϕ = ψ shall mean that (∀x)[ϕ(x)≃(ψx)], i.e., that ϕ and ψ are defined for the same arguments and are equal on those arguments. We consider partial recursive functions of one variable; applications of the paper to the case of several variables, or to the case of all partial recursive functions in any number of variables, can be made in the usual way using the coordinate functions (a)i of [K, p. 230]. It will furthermore be observed that we consider only concepts that are invariant with respect to general recursive functions; more limited notions of Gödel numbering, taking into account, say, primitive recursive structure, are beyond the scope of the present paper.

Splinters of recursive functions

1960 ◽  
Vol 25 (1) ◽  
pp. 33-38 ◽  
Author(s):  
J. S. Ullian
Keyword(s):  
Recursive Function ◽  
Partial Recursive Function ◽  
Recursive Functions ◽  
Basic Notation ◽  
One And Many

Basic notation in this paper is as in [3]. From [5] and [9] the following additional notation is derived, ϕi is the partial recursive function with index i, Wi its range. ∅ is the empty set. ‘≡’ denotes isomorphism between sets, ‘≡m’ many-one equivalence, ‘≡T’ Turing equivalence, ‘≦1’ and ‘≦m’ signify one-one and many-one reducibility respectively. ‘Recursive’ is used throughout for ‘general recursive’.

On homogeneity and definability in the first-order theory of the Turing degrees

1982 ◽  
Vol 47 (1) ◽  
pp. 8-16 ◽  
Author(s):  
Richard A. Shore
Keyword(s):  
Recursion Theory ◽  
Order Theory ◽  
Partial Ordering ◽  
Turing Degrees ◽  
First Order ◽  
Recursive Functions ◽  
First Order Theory ◽  
Partial Recursive Functions ◽  
Carry Over

Relativization—the principle that says one can carry over proofs and theorems about partial recursive functions and Turing degrees to functions partial recursive in any given set A and the Turing degrees of sets in which A is recursive—is a pervasive phenomenon in recursion theory. It led H. Rogers, Jr. [15] to ask if, for every degree d, (≥ d), the partial ordering of Turing degrees above d, is isomorphic to all the degrees . We showed in Shore [17] that this homogeneity conjecture is false. More specifically we proved that if, for some n, the degree of Kleene's (the complete set) is recursive in d(n) then ≇ (≤ d). The key ingredient of the proof was a new version of a result from Nerode and Shore [13] (hereafter NS I) that any isomorphism φ: → (≥ d) must be the identity on some cone, i.e., there is an a called the base of the cone such that b ≥ a ⇒ φ(b) = b. This result was combined with information about minimal covers from Jockusch and Soare [8] and Harrington and Kechris [3] to derive a contradiction from the existence of such an isomorphism if deg() ≤ d(n).

Preliminaries

2017 ◽  
Author(s):  
David J. Lobina
Keyword(s):  
Data Structures ◽  
Production Systems ◽  
Specific Data ◽  
Recursive Functions ◽  
Recursive Solution ◽  
Partial Recursive Functions ◽  
Recursive Structures ◽  
Recursive Data Structures ◽  
Recursive Data ◽  
The Relationship

Recursion, or the capacity of ‘self-reference’, has played a central role within mathematical approaches to understanding the nature of computation, from the general recursive functions of Alonzo Church to the partial recursive functions of Stephen C. Kleene and the production systems of Emil Post. Recursion has also played a significant role in the analysis and running of certain computational processes within computer science (viz., those with self-calls and deferred operations). Yet the relationship between the mathematical and computer versions of recursion is subtle and intricate. A recursively specified algorithm, for example, may well proceed iteratively if time and space constraints permit; but the nature of specific data structures—viz., recursive data structures—will also return a recursive solution as the most optimal process. In other words, the correspondence between recursive structures and recursive processes is not automatic; it needs to be demonstrated on a case-by-case basis.

Reducibility of partial recursive functions. II

1978 ◽  
Vol 18 (4) ◽  
pp. 541-548
Author(s):  
A. N. D�gtev
Keyword(s):  
Recursive Functions ◽  
Partial Recursive Functions
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