Recursively enumerable sets which are uniform for finite extensions

Donald A. Alton

doi:10.2307/2270262

Recursively enumerable sets which are uniform for finite extensions

Journal of Symbolic Logic ◽

10.2307/2270262 ◽

1971 ◽

Vol 36 (2) ◽

pp. 271-287 ◽

Cited By ~ 1

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

Journal of Symbolic Logic ◽

10.2307/2271984 ◽

1973 ◽

Vol 38 (4) ◽

pp. 579-593 ◽

Cited By ~ 44

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.

Variations on promptly simple sets

Journal of Symbolic Logic ◽

10.2307/2273796 ◽

1985 ◽

Vol 50 (1) ◽

pp. 138-148 ◽

Cited By ~ 3

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.

A note on frame extensions

Journal of Symbolic Logic ◽

10.2307/2272741 ◽

1972 ◽

Vol 37 (3) ◽

pp. 543-545

Author(s):

Louise Hay

Keyword(s):

Recursive Function ◽

Partial Recursive Function ◽

Recursively Enumerable ◽

Recursive Relations

In [2] “recursive frames” were introduced as a means of extending relations R on the nonnegative integers to relations RΛ on the isols. In [1], this extension procedure was generalized by the introduction of “partial recursive frames”; the resulting extended relation on the isols was called RΛ. It was shown in [1] that the two extension procedures agree for recursive relations R, while RΛ ⊇ RΛ if R is . The case when R is , nonrecursive was left open. We show in this note that the extension procedures in fact agree for all relations R.In the following, the notation and terminology is that of [1] and [2].Theorem. If R ⊆ XκQ is a recursively enumerable (r.e.) relation, then RΛ = RΛ.Proof. Clearly RΛ ⊆ RΛ, since every recursive frame is partial recursive. To prove RΛ ⊆ RΛ, we give a uniform effective method for expanding any partial recursiveR-frame F to a recursiveR-frame G such that F ⊆ G, so that So let F be a (nonempty) partial recursive R-frame, with CF(α) generated by . Let Rn denote the result of performing n steps in a fixed recursive enumeration of R. If g(α) is a partial recursive function, “g(α) is defined in n steps” means that in whichever coding of recursive computations is being used, a terminating computation for g with argument α has length ≤ n.

Three Special Topics

Recursion Theory for Metamathematics ◽

10.1093/oso/9780195082326.003.0015 ◽

1993 ◽

Author(s):

Raymond M. Smullyan

Keyword(s):

Recursive Function ◽

General Reader ◽

Recursively Enumerable ◽

Recursively Enumerable Sets ◽

Recursive Set

The topics of this chapter are of more specialized interest and are not necessary for the results of our final chapter. They will probably be of more interest to the specialist (particularly the results of Section III) than to the general reader. We know from Chapter 2 that if all recursive sets are representable in S, then S is undecidable. We also know from Chapter 4 that if all recursively enumerable sets are representable in S, then S is not only undecidable but generative. We might also ask the question, If all recursive sets are representable in S, is S necessarily generative? Shoenfield [1961] answered this question negatively. He constructed an axiomatizable system in which all recursive sets are representable, and so the system is undecidable, but he showed that the system is not creative. Let us say that all recursive sets are uniformly representable in S if there is a recursive function g(x) such that for any number i, if ωi is a recursive set, then g(i) is the Godel number of a formula which represents ωi in S. We will show that if all recursive sets are uniformly representable in S, then S is generative. We also showed in Chapter 2 that if all recursive sets are definable in S and S is consistent, then the pair (P,R) of its nuclei is recursively inseparable. Under the same hypothesis, is the pair (P,R) necessarily effectively inseparable? The answer is no. In Shoenfield’s system all recursive sets are not only representable, but definable. However, the set P is not creative. Hence the pair (P, R) of nuclei of the system, though recursively inseparable, is not effectively inseparable.

Torre models in the isols

Journal of Symbolic Logic ◽

10.2307/2275256 ◽

1994 ◽

Vol 59 (1) ◽

pp. 140-150 ◽

Cited By ~ 1

Author(s):

Joseph Barback

Keyword(s):

Number Theory ◽

Structural Property ◽

Recursive Function ◽

Peano Arithmetic ◽

The Other ◽

Close Connection ◽

New Variety ◽

Nonstandard Model ◽

Recursively Enumerable ◽

Recursively Enumerable Sets

AbstractIn [14] J. Hirschfeld established the close connection of models of the true AE sentences of Peano Arithmetic and homomorphic images of the semiring of recursive functions. This fragment of Arithmetic includes most of the familiar results of classical number theory. There are two nice ways that such models appear in the isols. One way was introduced by A. Nerode in [20] and is referred to in the literature as Nerode Semirings. The other way is called a tame model. It is very similar to a Nerode Semiring and was introduced in [6]. The model theoretic properties of Nerode Semirings and tame models have been widely studied by T. G. McLaughlin ([16], [17], and [18]).In this paper we introduce a new variety of tame model called a torre model. It has as a generator an infinite regressive isol with a nice structural property relative to recursively enumerable sets and their extensions to the isols. What is then obtained is a nonstandard model in the isols of the fragment of Peano Arithmetic with the following property: Let T be a torre model. Let f be any recursive function, and let fΛ be its extension to the isols. If there is an isol A with fΛ(A) ϵ T, then there is also an isol B ϵ T with fΛ(B) = fΛ(A).

When catalytic P systems with one catalyst can be computationally complete

Journal of Membrane Computing ◽

10.1007/s41965-021-00079-x ◽

2021 ◽

Author(s):

Artiom Alhazov ◽

Rudolf Freund ◽

Sergiu Ivanov

Keyword(s):

Computational Complexity ◽

P Systems ◽

Membrane Systems ◽

Computational Completeness ◽

Recursively Enumerable ◽

Recursively Enumerable Sets

AbstractCatalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst in the whole system: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining computational completeness even with only one catalyst. In this paper, we show that one catalyst is sufficient for obtaining computational completeness if either catalytic rules have weak priority over non-catalytic rules or else instead of the standard maximally parallel derivation mode, we use the derivation mode maxobjects, i.e., we only take those multisets of rules which affect the maximal number of objects in the underlying configuration.

Interpretability over peano arithmetic

Journal of Symbolic Logic ◽

10.2307/2586787 ◽

1999 ◽

Vol 64 (4) ◽

pp. 1407-1425

Author(s):

Claes Strannegård

Keyword(s):

Modal Logic ◽

Completeness Theorem ◽

Peano Arithmetic ◽

Embedding Theorem ◽

Compactness Theorem ◽

Partial Answer ◽

Recursively Enumerable ◽

Recursively Enumerable Sets ◽

Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).