On the reducibility of function classes

Abstract

N – the set of all natural numbers, F – the set of all total functions N→N, A , B⊆F. We say that A is m-reducible to B (A≤ m B), iff there is a recursive operator M such that f ∈A ↔ M (f)∈B for all f ∈F. Similarly, 1-reducibility, tt-, btt-, 1tt-and Turing reducibility can be introduced. Table of contents. 1. Introduction. 2. Definitions of reducibilities and their simplest properties. 3. m-reducibility and the arithmetical hierarchy. 4. m-reducibility on Σ_1^fn. 5. Special classes F−{ f }. 6. Comparing various reducibilities on Σ_1^fn. 7. Notes on reducibilities of classes of sets.

Original title: К.М.Подниекс. О сводимостях классов функций. Сб. Уравнения математической физики и теория алгоритмов, Рига, 1972, 120-139.

Full text

Karlis Podnieks. On the reducibility of function classes. In: Equations of Mathematical Physics and Theory of Algorithms, Riga, Latvia State
University, 1972, pp. 120–139 (in Russian, English translation: Automatic Control and Computer Sciences).
Abstract. N – the set of all natural numbers, F – the set of all total functions N→N,
A , B ⊆F
. We say that A is m-reducible to B (
A≤mB
), iff there is a
recursive operator M such that
f∈A ↔ M (f)∈B
for all
f∈F
. Similarly, 1-reducibility, tt-, btt-, 1tt- and Turing reducibility can be introduced.
Table of contents. 1. Introduction. 2. Definitions of reducibilities and their simplest properties. 3. m-reducibility and the arithmetical hierarchy. 4. m-reducibility on
Σ1
f n
. 5. Special classes
F−{ f}
. 6. Comparing various reducibilities on
Σ1
f n
. 7. Notes on reducibilities of classes of sets.
Keywords: recursive functions, reducibility, m-reducibility, tabular reducibility, Turing reducibility.