Work

Extending the Barmpalias–lewis-pye Limit Theorem to All Reals

Year: 2025

Type: article

Abstract: By a celebrated result of Kučera and Slaman [5], the Martin-Löf random left-c.e. reals form the highest left-c.e. Solovay degree. Barmpalias and Lewis-Pye [1] strengthened this result by showing that, for all left-c.e. reals $\alpha $ and $\beta $ such that $\beta $ is Martin-Löf random and all left-c.e. approximations $a_0,a_1,\dots $ and $b_0,b_1,\dots $ of $\alpha $ and $\beta $ , respectively, the limit $$ \begin{align*} \lim\limits_{n\to\infty}\frac{\alpha - a_n}{\beta - b_n} \end{align*} $$ exists and does not depend on the choice of the left-c.e. approximations to $\alpha $ and $\beta $ . Here we give an equivalent formulation of the result of Barmpalias and Lewis-Pye in terms of nondecreasing translation functions and generalize their result to the set of all (i.e., not necessarily left-c.e.) reals.

Source: Journal of Symbolic Logic

DOI: 10.1017/jsl.2025.10157

Author: IVAN TITOV

Institution: Université de Bordeaux

Language: English

Topic: Computability, Logic, AI Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550

Funder: Agence Nationale de la Recherche