On the computability of ordered fields

TL;DR

This paper explores the limits of computability in ordered fields, establishing that certain classes of computable real numbers lack computable presentations, and provides criteria for the computable presentability of archimedean ordered fields.

Contribution

It introduces new techniques for analyzing the computability of real number classes and proves the non-existence of computable presentations for specific classes, including polynomial time and Grzegorczyk hierarchy levels.

Findings

No computable presentations for polynomial time computable real numbers.

No computable presentations for $E_n$-computable real numbers, $n \\geq 2$.

Provides a criterion for computable presentability of archimedean ordered fields.

Abstract

In this paper we develop general techniques for classes of computable real numbers generated by subsets of total computable (recursive functions) with special restrictions on basic operations in order to investigate the following problems: whether a generated class is a real closed field and whether there exists a computable presentation of a generated class. We prove a series of theorems that lead to the result that there are no computable presentations neither for polynomial time computable no even for $E_n$-computable real numbers, where $E_n$ is a level in Grzegorczyk hierarchy, $n \geq 2$. We also propose a criterion of computable presentability of an archimedean ordered field.