Primitive Recursive Ordered Fields and Some Applications

TL;DR

This paper develops primitive recursive versions of key properties of computable ordered fields and applies these to establish primitive recursiveness in linear algebra and analysis problems, including real closures and PDE solutions.

Contribution

It introduces primitive recursive analogues of known computability results for ordered fields and applies them to various problems in linear algebra and analysis.

Findings

Primitive recursive real closures of ordered fields

Primitive recursive root-finding and matrix normal forms

Primitive recursive solution operators for linear PDE systems

Abstract

We establish primitive recursive versions of some known facts about computable ordered fields of reals and computable reals, and then apply them to proving primitive recursiveness of some natural problems in linear algebra and analysis. In particular, we find a partial primitive recursive analogue of Ershov-Madison's theorem about real closures of computable ordered fields, relate the corresponding fields to the primitive recursive reals, give sufficient conditions for primitive recursive root-finding, computing normal forms of matrices, and computing solution operators of some linear systems of PDE.