How incomputable is the separable Hahn-Banach theorem?

TL;DR

This paper analyzes the computational complexity of the Hahn-Banach Extension Theorem, linking it to reverse mathematics and computable analysis, and classifies its incomputability within a formal framework.

Contribution

It introduces a new classification of the Hahn-Banach theorem's computational complexity using multi-valued functions and reducibility, connecting it to WKL_0 in reverse mathematics.

Findings

The Hahn-Banach extension problem is Sep-complete.

A new framework relates WKL_0 to Sep-computable functions.

The study bridges reverse mathematics and computable analysis.

Abstract

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak Konig's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multi-valued function Sep and a natural notion of reducibility for multi-valued functions, we obtain a computational counterpart of the subsystem of second order arithmetic WKL_0. We study analogies and differences between WKL_0 and the class of Sep-computable multi-valued functions. Extending work of Brattka, we show that a natural multi-valued function associated with the Hahn-Banach Extension Theorem is Sep-complete.