Relative Computability and Uniform Continuity of Relations

TL;DR

This paper explores the relationship between relative computability and various notions of uniform continuity for relations, introducing a hierarchy of conditions necessary and sufficient for relative computability.

Contribution

It proposes and analyzes a hierarchy of uniform continuity notions for relations, establishing the Henkin Quantifier-based concept as necessary for relative computability.

Findings

A hierarchy of uniform continuity notions is necessary for relative computability.

The Henkin Quantifier-based uniform continuity is necessary for relative computability.

An omega-th level in the hierarchy is both necessary and sufficient for relative computability.

Abstract

A type-2 computable real function is necessarily continuous; and this remains true for relative, i.e. oracle-based computations. Conversely, by the Weierstrass Approximation Theorem, every continuous f:[0,1]->R is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multivalued functions f:[0,1]=>R (aka relations), Brattka and Hertling (1994) have considered two notions: weak continuity (which is weaker than relative computability) and strong continuity (which is stronger than relative computability). Observing that uniform continuity plays a crucial role in the Weierstrass Theorem, we propose and compare several notions of uniform continuity for relations. Here, due to the additional quantification over values y in f(x), new ways of (linearly) ordering quantifiers arise, yet none of them turn out as satisfactory. We are thus led to a notion of uniform continuity based on the Henkin Quantifier; and prove it necessary for relative computability. In fact iterating this condition yields a strict hierarchy of notions each necessary, and the omega-th level also sufficient, for relative computability.