Church, Cardinal and Ordinal Representations of Integers and Kolmogorov complexity

TL;DR

This paper explores effective representations of integers through Church, cardinal, and ordinal methods, linking their Kolmogorov complexities to hierarchies involving relativization and infinite computations.

Contribution

It formalizes effective versions of classical integer representations and establishes a hierarchy of their Kolmogorov complexities.

Findings

Kolmogorov complexities form a strict hierarchy.

Hierarchy coincides with relativization to jump oracles.

Infinite computations are also encompassed in this hierarchy.

Abstract

We consider classical representations of integers: Church's function iterators, cardinal equivalence classes of sets, ordinal equivalence classes of totally ordered sets. Since programs do not work on abstract entities and require formal representations of objects, we effectivize these abstract notions in order to allow them to be computed by programs. To any such effectivized representation is then associated a notion of Kolmogorov complexity. We prove that these Kolmogorov complexities form a strict hierarchy which coincides with that obtained by relativization to jump oracles and/or allowance of infinite computations.