Sub-computable Boundedness Randomness

TL;DR

This paper introduces a new concept of bounded computable randomness tailored for sub-computable classes like primitive recursive and PSPACE functions, establishing its robustness and exploring its theoretical properties.

Contribution

It defines and analyzes a novel form of randomness for sub-computable classes, providing multiple equivalent formulations and examining foundational theorems like van Lambalgen's.

Findings

Equivalent formulations via Martin-Löf tests, Kolmogorov complexity, and martingales.

Proves one direction of van Lambalgen's theorem holds, the other fails.

Discusses statistical properties of the new randomness notions.

Abstract

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for PSPACE functions. These new notions are robust in that there are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov complexity, and (3) martingales. We show these notions can be equivalently defined with prefix-free Kolmogorov complexity. We prove that one direction of van Lambalgen's theorem holds for relative computability, but the other direction fails. We discuss statistical properties of these notions of randomness.