Conditional automatic complexity and its metrics

TL;DR

This paper introduces two computable similarity metrics based on conditional automatic complexity, extending previous Kolmogorov complexity-based measures, and proves they satisfy all metric space axioms.

Contribution

It defines and analyzes two new similarity metrics based on conditional automatic complexity, demonstrating their metric properties.

Findings

Metrics are computable and satisfy metric axioms.

Analogous to Jaccard distance and Normalized Information Distance.

Extend Kolmogorov complexity-based similarity measures.

Abstract

Li, Chen, Li, Ma, and Vit\'anyi (2004) introduced a similarity metric based on Kolmogorov complexity. It followed work by Shannon in the 1950s on a metric based on entropy. We define two computable similarity metrics, analogous to the Jaccard distance and Normalized Information Distance, based on conditional automatic complexity and show that they satisfy all axioms of metric spaces.