Work

Algorithmically Optimal Outer Measures

Year: 2025

Type: article

Abstract: We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure coincide exactly with the algorithmic fractal dimensions. Our proof uses an especially convenient locally optimal outer measure κ defined in terms of Kolmogorov complexity. We discuss implications for point-to-set principles.

Source: ACM Transactions on Computation Theory

DOI: 10.1145/3733607

Authors: Jack H. Lutz, Neil Lutz

Institutions: Iowa State University, Swarthmore College

Language: English

Topic: Computability, Logic, AI Algorithms

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Funders: National Science Foundation, University of Pennsylvania, Iowa State University

Awards: CPS:Synergy: Safety-Aware Cyber-Molecular Systems, FET: Medium: Design for Dependability in Systems of Molecular Programs, CIRCLE: Catalyzing and Integrating Research, Collaboration, and Learning in Computing, Mathematics, and their Applications