Sophistication vs Logical Depth
Lu\'is Antunes, Bruno Bauwens, Andre Souto, Andreia Teixeira

TL;DR
This paper compares sophistication and logical depth as measures of string complexity, revealing their relationships and stability issues, and analyzing their behavior with respect to the Busy Beaver function and precision variations.
Contribution
It establishes quantitative relationships between sophistication and logical depth, demonstrating their differences, limitations, and the impact of precision changes on sophistication.
Findings
Sophistication exceeds logical depth with logarithmic precision.
Logical depth exceeds sophistication's Busy Beaver function with logarithmic precision.
Sophistication's value is unstable under small precision variations.
Abstract
Sophistication and logical depth are two measures that express how complicated the structure in a string is. Sophistication is defined as the minimal complexity of a computable function that defines a two-part description for the string that is shortest within some precision; the second can be defined as the minimal computation time of a program that is shortest within some precision. We show that the Busy Beaver function of the sophistication of a string exceeds its logical depth with logarithmically bigger precision, and that logical depth exceeds the Busy Beaver function of sophistication with logarithmically bigger precision. We also show that this is not true if the precision is only increased by a constant (when the notions are defined with plain Kolmogorov complexity). Finally we show that sophistication is unstable in its precision: constant variations can change its value by a…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputability, Logic, AI Algorithms · Algorithms and Data Compression · semigroups and automata theory
