Dimensionality and randomness

2024-05-11

arXiv.org

AI Summary

This paper investigates the systematic decay of Kolmogorov complexity in column sequences when a random bit string or real number is structured into a two-dimensional array or tree along a chosen dimension—a phenomenon termed “dimension-induced complexity loss.” Methodologically, the work integrates Kolmogorov complexity theory, algorithmic randomness, and multidimensional combinatorial coding techniques. The main contributions are threefold: (i) it establishes the first precise quantitative model for this decay, deriving tight upper and lower bounds on the decay rate; (ii) it proves the universality and structural invariance of these bounds across both array and tree representations; and (iii) it reveals a fundamental connection between the decay and the notion of “negligible classes” in algorithmic randomness, characterizing negligibility as the asymptotic threshold of complexity loss. These results extend the theoretical framework of algorithmic randomness to high-dimensional settings and provide a novel paradigm for modeling complexity behavior under structured representations.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Information Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and characterize this phenomenon for arrays and trees and its relationship to negligible classes.

Problem

• Quantifying Kolmogorov complexity loss in multidimensional arrays
• Characterizing complexity reduction when arranging strings into columns
• Analyzing relationship between dimensionality effects and negligible classes

Innovation

• Analyzes Kolmogorov complexity loss in arrays
• Quantifies dimensionality impact on random structures
• Characterizes relationship with negligible classes