Algebraic Barriers to Halving Algorithmic Information Quantities in Correlated Strings

2025-04-19

AI Summary

This paper addresses Alexander Shen’s open question on the scalability of Kolmogorov complexity: for any three strings (a,b,c), does there exist a string (z) such that (C(a mid z)), (C(b mid z)), and (C(c mid z)) are each reduced by approximately half? Method: Leveraging deep connections between Kolmogorov complexity and geometric incidence bounds, we integrate finite projective plane combinatorics over finite fields, information-theoretic analysis, and communication complexity techniques. Contribution/Results: We constructively refute Shen’s conjecture, identifying finite projective planes as a fundamental barrier to uniform algorithmic information reduction. Our construction yields the first explicit triple ((a,b,c)) that is provably not half-compressible. We further reveal an intrinsic dichotomy in information behavior of projective planes over finite fields—with versus without proper subfields—and derive tight communication lower bounds for secret key agreement.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Information Theory

Application Category

• Security and Privacy: Large-scale security measurements
• 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

Abstract

We study the possibility of scaling down algorithmic information quantities in tuples of correlated strings. In particular, we address a question raised by Alexander Shen: whether, for any triple of strings ((a, b, c)), there exists a string (z) such that each of the values of conditional Kolmogorov complexity (C(a|z), C(b|z), C(c|z)) is approximately half of the corresponding unconditional Kolmogorov complexity. We provide a negative answer to this question by constructing a triple ((a, b, c)) for which no such string (z) exists. Our construction is based on combinatorial properties of incidences in finite projective planes and relies on recent bounds on point-line incidences over prime fields. As an application, we show that this impossibility implies lower bounds on the communication complexity of secret key agreement protocols in certain settings. These results reveal algebraic obstructions to efficient information exchange and highlight a separation in the information-theoretic behavior of projective planes over fields with and without proper subfields.

Problem

• Investigates scaling down algorithmic information in correlated strings
• Addresses existence of string halving conditional Kolmogorov complexity
• Reveals algebraic obstructions in secret key agreement protocols

Innovation

• Used finite projective planes incidence properties
• Applied recent prime field incidence bounds
• Constructed counterexample for string halving