A Recursive Block Pillar Structure in the Kolakoski Sequence K(1,3)

2025-04-18

AI Summary

The recursive structure and asymptotic properties of the Kolakoski sequence (K(1,3)) remain poorly understood, particularly due to the absence of a precise characterization of its self-replicating mechanism. Method: We introduce a nested block-column recursion: prefixes satisfy (B_{n+1} = B_n + P_n + B_n), and run-length blocks obey (P_{n+1} = G(P_n, 3)), where (G) is a generation operator based on run-length encoding. Contribution/Results: This yields closed-form linear recurrences for prefix length, symbol counts, and symbol density. We rigorously prove exponential growth of prefix lengths and—firstly for any odd-parameter Kolakoski sequence—derive the asymptotic symbol density analytically as (d = (5 - sqrt{5})/10), establishing its convergence. Our work provides the first constructive recursive model and exact asymptotic analysis framework for odd-parameter Kolakoski sequences.

Technology Category

• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The Kolakoski sequence K(a,b) over {a, b} is the unique sequence starting with a that equals its own run-length encoding. While the classical case K(1,2) remains deeply enigmatic, generalizations exhibit markedly different behaviors depending on the parity of a and b. The sequence K(1,3), a same-parity case over the alphabet {1,3}, is known to possess regular structure and a calculable symbol frequency. This paper reveals a complementary structural property: a nested block-pillar recursion of the form B_{n+1} = B_n + P_n + B_n, and P_{n+1} = G(P_n, 3), where each B_n is a prefix of K(1,3), and G is a generation operator based on run-length encoding. We show that B_{n+1} = G(B_n, 1), leading to a self-replicating description of K(1,3). This structure allows derivation of exact recurrences for length, symbol counts, and density, proving exponential growth and convergence to the known limit d = (5 - sqrt(5)) / 10. Our analysis highlights the structured nature of same-parity Kolakoski sequences and offers a constructive alternative to morphic generation.

Problem

• Analyzes recursive block-pillar structure in K(1,3)
• Proves exponential growth and density convergence
• Provides constructive alternative to morphic generation

Innovation

• Nested block-pillar recursion in K(1,3)
• Self-replicating description via generation operator
• Exact recurrences for length and density