Tighter Bounds for Algorithmic Complexity Estimation Using a Reusable Code-Based Block Decomposition Method

2026-06-22

AI Summary

This work addresses the limitations of traditional Block Decomposition Method (BDM), which neglects inter-block dependencies, leading to redundant description lengths and an inability to capture shared structures. The authors propose an improved framework grounded in reusable program code, modeling inter-block dependencies as a descriptive resource optimization problem formalized through an “algorithmic attention” allocation mechanism. By integrating Coding Theorem Method (CTM), conditional algorithmic complexity, Shannon entropy, and combinatorial optimization, they construct a computable algorithmic attention model. Theoretically, the approach is shown to strictly outperform independent block descriptions in the presence of shared structures, with performance gains positively correlated to algorithmic mutual information, and the paper provides a feasible implementation pathway for practical application.

Abstract

The Block Decomposition Method (BDM) was introduced as an alternative to popular lossless compression methods such as LZW for estimating algorithmic complexity from the principles of algorithmic probability and classical information theory. It extends the Coding Theorem Method (CTM) from small objects to larger ones by combining local estimates of algorithmic complexity with a global account of repetition based on Shannon entropy. Here, we introduce a version of BDM in which dependencies between blocks are utilized to reduce the length of the description based on reusable program code in the decomposition of an object, and on conditional descriptions capable of accounting for shared structure between observations. We formalize this allocation of descriptive resources as algorithmic attention. Repeated or related components need not be described independently, and the resulting reduction in description length is governed by the amount of shared algorithmic information. We formulate this extension as a reuse optimization problem, show that exact optimization is NP-hard, derive conditions under which it improves upon independent descriptions, relate the achievable gains to algorithmic mutual information, prove the relationship with the previous BDM version, and provide a roadmap for its implementation using CTM-derived complexity and conditional complexity estimates.

Problem

• algorithmic complexity
• Block Decomposition Method
• algorithmic information
• description length
• reusable code

Innovation

• Block Decomposition Method
• algorithmic complexity
• code reuse
• algorithmic attention
• conditional description