[Paper Review] An Algorithmic Information Theory approach to the emergence of order using simple replication models

Sean D Devine|ArXiv.org|Jul 1, 2008

Computability, Logic, AI Algorithms21 references3 citations

TL;DR

This paper applies Algorithmic Information Theory (AIT) to simple replication models, showing that replicating systems generate and maintain low algorithmic entropy (high order) by exploiting external energy and ejecting disorder. The key contribution is that nested, interdependent replicators achieve high organizational complexity with minimal algorithmic entropy increase, enabling stable, adaptive far-from-equilibrium systems through evolutionary-like processes.

ABSTRACT

This paper applies Algorithmic Information Theory to simple examples of replication processes to illustrate how replicating structures can generate and maintain order in a non equilibrium system. Variation in replicating structures enhances the system's ability to maintain homeostasis in a changing environment by allowing it to evolve to a more restricted region of its state space. Stability is further enhanced when replicating systems develop dependencies, by sharing information or resources. Such systems co-evolve, becoming more independent of the external environment. Nested systems have a hierarchy of dependency but have low algorithmic entropy as they are in principle simpler to describe algorithmically. Nesting of replicating systems offsets the need for increased variety by allowing the structures to increase in organizational complexity with little increase in algorithmic entropy. Chaitin's d-diameter complexity provides a measure of the level of order in nested replicating systems.

Motivation & Objective

• To investigate how replication processes in non-equilibrium systems generate and sustain order using algorithmic information theory.
• To understand the role of variation and dependency in enhancing system stability and homeostasis in changing environments.
• To analyze how nested, interdependent replicators reduce algorithmic entropy while increasing organizational complexity.
• To establish a framework for measuring order in replicating systems using Chaitin’s d-diameter complexity and algorithmic entropy.
• To demonstrate that replication is a fundamental mechanism for the emergence of complex, ordered structures in physical systems.

Proposed method

• Uses algorithmic entropy as a measure of order, closely related to Shannon and Boltzmann entropy, to quantify structural complexity.
• Models replication processes using a logistic growth equation to represent homeostatic states where birth rate equals death rate.
• Applies Chaitin’s d-diameter complexity to assess the level of order in nested replicating systems.
• Decomposes the algorithmic entropy of mixed replicating and non-replicating structures into components: arrangement code, replicate specification, and non-replicate specification.
• Derives a composite entropy formula: $ H_{prov}(S) \cong \log_2 \mathcal{P} + R\log_2 \mathcal{M} + X\log_2 \mathcal{K} + \log_2 P_R + \log_2 P_X + \log_2 R + \log_2 X + \text{specification terms} $.
• Analyzes trade-offs between organizational complexity ($ D_{\text{org}} $) and algorithmic entropy, showing nesting reduces entropy cost of variation.

Experimental results

Research questions

• RQ1 How does replication generate and maintain low algorithmic entropy in non-equilibrium systems?
• RQ2 What is the role of variation in replicating structures in enabling system adaptation and homeostasis?
• RQ3 How do dependencies and co-evolution between replicators enhance system stability?
• RQ4 In what way does nesting of replicators reduce algorithmic entropy while increasing organizational complexity?
• RQ5 Can algorithmic information theory provide a robust measure of order in replicating systems, particularly in nested configurations?

Key findings

• Replicating systems generate highly ordered structures with low algorithmic entropy, making them more likely to emerge than random configurations.
• Variation in replicates increases algorithmic entropy but enables the system to evolve toward a more restricted, stable region of its state space.
• Coupled replicator systems co-evolve and achieve greater stability by sharing resources or information, reducing external dependence.
• Nested replicating systems exhibit high organizational complexity with minimal algorithmic entropy increase, due to efficient algorithmic description.
• Chaitin’s d-diameter complexity effectively measures the level of order in nested replicating systems, indicating a trade-off between variety and organization.
• The algorithmic entropy of a system with replicates and non-replicates is bounded by the sum of arrangement complexity, replicate and non-replicate specification, and structural specification terms.