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Synergy as the Failure of Distributivity.

Ivan Sevostianov1, Ofer Feinerman1

  • 1Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel.

Entropy (Basel, Switzerland)
|November 27, 2024
PubMed
Summary
This summary is machine-generated.

This study mathematically defines synergy by linking information theory and set theory, revealing it as a failure of distributivity. This framework allows for a rigorous understanding of emergence and information decomposition in complex systems.

Keywords:
decompositionemergenceinformation diagrams

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Area of Science:

  • Information Theory
  • Set Theory
  • Mathematical Physics
  • Complex Systems

Background:

  • The concept of emergence and synergy lacks a precise mathematical definition, hindering rigorous scientific study.
  • Existing frameworks struggle to reconcile the idea of 'more from less' with fundamental mathematical principles.
  • Set theory provides a basis for part-whole relationships, but its standard axioms preclude synergistic phenomena.

Purpose of the Study:

  • To establish a rigorous mathematical definition of synergy and emergence.
  • To explore the relationship between information theory, set theory, and synergistic phenomena.
  • To resolve contradictions in partial information decomposition and provide a foundation for studying emergent systems.

Main Methods:

  • Connecting information theory with set theory to analyze the mathematical properties of synergy.
  • Investigating discrete random variables as a model for information decomposition.
  • Formalizing the distinction between set-theoretic axioms and the behavior of random variables, identifying the failure of union/intersection distributivity.

Main Results:

  • Synergy is mathematically characterized as the failure of distributivity in information-theoretic contexts.
  • A multivariate theory is proposed that resolves self-contradictions in partial information decomposition.
  • Venn-type diagrams are adapted to represent information, accommodating synergistic effects.

Conclusions:

  • The study provides a rigorous mathematical foundation for understanding emergence and synergy.
  • Non-distributive variants of set theory offer a potential framework for describing emergent physical systems.
  • The proposed theory opens new avenues for quantifying and analyzing complex emergent phenomena.