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Randomly selected inputs in natural systems favor simple outputs. This study reveals that the probability of an output decays exponentially with its complexity, a bias predictable even with limited system knowledge.

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Area of Science:

  • Computational Biology
  • Complexity Science
  • Information Theory

Background:

  • Many natural systems operate as discrete input-output maps.
  • The distribution of outputs from random inputs is often assumed to be uniform without specific knowledge of the map.
  • Algorithmic information theory provides tools to analyze the inherent complexity of data and systems.

Purpose of the Study:

  • To investigate the a priori probability distribution of outputs generated by random inputs in real-world systems.
  • To demonstrate that these input-output maps exhibit a bias towards producing simpler outputs.
  • To derive a quantitative relationship between output probability and output complexity.

Main Methods:

  • Extension of fundamental results from algorithmic information theory.
  • Derivation of an upper bound for the probability of a given output P(x).
  • Analysis of various systems including RNA folding, ordinary differential equations, and financial models.

Main Results:

  • The a priori probability P(x) of generating output x decays exponentially with its approximate Kolmogorov complexity.
  • An upper bound P(x) ≲ 2^(-k(x)) is derived and shown to be tight for most inputs.
  • Key properties of P(x), including constants a and b, can be predicted with minimal map knowledge.

Conclusions:

  • Real-world input-output maps are fundamentally biased towards simplicity.
  • This bias is quantifiable and predictable, offering insights into diverse natural and artificial systems.
  • The findings have implications for understanding systems ranging from molecular biology to economics.