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Exploring Simplicity Bias in 1D Dynamical Systems.

Kamal Dingle1,2, Mohammad Alaskandarani1, Boumediene Hamzi3,4

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Summary
This summary is machine-generated.

Algorithmic information theory suggests simplicity bias, an inverse relationship between pattern probability and complexity. This study found simplicity bias in some dynamical systems, but not others, offering new prediction tools.

Keywords:
algorithmic probabilitydynamical systemssimplicity biastime series

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Area of Science:

  • Dynamical Systems
  • Algorithmic Information Theory
  • Complexity Science

Background:

  • Algorithmic information theory posits an inverse relationship between output pattern probability and complexity, known as simplicity bias.
  • Dynamical systems generate complex trajectories from parameter inputs, offering a framework to study this bias.

Purpose of the Study:

  • To investigate the presence and extent of simplicity bias in various discrete dynamical systems.
  • To determine if simplicity bias is a universal property across different types of maps.
  • To explore the predictive power of simplicity bias for output pattern probabilities.

Main Methods:

  • Examined five discrete dynamical systems: logistic map, Gauss map, sine map, Bernoulli map, and tent map.
  • Treated map parameters as inputs and digitized trajectories as outputs.
  • Sampled initial and parameter values to analyze output patterns and their probabilities.

Main Results:

  • The logistic map, Gauss map, and sine map exhibited simplicity bias.
  • The Bernoulli map and tent map did not show evidence of simplicity bias.
  • Simplicity bias provided surprisingly accurate a priori predictions for output pattern probabilities in some systems.

Conclusions:

  • Simplicity bias is not universally present in all discrete dynamical systems.
  • The study highlights the utility of probability-complexity relationships for analyzing patterns in dynamical systems.
  • This approach offers a novel method for predicting pattern probabilities with minimal system-specific detail.