Exploring Simplicity Bias in 1D Dynamical Systems

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.