Model Discovery and Discrete Inverse Problems with Cellular Automata and Boolean Networks

Abstract

We introduce and explore some methods based on algorithmic complexity and algorithmic probability that help address the challenge of empirical causal model discovery and inverse problems. These methods, based on Algorithmic Information Dynamics (AID), are designed to describe a possible pathway from observation to causal reconstruction of the dynamics and space-time evolution of discrete systems, with consideration given to inference cost. We apply these methods to two of the most popular discrete dynamical systems, cellular automata and Boolean networks. We show that an algorithmic-probability-guided simulation of dynamic properties of these discrete systems can connect back to fundamental questions of causality and scientific discovery, whereas complexity science has traditionally tended to obfuscate such connections or obviate them with informal concepts of emergence and self-organisation. In the inverse problem of phase-space reconstruction, we consider the cost trade-off between observation and simulation in the challenge of model inference. We combine both algorithmic complexity and Bayesian methods to characterise an observation as a sequence of small computable models allowing incremental scientific model discovery, thereby providing a complexity framework that contributes to the study of causation.