Solving the initial value problem for cellular automata by pattern decomposition

TL;DR

This paper demonstrates how to explicitly solve the initial value problem for certain cellular automata by analyzing and decomposing their spatiotemporal patterns, exemplified with elementary rule 156, enabling calculations of cell densities and block probabilities.

Contribution

It introduces a pattern decomposition method to solve initial value problems for cellular automata, exemplified with rule 156, providing explicit formulas for state evolution and statistical measures.

Findings

Explicit solution for rule 156's initial value problem

Method to compute density of ones after n iterations

Procedure to determine probabilities of longer symbol blocks

Abstract

For many cellular automata, it is possible to express the state of a given cell after $n$ iterations as an explicit function of the initial configuration. We say that for such rules the solution of the initial value problem can be obtained. In some cases, one can construct the solution formula for the initial value problem by analyzing the spatiotemporal pattern generated by the rule and decomposing it into simpler segments which one can then describe algebraically. We show an example of a rule when such approach is successful, namely elementary rule 156. Solution of the initial value problem for this rule is constructed and then used to compute the density of ones after $n$ iterations, starting from a random initial condition. We also show how to obtain probabilities of occurrence of longer blocks of symbols.