Explicit solution of the Cauchy problem for cellular automaton rule 172

TL;DR

This paper derives an explicit formula for the evolution of cellular automaton rule 172, enabling solutions to the initial value problem similar to PDEs, and analyzes expected values under probabilistic initial conditions.

Contribution

It provides the first explicit solution for rule 172's Cauchy problem and extends the analysis to probabilistic initial conditions, bridging CA and PDE solution methods.

Findings

Explicit formula for rule 172's state after n iterations

Expected cell value calculation under Bernoulli initial conditions

Applicability to finite and infinite lattices

Abstract

Cellular automata (CA) are fully discrete alternatives to partial differential equations (PDE). For PDEs, one often considers the Cauchy problem, or initial value problem: find the solution of the PDE satisfying a given initial condition. For many PDEs of the first order in time, it is possible to find explicit formulae for the solution at the time $t>0$ if the solution is known at $t=0$. Can something similar be achieved for CA? We demonstrate that this is indeed possible in some cases, using elementary CA rule 172 as an example. We derive an explicit expression for the state of a given cell after $n$ iteration of the rule 172, assuming that states of all cells are known at $n=0$. We then show that this expression ("solution of the CA") can be used to obtain an expected value of a given cell after $n$ iterations, provided that the initial condition is drawn from a Bernoulli distribution. This can be done for both finite and infinite lattices, thus providing an interesting test case for investigating finite size effects in CA.