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Cellular automata (CA) are fully discrete alternatives to partialdifferential equations (PDE). For PDEs, one often considers the Cauchy problem,or initial value problem: find the solution of the PDE satisfying a giveninitial condition. For many PDEs of the first order in time, it is possible tofind explicit formulae for the solution at the time $t>0$ if the solution isknown at $t=0$. Can something similar be achieved for CA? We demonstrate thatthis is indeed possible in some cases, using elementary CA rule 172 as anexample. 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 toobtain an expected value of a given cell after $n$ iterations, provided thatthe initial condition is drawn from a Bernoulli distribution. This can be donefor both finite and infinite lattices, thus providing an interesting test casefor investigating finite size effects in CA.

Explicit solution of the Cauchy problem for cellular automaton rule 172