On the Parity Problem in One-Dimensional Cellular Automata

TL;DR

This paper investigates the minimal neighborhood size needed for one-dimensional cellular automata to solve the parity problem, proving radius 2 is insufficient and designing a radius 4 rule that works, leaving radius 3 as an open question.

Contribution

The paper proves that radius 2 automata cannot solve the parity problem and introduces a radius 4 rule that successfully solves it, advancing understanding of cellular automata capabilities.

Findings

Radius 2 rules cannot solve the parity problem.

A radius 4 rule is constructed and proven to solve the problem.

The existence of a radius 3 solution remains open.

Abstract

We consider the parity problem in one-dimensional, binary, circular cellular automata: if the initial configuration contains an odd number of 1s, the lattice should converge to all 1s; otherwise, it should converge to all 0s. It is easy to see that the problem is ill-defined for even-sized lattices (which, by definition, would never be able to converge to 1). We then consider only odd lattices. We are interested in determining the minimal neighbourhood that allows the problem to be solvable for any initial configuration. On the one hand, we show that radius 2 is not sufficient, proving that there exists no radius 2 rule that can possibly solve the parity problem from arbitrary initial configurations. On the other hand, we design a radius 4 rule that converges correctly for any initial configuration and we formally prove its correctness. Whether or not there exists a radius 3 rule that solves the parity problem remains an open problem.