Simplified Game of Life: Algorithms and Complexity

TL;DR

This paper analyzes simplified versions of the Game of Life focusing on underpopulation and overpopulation rules, revealing polynomial-time solvability for the former and PSPACE-completeness for the latter.

Contribution

It introduces a formal analysis of two simple rule families in the Game of Life, establishing complexity results for configuration reachability problems.

Findings

Underpopulation rules are solvable in polynomial time.

Overpopulation rules lead to PSPACE-complete problems.

The study clarifies the computational complexity of basic dynamics in simplified Life models.

Abstract

Game of Life is a simple and elegant model to study dynamical system over networks. The model consists of a graph where every vertex has one of two types, namely, dead or alive. A configuration is a mapping of the vertices to the types. An update rule describes how the type of a vertex is updated given the types of its neighbors. In every round, all vertices are updated synchronously, which leads to a configuration update. While in general, Game of Life allows a broad range of update rules, we focus on two simple families of update rules, namely, underpopulation and overpopulation, that model several interesting dynamics studied in the literature. In both settings, a dead vertex requires at least a desired number of live neighbors to become alive. For underpopulation (resp., overpopulation), a live vertex requires at least (resp. at most) a desired number of live neighbors to remain alive. We study the basic computation problems, e.g., configuration reachability, for these two families of rules. For underpopulation rules, we show that these problems can be solved in polynomial time, whereas for overpopulation rules they are PSPACE-complete.