Simplified Game of Life: Algorithms and Complexity

Authors Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, Jakub Svoboda

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Author Details

Krishnendu Chatterjee
  • Institute of Science and Technology, Klosterneuburg, Austria
Rasmus Ibsen-Jensen
  • University of Liverpool, UK
Ismaël Jecker
  • Institute of Science and Technology, Klosterneuburg, Austria
Jakub Svoboda
  • Institute of Science and Technology, Klosterneuburg, Austria

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Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, and Jakub Svoboda. Simplified Game of Life: Algorithms and Complexity. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation
  • game of life
  • cellular automata
  • computational complexity
  • dynamical systems


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