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What Can Oracles Teach Us About the Ultimate Fate of Life?

Authors Ville Salo , Ilkka Törmä

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Author Details

Ville Salo
  • Department of Mathematics and Statistics, University of Turku, Finland
Ilkka Törmä
  • Department of Mathematics and Statistics, University of Turku, Finland


We thank the anonymous referees for their useful suggestions, and the nice people of the ConwayLife forum for their interest.

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Ville Salo and Ilkka Törmä. What Can Oracles Teach Us About the Ultimate Fate of Life?. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 131:1-131:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We settle two long-standing open problems about Conway’s Life, a two-dimensional cellular automaton. We solve the Generalized grandfather problem: for all n ≥ 0, there exists a configuration that has an nth predecessor but not an (n+1)st one. We also solve (one interpretation of) the Unique father problem: there exists a finite stable configuration that contains a finite subpattern that has no predecessor patterns except itself. In particular this gives the first example of an unsynthesizable still life. The new key concept is that of a spatiotemporally periodic configuration (agar) that has a unique chain of preimages; we show that this property is semidecidable, and find examples of such agars using a SAT solver. Our results about the topological dynamics of Game of Life are as follows: it never reaches its limit set; its dynamics on its limit set is chain-wandering, in particular it is not topologically transitive and does not have dense periodic points; and the spatial dynamics of its limit set is non-sofic, and does not admit a sublinear gluing radius in the cardinal directions (in particular it is not block-gluing). Our computability results are that Game of Life’s reachability problem, as well as the language of its limit set, are PSPACE-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Game of Life
  • cellular automata
  • limit set
  • symbolic dynamics


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