2026-07-22T07:15:42Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16472 2024-03-06T10:57:34Z ddc:004 open_access

What Can Oracles Teach Us About the Ultimate Fate of Life? Salo, Ville Törmä, Ilkka Game of Life cellular automata limit set symbolic dynamics 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ville Salo and Ilkka Törmä 2022 Is Part Of LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2022.131 urn:nbn:de:0030-drops-164721 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.131 eng https://creativecommons.org/licenses/by/4.0/legalcode