Structure and computability of preimages in the Game of Life

: Salo, Ville; Törmä, Ilkka

Publisher: Elsevier BV

: AMSTERDAM

: 2025

: Theoretical Computer Science

: THEOR COMPUT SCI

: 115237

: 1042

: 19

: 0304-3975

: 1879-2294

DOI: https://doi.org/10.1016/j.tcs.2025.115237

: https://doi.org/10.1016/j.tcs.2025.115237

: https://research.utu.fi/converis/portal/detail/Publication/491819155

Conway's Game of Life is a two-dimensional cellular automaton. As a dynamical system, it is well-known to be computationally universal, i.e. capable of simulating an arbitrary Turing machine. We show that in a sense taking a single backwards step of the Game of Life is a computationally universal process, by constructing patterns whose preimage computation encodes an arbitrary circuit-satisfaction problem, or, equivalently, any tiling problem. As a corollary, we obtain for example that the set of orphans is coNP-complete, exhibit a 6210 x 37800-periodic configuration whose preimage is nonempty but contains no periodic configurations, and prove that the existence of a preimage for a periodic point is undecidable. Our constructions were obtained by a combination of computer searches and manual design.

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