Structure and computability of preimages in the Game of Life

TL;DR

This paper demonstrates that computing preimages in Conway's Game of Life is as complex as solving arbitrary circuit-satisfaction problems, revealing deep computational universality in backward evolution.

Contribution

It establishes the computational universality of preimage computation in the Game of Life and shows its implications for complexity and decidability.

Findings

Preimage computation encodes arbitrary circuit-satisfaction problems.

The set of orphans is coNP-complete.

Existence of a preimage for a periodic point is undecidable.

Abstract

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 $6210 \times 37800$-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.