Gardens of Eden in the Game of Life

TL;DR

This paper establishes conditions under which certain patterns in Conway's Game of Life are not Gardens of Eden, linking pattern properties to computational complexity and preimage density.

Contribution

It proves that non-orphan zero-padded patterns imply non-Gardens of Eden and characterizes the preimages as dense semilinear configurations, connecting pattern structure to complexity.

Findings

Finite-support Gardens of Eden are in co-NP.

Preimages of finite-support configurations are dense semilinear.

Zero-padding patterns determine Garden of Eden status.

Abstract

We prove that in the Game of Life, if the thickness-four zero-padding of a rectangular pattern is not an orphan, then the corresponding finite-support configuration is not a Garden of Eden, and that the preimage of every finite-support configuration has dense semilinear configurations. In particular finite-support Gardens of Eden are in co-NP.