Garden of Eden (cellular automaton)

A Garden of Eden in Conway's Game of Life, discovered by R. Banks in 1971.[1] The cells outside the image are all dead (white).
An orphan in Life found by Achim Flammenkamp. Black squares are required live cells; blue x's are required dead cells.

In a cellular automaton, a Garden of Eden is a configuration that has no predecessor. It can be the initial configuration of the automaton but cannot arise in any other way. John Tukey named these configurations after the Garden of Eden in Abrahamic religions, which was created out of nowhere.[2]

A Garden of Eden is determined by the state of every cell in the automaton (usually a one- or two-dimensional infinite square lattice of cells). However, for any Garden of Eden there is a finite pattern (a subset of cells and their states, called an orphan) with the same property of having no predecessor, no matter how the remaining cells are filled in. A configuration of the whole automaton is a Garden of Eden if and only if it contains an orphan. For one-dimensional cellular automata, orphans and Gardens of Eden can be found by an efficient algorithm, but for higher dimensions this is an undecidable problem. Nevertheless, computer searches have succeeded in finding these patterns in Conway's Game of Life.

The Garden of Eden theorem of Moore and Myhill asserts that a cellular automaton on the square grid, or on a tiling of any higher dimensional Euclidean space, has a Garden of Eden if and only if it has twins, two finite patterns that have the same successors whenever one is substituted for the other.

  1. ^ Cite error: The named reference banks was invoked but never defined (see the help page).
  2. ^ Moore (1962).