A cellular automaton is a self-map \(F\) of \(S^{\mathbb Z^2}\) that is both translation-equivariant and local (the value at \((x,y)\) of \(F(c)\) depends only on the restriction of \(c\) to \(\{x-1,x,x+1\}\times\{y-1,y,y+1\}\)).
Two fundamental properties of cellular automata are: 1) pre-injectivity: two distinct configurations \(c,c'\) that differ in finitely many places must have distinct images under \(F\); 2) no Garden of Eden: the map \(F\) is surjective. Celebrated results by E. F. Moore [Proc. Sympos. Appl. Math. 14, 17–33 (1962; Zbl 0126.32408)] and J. Myhill [Proc. Am. Math. Soc. 14, 685–686 (1963; Zbl 0126.32501)] prove that these two properties are equivalent.
In this paper, the author considers in extreme detail the exact definition of pre-injectivity used by Moore and Myhill, which seem almost identical, and in fact turn out to be identical.