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  "body": "\n## Statement\n\nLet $A$ be a finite set, let $d \\ge 1$, and let $F \\colon A^{\\Z^d} \\to A^{\\Z^d}$ be a [[cellular automaton|5QXNC6NTVN]].\nCall two configurations $x, y$ asymptotic if they differ at only finitely many sites.\nCall $F$ pre-injective if $F(x) \\ne F(y)$ for all asymptotic configurations $x \\ne y$.\nThen $F$ is surjective if and only if $F$ is pre-injective.\n\n## Remarks\n\nA configuration outside the image of $F$ is called a Garden of Eden: it has no predecessor.\nSo the theorem says that a Garden of Eden exists exactly when two configurations that differ at finitely many sites have the same image.\nThe proof compares counts: in a large box of side $n$, the number of patterns grows like $\\abs{A}^{n^d}$, while the boundary of the box has only about $n^{d-1}$ sites.\nA missing pattern, or a pair of patterns with the same image, changes the exponential count by a factor that the boundary cannot compensate.\nThe theorem is due to Moore and Myhill.\nExample: rule 90 in one dimension is surjective, so it is pre-injective.\nIt is not injective, since the all-$0$ and the all-$1$ configurations both map to the all-$0$ configuration, but these are not asymptotic.\nConway's Game of Life is not surjective: it has Garden of Eden configurations.\nThe theorem extends to cellular automata on amenable groups, and it fails for the free group on two generators.\n\n## Explanations\n\n### L1 human\n\nA Garden of Eden is a picture that a cellular automaton can never produce, because nothing can lead to it.\nThe theorem says that such pictures exist exactly when the automaton loses information in a finite region.\nLosing information means that two different finite changes of a picture give the same next picture.\nSo counting how much information is lost tells whether every picture can be reached.\n\n### L2 human\n\nInjective implies pre-injective, so every injective cellular automaton on $\\Z^d$ is surjective.\nThis is a dynamical analogue of the fact that an injective map of a finite set to itself is onto.\nThe theorem is often used to show that a rule has Garden of Eden patterns by exhibiting two mutually erasable patterns.\nIts extension to amenable groups by Ceccherini-Silberstein, Machì and Scarabotti links it to [[amenable group actions|ZCG4DJ7GH3]].\nFor the free group of rank two there are surjective cellular automata that are not pre-injective.\n",
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