What is the distribution of Garden of Eden patterns in cellular automata with increasing pattern size?
as said by vzn, any pattern containing a Garden of Eden pattern is itself a Garden of Eden Pattern, the probability that a random nxn pattern is actually GoE goes to 1 as n goes to infinity (in the case of a non-surjective CA of course, otherwise it's 0). More precisely, it is lower bounded by $1 - \alpha^{n²}$ for some $0&lt;\alpha&lt;1$.
Indeed, consider a fixed GoE pattern P of size $k\times k$, then the probability for a random pattern of size $n\times n$ to be a GoE is at least $1 - (\frac{1}{q^{k^2}})^{(\frac{n}{k})^2}$, where $q$ is the number of states of the CA.
you can generate all the possible pre-image of a given pattern in finite time and you can also test whether a pattern is a GoE so your problem is clearly decidable. It's polynomial-time decidable in 1D and NP-hard in 2D.


in the 1D case, a CA can be seen as a finite state transducer. So the language $L_{GoE}$ of all GoE patterns is a regular language because it is the complement of $T(Q^\ast)$ (where $T$ is the transducer and $Q$ the set of states of the CA) which is itself a regular language as the image of a regular language by a transducer. Then $T(L_{GoE})$ is also regular and its the set you asked.
in the 2D case, I think we can show that it is NP-hard using the classical simulation of Turing machines by Wang tiles. The rough idea is the following. Consider any NP-complete problem $L$ and take a Turing machine $T$ which is a polynomial time verifier for the problem. Then construct a CA that basically checks that the input, if written in some special alphabet $A$, encodes a successful computation $T(x,y)$ of $T$ working on input $x$ and witness $y$. If so erases everything with a special state $a\not\in A$, but keep the input x. If some error is detected produce some special state $b\not\in A$. Finally, the CA does nothing on states which are not in $A$. Now consider a pattern formed by an input $x$ surrounded by many $a$. The pattern as always itself as a preimage, but it as a GoE preimage if and only if there is some $y$ such that $T(x,y)$ says "yes", i.e. iff $x\in L$. CQFD
At the time of writing I don't see how to prove that the problem is NP

as pointed-out Turing-universality is still possible without GoE patterns (in reversible CA for instance). But intrinsic universality requires GoE patterns. This comes from the fact that, first, a surjective CA only admits surjective CA as subautomata, and, second, grouping and iterating a CA doesn't change its surjectivity.