That’s true, but would I be right in saying that as long as there are no Garden of Eden states, you could in theory at least generate one possible prior state?
There is some terminological confusion in this thread: a Garden of Eden of a Cellular Automaton is a configuration which has no predecessor. (A configuration is a function assigning a state to every cell). An orphan is a finite subset of cells with their states having the property that any configuration containing the orphan is a Garden Of Eden configuration. Obviously every orphan can be extended to a Garden of Eden configuration (by choosing a state arbitrarily for the cells not in the orphan) but interestingly it is also true that every garden of eden contains an orphan. So the answer to your question is yes: if there are no orphans in the configuration then it is not a garden of eden, therefore, by definition, it has a predecessor.
That’s true, but would I be right in saying that as long as there are no Garden of Eden states, you could in theory at least generate one possible prior state?
There is some terminological confusion in this thread: a Garden of Eden of a Cellular Automaton is a configuration which has no predecessor. (A configuration is a function assigning a state to every cell). An orphan is a finite subset of cells with their states having the property that any configuration containing the orphan is a Garden Of Eden configuration. Obviously every orphan can be extended to a Garden of Eden configuration (by choosing a state arbitrarily for the cells not in the orphan) but interestingly it is also true that every garden of eden contains an orphan. So the answer to your question is yes: if there are no orphans in the configuration then it is not a garden of eden, therefore, by definition, it has a predecessor.