The constructibility of a configuration in a cellular automaton

A configuration is said to be with finite support if the states of all but fintely many cells in the array are quiescent. The results are as follows. It is recursively unsolvable when d≥2, for a configuration c with finite support in a d-dimensional cellular automaton, whether or not:o1.c is in the image of the parallel map for the cellular automaton.2.c is in the image of the parallel map for the cellular automaton restricted to the set of configurations with finite support.Further properties of parallel maps are also considered