Inverse and Injectivity of Parallel Relations Induced by Cellular Automata

Moore and Myhill showed that Garden-of-Eden theorem [2], [3]. A binary relation over the configurations is said to be "parallel" if it is induced by a cellular (tessellation) automaton. Richardson showed the equivalence between a parallel relation (a nondeterministic parallel map) with the quiescent state to be injective and its inverse to be parallel by the Garden-of-Eden theorem plus compactness of product topology [4]. This paper deals with the inverse and the injectivity when a cellular automaton is given that induces a parallel relation. We give an equivalent condition, concerning only the local map, for the inverse of a parallel relation to be parallel. Furthermore we show an equivalent condition, concerning only the local map, for the injectivity of a parallel map. Consequently, we note that these two conditions are represented by semirecursive predicates.