Simple Computation-Universal Cellular Spaces

The special izat ion of the theory of cellular spaces (cellular au tomata ) to those spaces which compute par t ia l recursive funct ions is presented. Neighborhood reduct ion and state-set reduct ion are shown to be pa r t i cu la r ly simple in this special theory, and one dimension is proved to be sufficient for computa t ion universa l i ty . Several computa t ion-un iversa l cellular spaces (CUCS's) are exhibi ted which are simple in the sense t h a t each cell has only a small number q of s ta tes and a small number p of neighbors. For example, a 1-dimensional CUCS with pq = 36 is presented. Two quite different proofs of the existence of a I -dimensional CUCS with only two neighbors are given. Final ly , one of the theorems derived is used to settle three open decidabi l i ty questions.