The Garden-of-eden Theorem for Finite Configurations

In [l] Moore showed that the existence of mutually erasable configurations in a two-dimensional tessellation space is sufficient for the existence of Garden-of-Eden configurations. In [2 ] Myhill showed that the existence of mutually indistinguishable configurations is both necessary and sufficient for the existence of Garden-of-Eden configurations. After redefining the basic concepts with some minor changes in terminology, and after restating the main results from [l] and [2], we shall establish the equivalence between the existence of mutually erasable configurations and the existence of mutually indistinguishable configurations. This implies that the converse of Moore's result is true as well. We then show that by limiting the universe to the set of all finite configurations of the tessellation array, both of the above conditions remain sufficient, but neither is then necessary. Finally, we establish a necessary and sufficient condition for the existence of Garden-of-Eden configurations when only finite configurations are considered. I. The tessellation structure and the Garden-of-Eden theorems. The tessellation array, which was first used by Von Neumann [3] in obtaining his results on machine self-reproduction, can be visualized as an infinite two-dimensional Euclidean space divided into square cells, in the fashion of a checkerboard, where each cell can hold any symbol from a finite set A. We use the set Z2 of ordered pairs of integers to name the cells in the tessellation array. An array configuration, i.e., a symbol placed in each cell, is formally a mapping c'.Z2—>A. The restriction of an array configuration c to a subset 5 of Z2 will be denoted by (c)sWe speak of this as the configuration of S in array configuration c. Each cell will behave like a deterministic and synchronous finite-state machine, and the symbol in cell (i, j) at time / will depend on the symbol in cell (i, j) at time t — 1 as well as the symbols in certain neighboring cells at time t — 1. In this paper, as in [l] and [2], we fix the neighbors of any cell to be those cells (including the cell itself) which have each of their coordinates differing by at most 1 from the coordinate of the given cell. Figure 1 shows the neighbors of cell (i, j). Received by the editors September 2, 1969 and, in revised form, November 3, 1969. A MS subject classifications. Primary 0288, 0280, 9440.