Quad Prize Submission: Simulating Elementary CAs with Trid CAs

The Trid neighbourhood is a three cell neighbourhood on a 2dimensional lattice, and is a subset of the Quad neighbourhood. I show that, with an appropriately prepared initial configuration of “diagonal stripes”, a given elementary CA can be simulated by the two state Trid neighbourhood CA whose local update rule is precisely that of the elementary CA. Thus, invoking Cook’s universality result for elementary CA rule 110, I exhibit a universal two state Trid neighbourhood CA. 1. Definitions and notation The Trid neighbourhood on a 2-dimensional square lattice is defined by the neighbour offsets 〈(0, 1), (0, 0), (1, 0)〉 . (1) In other words, the neighbourhood of a given cell consists of the cell itself, and the cells immediately adjacent to the north and to the east. A Trid CA is a CA with two states (i.e. with state set Z2 = {0, 1}) using the Trid neighbourhood. A local update rule for a Trid CA is a function Z2 → Z2. An elementary cellular automaton (ECA) is a 1-dimensional CA with two states and the three cell neighbourhood 〈−1, 0, 1〉 . (2) Thus a local update rule for an ECA is also a function Z2 → Z2. A configuration of a CA is a mapping of a state to each cell in the lattice. Thus for a D-dimensional two state CA, a configuration is a function Z → Z2. The global map for a CA is the extension of its local update rule to a function from configurations to configurations, in the usual way. 2. Simulating ECAs with Trid CAs First, define a mapping α from 1-D configurations to 2-D configurations. Let s : Z→ Z2 be a 1-D configuration. Then the 2-D configuration α(s) is defined by α(s)[x, y] = s[x− y] for all x, y ∈ Z . (3) Intuitively, α(s) is obtained by mapping the cell states of s along the diagonals which run southwest to northeast. This has the effect that the Trid neighbourhood can “see” the states of three adjacent ECA cells, as depicted in Figure 1. The mapping α is injective, but not surjective; thus α does not have an inverse. However, define a mapping β from 2-D configurations to 1-D configurations by β(s2)[x] = s2[x, 0] for all x ∈ Z (4) for all 2-D configurations s2 : Z → Z2. Then it is easy to see that β ◦ α is the identity on the set of 1-D configurations. Furthermore, α ◦ β is the identity on