CPSC 521 Assignment 9

Cellular automata (CAs) are inherently parallel as the cells evolve in parallel with inter-cell communication restricted to a local neighborhood. The fish and sharks problem from assignment 4 is an example of a 2D CA. In this assignment, we focus on 1D CA where each cell is in one of two states (0 or 1) and its value the next time step depends on the current cell value and the right and left neighbors. In the most general case, each cell i has a next state function fi : B × B × B → B, where B = {0, 1}, and cells 1 and n are adjacent, where n is the number of cells. Given a CA of size n, define g : B → B as the next state function over all cells. We say a CA is reversible if the mapping g is 1 to 1. A naive algorithm could check if a CA is reversible in O(n2) by applying g to each state in B and noting the image. The CA is reversible if and only if all images are unique. A new algorithm, proposed by Jesse Bingham, can make this decision in O(n) time. An informal outline of the algorithm follows. Let T = {000, 001, 010, 011, 100, 101, 110, 111}, the 8 possible states that a cell and its left and right neighbors can have, and let N = 1, 2, ..., n. Define digraph D to have the vertex set V = T ×N (so that each vertex can be written as (abc, i) 1 where a, b, c are binary digits and 1 ≤ i ≤ n). There is an arc from vertex (abc, i) to (xyz, j) if and only if b = x, c = y, and j = i + 1 or i = n and j = 1. Call an n-cycle in D c1, c2, ..., cn an assignment-cycle if all vertices have the second component of each ci is i. Note that D and all assignmentcycles depend only on n and not on any particular CA. A CA can be defined in terms of the functions for each cell i, fi. Given a CA f1, f2, ..., fn, define function L : V → B such that L((abc, i)) = fi(a, b, c). For an assignment cycle C = c1, ..., cn, let the label sequence of C be L(c1), ..., L(cn). Now observe that a CA is reversible if and only if no two assignment-cycles have the same label sequence. This is seen by noting that assignment-cycles naturally correspond to a CA state s and the label sequence to g(s). If two assignment-cycles s1 and s2 have the same label sequence, then g(s1) = g(s2) and the CA is not reversible. To determine if two assignment-cycles have the same label sequence, define a sequence of sets S1, S2, ..., Sn where each Si is a set of 4-tuples of elements of T. These sets will have the property that (w, x, y, z) ∈ Si if and only if there exist