Two-Player Games on Cellular Automata

Summary

ABSTRACT. Cellular automata games have traditionally been 0-player or solitaire games. We define a two-player cellular automata game played on a finite cyclic digraph G = (V,E). Each vertex assumes a weight. A move consists of selecting a vertex u with w(u) = 1 and firing it, i.e., complementing its weight and that of a selected neighborhood of u. The player first making all weights 0 wins, and the opponent loses. If there is no last move, the outcome is a draw. The main part of the paper consists of constructing a strategy. The 3-fold motivation for exploring these games stems from complexity considerations in combinatorial game theory, extending the hitherto 1-player cellular automata games to twoplayer games, and the theory of linear error correcting codes.

1. Introduction

Cellular Automata Games have traditionally been 0-player games such as Conway's Life, or solitaire games played on a grid or digraph G = (V, E). (This includes undirected graphs, since every undirected edge {u, v} can be interpreted as the pair of directed edges (u,v) and (v,u).) Each cell or vertex of the graph can assume a finite number of possible states. The set of all states is the alphabet. We restrict attention to the binary alphabet {0,1}. A position is an assignment of states to all the vertices. There is a local transition rule from one position to another: pick a vertex u and fire it, i.e., complement it together with its neighborhood. The aim is to move from a given position (such as all 0s) to a target position (such as all 0s). In many of these games any order of the moves produces the same result, so the outcome depends on the set of moves, not on the sequence of moves. Two commercial manifestations are Lights Out manufactured by Tiger Electronics, and Merlin Magic Square by Parker Brothers (but Arthur-Merlin games are something else again). Quite a bit is known about such solitaire games.