Occurrence of Gliders in an Infinite Class of Life-Like Cellular Automata

Abstract

The Game of Life (GL), Larger Than Life (LtL), and the Kaleidoscope of Life (KL) are cellular automaton (CA) models with a rich palette of configurations, some of which facilitate universal computation. Common to all these models is that the transition rules by which they are governed are outer-totalistic. The KL distinguishes itself by the striking simplicity of its transition rule, which does not even take into account a cell’s state itself for its update. This paper investigates an infinite class of CA, all of which are similar to KL except for their differently sized neighborhoods. Characterized by a discrete parameter d, a neighborhood in such a CA consists of the cells at Moore distances 1, 2,..., or d of a cell. We show that signal-carrying configurations (“gliders”) occur in infinitely many of these CA models. We also show that the probability of convergence of a random configuration toward a dead cellular space increases with the increase in parameter d. These seemingly contradictory results suggest that the presence of gliders are not necessarily a reliable benchmark for the sustainability of Life in cellular space.