Exploring Wolfram’s Notion of Computational Irreducibility with a Two-Dimensional Cellular Automaton

Abstract

The notion of computational irreducibility says that a problem is computationally irreducible when the only way to solve it is to traverse a trajectory through a state space step by step using no shortcut. In this paper, we will explore this notion by addressing whether computational irreducibility is a consequence of how a particular problem is represented. To do so, we will examine two versions of a given game that are isomorphic representations of both the play space and the transition rules underlying the game. We will then develop a third isomorph of the play space with transition rules that seem to only be determined in a computationally irreducible manner. As a consequence, it would seem that representing the play space differently in the third isomorph introduces computational irreducibility into the game where it was previously lacking. If so, we will have shown that, in some cases at least, computational irreducibility depends on the representation of a given problem.