Cellular Space Models: New Formalism for Simulation and Science

Abstract

Arthur Burks, when explicating what is meant by a “Law of Nature” states that it has three main features: uniqueness, modality, and uniformity. (Burks, 1977) (p. 425) Very briefly, “uniqueness” means that the law makes unequivocal assertions, “modality”, that it holds not only for actual, but also possible, situations, and “uniformity”, that it applies uniformly to all points in space and time. Proceeding to formalize these concepts, Burks presents cellular automata as model systems in which such laws of this nature are readily comprehended. (Burks, 1977, p. 562). A cellular automaton¹ is characterized by specifying three parameters: a set of states S, a neighborhood N, and a transition function T. The interpretation of the triple <S,N,T> is as follows:

One imagines a checkerboard stretching out towards infinity in both north-south and east-west axes. At each square is located a cell with state set S. The neighbors of a cell located at square (i,j) (where as will be apparent “neighbors” is intended in an informational sense) are simply determined from the neighborhood N: in fact, N is a finite ordered set of integer pairs and the neighbors of this cell are located at the squares obtained by adding (i,j) to each pair of N (vector addition). Now imagine a global state of this system pertaining at some time instant t, i.e., assign to each cell a state from the set S. Then the system will move to a succeeding global state at time instant t+1 which is determined as follows: Simultaneously, for each cell apply the transition functionn T to the states of its neighbors and let the result be the state of the cell at t+1.