Flexible Time and Ether in One-dimensional Cellular Automata

Abstract

Flexible Time is a new formalism for calculations about one-dimensional cellular automata. It unifies the states of a finite number of cells into a single object, even if they occur at different times. This gives greater flexibility to handle the structures that occur in the development of a cellular automaton. An ether is a periodic pattern of cells that arises in some cellular automata from almost all random initial configurations. In this thesis, the formalism is developed in detail and then applied to the problem of ether formation. For the elementary cellular automaton Rule 54, a partial result is proved: There is a fragment of the ether that arises with probability 1 from every random initial configuration and is propagated with probability 1 to any later time. This is a strong argument that the ether under Rule 54 indeed arises from almost all input configurations.