Cellular automata with non-working cells

Abstract

A cellular automaton is an array of interconnected finite-state machines, each such machine being referred to as a cell. The interconnections form ,a neighborhood for each cell, and the state of each cell can change at discrete time steps based upon the states of the cells in the neighborhood. It has been shown by Von Neumann, Codd and others that automata of this type can function as universal computers, or as constructors capable of creating a given pattern of states in the array. The construction power is sufficient for an automaton to self-reproduce, or build a copy of itself elsewhere in the space.

The last sections of the thesis contain suggestions for alternate models that might be developed to work in faulty spaces, and a list of some open problems for further research. The appendices list the transition rules for the model presented, and contain computer output for the simulations of the various basic operations.

The existence of these automata is demonstrated by giving the rules for a particular automaton that has the desired properties. This model generally uses double paths for the transfer of information. Signals are propagated along such paths at the rate of one cell per two time steps. The underlying space has the usual 5-cell neighborhood. The model is presented by showing how the various basic operations for transfer of information, logical control, and construction of new automata are performed. Verification of the proper working of the model is done by extensive computer simulation.

This thesis is concerned with the logical power of cellular automata given that some of the cells in the array do not function properly. A restricted definition for allowable faults in the cells is given, such cells being called "non-working." A non-working cell is assumed to fail in such a manner that it enters a state (or one of a set of states) that is recognizable by the cells of the neighborhood, but no longer changes state at subsequent time steps. Under the assumption that such non-working cells are sufficiently separated it is shown that automata with the power of a universal computer and a universal constructor can be created.