Fuzzy Cellular Automata in Conjunctive Normal Form

Abstract

Cellular automata (CA) are discrete dynamical systems comprised of a lattice of finite-state cells. At each time step, each cell updates its state as a function of the previous state of itself and its neighbours.

Fuzzy cellular automata (FCA) are a real-valued extension of Boolean cellular automata which "fuzzifies" Boolean logic in the transition function using real values between zero and one (inclusive). To date, FCA have only been studied in disjunctive normal form (DNF).

In this thesis, we study FCA in conjunctive normal form (CNF). We classify FCA in CNF both analytically and empirically. We compare these classes to their DNF counterparts. We prove that certain FCA exhibit chaos in CNF, in contrast to the periodic behaviours of DNF FCA. We also briefly explore five different forms of fuzzy logic, and suggest further study. In support of this research, we introduce novel methods of simulating and visualizing FCA.