An Overview of Recent Solutions to and Lower Bounds for the Firing Synchronization Problem

Complex systems in a wide variety of areas such as biological modeling, image processing, and language recognition can be modeled using networks of very simple machines called finite automata. Connecting subsystems modeled using finite automata into a network allows for more computational power. One such network, called a cellular automaton, consists of an n-dimensional array for n > 1 with a single finite automaton located at each point of the array. One of the oldest problems associated with cellular automata is the firing synchronization problem, originally proposed by John Myhill in 1957. As with any long-standing problem, there are a large number of solutions to the firing synchronization problem. Our goal, and the contribution of this work, is to summarize recent solutions to the problem. We focus primarily on solutions to the original problem, that is, the problem where the network is a one-dimensional array and there is a single initiator located at one of the ends. We summarize both minimal-time and non-minimal-time solutions, with an emphasis on solutions that were published after 1998. We also focus on solutions that minimize the number of states required by the finite automata. In the process we also identify open problems that remain in terms of finding minimal-state solutions to the firing synchronization problem.