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. A finite automaton is an idealized machine that follows a fixed set of rules and changes its state to recognize a pattern, accepting or rejecting the input based on the state in which it finds itself at the conclusion of the computation. In general a subsystem can be modeled using a finite automaton if it can be described by a finite set of states and rules. 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. Each automaton reads the current state of its neighbors to determine how to modify its own state. The interest in studying cellular automata is not in the description of the local state transitions of each machine, but rather in studying the global behavior across the entire array. One of the oldest problems associated with cellular automata is the firing synchronization problem, originally proposed by John Myhill in 1957. In the firing synchronization problem each machine of the cellular automata is identical, except for a single designated machine called the general or initiator. The synchronization starts with every cell in a quiescent state, except the initiator which starts in the general state and is the one initiating the synchronization. The next state of each cell is determined by the cell’s state and the state of its neighbors, with a required rule being that a machine that is quiescent and has two neighbors that are quiescient must remain so in the next round of the computation. The goal of the problem is to synchronize the array of automata, which means that each machine enters a unique designated firing state for the first time and simultaneously at some point during the execution. As might be expected with a problem that has existed as long as the firing synchronization problem, there are a wide range of variants for the network and its initial configuration. In the original problem the initiator is placed at one end of the array, either on the left or the right. A problem that is more complex than the original problem is the generalized problem, defined by Moore and Langdon in 1968 [12]. In this version the initiator can be placed anywhere in the array, although there is a restriction that there is a single initiator. The problem can be made more