Synchronization of Cellular Arrays: The Firing Squad Problem in Two Dimensions

are copies of a single synchronous finite-state machine M. The internal state of a cell at time t @ 1 depends on its internal state and on the internal states of its left and right neighbors at time t. At time t 0, one of the end cells (called "the general") enters an internal state s, ("fire when ready"), while all other cells (called "the soldiers") are in a quiescent state sq. The array must behave as follows: at some instant t = t f , all cells ("the general" and "the soldiers") must execute a transition to an internal state s I ("fire"), and no cell must have entered s I prior to time t s . The problem consists of defining the structure of the cell M , with the stipulation that its complexity must not depend on the number of cells in the array. The first (unpublished) solution of the firing squad problem was found by McCarthy and Minsky. I t can easily be shown that the minimal time