A Linear Speed-Up Theorem for Cellular Automata

Abstract Ibarra (1985) showed that, given a cellular automaton of range 1 recognizing some language in time n +1+ R ( n ), we can obtain another CA of range 1 recognizing exactly the same language but in time n +1+ R ( n )/ k ( k ⩾2 arbitrary). Their proof proceeds indirectly (through the simulation of CAs by a special kind of sequential machines, the STMs) and we think it misses that way some of the deep intuition of the problem. We, therefore, provide here a direct proof of this result (extended to the case of CAs of arbitrary range) involving the explicit construction of a CA working in time n +1+ R ( n )/ k . This speeded-up automaton first gathers the cells of the line k by k in n +1 steps which then enables it to start computing by “leaps” of k steps, thus completing the R ( n ) remaining steps in time R ( n )/ k . The major problem arising from the obligation to pass from one phase to the other synchronously is solved using a synchronization process derived from the solutions of the well-known “firing-squad synchronization problem” (FSSP).