A linear speed-up theorem for cellular automata

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).