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Compositionality in the Synchronization of Processors Exchanging a Minimal Amount of Information

We consider the problem of synchronizing a network of identical processors that work synchronously at discrete steps. Processors are arranged in an array of m rows and n columns. Solutions at a given time t(m; n) are presented for some interesting class of functions t(m; n). An interesting feature of our approach is a modular description of solutions in terms of \fragments" of cellular automata that we call signals. A solution is thus a special signal obtained as a composition of many simpler signals. Compositional rules to obtain new signals (and thus new solutions) starting from known ones are given. In particular we apply these rules to obtain new solutions, using some elementary signals and minimal time solutions.