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We study cellular automata where the state at each site is decided by amajority vote of the sites in its neighborhood. These are equivalent, for arestricted set of initial conditions, to non-zero probability transitions insingle spin-flip dynamics of the Ising model at zero temperature.  We show that in three or more dimensions these systems can simulate Booleancircuits of AND and OR gates, and are therefore P-complete. That is, predictingtheir state t time-steps in the future is at least as hard as any other problemthat takes polynomial time on a serial computer.  Therefore, unless a widely believed conjecture in computer science is false,it is impossible even with parallel computation to predict majority-votecellular automata, or zero-temperature single spin-flip Ising dynamics,qualitatively faster than by explicit simulation.Comment: 10 pages with figure

Majority-Vote Cellular Automata, Ising Dynamics, and P-Completeness