Computational Irreducibility and the Predictability of Complex Physical Systems

Abstract

Using elementary cellular automata (CA) as an example, we show how to coarse-grain CA in all classes of Wolfram's classification. We find that computationally irreducible (CIR) physical processes can be predictable and even computationally reducible at a coarse-grained level of description. The resulting coarse-grained CA which we construct emulate the large-scale behavior of the original systems without accounting for small-scale details. At least one of the CA that can be coarsegrained is irreducible and known to be a universal Turing machine.PACS numbers: 05.45. Ra, 05.10.Cc, 47.54.+r Can one predict the future evolution of a physical process which is described or modeled by a computationally irreducible (CIR) mathematical algorithm? For such systems, in order to know the system's state after (e.g.) one million time steps, there is no faster algorithm than to solve the equation of motion a million time steps into the future. Wolfram has suggested that the existence of CIR systems in nature is at the root of our apparent inability to model and understand complex systems [1, 2,3,4].Complex physical systems that are CIR might therefore seem to be inherently unpredictable. It is tempting to conclude from this that the enterprise of physics itself is doomed from the outset; rather than attempting to construct solvable mathematical models of physical processes, computational models should be built, explored and empirically analyzed. This argument, however, assumes that infinite precision is required for the prediction of future evolution. Usually coarse-grained or even statistical information is sufficient: indeed, a physical model is usually correct only to a certain level of resolution, so that there is little interest in predictions from such a model on a scale outside its regime of validity.In this Letter, we report on experiments with nearest neighbour one-dimensional cellular automata, which show that because in practice one only seeks coarsegrained information, complex physical systems can be predictable and even computationally reducible at some level of description. The implication of these results is that, at least for systems whose complexity is the outcome of very simple rules, useful approximations can be made that enable predictions about future behavior.

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