On the Computational Capabilities of Physical Systems

In this paper strong limits on the accuracy of real-world physical computation are established. First it is proven that there cannot be a physical computer C to which one can pose any and all computational tasks concerning the physical universe. Next it is proven that no physical computer C can correctly carry out any computational task in the subset of such tasks that can be posed to C. This means that there cannot be a physical computer that can be assured of correctly “processing information faster than the universe does”. Because this result holds independent of how or if the computer is physically coupled to the rest of the universe, it also means that there cannot exist an infallible, general-purpose observation apparatus, nor an infallible, generalpurpose control apparatus. These results do not rely on systems that are infinite, and/or non-classical, and/or obey chaotic dynamics. They also hold even if one could use an infinitely fast, infinitely dense computer, with computational powers greater than that of a Turing Machine (TM). In fact a non-TM formulation of computation is needed to address the issues considered in this paper. Analogues of results concerning universal Turing Machines and the Halting theorem are derived for this novel kind of computer, as are results concerning the (im)possibility of certain kinds of error-correcting codes. In addition, an analogue of algorithmic information complexity, “prediction complexity”, is elaborated. A task-independent bound is derived on how much the prediction complexity of a computational task can differ for two different reference universal physical computers used to solve that task. This is analogous to the “encoding” bound governing