On the computational capabilities of physical systems part II: relationship with conventional computer science

Abstract

In the first of this pair of papers, it was proven that that no physical computer can correctly carry out all computational tasks that can be posed to it. The generality of this result follows from its use of a novel definition of computation, ``physical computation''. This second paper of the pair elaborates the mathematical structure and impossibility results associated with physical computation. Analogues of Chomsky hierarcy results concerning universal Turing Machines and the Halting theorem are derived, 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 universal physical computers used to solve that task, a bound similar to the ``encoding'' bound governing how much the algorithm information complexity of a Turing machine calculation can differ for two universal Turing machines. Finally, it is proven that either the Hamiltonian of our universe proscribes a certain type of computation, or prediction complexity is unique (unlike algorithmic information complexity).