Towards formal verification of computations and hypercomputations in relativistic physics

Abstract

It is now more than 15 years since Copeland and Proudfoot introduced the term hypercomputation. Although no hypercomputer has yet been built (and perhaps never will be), it is instructive to consider what properties any such device should possess, and whether these requirements could ever be met. Aside from the potential benefits that would accrue from a positive outcome, the issues raised are sufficiently disruptive that they force us to re-evaluate existing computability theory. From a foundational viewpoint the questions driving hypercomputation theory remain the same as those addressed since the earliest days of computer science, viz. what is computation? and what can be computed? Early theoreticians developed models of computation that are independent of both their implementation and their physical location, but it has become clear in recent decades that these aspects of computation cannot always be neglected. In particular, the computational power of a distributed system can be expected to vary according to the spacetime geometry in which the machines on which it is running are located. The power of a computing system therefore depends on its physical environment and cannot be specified in absolute terms. Even Turing machines are capable of super-Turing behaviour, given the right environment.