Non-Turing computations via Malament-Hogarth space-times

TL;DR

This paper explores how classical general relativity might allow for non-Turing computations, challenging traditional limits on computability and examining the implications for the arithmetical and analytical hierarchies.

Contribution

It demonstrates that certain limitations of computation depend on the physical background theory and shows how relativistic phenomena can enable non-recursive functions to be computed.

Findings

Classical general relativity can invalidate some Church-Turing theses.

Relativistic effects can be used to compute non-recursive functions.

Obstacles to relativistic computation can be overcome with better design.

Abstract

We investigate the Church-Kalm\'ar-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church-Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various ``obstacles'' to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the ``design'' of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.