Relativistic Computers and Non-uniform Complexity Theory

Recent research in theoretical physics on ‘Malament-Hogarth space-times’ indicates that so-called relativistic computers can be conceived that can carry out certain classically undecidable queries in finite time. We observe that the relativistic Turing machines which model these computations recognize precisely theΔ2−sets of the Arithmetical Hierarchy. In a complexity-theoretic analysis, we show that the (infinite) computations of S(n)-space bounded relativistic Turing machines are equivalent to (finite) computations of Turing machines that use a S(n)-bounded advice f , where f itself is computable by a S(n)-space bounded relativistic Turing machine. This bounds the power of polynomial-space bounded relativistic Turing machines by TM/poly. We also show that S(n)-space bounded relativistic Turing machines can be limited to one or two relativistic phases of computing. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics. in: D. Deutsch, The Fabric of Reality, 1998.