A Relative Church-Turing-Deutsch Thesis from Special Relativity and Undecidability

TL;DR

This paper explores the limits of simulation and computation within the framework of special relativity, demonstrating that certain properties are undecidable and proposing a relativistic extension of the Church-Turing-Deutsch thesis linking classical and quantum computation.

Contribution

It introduces a relative model of computation based on special relativity, showing undecidability of simulation properties and extending the Church-Turing-Deutsch thesis to relativistic quantum mechanics.

Findings

Simulation properties are undecidable, similar to the Halting problem.

Relativistic effects lead to undecidability in local vs. global computation.

A relativistic model supports a version of the Church-Turing-Deutsch thesis.

Abstract

Beginning with Turing's seminal work in 1950, artificial intelligence proposes that consciousness can be simulated by a Turing machine. This implies a potential theory of everything where the universe is a simulation on a computer, which begs the question of whether we can prove we exist in a simulation. In this work, we construct a relative model of computation where a computable \textit{local} machine is simulated by a \textit{global}, classical Turing machine. We show that the problem of the local machine computing \textbf{simulation properties} of its global simulator is undecidable in the same sense as the Halting problem. Then, we show that computing the time, space, or error accumulated by the global simulator are simulation properties and therefore are undecidable. These simulation properties give rise to special relativistic effects in the relative model which we use to construct a relative Church-Turing-Deutsch thesis where a global, classical Turing machine computes quantum mechanics for a local machine with the same constant-time local computational complexity as experienced in our universe.