Implications of computer science theory for the simulation hypothesis
David H. Wolpert
TL;DR
This paper formalizes the simulation hypothesis within computer science by coupling it to the Physical Church-Turing Thesis and its reverse, then derives rigorous results using Kleene's recursion theorem and Rice's theorem. It proves a Simulation Lemma: a universe with RPCT can simulate any RPCT-obeying universe that satisfies the PCT; and a Self-Simulation Lemma: under the stronger pristine RPCT and PCT, a universe can freely simulate itself, implying the possibility of existing inside a simulation run on a computer we ourselves operate. The work introduces a simulation graph to study how universes can simulate one another, analyzes time-delay and shielding to prevent cheating in self-simulation, and discusses the philosophical and mathematical consequences, including undecidability results about which universes can simulate which. The findings have broad implications for the foundations of physics, the nature of identity, and the limits of meta-theoretical reasoning about simulations, while remaining applicable to any computable universe rather than being tied to our actual physics. The study also opens avenues for extending the framework to quantum and relativistic contexts and for exploring the practical and normative implications of simulation hierarchies.
Abstract
The simulation hypothesis has recently excited renewed interest in the physics and philosophy communities. However, the hypothesis specifically concerns {\textit{computers}} that simulate physical universes. So to formally investigate the hypothesis, we need to understand it in terms of computer science (CS) theory. In addition we need a formal way to couple CS theory with physics. Here I couple those fields by using the physical Church-Turing thesis. This allow me to exploit Kleene's second recursion, to prove that not only is it possible for {us} to be a simulation being run on a computer, but that we might be in a simulation being run a computer \emph{by us}. In such a ``self-simulation'', there would be two identical instances of us, both equally ``real''. I then use Rice's theorem to derive impossibility results concerning simulation and self-simulation; derive implications for (self-)simulation if we are being simulated in a program using fully homomorphic encryption; and briefly investigate the graphical structure of universes simulating other universes which contain computers running their own simulations. I end by describing some of the possible avenues for future research. While motivated in terms of the simulation hypothesis, the results in this paper are direct consequences of the Church-Turing thesis. So they apply far more broadly than the simulation hypothesis.