Implications of computer science theory for the simulation hypothesis
It addresses foundational questions about reality and computation for physics and philosophy, with broad implications beyond the simulation hypothesis.
The paper investigates the simulation hypothesis using computer science theory, proving that self-simulation is possible and deriving impossibility results via Rice's theorem, with implications for encrypted simulations and graphical structures of nested universes.
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.