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.

翻译：模拟假说近来在物理学和哲学界重新激发了广泛兴趣。然而，该假说特指模拟物理宇宙的计算机。因此，为形式化研究这一假说，我们需要从计算机科学理论的角度来理解它。此外，我们还需要一种形式化的方法将计算机科学理论与物理学相结合。本文通过物理丘奇-图灵论题实现了这两个领域的耦合。这使得我能够利用克莱尼第二递归定理证明：我们不仅可能是运行在计算机上的模拟存在，甚至可能处于一个由我们自身运行的计算机所执行的模拟之中。在这样的“自我模拟”中，将存在两个完全相同的“我们”的实例，两者具有同等的“真实性”。随后，我运用莱斯定理推导出关于模拟与自我模拟的不可能性结果；分析了若我们处于采用全同态加密程序中的模拟（或自我模拟）所产生的影响；并简要探讨了宇宙模拟其他包含自主运行模拟的计算机的宇宙所形成的图结构。最后，我指出了若干未来可能的研究方向。虽然本文的出发点是模拟假说，但所得结论实质上是丘奇-图灵论题的直接推论，因此其适用范围远超出模拟假说本身。