 Although the original Church-Turing thesis involves the abstract mathematical notion of computability, physicists as well as computer scientists often interpret it as saying something about the scope and limitations of physical computing machines. Wolfram (1985) claims that any physical system can be simulated (to any degree of approximation) by a universal Turing machine, and that complexity bounds on Turing machine simulations have physical significance. For example, if the computation of the minimum energy of some system of \(n\) particles requires at least an exponentially increasing number of steps in \(n\), then the actual relaxation of this system to its minimum energy state will also take exponential time. Aharonov (1999) strengthens this thesis (in the context of showing its putative incompatibility with quantum mechanics) when she says that a PTM can simulate any reasonable physical device at polynomial cost. In order for the physical Church-Turing thesis to make sense we have to relate physical space and time parameters to their computational counterparts: memory capacity and number of computation steps, respectively. There are various ways to do that, leading to different formulations of the thesis (Pitowsky 1990). For example, one can encode the set of instructions of a universal Turing machine and the state of its infinite tape in the binary development of the position coordinates of a single particle. Consequently, one can physically ‘realise’ a universal Turing machine as a billiard ball with hyperbolic mirrors (Moore 1990; Pitowsky 1996). For the most intuitive connection between abstract Turing machines and physical devices see the pioneering work of Gandy (1980), simplified later by Sieg and Byrnes (1999), and discussed, for example, in Copeland (2018). It should be stressed that there is no relation between the original Church-Turing thesis and its physical version (Pitowsky and Shagrir 2003), and while the former concerns the concept of computation that is relevant to logic (since it is strongly tied to the notion of proof which requires validation), it does not analytically entail that all computations should be subject to validation. Indeed, there is a long historical tradition of analog computations which use continuous physical processes (Dewdney 1984), and the output of these computations is validated either by repetitive “runs” or by validating the physical theory that presumably governs the behaviour of the analog computer.