By Barus C.

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**Example text**

2 Physics This physical version of the Church–Turing thesis (PCTT) says that any physical system can be simulated (to any degree of approximation) by a universal Turing machine [Wol85], 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 an exponential time.

See [Hag06] for an accessible review. It should be stressed that there is no relation between the original CTT and its physical version [SP03], 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. For the most intuitive connection between abstract Turing machines and physical devices see the pioneering work of Gandy [Gan80], simplified later by Sieg and Byrnes [SB99].

Admittedly, Silberstein discusses only one spatial dimension, and so in this particular case the Euclidean and the discrete distance functions coincide. Differential equations are replaced with difference equations, and finite differences are derived with Taylor series expansions. The construction is rich enough to derive most of classical physics. Silberstein goes further to apply it to the special theory of relativity, suggesting a discrete spacetime analog, which, while unknown to most working physicists in Europe and the USA at that time, had a certain influence on the attempts of the late 1940s to establish a quantum mechanics based on discrete spacetime (see Chapter 5).

### An Example of Torsional Viscous Retrogression (1919)(en)(3s) by Barus C.

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