Computations via Newtonian and relativistic kinematic systems
We are developing a rigorous methodology to analyse experimental computation, by which we mean the idea of computing a set or function by experimenting with some physical equipment. Here we consider experimental computation by kinematic systems under both Newtonian and relativistic kinematics. An experimental procedure, expressed in a language similar to imperative programming languages, is applied to equipment, having the form of a bagatelle, and is interpreted using the two theories. We prove that for any set A of natural numbers there exists a two-dimensional kinematic system B"A with a single particle P whose observable behaviour decides [email protected]?A for all [email protected]?N. The procedure can operate under (a) Newtonian mechanics or (b) relativistic mechanics. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on the methodology, which seeks a formal theory for performing abstract experiments with physical restrictions on the construction of systems. We conclude with some open problems.