Narrow Results

We argue that dynamical systems involving discrete and continuous data can be modelled by Turing machines with oracles that are physical processes. Using the theory introduced in Beggs et al. [2,3], we consider the scope and limits of polynomial time computations by such systems. We propose a general polynomial time Church-Turing Thesis for feasible computations by analogue-digital systems, having the non-uniform complexity class BPP//log* as theoretical upper bound. We show why BPP//log* should be replace P/poly, which was proposed by Siegelmann for neural nets [23,24]. Then we examine whether other sources of hypercomputation can be found in analogue-digital systems besides the oracle itself. We prove that the higher polytime limit P/poly can be attained via non-computable analogue-digital interface protocols.

We consider the experimenter (e.g. the experimental physicist) as a Turing machine — the digital component — and the experiment of measurement — the analog component — as an oracle to the Turing machine. The algorithm running in the machine abstracts the experimental method of measurement (encoding the recursive structure of experimental actions) chosen by the experimenter. In this paper we prove that the central analogue-digital complexity classes $P$, $P/poly$ and $P/poly\cap REC$ can be characterized in terms of protocols to perform measurements controlled by standard Turing machines.