Classifying the computational power of stochastic physical oracles

Consider a computability and complexity theory in which the classical set-theoretic oracle to a Turing machine is replaced by a physical process, and oracle queries return measurements of physical behaviour. The idea of such physical oracles is relevant to many disparate situations, but research has focussed on physical oracles that were classic deterministic experiments which measure physical quantities. In this paper, we broaden the scope of the theory of physical oracles by tackling non-deterministic systems. We examine examples of three types of non-determinism, namely systems that are: (1) physically nondeterministic, as in quantum phenomena; (2) physically deterministic but whose physical theory is non-deterministic, as in statistical mechanics; and (3) physically deterministic but whose computational theory is non-deterministic caused by error margins. Physical oracles that have probabilistic theories we call stochastic physical oracles. We propose a set SPO of axioms for a basic form of stochastic oracles. We prove that Turing machines equipped with a physical oracle satisfying the axioms SPO compute precisely the non-uniform complexity class BPP//log* in polynomial time. This result of BPP/log* is a computational limit to a great range of classical and non-classical measurement, and of analogue-digital computation in polynomial time under general conditions.info:eu-repo/semantics/publishedVersio

Classifying the computational power of stochastic physical oracles

Consider a computability and complexity theory in which theclassical set-theoretic oracle to a Turing machine is replaced bya physical process, and oracle queries return measurements ofphysical behaviour. The idea of such physical oracles is relevantto many disparate situations, but research has focussed on physicaloracles that were classic deterministic experiments whichmeasure physical quantities. In this paper, we broaden the scopeof the theory of physical oracles by tackling non-deterministicsystems. We examine examples of three types of non-determinism,namely systems that are: (1) physically nondeterministic,as in quantum phenomena; (2) physically deterministic butwhose physical theory is non-deterministic, as in statistical mechanics;and (3) physically deterministic but whose computationaltheory is non-deterministic caused by error margins. Physicaloracles that have probabilistic theories we call stochasticphysical oracles. We propose a set SPO of axioms for a basicform of stochastic oracles. We prove that Turing machinesequipped with a physical oracle satisfying the axioms SPO computeprecisely the non-uniform complexity class BPP//log* inpolynomial time. This result of BPP//log* is a computationallimit to a great range of classical and non-classical measurement,and of analogue-digital computation in polynomial time undergeneral conditions