Consider modelling an experiment intended to measure a quantity, such as position or mass in Classical Mechanics. A theoretical analysis of the process may examine the operations used in the experimental procedure and equipment, and explore relationships between the accuracy of the measurement and physical properties of the operations, such as their precision, and their use of time, space, and energy. How does the accuracy of measurements depend on resources? In a series of studies [5–7, 2, 1, 3] we have developed a methodology to investigate some fundamental notions about experimental methods in order to answer the question, What can one compute with a physical system? The methodology is based on defining experiments formally over some tightly specified theory T and analysing observable behaviours as the input-output of computations. The idea is (i) to understand the notion of computation inside a physical theory T in a way that is independent of classical computability theory of algorithms; and

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We develop the idea of using a physical experiment as an oracle to an algorithm. Of particular interest are protocols that manage oracle queries and count the resources involved. We investigate the computational power of deterministic and non-deterministic Turing machines connected to two physical oracles, namely, the scatter machine experiment and the collider machine experiment, which were introduced and studied in some depth earlier. Then we prove relativisation theorems for the conjectures concerning P, NP, PSPACE relative to these two physical oracles. Finally, we reflect generally on physical oracles for complexity theory.

Earlier we developed a theory of combining algorithms with physical systems based upon using physical experiments as oracles to algorithms. Although our concepts and methods are general, each physical oracle requires its own analysis based upon some fragment of physical theory that specifies the equipment and its behaviour. For specific examples of physical system (mechanical, optical, electrical) the computational power has been characterised using non-uniform complexity classes. The power of the known examples vary according to assumptions on precision and timing but seem to lead to the same complexity classes, namely P / log ⋆ and BP P/ / log ⋆. In this paper we develop sets of axioms for the interface between physical equipment and algorithms that allow us to prove general characterisations, in terms of P / log ⋆ and BP P/ / log ⋆, for large classes of physical oracles, in a uniform way. Sufficient conditions on physical equipment are given that ensure a physical

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