Physical Oracles: The Turing Machine and the Wheatstone Bridge

Int. J. Found. Comput. Sci.

Poças

et al. 2014

Self Cite

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.

Section: The Physical Oraclementioning

confidence: 99%

“…In [2,4,5,9,12,13] we have analysed a variety of physical experiments. They have in common (a) some initial conditions that can be tuned to some specific values; (b) a physical process, depending on the initial conditions, which takes a (possibly infinite) amount of time; and (c) a finite set of possible results or outcomes.…”

Section: On Physical Experimentsmentioning

confidence: 99%

An Analogue-Digital Church-Turing Thesis

Beggs

Int. J. Found. Comput. Sci.

Poças

et al. 2014

Self Cite

“…In subsequent work, we studied several other examples of experiments as oracles, including the collider machine experiment (CME) [14] and the Wheatstone bridge [15]. 6 The computational power of the physical oracles we have studied varies according to assumptions on precision and timing.…”

Section: (B) Physical Oraclesmentioning

confidence: 99%

“…This is designed to measure an electrical resistance x. A resistance y is prepared and compared with the original resistance [15]. -The general scatter machine.…”

Section: Then There Is a Physical Oracle To A Turing Machine Using Tmentioning

confidence: 99%

Axiomatizing physical experiments as oracles to algorithms

Beggs

Phil. Trans. R. Soc. A.

Tucker

2012

Self Cite

We developed earlier a theory of combining algorithms with physical systems, on the basis of using physical experiments as oracles to algorithms. Although our concepts and methods are general, each physical oracle requires its own analysis, on the basis of some fragment of physical theory that specifies the equipment and its behaviour. For specific examples of physical systems (mechanical, optical, electrical), the computational power has been characterized using non-uniform complexity classes. The powers 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 BPP// log . In this study, we develop sets of axioms for the interface between physical equipment and algorithms that allow us to prove general characterizations, in terms of P/ log and BPP// log , for large classes of physical oracles, in a uniform way. Sufficient conditions on physical equipment are given that ensure a physical system satisfies the axioms.

“…We call such experiments two-sided. Two-sided measurement is the focus of our previous work (see [9,12]), and that of axiomatic measurement theory (see [15,16,18]).…”

Section: Introductionmentioning

confidence: 99%

Oracles that measure thresholds: the Turing machine and the broken balance

Beggs¹,

Journal of Logic and Computation

Poças

et al. 2013

What can algorithms compute with the help of information provided by an oracle that is a physical system? We have developed a theory that combines Turing machines with experiments that perform physical measurements in which queries are governed by subtle timing protocols and provide the equipment with numerical data with (i) infinite precision, (ii) finite but unbounded precision or (iii) finite but fixed precision. Here, we consider the measurement of physical quantities that are thresholds, whose values are obtained by a sequence of approximate measurements that converge either from above or from below. The thresholds may be authentic physical properties or artefacts of the methods and equipment that performs the measurement. Using a canonical example of a threshold oracle, the broken beam balance for measuring mass, we develop methods to cope with thresholds and classify the computational power in polynomial time of this physical oracle using non-uniform complexity classes. Surprisingly, new complexity classes arise illuminating the influence of the operation of the equipment. All classes break the Turing Barrier.