The impact of models of a physical oracle on computational power

Beggs, Edwin J.; Costa, José Félix; Tucker, John V.

doi:10.1017/s0960129511000557

Cited by 20 publications

(40 citation statements)

References 26 publications

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“…One problem with our framework for computation systems thus far is that it does not adequately account for how the time a measurement takes to be carried out may vary. Indeed, as discussed by Beggs and Tucker [2][3][4] the time it takes to measure a quantity may depend on the quantity itself.…”

Section: Timed Computation Systemsmentioning

confidence: 99%

“…In [5] Beggs et al describe how a Newtonian kinematic system can be used to tackle a problem that's uncomputable for a Turing machine; computing the characteristic function of any given subset of N. Similarly, they achieve oracle-like results using "experiments" consisting of either a precise set of scales [2], or of a cannon and a wedge [4], calling a Turing machine combined with such classical physical experiments an "analogue-digital device".…”

Section: Introductionmentioning

confidence: 99%

“…for some constants c, d ∈ (0, ∞). Indeed in[4] Beggs et al demonstrate that the time take for all measurements utilising a cannon and a smooth wedge should be bounded above and below by an inverse polynomial function of the distance between the position of the height of the cannon (which is what is being measured) and the cusp of the wedge.This leads us to the following suggestion of what a reasonable requirement on the measurement time function of a CPCS should be. Let ∂ (A) be the boundary of a set A ⊆ X in the Euclidean metric, the set of boundary elements of a partition α is then ∂ (α) = A∈α ∂ (A).…”

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confidence: 99%

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Physical Computation, P/poly and P/log*

Whyman

2016

Electron. Proc. Theor. Comput. Sci.

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In this paper we give a framework for describing how abstract systems can be used to compute if no randomness or error is involved. Using this we describe a class of classical "physical" computation systems whose computational capabilities in polynomial time are equivalent to P/poly. We then extend our framework to describe how measurement and transformation times may vary depending on their input. Finally we describe two classes of classical "physical" computation systems in this new framework whose computational capabilities in polynomial time are equivalent to P/poly and P/log*.Comment: In Proceedings PC 2016, arXiv:1606.0651

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Section: Timed Computation Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Physical Computation, P/poly and P/log*

Whyman

2016

Electron. Proc. Theor. Comput. Sci.

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“…-The general scatter machine. This is designed to measure a position x on an interval, by comparing it with a dyadic position y [18].…”

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

confidence: 99%

Axiomatizing physical experiments as oracles to algorithms

Beggs

Costa

Tucker

2012

Phil. Trans. R. Soc. A.

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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.

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“…An early analysis on the computability of physical constants is (Geroch & Hartle 1986), where an informal attempt at defining measurement can be found. We are developing a new theory that combines measurement and computation, in a series of papers that examines a range of experiments and their effects on computation: (Beggs, Costa, Loff & Tucker 2008;Beggs, Costa, Loff & Tucker 2009;Beggs, Costa & Tucker 2010a;Beggs, Costa & Tucker 2010b;Beggs, Costa & Tucker 2012b;Beggs, Costa & Tucker 2012a;Beggs, Costa & Tucker 2014). Like Geroche and Hartle, our study is rooted in physics but it uses a methodology that combines physical theories with the rich concepts and results of computability and complexity theory.…”

Section: Introductionmentioning

confidence: 99%

Computations with oracles that measure vanishing quantities

Beggs

Costa

Poças

et al. 2016

Math. Struct. Comp. Sci.

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We consider computation with real numbers that arise through a process of physical measurement. We have developed a theory in which physical experiments that measure quantities can be used as oracles to algorithms and we have begun to classify the computational power of various forms of experiment using non-uniform complexity classes. Earlier, in Beggs et al. (2014 Reviews of Symbolic Logic7(4) 618–646), we observed that measurement can be viewed as a process of comparing a rational number z – a test quantity – with a real number y – an unknown quantity; each oracle call performs such a comparison. Experiments can then be classified into three categories, that correspond with being able to return test results $$\begin{eqnarray*} z < y\text{ or }z > y\text{ or }\textit{timeout},\\ z < y\text{ or }\textit{timeout},\\ z \neq y\text{ or }\textit{timeout}. \end{eqnarray*} $$ These categories are called two-sided, threshold and vanishing experiments, respectively. The iterative process of comparing generates a real number y. The computational power of two-sided and threshold experiments were analysed in several papers, including Beggs et al. (2008 Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences)464 (2098) 2777–2801), Beggs et al. (2009 Proceedings of the Royal Society, Series A (Mathematical, Physical and Engineering Sciences)465 (2105) 1453–1465), Beggs et al. (2013a Unconventional Computation and Natural Computation (UCNC 2013), Springer-Verlag 6–18), Beggs et al. (2010b Mathematical Structures in Computer Science20 (06) 1019–1050) and Beggs et al. (2014 Reviews of Symbolic Logic, 7 (4):618-646). In this paper, we attack the subtle problem of measuring physical quantities that vanish in some experimental conditions (e.g., Brewster's angle in optics). We analyse in detail a simple generic vanishing experiment for measuring mass and develop general techniques based on parallel experiments, statistical analysis and timing notions that enable us to prove lower and upper bounds for its computational power in different variants. We end with a comparison of various results for all three forms of experiments and a suitable postulate for computation involving analogue inputs that breaks the Church–Turing barrier.

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The impact of models of a physical oracle on computational power

Cited by 20 publications

References 26 publications

Physical Computation, P/poly and P/log*

Physical Computation, P/poly and P/log*

Axiomatizing physical experiments as oracles to algorithms

Computations with oracles that measure vanishing quantities

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