Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

Ziegler, Martin

doi:10.1016/j.amc.2009.04.062

Cited by 19 publications

(15 citation statements)

References 83 publications

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“…Once again, to avoid singularity of u m (s) at s = 0, we begin with the following estimate: Let 0 < ǫ < t n . Then it follows from Fact 5-(1), (2), (5), (27), (30), and a similar calculation as performed in Claim I that → 0 effectively as n → ∞ (recall that t n = t/2 n ). The proof for the claim II, and thus for the lemma is now complete.…”

Section: Proof Of Propositionmentioning

confidence: 79%

“…Its validity had been challenged, though, in the sound setting of Recursive Analysis: with a computable C 1 initial condition to the Wave Equation leading to an incomputable solution [11,13]. The controversy was later resolved by demonstrating that, in both physically [30,1] and mathematically more appropriate Sobolev space settings, the solution is computable uniformly in the initial data [23]. Recall that functions f in a Sobolev space are not defined pointwise but by local averages in the L q sense 4 (in particular q = 2 corresponding to energy) with derivatives understood in the distributional sense.…”

Section: Introductionmentioning

confidence: 99%

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Computability of the Solutions to Navier-Stokes Equations via Effective Approximation

Sun¹,

Zhong²,

Ziegler³

2020

Complexity and Approximation

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As one of the seven open problems in the addendum to their 1989 book Computability in Analysis and Physics, Pour-El and Richards proposed "... the recursion theoretic study of particular nonlinear problems of classical importance. Examples are the Navier-Stokes equation, the KdV equation, and the complex of problems associated with Feigenbaum's constant." In this paper, we approach the question of whether the Navier-Stokes Equation admits recursive solutions in the sense of Weihrauch's Type-2 Theory of Effectivity. A natural encoding ("representation") is constructed for the space of divergence-free vector fields on 2-dimensional open square Ω = (− 1, 1) 2 . This representation is shown to render first the mild solution to the Stokes Dirichlet problem and then a strong local solution to the nonlinear inhomogeneous incompressible Navier-Stokes initial value problem uniformly computable. Based on classical approaches, the proofs make use of many subtle and intricate estimates which are developed in the paper for establishing the computability results.

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Section: Proof Of Propositionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Computability of the Solutions to Navier-Stokes Equations via Effective Approximation

Sun¹,

Zhong²,

Ziegler³

2020

Complexity and Approximation

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“…The theory T is everywhere and so we are actually studying T -computability and T -computational complexity. Our approach has been applied in a new discussion of the physical basis of the Church-Turing Thesis in Ziegler [17].…”

Section: Principle 3 Mapping the Border Between Computer And Hypercomentioning

confidence: 99%

Physical Oracles: The Turing Machine and the Wheatstone Bridge

2010

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Earlier, we have studied computations possible by physical systems and by algorithms combined with physical systems. In particular, we have analysed the idea of using an experiment as an oracle to an abstract computational device, such as the Turing machine. The theory of composite machines of this kind can be used to understand (a) a Turing machine receiving extra computational power from a physical process, or (b) an experimenter modelled as a Turing machine performing a test of a known physical theory T .Our earlier work was based upon experiments in Newtonian mechanics. Here we extend the scope of the theory of experimental oracles beyond Newtonian mechanics to electrical theory. First, we specify an experiment that measures resistance using a Wheatstone bridge and start to classify the computational power of this experimental oracle using non-uniform complexity classes. Secondly, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on our ability to measure resistance by the Wheatstone bridge.The connection between the algorithm and physical test is mediated by a protocol controlling each query, especially the physical time taken by the experimenter. In our studies we find that physical experiments have an exponential time protocol; this we formulate as a general conjecture. Our theory proposes that measurability in Physics is subject to laws which are co-lateral effects of the limits of computability and computational complexity.

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“…These are large questions with controversial answers -see, for instance, Geroch and Hartle (1986), Kreisel (1974), Moore (1990), Yao (2003), Németi and Dávid (2006), Odifreddi (1989) and Ziegler (2009). In the restricted case of our theory of physical oracles, having investigated physical systems drawn from several branches of Physics, we currently expect that physical experiments with continuous behaviour exhibit a computational power in polynomial time of P /log * .…”

Section: Implications For the Stability Of Computation By Physical Momentioning

confidence: 99%

“…The methodology of our programme has been used to analyse Church-Turing theses in Ziegler (2009). Our methodology, which is based on four principles and a carefully specified theory T , requires a reformulation of the informal thesis in Section 8.2 in which the theory T is explicit.…”

Section: In Search Of a Church-turing Thesis For Physical Systemsmentioning

confidence: 99%

The impact of models of a physical oracle on computational power

Beggs

Costa

Tucker

2012

Math. Struct. Comp. Sci.

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Link to this article: http://journals.cambridge.org/abstract_S0960129511000557How to cite this article: EDWIN J. BEGGS, JOSÉ FÉLIX COSTA and JOHN V. TUCKER (2012). The impact of models of a physical oracle on computational power.Using physical experiments as oracles for algorithms, we can characterise the computational power of classes of physical systems. Here we show that two different physical models of the apparatus for a single experiment can have different computational power. The experiment is the scatter machine experiment (SME), which was first presented in Beggs and Tucker (2007b). Our first physical model contained a wedge with a sharp vertex that made the experiment non-deterministic with constant runtime. We showed that Turing machines with polynomial time and an oracle based on a sharp wedge computed the non-uniform complexity class P /poly. Here we reconsider the experiment with a refined physical model where the sharp vertex of the wedge is replaced by any suitable smooth curve with vertex at the same point. These smooth models of the experimental apparatus are deterministic. We show that no matter what shape is chosen for the apparatus:(i) the time of detection of the scattered particles increases at least exponentially with the size of the query; and (ii) Turing machines with polynomial time and an oracle based on a smooth wedge compute the non-uniform complexity class P /log * $ P /poly.We discuss evidence that many experiments that measure quantities have exponential runtimes and a computational power of P /log * . † The authors would like to thank EPSRC for their support under grant EP/C525361/1. The research of José Félix Costa is also supported by Fundação para a Ciência e Tecnologia, Financiamento Base 2010-ISFL-1-209.

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Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

Cited by 19 publications

References 83 publications

Computability of the Solutions to Navier-Stokes Equations via Effective Approximation

Computability of the Solutions to Navier-Stokes Equations via Effective Approximation

Physical Oracles: The Turing Machine and the Wheatstone Bridge

The impact of models of a physical oracle on computational power

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