Computability in Analysis and Physics

Pour-El, Marian Boykan; Richards, J. Ian

doi:10.1007/978-3-662-21717-7

Cited by 603 publications

(205 citation statements)

References 49 publications

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“…To describe this result -originally proven by V. Lifschitz [14] -in precise terms, let us recall the definitions of computable numbers, computable functions, and computable compact sets; see, e.g., [16,18] (see also [1-6, 11, 12]). …”

Section: Second Result: Computability From Uniqueness To Approximate mentioning

confidence: 99%

Towards Formalizing Non-monotonic Reasoning in Physics: Logical Approach Based on Physical Induction and Its Relation to Kolmogorov Complexity

Kreinovich

2012

Correct Reasoning

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Section: Second Result: Computability From Uniqueness To Approximate mentioning

confidence: 99%

Towards Formalizing Non-monotonic Reasoning in Physics: Logical Approach Based on Physical Induction and Its Relation to Kolmogorov Complexity

Kreinovich

2012

Correct Reasoning

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“…See [Wei00] for an up-to-date monograph on computable analysis from the computability point of view, [Ko91] for a presentation from a complexity point of view, or [PER89] for a general introduction to the subject.…”

Section: The Gpac Polynomial Differential Equations and Computable mentioning

confidence: 99%

“…The most prominent undecidable problem is the Halting Problem: given a universal Turing machine and some input to it, decide whether the machine eventually halts or not. To address this kind of questions for IVPs, we use the computable analysis approach [PER89], [Ko91], [Wei00], which we presented in the end of Section 2. Using that approach, it was shown in [GZB07] that given an analytic IVP (17), defined with computable data, its corresponding maximal interval may be non-computable.…”

Section: Application -Undecidability For Pivps With Comparable Paramementioning

confidence: 99%

Computational bounds on polynomial differential equations

Graça

Buescu

Campagnolo

2009

Applied Mathematics and Computation

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In this paper we study from a computational perspective some properties of the solutions of polynomial ordinary differential equations.We consider elementary (in the sense of Analysis) discrete-time dynamical systems satisfying certain criteria of robustness. We show that those systems can be simulated with elementary and robust continuous-time dynamical systems which can be expanded into fully polynomial ordinary differential equations with coefficients in Q[π]. This sets a computational lower bound on polynomial ODEs since the former class is large enough to include the dynamics of arbitrary Turing machines.We also apply the previous methods to show that the problem of determining whether the maximal interval of definition of an initial-value problem defined with polynomial ODEs is bounded or not is in general undecidable, even if the parameters of the system are computable and comparable and if the degree of the corresponding polynomial is at most 56.Combined with earlier results on the computability of solutions of polynomial ODEs, one can conclude that there is from a computational point of view a close connection between these systems and Turing machines.

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“…Although computable analysis can be adapted to other topological spaces, here we restrict it to R n , which is our case of interest. For more details the reader is referred to [11], [12], [13]. The idea underlying computable analysis to compute over a set A is to encode each element a of A into a countable sequence of symbols, called ρ-name.…”

Section: Computable Analysismentioning

confidence: 99%

Computing Domains of Attraction for Planar Dynamics

Graça

Zhong

2009

Lecture Notes in Computer Science

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Abstract. In this note we investigate the problem of computing the domain of attraction of a flow on R 2 for a given attractor. We consider an operator that takes two inputs, the description of the flow and a cover of the attractors, and outputs the domain of attraction for the given attractor. We show that: (i) if we consider only (structurally) stable systems, the operator is (strictly semi-)computable; (ii) if we allow all systems defined by C 1 -functions, the operator is not (semi-)computable. We also address the problem of computing limit cycles on these systems.Many problems about dynamical systems (DSs) are concerned with their long term behavior. For example, given some trajectory, where will it end up? Which are the invariant sets of a DS? Which are its attractors?Recently, with the advent of increasingly powerful digital computers, numerous new ideas and concepts related to these question have appeared (e.g. sensitive dependence on initial conditions, chaos, strange attractors, Mandelbrot set). However it is interesting to note that the computer, while being an invaluable tool to get some intuition about a DS, is rarely used to prove results. Usually the formal analysis of DSs is done analytically (but often relies on information provided by numerical simulations), using heavy mathematics with little reliance on the computer. A notable exception is the proof that the Lorenz strange attractor exists and is robust under small perturbations [1], [2].One of the reasons for this phenomenon is that computers introduce truncation errors which, in conjunction with other properties such as sensitive dependency on initial conditions, is likely to produce simulated trajectories that cannot be proved accurate. Of course, there are many results exhibiting that these simulations are valuable; the foremost of such results is perhaps the Shadowing Lemma [3], [4]. However the accuracy of a particular simulation, especially when we are interested in global properties, can usually be put into question.In this paper we deal with a particular type of the problems mentioned above: is it possible to conceive a computer program that, given an input that describes a dynamical system (DS) as well as an attractor of this DS, computes (rigourously) the basin of attraction of the given attractor?

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Computability in Analysis and Physics

Cited by 603 publications

References 49 publications

Towards Formalizing Non-monotonic Reasoning in Physics: Logical Approach Based on Physical Induction and Its Relation to Kolmogorov Complexity

Towards Formalizing Non-monotonic Reasoning in Physics: Logical Approach Based on Physical Induction and Its Relation to Kolmogorov Complexity

Computational bounds on polynomial differential equations

Computing Domains of Attraction for Planar Dynamics

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