## Is wave propagation computable or can wave computers beat the Turing machine? (2002)

Venue: | PROC. LONDON MATH SOC |

Citations: | 24 - 5 self |

### BibTeX

@INPROCEEDINGS{Weihrauch02iswave,

author = {Klaus Weihrauch and Ning Zhong},

title = {Is wave propagation computable or can wave computers beat the Turing machine?},

booktitle = {PROC. LONDON MATH SOC},

year = {2002},

pages = {312--332},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

By the Church-Turing Thesis a numerical function is computable by a physical device if and only if it is computable by a Turing machine. The `if'-part is plausible since every (sufficiently small) Turing machine can be simulated by a computer program which operates correctly as long as sufficient time and storage

### Citations

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Citation Context ...U…1; ? †, of the Cauchy problem (3.5) at time t ˆ 1 belongs to C k ‡ 1 …R 3 †nC k ‡ 2 …R 3 †. Proof. By the classical theory of elliptic equations (see, for example, the book by Gilbarg and Trudinger =-=[5]-=-), the Poisson equation Du…x† ˆf …x†; x 2 R 3 ; with the forcing term f 2 C k; a …R 3 †, has a solution in C k ‡ 2; a …R 3 † (elliptic gain of 2). Let F 2 C k ‡ 2; a …R 3 † be such a solution. Then th... |

1305 |
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Citation Context ... [12; 16, Chapter 9]. In this paper we use `Type-2 Theory of Effectivity', TTE for short [7, 16], a computability concept for analysis, which is based on Turing's de®nition of computable real numbers =-=[15]-=- and Grzegorczyk's de®nition of computable real functions [6]. This model of computation is realistic, since the TTE-computable functions can be realized by Turing machines. TTE is consistent with the... |

372 |
Complexity and Real Computation
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(Show Context)
Citation Context ...alysis which have been proposed [16, Chapter 9] we have chosen `Type 2 Theory of Effectivity' (TTE), which is conceptually simple, very ¯exible and very realistic. In contrast to `real number models' =-=[14, 1]-=- where programs handle real numbers as entities, in TTE programs operate on `®nite properties' of `in®nite objects' like real numbers or continuous real functions. This is a much more realistic model,... |

260 |
An Introduction to Computable Analysis
- Weihrauch
- 2000
(Show Context)
Citation Context ...al functions and other functions from analysis several non-equivalent computability concepts have been proposed [12; 16, Chapter 9]. In this paper we use `Type-2 Theory of Effectivity', TTE for short =-=[7, 16]-=-, a computability concept for analysis, which is based on Turing's de®nition of computable real numbers [15] and Grzegorczyk's de®nition of computable real functions [6]. This model of computation is ... |

35 |
The Church-Turing thesis as a guiding principle for physics
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(Show Context)
Citation Context ...erformed on digital computers, and hence on Turing machines. Nevertheless, there might exist physical processes which are not Turing computable in this way. For discussions and further references see =-=[4, 13]-=-. Is Turing's computability concept suf®ciently powerful to model all kinds of physical processes? If the answer is `yes', then perhaps the Church±Turing Thesis could be derivable from the laws of phy... |

30 | Concrete models of computation for topological algebras - Stoltenberg-Hansen, Tucker - 1999 |

22 |
Computability and physical theories
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Citation Context ...erformed on digital computers, and hence on Turing machines. Nevertheless, there might exist physical processes which are not Turing computable in this way. For discussions and further references see =-=[4, 13]-=-. Is Turing's computability concept suf®ciently powerful to model all kinds of physical processes? If the answer is `yes', then perhaps the Church±Turing Thesis could be derivable from the laws of phy... |

15 |
Computable functionals, Fund
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(Show Context)
Citation Context ...ffectivity', TTE for short [7, 16], a computability concept for analysis, which is based on Turing's de®nition of computable real numbers [15] and Grzegorczyk's de®nition of computable real functions =-=[6]-=-. This model of computation is realistic, since the TTE-computable functions can be realized by Turing machines. TTE is consistent with the approach used by Pour-El and Richards [10] but more expressi... |

11 |
Computability and noncomputability in classical analysis
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Citation Context ... Sobolev spaces H s …R 3 † are naturally associated with the wave equation. We have seen in the previous section that the wave operator SW does not propagate C k ´ C k 1 regularity. Littman's theorem =-=[11]-=- asserts that SW does not propagate norms based on L p for p 6ˆ 2 in dimensions greater than 1. However, if one measures regularity in the H s sense, there is no loss of derivatives in any dimension. ... |

10 |
Theory of representations, Theoret
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(Show Context)
Citation Context ...al functions and other functions from analysis several non-equivalent computability concepts have been proposed [12; 16, Chapter 9]. In this paper we use `Type-2 Theory of Effectivity', TTE for short =-=[7, 16]-=-, a computability concept for analysis, which is based on Turing's de®nition of computable real numbers [15] and Grzegorczyk's de®nition of computable real functions [6]. This model of computation is ... |

6 |
Computable invariance, Theoret
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(Show Context)
Citation Context ...-computable function f 2 C 1 …R 3 † such that the solution u of the Cauchy problem (1.1) is not d 3-computable in any neighborhood of any point of the set D. According to a general theorem by Brattka =-=[2]-=- any function of suf®ciently high degree of discontinuity with some weak computability properties maps some computable elements to non-computable ones. For the instances we have considered, the soluti... |

1 |
Bing-Yu Zhang and Ning Zhong, `Computability of solutions of the Korteweg± de Vries equation
- Gay
(Show Context)
Citation Context ... SchroÈdinger equation [19, 20]. In both cases the propagator is computable. Furthermore, for the Korteweg±de Vries equation the periodic solution is computable if the initial condition is computable =-=[3]-=-. However, computability of many other physical processes, for example, boundary value problems, has not yet been investigated. Therefore, there still might be a physical device which beats the Turing... |

1 |
A recursive function de®ned on a compact interval and having a continuous derivative that is not recursive
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- 1971
(Show Context)
Citation Context ...and is …t 1 3 ; t 3†-continuous but not …t 3; t 3†-continuous on C 1 …R 3 †. Parallel to these topological irregularities of the solution operator there are computational ones. Myhill gave an example =-=[8]-=- ofad 1-computable differentiable function f 2 C 1 …R† such that the derivative f 0 2 C…R† is not d 1-computable. (Notice that the differentiation operator D on C 1 …R† is not …t 1; t 1†-continuous an... |

1 |
Ning Zhong, `The wave equation with computable initial data whose unique solution is nowhere computable
- and
- 1997
(Show Context)
Citation Context ...is paper we shall concentrate on a special type of physical process, namely one on scalar waves in Euclidean space. We start from remarkable results by Pour-El and Richards [11] and Pour-El and Zhong =-=[9]-=-, who constructed computable initial conditions f for the threedimensional wave equation utt ˆ Du; u…0; x† ˆf …x†; ut…0; x† ˆ0; t 2 R; x 2 R 3 …1:1† ; such that the unique solutions are not computable... |

1 |
Weihrauch and Ning Zhong, `The wave propagator is Turing computable', Computability and complexity
- Klaus
- 1998
(Show Context)
Citation Context ...er the Pour-El±Richards paradox can be used to design a `wave computer' which beats the Turing machine. It turns out that at least there is no obvious way to construct such a wave computer. Note that =-=[17, 18]-=- are preliminary versions of this paper. 2. Computability on second countable spaces This section summarizes some basic concepts from Type-2 Theory of Effectivity (TTE) and introduces the computabilit... |