## Computing over the reals: Foundations for scientific computing

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Venue: | Notices of the AMS |

Citations: | 31 - 3 self |

### BibTeX

@ARTICLE{Braverman_computingover,

author = {Mark Braverman and Stephen Cook},

title = {Computing over the reals: Foundations for scientific computing},

journal = {Notices of the AMS},

year = {},

volume = {53},

pages = {2006}

}

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### Abstract

We give a detailed treatment of the “bit-model ” of computability and complexity of real functions and subsets of R n, and argue that this is a good way to formalize many problems of scientific computation. In Section 1 we also discuss the alternative Blum-Shub-Smale model. In the final section we discuss the issue of whether physical systems could defeat the Church-Turing Thesis. 1

### Citations

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- 1979
(Show Context)
Citation Context ..., ai2, . . .,aik } that adds up to A, i.e. ai1 + ai2 + . . . + aik = A. This problem is widely believed not to have an efficient solution in the discrete case. In fact it is NP-complete in this case (=-=[GJ79]-=-), and having a polynomial time algorithm for it would imply that P = NP, which is believed to be unlikely. There is no reason to think that it should be any easier in the continuous setting than in t... |

1279 |
On Computable Numbers, with an Application to the Entscheidungsproblem
- Turing
(Show Context)
Citation Context ...∗ Partially supported by an NSERC Post-graduate Scholarship † Partially supported by an NSERC Discovery grant 1sTuring defined the notion of a single computable real number in his landmark 1936 paper =-=[Tur36]-=-: a real number is computable if its decimal expansion can be computed in the discrete sense (i.e. output by some Turing machine). But he did not go on to define the notion of computable real function... |

931 | Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer
- Shor
- 1997
(Show Context)
Citation Context ...ned, in particular by advancements in the theory of quantum computation. In principle, if a quantum computer could be implemented, it would allow us to factor an integer N in time polynomial in log N =-=[Shor97]-=-. This would probably violate the ECT, since factoring is believed to require superpolynomial time on a classical computer. On the other hand there is no apparent way in which quantum computation woul... |

389 |
On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines
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- 1989
(Show Context)
Citation Context .... Algebraic complexity theory goes back to the 1950s (see [BM75, BCS97] for surveys). For scientific computing the most influential model in the algebraic setting is due to Blum, Shub and Smale (BSS) =-=[BSS89]-=-. The model and its theory are thoroughly developed in the book [BCSS98] (see also the article [Blum04] in the AMS Notices for an exposition). In the BSS model, the computer has registers which can ho... |

257 |
Computable Analysis
- Weihrauch
- 2000
(Show Context)
Citation Context ...is: are there physical systems that can compute functions which are uncomputable in the standard computer model? Some of the material presented here is given in more detail in [Brv05]. See [Ko91] and =-=[Wei00]-=- for general references on bit-computability models. Acknowledgments The authors are grateful to the following people for helpful comments on a preliminary version of this paper: Eric Allender, Lenore... |

189 | Algebraic Complexity Theory - Bürgisser, Clausen, et al. - 1997 |

177 |
Complexity Theory of Real Functions
- Ko
- 1991
(Show Context)
Citation Context ...Turing thesis: are there physical systems that can compute functions which are uncomputable in the standard computer model? Some of the material presented here is given in more detail in [Brv05]. See =-=[Ko91]-=- and [Wei00] for general references on bit-computability models. Acknowledgments The authors are grateful to the following people for helpful comments on a preliminary version of this paper: Eric Alle... |

154 |
Dynamics in one complex variable. Introductory lectures
- Milnor
- 1999
(Show Context)
Citation Context ...The existence of such a machine and the amount of 14stime the computation takes depends on the properties of the particular Julia set. More information on the properties of Julia sets can be found in =-=[Mil00]-=-. Computationally, the “easiest” case is that of the hyperbolic Julia sets. These are the sets for which the orbit of the point 0 either diverges to ∞ or converges to an attracting orbit. Equivalently... |

149 |
Hilbert’s Tenth Problem
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- 1993
(Show Context)
Citation Context ...the standard bit computer model in the discrete setting. One of the big successes of discrete computability theory is is the uncomputability results; especially the solution of Hilbert’s 10th problem =-=[Mat93]-=-. The theorem states that there is no procedure (e.g. no Turing machine) which always correctly determines whether a given Diophantine equation has a solution. The result is convincing because of gene... |

102 | The Computational Complexity of Algebraic and Numeric Problems - BORODIN, MUNRO - 1975 |

70 | The fundamental theorem of algebra in terms of computational complexity - Schönhage - 1981 |

61 |
Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles
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(Show Context)
Citation Context .... Because of the Second World War the results were only published many years later [Maz63]. A definition which is equivalent to bit-computability was first proposed by Grzegorczyk [Grz55] and Lacombe =-=[Lac55]-=-. It has been since developed and generalized. More recent references on the subject include [Ko91], [PR89], and [Wei00]. Let us see some examples to illustrate this notion. Examples of Bit-computabil... |

56 |
Computational complexity of real functions
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- 1982
(Show Context)
Citation Context ...near factors Lj(z)=ujz +vj (1≤j≤n) such that |P−L1L2···Ln|<2−s holds. Some of the early work regarding the computational complexity of operators such as taking derivatives and integration was done in =-=[KF82]-=-. A more detailed exposition of the results can be found in [Ko91]. 324 NOTICES OF THE AMS VOLUME 53, NUMBER 3sThe complexity of computing a set is the time T(n) it takes to decide one pixel. More for... |

51 | Computability on subsets of euclidean space i: Closed and compact subsets - Brattka, Weihrauch - 1999 |

34 | Complexity theory and numerical analysis
- Smale
- 1997
(Show Context)
Citation Context ...am to approximate the exact output values, so that functions like e x become computable. However formulating a good general definition in the BSS model 2salong these lines is not straightforward: see =-=[Sma97]-=- for an informal treatment and [Brv05] for a discussion and a possible formal model. For uncomputability results, BSS theory concentrates on set decidability rather than function computation. A set C ... |

27 | Yampolski M.: Non-computable Julia sets
- Braverman
- 2006
(Show Context)
Citation Context ...e usually drawn, such as the ones on Figure 2. This raises the question of whether all Julia sets are computable so efficiently. The answer to this question is negative. In fact, it has been shown in =-=[BY04]-=- that there are some values of c for which Jc cannot be computed (even with oracle access to c). The construction is based on Julia sets with Siegel disks. A parameter c which “fools” all the machines... |

24 | Is wave propagation computable or can wave computers beat the Turing machine
- Weihrauch, Zhong
- 2002
(Show Context)
Citation Context ...robability distribution of the outcome: 50% “heads” and 50% “tails”. Another example where “theoretical hardness” of the wave equation does not immediately imply a violation of the CT is presented in =-=[WZ02]-=-. This leads to a question that is essentially equivalent to the CT: (∗) Is there a robust physical system that is hard to simulate numerically? This is a question that can be formulated in the framew... |

23 | The Computational Complexity of Some Julia Sets - Rettinger, Weihrauch |

21 | Classical physics and the Church-Turing thesis
- Yao
(Show Context)
Citation Context ...ful than the standard computer. This provides further motivation for exploring the computability and computational complexity of physical problems in the bit model. The discussion is based in part on =-=[Yao02]-=-. The Church-Turing thesis (CT), in its common interpretation, is the belief that the Turing machine, which is computationally equivalent to the common computer, is the most general model of computati... |

19 |
Les ensembles récursivement ouverts ou fermés, et leurs applications à l’Analyse récursive
- Lacombe
- 1958
(Show Context)
Citation Context ...y Theorem 3, S is a bit-computable set. This is despite the fact (pointed out in [Blum04]) that the only rational points in S are (0, 1) and (1, 0). The bit computability notion dates back to Lacombe =-=[Lac58]-=-. We refer the reader to [BW99, Wei00, Brv05] for a more detailed discussion. 3.4 Computational complexity in the bit model Since the basic object in the discussion above is a Turing Machine, the comp... |

16 | A fast algorithm for Julia sets of hyperbolic rational functions - Rettinger |

15 |
Computable functionals, Fund
- Grzegorczyk
- 1955
(Show Context)
Citation Context ...machine Mφ g (n) The running time T(n) of M φ f for computing g(x), we can compute f ◦ g(x) by M Mφ g f (n). 8sThis is the bit-computability notion for functions. It was first proposed by Grzegorczyk =-=[Grz55]-=- and Lacombe [Lac55]. It has been since developed and generalized. More recent references on the subject include [Ko91], [PR89], and [Wei00]. Let us see some examples to illustrate this notion. 3.2 Ex... |

10 | Is the Mandelbrot set computable
- Hertling
(Show Context)
Citation Context ... 2 −n -far from M, the orbit of c escapes in at most T(n) steps. In fact, no such estimates are known in general, and the questions of their existence is equivalent to the bit computability of M (cf. =-=[Hert05]-=-). Some of the most fundamental properties of M remain open. For example, it is conjectured that it is locally connected, but with no proof so far. 13sFigure 4: The naïve algorithm for computing M, an... |

9 | Computing over the reals: Where Turing meets Newton
- Blum
(Show Context)
Citation Context ...omputing the most influential model in the algebraic setting is due to Blum, Shub and Smale (BSS) [BSS89]. The model and its theory are thoroughly developed in the book [BCSS98] (see also the article =-=[Blum04]-=- in the AMS Notices for an exposition). In the BSS model, the computer has registers which can hold arbitrary elements of the underlying ring R and performs exact arithmetic (+, −, ·, and ÷ in the cas... |

9 |
Classes récursivement fermés et fonctions majorantes. Comptes Rendus Académie des Sciences
- Lacombe
- 1955
(Show Context)
Citation Context ... running time T(n) of M φ f for computing g(x), we can compute f ◦ g(x) by M Mφ g f (n). 8sThis is the bit-computability notion for functions. It was first proposed by Grzegorczyk [Grz55] and Lacombe =-=[Lac55]-=-. It has been since developed and generalized. More recent references on the subject include [Ko91], [PR89], and [Wei00]. Let us see some examples to illustrate this notion. 3.2 Examples of bit-comput... |

8 |
Computable Analysis
- Mazur
- 1963
(Show Context)
Citation Context ...ity notion for functions. Early work on the computability of real functions was done by Banach and Mazur in 1936–1939. Because of the Second World War the results were only published many years later =-=[Maz63]-=-. A definition which is equivalent to bit-computability was first proposed by Grzegorczyk [Grz55] and Lacombe [Lac55]. It has been since developed and generalized. More recent references on the subjec... |

7 | The emperor’s new recursiveness: the epigraph of the exponential function in two models of computability
- Brattka
- 2000
(Show Context)
Citation Context ...dability results in terms of practical computing, because simple subsets of R 2 which can be easily “drawn”, such as the Koch snowflake and the graph of y = e x (Fig. 1) are undecidable in this sense =-=[Brt03]-=-. Figure 1: The Koch snowflake and the graph of the equation y = e x In the bit model there is a nice definition of decidability (bit-computability) for bounded subsets of R n (Section 3.3). For the c... |

7 | Hyperbolic Julia Sets are Poly-Time Computable - Braverman - 2005 |

6 | Computability of Subsets of Euclidean Space I - Brattka, Weihrauch - 1999 |

6 |
Computing Over the Reals: Where Turing Meets
- Blum
- 2004
(Show Context)
Citation Context ...model and its theory are thoroughly developed in the book [BCSS98] 318 NOTICES OF THE AMS VOLUME 53, NUMBER 3sFigure 1. The Koch snowflake and the graph of the equation y =e x . (see also the article =-=[Blum04]-=- in the Notices for an exposition). In the BSS model, the computer has registers which can hold arbitrary elements of the underlying ring R. Computer programs perform exact arithmetic (+,−,·, and ÷ in... |

5 |
Complexity theory and numerical analysis. Acta Numer
- Smale
- 1997
(Show Context)
Citation Context ...gram to approximate the exact output values, so that functions like e x become computable. However formulating a good general definition in the BSS model along these lines is not straightforward: see =-=[Sma97]-=- for an informal treatment and [Brv05] for a discussion and a possible formal model. For uncomputability results, BSS theory concentrates on set decidability rather than function computation. A set C⊆... |

2 |
2000 Dynamics in One Complex Variable—Introductory Lectures 2nd edn
- Milnor
- 2000
(Show Context)
Citation Context ... existence of such a machine and the amount of time the computation takes depend on the properties of the particular Julia set. An excellent exposition on the properties of Julia sets can be found in =-=[Mil00]-=-. Computationally, the “easiest” case is that of the hyperbolic Julia sets. These are the sets for which the orbit of the point 0 either diverges to ∞ or converges to an attracting orbit. Equivalently... |

1 |
On computational complexity of Siegel Julia sets, arXiv.org e-Print archive, 2005. Available from http://www.arxiv.org/abs/math.DS/0502354
- Binder, Braverman, et al.
(Show Context)
Citation Context ...efficiently. The answer to this question is negative. In fact, it has been shown in [BY04] that there are some values of c for which Jc cannot be computed (even with oracle access to c). Moreover, in =-=[BBY05]-=- it has been shown that a computable Jc with an explicitly computable c can have an arbitrarily high computational complexity. The constructions are based on Julia sets with Siegel disks. A parameter ... |

1 |
Non-computable Julia sets, arXiv.org e-Print archive
- Braverman, Yampolsky
- 2004
(Show Context)
Citation Context ...are usually drawn, such as the ones on Fig. 2. This raises the question of whether all Julia sets are computable so efficiently. The answer to this question is negative. In fact, it has been shown in =-=[BY04]-=- that there are some values of c for which Jc cannot be computed (even with oracle access to c). Moreover, in [BBY05] it has been shown that a computable Jc with an explicitly computable c can have an... |

1 |
On the Complexity of Real Functions. arXiv.org e-Print archive, 2005. Available from http://www.arxiv.org/abs/cs.CC/0502066
- Braverman
- 2005
(Show Context)
Citation Context ...ues, so that functions like e x become computable. However formulating a good general definition in the BSS model 2salong these lines is not straightforward: see [Sma97] for an informal treatment and =-=[Brv05]-=- for a discussion and a possible formal model. For uncomputability results, BSS theory concentrates on set decidability rather than function computation. A set C ⊆ R n is decidable if some BSS compute... |

1 |
Itération des polynomes quadratiques complexes
- Douadyand
- 1982
(Show Context)
Citation Context ...dioid, the orbit of c converges to an attracting point. These components are called hyperbolic components because they index the hyperbolic Julia sets that will be discussed below. Douady and Hubbard =-=[DH82]-=- have shown that Conjecture 4 implies that the interior of M consists entirely of hyperbolic components. Conjecture 5 The interior of M is equal to the union of its hyperbolic components. The latter c... |

1 | Equation solving in terms of computational complexity - Schöhage - 1986 |

1 |
Relativistic Quantum Fields" by J.D. Bjorken and S.D
- unknown authors
- 1965
(Show Context)
Citation Context ...alues, so that functions like e x become computable. However formulating a good general definition in the BSS model along these lines is not straightforward: see [Sma97] for an informal treatment and =-=[Brv05]-=- for a discussion and a possible formal model. For uncomputability results, BSS theory concentrates on set decidability rather than function computation. A set C⊆ R n is decidable if some BSS computer... |