Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Computer Science for Continuous Data

Survey, Vision, Theory, and Practice of a Computer Analysis System

  • Conference paper
  • First Online:
Computer Algebra in Scientific Computing (CASC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13366))

Included in the following conference series:

  • 687 Accesses

Abstract

Building on George Boole’s work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied “his” machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.

The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics.

This work was supported by the National Research Foundation of Korea (grant 2017R1E1A1A03071032) and by the International Research & Development Program of the Korean Ministry of Science and ICT (grant 2016K1A3A7A03950702) and by the European Union’s Horizon 2020 MSCA IRSES project #731143.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Arguably this also applies to the discrete case, where every function is trivially continuous.

  2. 2.

    One could replace it with some real error bound \(\varepsilon >0\).

  3. 3.

    Without the second-order property of being topologically complete.

References

  1. Ambos-Spies, K., Brandt, U., Ziegler, M.: Real benefit of promises and advice. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 1–11. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39053-1_1

    Chapter  MATH  Google Scholar 

  2. Avigad, J., Yin, Y.: Quantifier elimination for the reals with a predicate for the powers of two. Theor. Comput. Sci. 370(1–3), 48–59 (2007). https://doi.org/10.1016/j.tcs.2006.10.005

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauer, A.: Clerical. https://github.com/andrejbauer/clerical (2017)

  4. Boldo, S., Jourdan, J.H., Leroy, X., Melquiond, G.: Verified compilation of floating-point computations. J. Autom. Reason. 54(2), 135–163 (2015)

    Article  MathSciNet  Google Scholar 

  5. Bornemann, F., Laurie, D., Wagon, S., Waldvogel, J.: The SIAM 100-Digit Challenge. SIAM (2004). http://www.siam.org/books/100digitchallenge/

  6. Brattka, V.: The emperor’s new recursiveness: the epigraph of the exponential function in two models of computability. In: Ito, M., Imaoka, T. (eds.) Words, Languages & Combinatorics III, pp. 63–72. World Scientific Publishing, Singapore (2003), iCWLC 2000, Kyoto, Japan, March 14–18 (2000)

    Google Scholar 

  7. Brattka, V., Hertling, P.: Feasible real random access machines. J. Complex. 14(4), 490–526 (1998)

    Article  MathSciNet  Google Scholar 

  8. Brattka, V., Pauly, A.: Computation with advice. In: Zheng, X., Zhong, N. (eds.) CCA 2010, Proceedings of the Seventh International Conference on Computability and Complexity in Analysis. Electronic Proceedings in Theoretical Computer Science, vol. 24, pp. 41–55 (2010)

    Google Scholar 

  9. Brattka, V., Schröder, M.: Computing with sequences, weak topologies and the axiom of choice. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 462–476. Springer, Heidelberg (2005). https://doi.org/10.1007/11538363_32

    Chapter  Google Scholar 

  10. Brauße, F., et al.: Semantics, logic, and verification of “exact real computation". Tech. rep., arXiv (2021)

    Google Scholar 

  11. Braverman, M., Cook, S.A.: Computing over the reals: foundations for scientific computing. Notice AMS 53(3), 318–329 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Cho, J., Park, S., Ziegler, M.: Computing periods \({\ldots }\). In: Proceedings of the WALCOM: Algorithms and Computation - 12th International Conference, WALCOM 2018, Dhaka, Bangladesh, 3–5 March 2018, pp. 132–143 (2018). https://doi.org/10.1007/978-3-319-75172-6_12

  13. Cook, S.A.: Soundness and completeness of an axiom system for program verification. SIAM J. Comput. 7(1), 70–90 (1978). https://doi.org/10.1137/0207005

    Article  MathSciNet  MATH  Google Scholar 

  14. Dries, L.v.d.: The field of reals with a predicate for the powers of two. Manus. Math.54, 187–196 (1986), http://eudml.org/doc/155108

  15. Dyer, M.E., Frieze, A.M., Kannan, R.: A random polynomial time algorithm for approximating the volume of convex bodies. J. ACM 38(1), 1–17 (1991). https://doi.org/10.1145/102782.102783

    Article  MathSciNet  MATH  Google Scholar 

  16. Férée, H., Ziegler, M.: On the computational complexity of positive linear functionals on \(\cal{C}[0;1]\). In: Kotsireas, I.S., Rump, S.M., Yap, C.K. (eds.) MACIS 2015. LNCS, vol. 9582, pp. 489–504. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-32859-1_42

    Chapter  MATH  Google Scholar 

  17. Friedman, H.: The computational complexity of maximization and integration. Adv. Math. 53, 80–98 (1984). https://doi.org/10.1016/0001-8708(84)90019-7

    Article  MathSciNet  MATH  Google Scholar 

  18. Giga, Y., Miyakawa, T.: Solutions in \(l^r\) of the Navier-stokes initial value problem. Arch. Ration. Mech. Anal. 89(3), 267–281 (1985)

    Article  Google Scholar 

  19. Goldreich, O.: On promise problems: a survey. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical Computer Science. LNCS, vol. 3895, pp. 254–290. Springer, Heidelberg (2006). https://doi.org/10.1007/11685654_12

    Chapter  Google Scholar 

  20. Hertling, P.: Topological complexity with continuous operations. J. Complex. 12, 315–338 (1996). https://doi.org/10.1006/jcom.1996.0021

    Article  MathSciNet  MATH  Google Scholar 

  21. Hertling, P.: Is the Mandelbrot set computable? Math. Log. Q. 51(1), 5–18 (2005)

    Article  MathSciNet  Google Scholar 

  22. Hertling, P., Spandl, C.: Computing a solution of Feigenbaum’s functional equation in polynomial time. Log. Methods Comput. Sci. 10(4), 4:7, 9 (2014). https://doi.org/10.2168/LMCS-10(4:7)2014

  23. Hida, Y., Li, X.S., Bailey, D.H.: Library for double-double and quad-double arithmetic. Tech. rep, Lawrence Berkeley National Laboratory (2007)

    Google Scholar 

  24. Hoyrup, M., Rute, J.: Computable measure theory and algorithmic randomness. In: Handbook of Computability and Complexity in Analysis. TAC, pp. 227–270. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-59234-9_7

    Chapter  MATH  Google Scholar 

  25. Jiman, H.: Real computation: from computability via efficiency to practice. M.sc. thesis, School of Computing (2021)

    Google Scholar 

  26. Kanada, Y.J.: . Math. Cult. 1(1), 72–83 (2003). Calculation of circumferential ratio by computer

    Google Scholar 

  27. Kawamura, A.: Lipschitz continuous ordinary differential equations are polynomial-space complete. Comput. Complex. 19(2), 305–332 (2010). https://doi.org/10.1007/s00037-010-0286-0

    Article  MathSciNet  MATH  Google Scholar 

  28. Kawamura, A., Müller, N., Rösnick, C., Ziegler, M.: Computational benefit of smoothness: parameterized bit-complexity of numerical operators on analytic functions and Gevrey’s hierarchy. J. Complex. 31(5), 689–714 (2015). https://doi.org/10.1016/j.jco.2015.05.001

    Article  MathSciNet  MATH  Google Scholar 

  29. Kawamura, A., Steinberg, F., Ziegler, M.: Towards computational complexity theory on advanced function spaces in analysis. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds.) CiE 2016. LNCS, vol. 9709, pp. 142–152. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40189-8_15

    Chapter  Google Scholar 

  30. Kawamura, A., Steinberg, F., Ziegler, M.: On the computational complexity of the Dirichlet problem for Poisson’s equation. Math. Struct. Comput. Sci. 27(8), 1437–1465 (2017). https://doi.org/10.1017/S096012951600013X

    Article  MathSciNet  MATH  Google Scholar 

  31. Ko, K.I.: The maximum value problem and NP real numbers. J. Comput. Syst. Sci. 24, 15–35 (1982)

    Article  MathSciNet  Google Scholar 

  32. Ko, K.I.: Complex. Theory Real Funct. Progress in Theoretical Computer Science, Birkhäuser, Boston (1991)

    Book  Google Scholar 

  33. Ko, K.I., Friedman, H.: Computational complexity of real functions. Theoret. Comput. Sci. 20, 323–352 (1982)

    Article  MathSciNet  Google Scholar 

  34. Køber, P.K.: Uniform domain representations of \(\ell _p\)-spaces. Math. Log. Q. 180(2), 180–205 (2007)

    Article  Google Scholar 

  35. Koswara, I., Pogudin, G., Selivanova, S., Ziegler, M.: Bit-complexity of solving systems of linear evolutionary partial differential equations. In: Santhanam, R., Musatov, D. (eds.) CSR 2021. LNCS, vol. 12730, pp. 223–241. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79416-3_13

    Chapter  Google Scholar 

  36. Kreisel, G., Macintyre, A.: Constructive logic versus algebraization, I. In: Troelstra, A., van Dalen, D. (eds.) The L. E. J. Brouwer Centenary Sympos. Studies in Logic and the Foundations of Mathematics, vol. 110, pp. 217–260. North-Holland, Amsterdam (1982), (Noordwijkerhout, June 8–13 1981)

    Google Scholar 

  37. Kreitz, C., Weihrauch, K.: Theory of representations. Theoret. Comput. Sci. 38, 35–53 (1985)

    Article  MathSciNet  Google Scholar 

  38. Lambov, B.: RealLib: an efficient implementation of exact real arithmetic. Math. Struct. Comput. Sci. 17, 81–98 (2007)

    Article  MathSciNet  Google Scholar 

  39. Le Roux, S., Ziegler, M.: Singular coverings and non-uniform notions of closed set computability. In: Dillhage, R., Grubba, T., Sorbi, A., Weihrauch, K., Zhong, N. (eds.) Proceedings of the Fourth International Conference on Computability and Complexity in Analysis (CCA 2007). Electronic Notes in Theoretical Computer Science, vol. 202, pp. 73–88. Elsevier (2008), CCA 2007, Siena, Italy, 6–18 June 2007

    Google Scholar 

  40. Lee, H.: Random sampling of continuous objects. Ph.D. thesis, School of Computing (2020)

    Google Scholar 

  41. Lim, D., Ziegler, M.: Quantitative coding and complexity theory of continuous data. Tech. rep., arXiv (2021)

    Google Scholar 

  42. Luckhardt, H.: A fundamental effect in computations on real numbers. Theoret. Comput. Sci. 5(3), 321–324 (1977)

    Article  MathSciNet  Google Scholar 

  43. McNicholl, T.H.: A note on the computable categoricity of \(\ell ^p\) spaces. In: Beckmann, A., Mitrana, V., Soskova, M. (eds.) CiE 2015. LNCS, vol. 9136, pp. 268–275. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-20028-6_27

    Chapter  Google Scholar 

  44. Mori, T., Tsujii, Y., Yasugi, M.: Computability of probability distributions and characteristic functions. Log. Methods Comput. Sci. 9, 3:9, 11 (2013). https://doi.org/10.2168/LMCS-9(3:9)2013

  45. Mostowski, A.: On computable sequences. Fundam. Math. 44, 37–51 (1957)

    Article  MathSciNet  Google Scholar 

  46. Müller, N.T.: The iRRAM: exact arithmetic in C++. In: Blanck, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45335-0_14

    Chapter  Google Scholar 

  47. Nakao, M.T., Plum, M., Watanabe, Y.: Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations. Springer Series in Computational Mathematics, Springer (2019). https://doi.org/10.1007/978-981-13-7669-6

  48. Neumann, E., Ouaknine, J., Worrell, J.: On ranking function synthesis and termination for polynomial programs. In: Konnov, I., Kovács, L. (eds.) 31st International Conference on Concurrency Theory, CONCUR 2020, September 1–4, 2020, Vienna, Austria (Virtual Conference). LIPIcs, vol. 171, pp. 15:1–15:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.CONCUR.2020.15

  49. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley (1994)

    Google Scholar 

  50. Park, S., Ziegler, M.: Reliable degenerate matrix diagonalization. Tech. Rep. CS-TR-2018-415, KAIST (2018)

    Google Scholar 

  51. Pauly, A., Seon, D., Ziegler, M.: Computing Haar measures. In: Fernández, M., Muscholl, A. (eds.) 28th EACSL Annual Conference on Computer Science Logic, CSL 2020, January 13–16, 2020, Barcelona, Spain. LIPIcs, vol. 152, pp. 34:1–34:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPIcs.CSL.2020.34

  52. Pauly, A., Ziegler, M.: Relative computability and uniform continuity of relations. J. Log. Anal. 5(7), 1–39 (2013)

    Article  MathSciNet  Google Scholar 

  53. Pour-El, M.B., Richards, J.I.: The wave equation with computable initial data such that its unique solution is not computable. Advances in Math. 39, 215–239 (1981)

    Article  MathSciNet  Google Scholar 

  54. Rettinger, R.: Bloch’s constant is computable. J. Univ. Comput. Sci. 14(6), 896–907 (2008)

    MathSciNet  MATH  Google Scholar 

  55. Ryu, S., Park, J., Park, J.: Toward analysis and bug finding in javascript web applications in the wild. IEEE Softw. 36(3), 74–82 (2019). https://doi.org/10.1109/MS.2018.110113408

    Article  Google Scholar 

  56. Schröder, M.: Admissible representations in computable analysis. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 471–480. Springer, Heidelberg (2006). https://doi.org/10.1007/11780342_48

    Chapter  Google Scholar 

  57. Schröder, M.: Admissibly Represented Spaces and Qcb-Spaces. In: Handbook of Computability and Complexity in Analysis. TAC, pp. 305–346. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-59234-9_9

    Chapter  Google Scholar 

  58. Selivanova, S., Steinberg, F., Thies, H., Ziegler, M.: Exact real computation of solution operators for linear analytic systems of partial differential equations. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds.) CASC 2021. LNCS, vol. 12865, pp. 370–390. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-85165-1_21

    Chapter  Google Scholar 

  59. Specker, E.: The fundamental theorem of algebra in recursive analysis. In: Dejon, B., Henrici, P. (eds.) Constructive Aspects of the Fundamental Theorem of Algebra, pp. 321–329. Wiley-Interscience, London (1969)

    Google Scholar 

  60. Steinberg, F.: Complexity theory for spaces of integrable functions. Logical Methods in Computer Science 13(3), Paper No. 21, 39 (2017). https://doi.org/10.23638/LMCS-13(3:21)2017

  61. Sun, S.-M., Zhong, N., Ziegler, M.: Computability of the solutions to Navier-Stokes equations via effective approximation. In: Du, D.-Z., Wang, J. (eds.) Complexity and Approximation. LNCS, vol. 12000, pp. 80–112. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-41672-0_7

    Chapter  MATH  Google Scholar 

  62. Triebel, H.: Theory of Function Spaces I, II, III. Birkhäuser (1983, 1992, 2006). https://doi.org/10.1007/978-3-0346-0416-1

  63. Weihrauch, K.: Computable Analysis. Springer, Berlin (2000). https://doi.org/10.1007/978-3-642-56999-9

  64. Weihrauch, K.: The computable multi-functions on multi-represented sets are closed under programming. J. Univ. Comput. Sci. 14(6), 801–844 (2008)

    MathSciNet  MATH  Google Scholar 

  65. Weihrauch, K., Tavana-Roshandel, N.: Representations of measurable sets in computable measure theory. Logical Methods Comput. Sci. 10, 3:7,21 (2014). https://doi.org/10.2168/LMCS-10(3:7)2014

  66. Weihrauch, K., Zhong, N.: Is wave propagation computable or can wave computers beat the Turing machine? Proc. Lond. Math. Soc. 85(2), 312–332 (2002)

    Article  MathSciNet  Google Scholar 

  67. Yap, C., Sagraloff, M., Sharma, V.: Analytic root clustering: a complete algorithm using soft zero tests. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 434–444. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39053-1_51

    Chapter  Google Scholar 

  68. Yap, C.K.: On guaranteed accuracy computation. In: Geometric Computation, pp. 322–373. World Scientific Publishing, Singapore (2004)

    Google Scholar 

  69. Yu, J., Yap, C., Du, Z., Pion, S., Brönnimann, H.: The design of Core 2: a library for exact numeric computation in geometry and algebra. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 121–141. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15582-6_24

    Chapter  Google Scholar 

  70. Ziegler, M.: Real computation with least discrete advice: a complexity theory of nonuniform computability with applications to effective linear algebra. Ann. Pure Appl. Logic 163(8), 1108–1139 (2012). https://doi.org/10.1016/j.apal.2011.12.030

    Article  MathSciNet  MATH  Google Scholar 

  71. Ziegler, M., Brattka, V.: Computability in linear algebra. Theoret. Comput. Sci. 326(1–3), 187–211 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Manchester, Manchester, UK

    Franz Brauße

  2. Maastricht University, Maastricht, The Netherlands

    Pieter Collins

  3. KAIST, Daejeon, Republic of Korea

    Martin Ziegler

Authors
  1. Franz Brauße
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pieter Collins
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Martin Ziegler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Ziegler .

Editor information

Editors and Affiliations

  1. Université de Lille, Villeneuve d'Ascq, France

    François Boulier

  2. Coventry University, Coventry, UK

    Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Brauße, F., Collins, P., Ziegler, M. (2022). Computer Science for Continuous Data. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2022. Lecture Notes in Computer Science, vol 13366. Springer, Cham. https://doi.org/10.1007/978-3-031-14788-3_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-14788-3_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-14787-6

  • Online ISBN: 978-3-031-14788-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation