Skip to main content

Optical Computing and Computational Complexity

  • Conference paper
Unconventional Computation (UC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4135))

Included in the following conference series:

Abstract

This work concerns the computational complexity of a model of computation that is inspired by optical computers. The model is called the continuous space machine and operates in discrete timesteps over a number of two-dimensional images of fixed size and arbitrary spatial resolution. The (constant time) operations on images include Fourier transformation, multiplication, addition, thresholding, copying and scaling. We survey some of the work to date on the continuous space machine. This includes a characterisation of the power of an important discrete restriction of the model. Parallel time corresponds, within a polynomial, to sequential space on Turing machines, thus satisfying the parallel computation thesis. A characterisation of the complexity class NC in terms of the model is also given. Thus the model has computational power that is (polynomially) equivalent to that of many well-known parallel models. Such characterisations give a method to translate parallel algorithms to optical algorithms and facilitate the application of the complexity theory toolbox to optical computers. In the present work we improve on these results. Specifically we tighten a lower bound and present some new resource trade-offs.

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Arsenault, H.H., Sheng, Y.: An introduction to optics in computers. Tutorial texts in optical engineering, vol. TT 8. SPIE (1992)

    Google Scholar 

  2. Balcázar, J.L., Díaz, J., Gabarró, J.: Structural complexity II. EATCS Monographs on Theoretical Computer Science, vol. 22. Springer, Berlin (1988)

    Google Scholar 

  3. Bracewell, R.N.: The Fourier transform and its applications, 2nd edn. Electrical and electronic engineering series. McGraw-Hill, New York (1978)

    MATH  Google Scholar 

  4. Caulfield, H.J.: Space-time complexity in optical computing. In: Javidi, B. (ed.) Optical information-processing systems and architectures II, vol. 1347, pp. 566–572 (July 1990)

    Google Scholar 

  5. Chandra, A.K., Stockmeyer, L.J.: Alternation. In: 17th Annual Symposium on Foundations of Computer Science, Houston, Texas, pp. 98–108. IEEE, Los Alamitos (1976)

    Chapter  Google Scholar 

  6. Feitelson, D.G.: Optical Computing: A survey for computer scientists. MIT Press, Cambridge (1988)

    Google Scholar 

  7. Goldschlager, L.M.: Synchronous parallel computation. PhD thesis, University of Toronto, Computer Science Department (December 1977)

    Google Scholar 

  8. Goldschlager, L.M.: A universal interconnection pattern for parallel computers. Journal of the ACM 29(4), 1073–1086 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Goodman, J.W.: Introduction to Fourier optics, 2nd edn. McGraw-Hill, New York (1996)

    Google Scholar 

  10. Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to parallel computation: P-completeness theory. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  11. Karp, R.M., Ramachandran, V.: Parallel algorithms for shared memory machines. vol. A. Elsevier, Amsterdam (1990)

    Google Scholar 

  12. Lee, J.N.: Design issues in optical processing. In: Cambridge studies in modern optics. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  13. Louri, A., Post, A.: Complexity analysis of optical-computing paradigms. Applied optics 31(26), 5568–5583 (1992)

    Article  Google Scholar 

  14. McAulay, A.D.: Optical computer architectures. Wiley, Chichester (1991)

    Google Scholar 

  15. Naughton, T., Javadpour, Z., Keating, J., Klíma, M., Rott, J.: General-purpose acousto-optic connectionist processor. Optical Engineering 38(7), 1170–1177 (1999)

    Article  Google Scholar 

  16. Naughton, T.J.: Continuous-space model of computation is Turing universal. In: Yeung, D.-Y., Kwok, J.T., Fred, A., Roli, F., de Ridder, D. (eds.) SSPR 2006 and SPR 2006. LNCS, vol. 4109, pp. 121–128. Springer, Heidelberg (2006)

    Google Scholar 

  17. Naughton, T.J.: A model of computation for Fourier optical processors. In: Löwe, W., Südholt, M. (eds.) SC 2006. LNCS, vol. 4089, pp. 24–34. Springer, Heidelberg (2006)

    Google Scholar 

  18. Naughton, T.J., Woods, D.: On the computational power of a continuous-space optical model of computation. In: Margenstern, M., Rogozhin, Y. (eds.) MCU 2001. LNCS, vol. 2055, pp. 288–299. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  19. Parberry, I.: Parallel complexity theory. Wiley, Chichester (1987)

    MATH  Google Scholar 

  20. Pratt, V.R., Rabin, M.O., Stockmeyer, L.J.: A characterisation of the power of vector machines. In: Proc. 6th Annual ACM Symposium on Theory of Computing, pp. 122–134. ACM Press, New York (1974)

    Chapter  Google Scholar 

  21. Pratt, V.R., Stockmeyer, L.J.: A characterisation of the power of vector machines. Journal of Computer and Systems Sciences 12, 198–221 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reif, J.H., Tyagi, A.: Efficient parallel algorithms for optical computing with the discrete Fourier transform (DFT) primitive. Applied optics 36(29), 7327–7340 (1997)

    Article  Google Scholar 

  23. van Emde Boas, P.: Machine models and simulations. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, ch. 1, vol. A. Elsevier, Amsterdam (1990)

    Google Scholar 

  24. VanderLugt, A.: Optical Signal Processing. Wiley Series in Pure and Applied Optics. Wiley, New York (1992)

    Google Scholar 

  25. Weihrauch, K.: Computable Analysis: An Introduction. Texts in Theoretical Computer Science. Springer, Berlin (2000)

    MATH  Google Scholar 

  26. Woods, D.: Computational complexity of an optical model of computation. PhD thesis, National University of Ireland, Maynooth (2005)

    Google Scholar 

  27. Woods, D.: Upper bounds on the computational power of an optical model of computation. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 777–788. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  28. Woods, D., Gibson, J.P.: Complexity of continuous space machine operations. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 540–551. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  29. Woods, D., Gibson, J.P.: Lower bounds on the computational power of an optical model of computation. In: Calude, C.S., Dinneen, M.J., Păun, G., Jesús Pérez-Jímenez, M., Rozenberg, G. (eds.) UC 2005. LNCS, vol. 3699, pp. 237–250. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  30. Woods, D., Naughton, T.J.: An optical model of computation. Theoretical Computer Science 334(1–3), 227–258 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yu, F.T.S., Jutamulia, S., Yin, S. (eds.): Introduction to information optics. Academic Press, London (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Woods, D. (2006). Optical Computing and Computational Complexity. In: Calude, C.S., Dinneen, M.J., Păun, G., Rozenberg, G., Stepney, S. (eds) Unconventional Computation. UC 2006. Lecture Notes in Computer Science, vol 4135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11839132_4

Download citation

  • DOI: https://doi.org/10.1007/11839132_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-38593-6

  • Online ISBN: 978-3-540-38594-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics