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A Survey of Continuous-Time Computation Theory

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Advances in Algorithms, Languages, and Complexity

Abstract

Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuous-time computation. However, while special-case algorithms and devices are being developed, relatively little work exists on the general theory of continuous- time models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions.

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References

  1. L. M. Adleman, Molecular computation of solutions to combinatorial problems. Science 266 (11 Nov. 1994 ), 1021 – 1024.

    Article  Google Scholar 

  2. J. A. Anderson and E. Rosenfeld (eds.), Neurocomputing: Foundations of Research. The MIT Press, Cambridge, MA, 1988.

    Google Scholar 

  3. E. Asarin, O. Maler, On some relations between dynamical systems and transition systems. Proc. 21st Internat. Colloq. on Automata, Languages, and Programming, 59-72. Lecture Notes in Computer Science 820, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  4. C. H. Bennett, Logical reversibility of computation. IBM J. Res. Develop. 17 (1973), 525 – 532.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. H. Bennett, Time/space trade-offs for reversible computation. SIAM J. Comput. 18 (1989), 766–776.

    Article  MathSciNet  MATH  Google Scholar 

  6. E. Bernstein, U. Vazirani, Quantum complexity theory. Proc. 25th ACM Symp. on Theory of Computation, 11–20. ACM Press, New York, NY, 1993.

    Google Scholar 

  7. L. Blum, M. Shub, S. Smale, On a theory of computation over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the Amer. Math. Soc. 21 (1989), 1–46.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Branicky, Analog computation with continuous ODE’ss. Proc. Workshop on Physics and Computation 1994, 265–274. IEEE Computer Society Press, Los Alamitos, CA, 1994.

    Google Scholar 

  9. M. Branicky, Universal computation and other capabilities of hybrid and continuous dynamical systems. Theoret. Comput. Sci. 138 (1995), 67–100.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. W. Brockett, Smooth dynamical systems which realize arithmetical and logical operations. Three Decades of Mathematical System Theory (H. Nijmeijer, J. M. Schumacher, eds.) Lecture Notes in Control and Information Sciences 135, Springer-Verlag, Berlin, 1989.

    Google Scholar 

  11. R. W. Brockett, Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems. Linear Algebra and Its Applications 146 (1991), 79–91.

    Article  MathSciNet  MATH  Google Scholar 

  12. V. Bush, The differential analyzer, a new machine for solving differential equations. J. Franklin Inst. 212 (1931), 447–488.

    Article  Google Scholar 

  13. M. Casey, The dynamics of discrete-time computation, with application to recurrent neural networks and finite state machine extraction. Neural Computation 8 (1996), 1135–1178.

    Article  Google Scholar 

  14. A. Cichocki, R. Unbehauen, Neural Networks for Optimization and Signal Processing. Wiley/Teubner, Stuttgart, 1993.

    MATH  Google Scholar 

  15. Hopfield, J. J. and Tank, D. W. Neural computation of decisions in optimization problems. Biological Cybernetics 52 (1985), 141–152.

    MathSciNet  MATH  Google Scholar 

  16. D. G. Feitelson, Optical Computing: A Survey for Computer Scientists. The MIT Press, Cambridge, MA, 1988.

    Google Scholar 

  17. M. Garzon, Models of Massive Parallelism: Analysis of Cellular Automata and Neural Networks. Springer-Verlag, Berlin, 1995.

    MATH  Google Scholar 

  18. A. Hausner, Analog and Analog/Hybrid Computer Programming. Prentice-Hall, Englewood Cliffs, NJ, 1971.

    MATH  Google Scholar 

  19. M. W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, San Diego, CA, 1974.

    MATH  Google Scholar 

  20. J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Nat. Acad. Sci. USA 81 (1984), 3088–3092. Reprinted in [2], pp. 579–583.

    Google Scholar 

  21. J. J. Hopfield, D. W. Tank, Neural computation of decisions in optimization problems. Biological Cybernetics 52 (1985), 141–152.

    MathSciNet  MATH  Google Scholar 

  22. A. S. Jackson, Analog Computation. McGraw-Hill, New York, NY, 1960.

    MATH  Google Scholar 

  23. C. L. Johnson, Analog Computer Techniques, 2nd Ed. McGraw-Hill, New York, NY, 1963.

    Google Scholar 

  24. P. Koiran, Dynamics of discrete time, continuous state Hopfield networks. Neural Computation 6 (1994), 459–468.

    Article  Google Scholar 

  25. P. Koiran, M. Cosnard, M. Garzon, Computability with low- dimensional dynamical systems. Theoret. Comput. Sci. 132 (1994), 113–128.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. A. Korn, T. M. Korn, Electronic Analog and Hybrid Computers, 3rd Ed. McGraw-Hill, New York, NY, 1964.

    Google Scholar 

  27. L. Lipshitz, L. Rubel, A differentially algebraic replacement theorem, and analog computability. Proc. Amer. Math. Soc. 99 (1987), 367–372.

    Article  MathSciNet  MATH  Google Scholar 

  28. R. J. Lipton, DNA solution of hard computational problems. Science 268 (28 Apr. 1995 ), 542–545.

    Google Scholar 

  29. W. Maass, P. Orponen, On the effect of analog noise in discrete- time analog computation. Proc. Neural Information Processing Systems 1996, to appear.

    Google Scholar 

  30. C. Mead, Analog VLSI and Neural Systems. Addison-Wesley, Reading, MA, 1989.

    MATH  Google Scholar 

  31. C. Moore, Unpredictability and undecidability in physical systems. Phys. Review Letters 64 (1990), 2354–2357.

    Article  MATH  Google Scholar 

  32. C. Moore, Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4 (1991), 199–230.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Omohundro, Modelling cellular automata with partial differential equations. Physica 10D (1984), 128–134.

    MathSciNet  Google Scholar 

  34. P. Orponen, On the Computational Power of Continuous Time Neural Networks. Project NeuroCOLT Report NC-TR-95-051, Royal Holloway College, Univ. of London, Dept. of Computer Science, 1995. 18 pp.

    Google Scholar 

  35. P. Orponen, The computational power of discrete Hopfield nets with hidden units. Neural Computation 8 (1996), 403–415.

    Article  Google Scholar 

  36. P. Orponen, M. Matamala, Universal computation by finite two- dimensional coupled map lattices. Proc. Workshop on Physics and Computation 1996, to appear.

    Google Scholar 

  37. J. Palis, Jr., W. de Melo, Geometric Theory of Dynamical Systems: An Introduction. Springer-Verlag, New York, NY, 1982.

    Google Scholar 

  38. M. B. Pour-El, Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers). Trans. Amer. Math. Soc. 199 (1974), 1–28.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. B. Pour-El, J. I. Richards, Computability in Analysis and Physics. Springer-Verlag, Berlin, 1989.

    MATH  Google Scholar 

  40. P. Pudlák, Complexity theory and genetics. Proc. 9th Ann. IEEE Conf. on Structure in Complexity Theory, 383–395. IEEE Computer Society Press, Los Alamitos, CA, 1994.

    Google Scholar 

  41. J. H. Reif, J. D. Tygar, A. Yoshida, The computability and complexity of optical beam tracing. Proc. 31st Ann. IEEE Symp. on Foundations of Computer Science, 106–114. IEEE Computer Society Press, Los Alamitos, CA, 1990.

    Google Scholar 

  42. D. Rooß, K. Wagner, On the Power of Bio-Computers. Technical Report, Universität Würzburg, Inst, für Informatik, Feb. 1995.

    Google Scholar 

  43. L. A. Rubel, Some mathematical limitations of the general-purpose analog computer. Adv. in Appl. Math. 9 (1988), 22–34.

    Article  MathSciNet  MATH  Google Scholar 

  44. L. A. Rubel, The extended analog computer. Adv. in Appl. Math. 14 (1993), 39–50.

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Sánchez-Sinencio, C. Lau, Artificial Neural Networks: Paradigms, Applications, and Hardware Implementations. IEEE Press, New York, 1992.

    Google Scholar 

  46. C. E. Shannon, Mathematical theory of the differential analyzer. J. Math. Phys. MIT 20 (1941), 337–354. Reprinted in [48], 496–513.

    Google Scholar 

  47. C. E. Shannon, The theory and design of linear differential equation machines. Report to the National Defense Research Council, January 1942. Reprinted in [48], pp. 514–559.

    Google Scholar 

  48. C. E. Shannon, Collected Papers (N. J. A. Sloane, A. Wyner, eds.). IEEE Press, Piscataway, NJ, 1993.

    Google Scholar 

  49. P. Shor, Algorithms for quantum computation: discrete logarithms and factoring. Proc. 35th Ann. IEEE Symp. on Foundations of Computer Science, 124–134. IEEE Computer Society Press, Los Alamitos, CA, 1994.

    Google Scholar 

  50. H. T. Siegelmann, S. Fishman, Analog computing and dynamical systems. Manuscript, April 1996. 34 pp.

    Google Scholar 

  51. H. T. Siegelmann, E. D. Sontag, On the computational power of neural nets. J. Comput. System Sciences 50 (1995), 132–150.

    Article  MathSciNet  MATH  Google Scholar 

  52. A. Vergis, K. Steiglitz, B. Dickinson, The complexity of analog computation. Math, and Computers in Simulation 28 (1986), 91–113.

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Jyväskylä, P. O. Box 35, FIN-40351, Jyväskylä, Finland

    Pekka Orponen

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  1. Pekka Orponen
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Editor information

Editors and Affiliations

  1. University of Minnesota, USA

    Ding-Zhu Du

  2. State University of New York, Stony Brook, USA

    Ker-I Ko

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© 1997 Kluwer Academic Publishers

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Orponen, P. (1997). A Survey of Continuous-Time Computation Theory. In: Du, DZ., Ko, KI. (eds) Advances in Algorithms, Languages, and Complexity. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3394-4_11

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  • DOI: https://doi.org/10.1007/978-1-4613-3394-4_11

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-3396-8

  • Online ISBN: 978-1-4613-3394-4

  • eBook Packages: Springer Book Archive

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