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Introduction to Information-Based Complexity

  • J. F. Traub

Abstract

Information-based complexity is based on three assumptions: information is partial, contaminated, and it costs. Its goal is to create a general theory about problems with such information, and to apply the results to solving specific problems in varied disciplines. Some of the questions asked in information- based complexity are listed and some results are indicated. Information-based complexity is contrasted with combinatorial complexity which studies problems such as the traveling salesman problem and linear programming. In combinatorial complexity, information is assumed to be complete, exact, and free. The relation between information theory and information-based complexity is briefly discussed.

Keywords

Travel Salesman Problem Human Visual System Clock Synchronization Geophysical Exploration Intrinsic Uncertainty 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Boult, T. and Sikorski, K. (1986), Complexity of Computing Topological Degree of Lipschitz Functions in n Dimensions, to appear in Journal of Complexity, 2.Google Scholar
  2. [2]
    Brillouin, L. (1956), Science and Information Theory, Academic Press, New York.MATHGoogle Scholar
  3. [3]
    Fisher, R. A. (1950), Contributions to Mathematical Statistics, John Wiley and Sons, New York.MATHGoogle Scholar
  4. [4]
    Gallager, R. G. (1968), Information Theory and Reliable Communication, John Wiley and Sons, New York.MATHGoogle Scholar
  5. [5]
    Grimson, W. E. L. (1981), From Images to Surfaces: A Computational Study of the Human Early Visual System, MIT Press, Cambridge, Mass.Google Scholar
  6. [6]
    Kacewicz, B. Z. (1984), How to Increase the Order to Get Minimal-Error Algorithms for Systems of ODE, Numer. Math.,45, 93–104.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Kadane, J. B., G. W. Wasilkowski, and H. Woźniakowski (1984), Can Adaption Help on the Average for Stochastic Information?, Report, Columbia University.Google Scholar
  8. [8]
    Kolmogorov, A. N. and V. M. Tihomirov, ɛ-Entropy and ɛ-Capacity of sets in Functional Spaces, Usp. Mat. Nauk. 14, 3–80 (in Russian), (1959). Translation in Amer. Math. soc. Trans. 17, 277–364, (1961).MathSciNetMATHGoogle Scholar
  9. [8a]
    Kolmogorov, A. N. and V. M. Tihomirov, ɛ-Entropy and ɛ-Capacity of sets in Functional Spaces, Usp. Mat. Nauk. 14, 3–80 (in Russian), (1959). Translation in Amer. Math. Soc. Transl. 17, 277 – 364, (1961).MathSciNetGoogle Scholar
  10. [9]
    Kuczyński, J. (1986), On the Optimal Solution of Large Eigenpair Problems, to appear in Journal of Complexity, >2.Google Scholar
  11. [10]
    Kullback, S. (1961), Information Theory and Statistics, John Wiley and Sons, New York.Google Scholar
  12. [11]
    Lee, D. (1985), Optimal Algorithms for Image Understanding: Current Status and Future Plans, Journal of Complexity, 1, 138–146.MathSciNetMATHCrossRefGoogle Scholar
  13. [12]
    Lee, D. and G. W. Wasilkowski (1985), Approximation of Linear Functionals on a Banach Space with a Gaussian Measure, Report, Columbia University. To appear in Journal of Complexity, 2 (1986).Google Scholar
  14. [13]
    Marr, D. (1981), VISION: A Computational Investigation in the Human Representation and Processing of Visual Information, W. H. Freeman, San Francisco.Google Scholar
  15. [14]
    Marr, D. and T. Poggio (1977), From Understanding Computation to Understanding Neural Circuitry, Neuroscience Research Program Bulletin 15, 470–488.Google Scholar
  16. [15]
    Marschak, J. and R. Radner (1972), Economic Theory of Teams, Yale University Press, New Haven.MATHGoogle Scholar
  17. [16]
    Milanese, M., R. Tempo, and A. Vicino (1986), Strongly Optimal Algorithms and Optimal Information in Estimation Problems, to appear in Journal of Complexity, 2.Google Scholar
  18. [17]
    Nemirovsky, A. S. and D. B. Yudin (1983), Problem Complexity and Method Efficiency in Optimization, Wiley-Interscience, New York.MATHGoogle Scholar
  19. [18]
    Pinsker, M. S. (1964), Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco.MATHGoogle Scholar
  20. [19]
    Shannon, C. E. (1948), Mathematical Theory of Communication, Bell Syst. Tech. J. 27, 379–423, 623–658.MathSciNetMATHGoogle Scholar
  21. [19a]
    Shannon, C. E. (1948), Mathematical Theory of Communication, Bell Syst. Tech. J. 2, 379–423, 623–658.MathSciNetMATHGoogle Scholar
  22. [20]
    Sikorski, K. (1984), Optimal Solution of Nonlinear Equations Satisfying a Lipschitz Condition, Numer. Math. 43, 225–240.MathSciNetMATHCrossRefGoogle Scholar
  23. [21]
    Sikorski, K. (1985), Optimal Solutions of Nonlinear Equations, Journal of Complexity, 1.Google Scholar
  24. [22]
    Theil, H. (1967), Economics and Information Theory, North-Holland, Amsterdam.Google Scholar
  25. [23]
    Traub, J. F. (1985), Information, Complexity, and the Sciences, University Lecture, Columbia University.Google Scholar
  26. [24]
    Traub, J. F., G. W. Wasilkowski, and H. Woźniakowski (1983), Information, Uncertainty, Complexity, Addison-Wesley, Reading, Mass.MATHGoogle Scholar
  27. [25]
    Traub, J. F., G. W. Wasilkowski, and H. Woźniakowski (1984a), Average Case Optimality for Linear Problems, Journal Theoret. Comp. Sci., 29, 1–25.CrossRefGoogle Scholar
  28. [26]
    Traub, J. F., G. W. Wasilkowski, and H. Woźniakowski (1984b), When is Non-adaptive Information as Powerful as Adaptive Information?, Proceedings of the 23rd IEEE Conference on Decision and Control, 1536–1540.Google Scholar
  29. [27]
    Traub, J. F. and H. Woźniakowski (1984a), A General Theory of Optimal Algorithms, Academic Press, New York, NY.MATHGoogle Scholar
  30. [28]
    Traub, J. F. and H. Woźniakowski (1984a), Information and Computation, chapter in Advances in Computers 23, M. C. Yovits, editor, Academic Press, New York 35–92.Google Scholar
  31. [29]
    Traub, J. F. and H. Woźniakowski (1984b), On the Optimal Solution of Large Linear Systems, Journal of the ACM, 31, 545–559.MATHCrossRefGoogle Scholar
  32. [30]
    Traub, J. F. and H. Woźniakowski (1986a), Information-based Complexity, to appear in Annual Review of Computer Science 1, Annual Reviews, Inc., Palo Alto.Google Scholar
  33. [31]
    Traub, J. F. and H. Woźniakowski (1986b), Measures of Uncertainty and Information, in progress.Google Scholar
  34. [32]
    Twomey, S. (1977),Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurement, Developments in Geomathematics 3, Elsevier, Amsterdam.Google Scholar
  35. [33]
    Wasilkowski, G. W. (1985), Average Case Optimality, Journal of Complexity 1, 107–117.MathSciNetMATHCrossRefGoogle Scholar
  36. [34]
    Wasilkowski, G. W. (1985), Clock Synchronization Problem with Random Delays, Report, Columbia University.Google Scholar
  37. [35]
    Wasilkowski, G. W. and H. Woźniakowski (1984), Can Adaption Help on Average?, Numer. Math., 44, 169 – 190.MathSciNetMATHCrossRefGoogle Scholar
  38. [36]
    Werschulz, A. G. (1985a), Complexity of Differential and Integral Equations, Journal of Complexity, 1.Google Scholar
  39. [37]
    Werschulz, A. G. (1985b), What is the Complexity of Ill-posed Problems?, Report, Columbia University.Google Scholar
  40. [38]
    Wiener, N. (1948), Cybernetics, John Wiley and Sons, New York.Google Scholar
  41. [39]
    Woźniakowski, H. (1985), A Survery of Information-Based Complexity, Journal of Complexity 1, 11 – 44.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1998

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  • J. F. Traub

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