The Search for the Universal Concept of Complexity and a General Optimality Condition of Cooperative Agents

  • Victor Korotkich
Chapter
Part of the Cooperative Systems book series (COSY, volume 1)

Abstract

There are many different notions of complexity. However, complexity does not have a generally accepted universal concept. It is becoming more clear that the search for the universal concept must be done within a final theory. In this chapter a concept of structural complexity for the first time suggests the real opportunity to search the universal concept of complexity within a final theory. Experimental facts given in this chapter allow to suggest a general optimality condition of cooperative agents in terms of structural complexity. The optimality condition says that cooperative agents show their best performance for a particular problem when their structural complexity equals the structural complexity of the problem. According to the optimality condition to control a complex system efficiently means to equate its structural complexity with the structural complexity of the problem.

Keywords

Complexity cooperative agents optimality condition travelling salesman problem 
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References

  1. [1]
    C. Bennett, “On the nature and origin of complexity in discrete, homogeneous, locally-interacting systems”, Foundations of Physics, 16:585–592, 1986.MathSciNetCrossRefGoogle Scholar
  2. [2]
    L. Brillouin, Science and Information Theory, Academic, London, 1962.MATHGoogle Scholar
  3. [3]
    G. J. Chaitin, “On the length of programs for computing binary sequences”, Journal of ACM, 13: 547–569, 1966.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    G. J. Chaitin, Exploring Randomness, Springer-Verlag, 2001.MATHCrossRefGoogle Scholar
  5. [5]
    J. P. Crutchfield, “The calculi of emergence”, Physica D, 75: 11–54, 1994.MATHCrossRefGoogle Scholar
  6. [6]
    H. C. Fogedby, 7. Stat. Phys., 69: 411, 1992.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP Completeness, Freeman, San Franisco, 1979.MATHGoogle Scholar
  8. [8]
    M. Gell-Mann, The Quark and the Jaguar: Adventures in the Simple and Complex, W. H. Freeman and Company, New York, 1994.MATHGoogle Scholar
  9. [9]
    M. Gell-Mann and S. Lloyd, Complexity, 2: 44, 1996.MathSciNetCrossRefGoogle Scholar
  10. [10]
    P. Grassberger, Int. J. Theor. Phys., 25: 907, 1986.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    J. P. Crutchfield and D. P. Feldman, Phys. Rev. E, 55: 1239, 1997.CrossRefGoogle Scholar
  12. [12]
    J. H. Holland, Hidden Order: How Adaptation Builds Complexity, Addison-Wesley, Reading, MA, 1995.Google Scholar
  13. [13]
    J. H. Holland, Emergence: From Chaos to Order, Perseus Books, Reading, Massachusetts, 1998.Google Scholar
  14. [14]
    J. Horgan, The End of Science, Broadway Books, New York, 1996.MATHGoogle Scholar
  15. [15]
    B. A. Huberman and T. Hogg, Physica D, 22: 376, 1986.MathSciNetGoogle Scholar
  16. [16]
    S. Kauffman, At Home in the Universe: the Search for Laws of Self-organization and Complexity, Oxford University Press, New York, 1995.Google Scholar
  17. [17]
    A. Kolmogorov, “Three approaches to the definition of the concept ‘Quantity of Information’”, Problems of Information Transmission, 1: 1–7, 1965.Google Scholar
  18. [18]
    N. M. Korobov, Trigonometric Sums and their Applications, Nauka, Moscow, 1989.MATHGoogle Scholar
  19. [19]
    V. Korotkich, A Mathematical Structure for Emergent Computation, Kluwer Academic Publishers, 1999.MATHGoogle Scholar
  20. [20]
    V. Korotkich, “On complexity and optimization in emergent computation”, In P. Pardalos, editor, Approximation and Complexity in Numerical Optimization, pp. 347–363, Kluwer Academic Publishers, 2000.Google Scholar
  21. [21]
    V. Korotkich, “On optimal algorithms in emergent computation”, In A. Rubinov and B. Glover, editors, Optimization and Related Topics, pp. 83–102, Kluwer Academic Publishers, 2001.Google Scholar
  22. [22]
    V. Korotkich, “On self-organization of cooperative systems and fuzzy logic”, In V. Dimitrov and V. Korotkich, editors, Fuzzy Logic: A Framework for the New Millennium, pp. 147–167, Springer-Verlag, 2002.Google Scholar
  23. [23]
    R. Landauer, IBM J. Res. Dev. 3: 183, 1961.MathSciNetCrossRefGoogle Scholar
  24. [24]
    C. Langton, Physica D, 42: 12, 1990.MathSciNetCrossRefGoogle Scholar
  25. [25]
    M. Li and P. Vitanyi, An Introduction to Kolmogorov Complexity and its Applications, Springer-Verlag, 1997.MATHCrossRefGoogle Scholar
  26. [26]
    S. Lloyd and H. Pagels, “Complexity as thermodynamic depth”, Annals of Physics, 188: 186–213, 1988.MathSciNetCrossRefGoogle Scholar
  27. [27]
    P. G. Mezey, Potential Energy Hypersurfaces, Elsevier, Amsterdam, 1987.Google Scholar
  28. [28]
    L. Mordell, “On a sum analogous to a Gauss’ sum”, Quart. J. Math., 3: 161–167, 1932.MATHCrossRefGoogle Scholar
  29. [29]
    M. Morse, “Recurrent geodesics on a surface of negative curvature”, Trans. Amer. Math. Soc, 22: 84, 1921.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    R. Nadeau and M. Kafatos. The Non-local Universe, Oxford University Press, New York, 2001.Google Scholar
  31. [31]
    G. Nicolis and I. Prigogine, Exploring Complexity, New York, Freeman, 1989.Google Scholar
  32. [32]
    C. Papadimitriou, Computational Complexity, Addison-Wesley, Reading, MA, 1994.MATHGoogle Scholar
  33. [33]
    A.S. Perelson and S.A. Kauffman, editors, Molecular Evolution on Rugged Landscapes: Proteins, RNA, and the Immune System, vol. 9 of Santa Fe Studies, Reading, MA, Addison-Wesley, 1991.Google Scholar
  34. [34]
    E. Prouhet, “Memoire sur quelques relations entre les puissances des nombres”, CR. Acad. Sci., Paris, 33: 225, 1851.Google Scholar
  35. [35]
    S. Ramanujan, “Note on a set of simultaneous equations”, J. Indian Math. Soc, 4: 94–96, 1912.MATHGoogle Scholar
  36. [36]
    G. Reinelt, TSPLIB 1.2 [Online] Available from: URL ftp://ftp.wiwi.uni-frankfurt.de/pub/TSPLIB 1.2, 2000.
  37. [37]
    C. E. Shannon, “A mathematical theory of communication”, Bell System Technical Journal, July and October, 1948,Google Scholar
  38. [37]a
    C. E. Shannon, “A mathematical theory of communication”, reprinted in C. E. Shannon and W. Weaver, editors, A Mathematical Theory of Communication, University of Illinois Press, Urbana, 1949.Google Scholar
  39. [38]
    R. Solomonoff, “A formal theory of inductive inference”, Information and Control, 7:1–22, 1964.MathSciNetMATHCrossRefGoogle Scholar
  40. [39]
    L. Szilard, Z Phys., 53: 840, 1929.MATHCrossRefGoogle Scholar
  41. [40]
    A. Thue, “Uber unendliche zeichenreihen”, Norske vid. Selsk. Skr. I. Mat. Nat. Kl. Christiana, 7: 1, 1906.Google Scholar
  42. [41]
    J. Traub, G. Wasilkowski and H. Wozniakowski, Information-based Complexity, Academic Press, 1988.MATHGoogle Scholar
  43. [42]
    S. Weinberg, Dreams of a Final Theory, Pantheon, New York, 1992.Google Scholar
  44. [43]
    H. Weyl, “Uber die gleichverteilung von zahlen mod”, Eins. Math. Ann., 77: 313–352, 1915/16.MathSciNetCrossRefGoogle Scholar
  45. [44]
    J. A. Wheeler, A Journey into Gravity and Spacetime, Scientific American Library, New York, 1990.Google Scholar
  46. [45]
    E. Witten, “Reflections on the fate of spacetime”, Physics Today, 49: 24–30, 1996.MathSciNetCrossRefGoogle Scholar
  47. [46]
    S. Wolfram, Cellular automata and complexity, Addison-Wesley, 1994.MATHGoogle Scholar
  48. [47]
    W. H. Zurek, “Algorithmic randomness and physical entropy”, Physical Review A, 40: 4731–4751, 1989.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Victor Korotkich
    • 1
  1. 1.Faculty of Informatics and CommunicationCentral Queensland UniversityMackayAustralia

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