Advertisement

Algorithmic Complexity

  • Ming Li
  • Paul Vitányi
Part of the Graduate Texts in Computer Science book series (TCS)

Abstract

The most natural approach to defining the quantity of information is clearly to define it in relation to the individual object (be it Homer’s Odyssey or a particular type of dodo) rather than in relation to a set of objects from which the individual object may be selected. To do so, one could define the quantity of information in an object in terms of the number of bits required to describe it. A description of an object is evidently only useful if we can reconstruct the object from this description.

Keywords

Turing Machine Binary String Recursive Function Infinite Sequence Kolmogorov Complexity 
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.

History and References

  1. R.J. Solomonoff, 1960/1964, A.N. Kolmogorov, 1965, and G.J. Chaitin, 1969.Google Scholar
  2. A.A. Markov [Soviet Math. Dokl, 5(1964), 922–924]zbMATHGoogle Scholar
  3. G.J. Chaitin [J. ACM, 13(1966), 547–569].MathSciNetzbMATHCrossRefGoogle Scholar
  4. M. Koppel and H. Atlan [Unpublished Manuscript] and M. Koppel [Complex Systems, 1 (1987), 1087-1091.Google Scholar
  5. The Universal Turing Machine: A Half-Century Survey, R. Herken, ed., Oxford Univ. Press, 1988, 435-452].Google Scholar
  6. A.N. Kolmogorov, Lecture Notes in Mathematics, vol. 1021, Springer-Verlag, 1983, 1-5.Google Scholar
  7. A.N. Kolmogorov, V.A. Uspensky, Theory Probab. Appl., 32(1987), 389–412.zbMATHCrossRefGoogle Scholar
  8. a.Kh. Shen’, Soviet Math. Dokl., 28(1983), 295–299.zbMATHGoogle Scholar
  9. V.V. V’yugin, Theory Probab. Appl., 32(1987), 508–512.MathSciNetCrossRefGoogle Scholar
  10. T.M. Cover and J.A. Thomas, Elements of Information Theory, Wiley, 1991, 169-182.Google Scholar
  11. L. Löfgren [Automata Theory, E. Caianiello, Ed., Academic Press, 1966, 251-268.Google Scholar
  12. Computer and Information Sciences II, J. Tou (Ed.), Academic Press, 1967, 165-175].Google Scholar
  13. D.W. Loveland introduced n-strings in [Inform. Contr., 15(1969), 510–526; Proc. ACM 1st Symp. Theory Comput., 1969, 61-65]MathSciNetzbMATHCrossRefGoogle Scholar
  14. L.A. Levin and A.K. Zvonkin [Russ. Math. Surv., 25:6(1970), 83–124]MathSciNetzbMATHCrossRefGoogle Scholar
  15. P. Gács’s [Komplexität und Zufälligkeit, Ph.D. Thesis, J.W. Goethe Univ., Frankfurt am Main, 1978.Google Scholar
  16. unpublished; Lecture Notes on Descriptional Complexity and Randomness, Manuscript, Boston University, 1987].Google Scholar
  17. V.V. V’yugin, Selecta Mathematica formerly Sovietica, 13:4(1994), 357–389.MathSciNetGoogle Scholar
  18. (Translated from the Russian Semiotika and Informatika, 16(1981), 14–43).Google Scholar
  19. [A.K. Zvonkin and L.A. Levin, Russ. Math. Surv., 25:6(1970), 83–124].MathSciNetzbMATHCrossRefGoogle Scholar
  20. [M. Sipser, Lecture Notes on Complexity Theory, MIT Lab Computer Science, 1985, unpublished].Google Scholar
  21. [R.J. Solomonoff, Inform. Contr., 7(1964), 1–22, 224-254].MathSciNetzbMATHCrossRefGoogle Scholar
  22. M. Sipser [Theoret. Comput. Sci., 15(1981), 291–309].MathSciNetzbMATHCrossRefGoogle Scholar
  23. P. Martin-Löf [Inform. Contr., 9(1966), 602–619; Z. Wahrsch. Verw. Geb., 19(1971), 225-230].CrossRefGoogle Scholar
  24. G.J. Chaitin [J. ACM, 16(1969), 145–159].MathSciNetzbMATHCrossRefGoogle Scholar
  25. [A.K. Zvonkin and L.A. Levin, Russ. Math. Surv., 25:6(1970), 83–124; P. Gács, Lecture Notes on Descriptional Complexity and Randomness, Manuscript, Boston University, 1987].MathSciNetzbMATHCrossRefGoogle Scholar
  26. [A.K. Zvonkin and L.A. Levin, Russ. Math. Surv., 25:6(1970), 83–124]MathSciNetzbMATHCrossRefGoogle Scholar
  27. C.P. Schnorr [Lecture Notes in Mathematics, vol. 218, Springer-Verlag, 1971].Google Scholar
  28. D.E. Knuth, Seminumerical Algorithms, Addison-Wesley, 1981, pp. 142-169.Google Scholar
  29. [A.N. Kolmogorov and V.A. Uspensky, Theory Probab. Appl., 32(1987), 389–412.zbMATHCrossRefGoogle Scholar
  30. V.A. Uspensky, A.L. Semenov and A. Kh. Shen’, Russ. Math. Surv., 45:1(1990), 121–189].CrossRefGoogle Scholar
  31. [M. Li and P.M.B. Vitányi, Math. Systems Theory, 27(1994), 365–376].MathSciNetzbMATHCrossRefGoogle Scholar
  32. [A.K. Zvonkin and L.A. Levin, Russ. Math. Surv., 25:6(1970), 83–124].MathSciNetzbMATHCrossRefGoogle Scholar
  33. [J. ACM, 21(1974), 403-423; Scientific American, 232:5(1975), 47-52].Google Scholar
  34. G.J. Chaitin and C.H. Bennett [C.H. Bennett and M. Gardner, Scientific American, 241:5(1979), 20–34].CrossRefGoogle Scholar
  35. [P. Gács, Lecture Notes on De-scriptional Complexity and Randomness, Manuscript, Boston University, 1987].Google Scholar
  36. M. van Lambalgen [J. Symb. Logic, 54(1989), 1389–1400].zbMATHCrossRefGoogle Scholar
  37. [J.M. Barzdins, Soviet Math. Dokl., 9(1968), 1251–1254, and D.W. Loveland, Proc. ACM 1st Symp. Theory Comput., 1969, 61-65].Google Scholar
  38. [Russ. Math. Surv., 38:4(1983), 27-36]Google Scholar
  39. [L.A. Levin, Problems Inform. Transmission, 10:3(1974), 206–210].Google Scholar
  40. A.N. Kolmogorov [Problems Inform. Transmission, 1:1(1965), 1–7; IEEE Trans. Inform. Theory, IT-14(1968), 662-665.MathSciNetGoogle Scholar
  41. Russ. Math. Surv., 38:4(1983), 27-36; Lecture Notes in Mathematics, vol. 1021, Springer-Verlag, 1983, 1-5]Google Scholar
  42. [A.K. Zvonkin and L.A. Levin, Russ. Math. Surv., 25:6(1970), 83–124].MathSciNetzbMATHCrossRefGoogle Scholar
  43. [A.K. Zvonkin and L.A. Levin, Russ. Math. Surv., 25:6(1970), 83–124.MathSciNetzbMATHCrossRefGoogle Scholar
  44. A.N. Kolmogorov, Russ. Math. Surv., 38:4(1983), 27–36].MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Ming Li
    • 1
  • Paul Vitányi
    • 2
  1. 1.Department of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Centrum voor Wiskunde en InformaticaSJ AmsterdamThe Netherlands

Personalised recommendations