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A Note on Blum Static Complexity Measures

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Book cover Computation, Physics and Beyond (WTCS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7160))

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Abstract

Dual complexity measures have been developed by Burgin, under the influence of the axiomatic system proposed by Blum in [3]. The concept of dual complexity measure is a generalization of Kolmogorov/Chaitin complexity, also known as algorithmic or static complexity. In this paper we continue this effort by extending some of the well known results for plain and prefix-free complexities to the general case of Blum universal static complexity. We also extend some results obtained by Calude in [9] to a larger class of computable measures, proving that transducer complexity is a dual (Blum static) complexity measure.

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References

  1. Balcázar, J.L., Ronald, V.: Book On Generalized Kolmogorov Complexity. In: Monien, B., Vidal-Naquet, G. (eds.) STACS 1986. LNCS, vol. 210, pp. 334–340. Springer, Heidelberg (1985)

    Chapter  Google Scholar 

  2. Blum, M.: A machine-independent theory of the complexity of recursive functions. Journal of the ACM 14(2), 322–336 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blum, M.: On the size of machines. Information and Control 11, 257–265 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burgin, M.: Generalized Kolmogorov complexity and other dual complexity measures. Translated from Kibernetica 4, 21–29 (1990); Original article submitted June 19 (1986)

    MATH  Google Scholar 

  5. Burgin, M.: Algorithmic complexity of recursive and inductive algorithms. Theoretical Computer Science 317, 31–60 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burgin, M.: Algorithmic complexity as a criterion of unsolvability. Theoretical Computer Science 383, 244–259 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Calude, C.: Information and Randomness - An Algorithmic Perspective. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  8. Calude, C.: Theories of Computational Complexity. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  9. Calude, C., Salomaa, K., Roblot, T.K.: Finite State Complexity and Randomness, Technical Report CDMTCS 374 (December 2009/revised June 2010)

    Google Scholar 

  10. Chaitin, G.J.: On the Length of Programs for Computing Finite Binary Sequences. J. ACM 13(4), 547–569 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chaitin, G.J.: On the Length of Programs for Computing Finite Binary Sequences: statistical considerations. J. ACM 16(1), 145–159 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chaitin, G.J.: A Theory of Program Size Formally Identical to Information Theory. J. ACM 22(3), 329–340 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chaitin, G.J.: A Theory of Program Size Formally Identical to Information Theory. J. ACM 22(3), 329–340 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chaitin, G.J.: Algorithmic Information Theory, Cambridge Tracts in Theoretical Computer Science, vol. I. Cambridge University Press (1987)

    Google Scholar 

  15. Davis, M., Sigal, R., Weyuker, E.: Computability, Complexity, and Languages, 2nd edn. Academic Press, New York (1994)

    Google Scholar 

  16. Gacs, P.: On the symmetry of algorithmic information. Soviet Mathematics Doklady 15, 1477–1480 (1974)

    MATH  Google Scholar 

  17. Kolmogorov, A.N.: Problems Inform. Transmission 1, 1–7 (1965)

    Google Scholar 

  18. Kolmogorov, A.N.: Three approaches to the definition of the quantity of information. Problems of Information Transmission 1, 3–11

    Google Scholar 

  19. Kolmogorov, A.N.: Combinatorial foundations of information theory and the calculus of probabilities. Russian Mathematical Surveys 38(4), 27–36 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Levin, L.A.: Laws of information conservation (non-growth) and aspects of the foundation of probability theory. Problems of Information Transmission 10(3), 206–210 (1974)

    Google Scholar 

  21. Levin, L.A., Zvonkin, A.K.: The Complexity of finite objects and the Algorithmic Concepts of Information and Randomness. Russian Math. Surveys 25(6), 83–124 (1970)

    Article  MATH  Google Scholar 

  22. Loveland, D.A.: On Minimal-Program Complexity Measures. In: STOC, pp. 61–65 (1969)

    Google Scholar 

  23. Papadimitriou, C.H., Lewis, H.: Elements of the theory of computation. Prentice-Hall (1982); 2nd edn. (September 1997)

    Google Scholar 

  24. Schmidhuber, J.: Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit. International Journal of Foundations of Computer Science 13(4), 587–612 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Oliver, B.M., Pierce, J.R., Shannon, C.E.: The Philosophy of PCM. Proceedings Institute of Radio Engineers 36, 1324–1331 (1948)

    Google Scholar 

  26. Schnorr, C.-P.: Process complexity and effective random tests. Journal of Computer and System Sciences 7(4), 376–388 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  27. Solomonoff, R.J.: A Formal Theory of Inductive Inference, Part I. Information and Control 7(1), 1–22 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  28. Solomonoff, R.J.: A Formal Theory of Inductive Inference, Part II. Information and Control 7(2), 224–254 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  29. Solomonoff, R.J.: Complexity-Based Induction Systems: Comparisons and Convergence Theorems. IEEE Trans. on Information Theory IT-24(4), 422–432 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  30. Solovay, R.M.: Draft of paper (or series of papers) on Chaitin’s work. Unpublished notes, 1–215 (May 1975)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science and Information Technology, The University of Prince Edward Island, Canada

    Cezar Câmpeanu

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  1. Cezar Câmpeanu
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Editors and Affiliations

  1. Department of Computer Science, University of Auckland, Private Bag 92019, 1142, Auckland, New Zealand

    Michael J. Dinneen , Bakhadyr Khoussainov  & André Nies ,  & 

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Câmpeanu, C. (2012). A Note on Blum Static Complexity Measures. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds) Computation, Physics and Beyond. WTCS 2012. Lecture Notes in Computer Science, vol 7160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27654-5_6

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  • DOI: https://doi.org/10.1007/978-3-642-27654-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27653-8

  • Online ISBN: 978-3-642-27654-5

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