Skip to main content
SpringerLink
Log in

Decreasing Complexity in Inductive Computations

  • Chapter
  • First Online:
Book cover Advances in Unconventional Computing

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 22))

  • 1476 Accesses

Abstract

Kolmogorov or algorithmic complexity has found applications in many areas including medicine, biology, neurophysiology, physics, economics, hardware and software engineering. Conventional Kolmogorov/algorithmic complexity and its modifications are based on application of conventional, i.e., recursive, algorithms, such as Turing machines. Inductive complexity studied in this paper is based on application of unconventional algorithms such as inductive Turing machines, which are super-recursive as they can compute much more than recursive algorithm can. It is possible to apply inductive complexity in all cases where Kolmogorov complexity is used. In particular, inductive complexity has been used in the study of mathematical problem complexity. The main goal of this work is to show how inductive algorithms can reduce complexity of programs and problems. In Sect. 8.2, we build the constructive hierarchy of inductive Turing machines and study the corresponding hierarchy of inductively computable functions. Inductive Turing machines from the constructive hierarchy are very powerful because they can build (compute) the whole arithmetical hierarchy. In Sect. 8.3, it is proved that inductive algorithms from the constructive hierarchy can essentially reduce complexity of programs and problems and the more powerful inductive algorithms are utilized the larger reduction of complexity is achievable.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Benci, V., Bonanno, C., Galatolo, S., Menconi, G., Virgilio, M. Dynamical systems and computable information, Preprint in Physics cond-mat/0210654, 2002 (electronic edition: http://arXiv.org)

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Burgin, M.S.: Inductive Turing machines. Not. Acad. Sci. USSR 27(3), 1289–1293 (1983)

    Google Scholar 

  4. Burgin, M.S.: Generalized Kolmogorov complexity and other dual complexity measures. Cybern. Syst. Anal. 26(4), 481–490 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burgin, M.: Super-recursive algorithms as a tool for high performance computing. In: Proceedings of the High Performance Computing Symposium, pp. 224–228. San Diego (1999)

    Google Scholar 

  6. Burgin, M.: Nonlinear phenomena in spaces of algorithms. Int. J. Comput. Math. 80(12), 1449–1476 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burgin, M.: Algorithmic complexity of recursive and inductive algorithms. Theor. Comput. Sci. 317(1/3), 31–60 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burgin, M.: Superrecursive Algorithms. Springer, New York (2005)

    MATH  Google Scholar 

  9. Burgin, M. Superrecursive hierarchies of algorithmic problems. In: Proceedings of the 2005 International Conference on Foundations of Computer Science, pp. 31–37. CSREA Press, Las Vegas (2005)

    Google Scholar 

  10. Burgin, M.: Algorithmic complexity as a criterion of insolvability. Theor. Comput. Sci. 383(2/3), 244–259 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Burgin, M.: Algorithmic complexity of computational problems. Int. J. Comput. Inf. Technol. 2(1), 149–187 (2010)

    Google Scholar 

  12. Burgin, M.: Measuring Power of Algorithms, Computer Programs, and Information Automata. Nova Science Publishers, New York (2010)

    Google Scholar 

  13. Burgin, M., Calude, C.S., Calude, E.: Inductive Complexity Measures for Mathematical Problems. CDMTCS Research, Report 416 (2011)

    Google Scholar 

  14. Burgin, M., Debnath, N.C.: Complexity of algorithms and software metrics. In: Proceedings of the ISCA 18th International Conference “Computers and their Applications, pp. 259–262. International Society for Computers and their Applications, Honolulu, Hawaii (2003)

    Google Scholar 

  15. Calude, C.S., Calude, E., Queen, M.S.: Inductive complexity of P versus NP Problem. Unconv. Comput. Nat. Comput. Lect. Notes Comput. Sci. 7445, 2–9 (2012)

    MATH  Google Scholar 

  16. Crosby, S.A., Wallach, D.S.: Denial of Service via Algorithmic Complexity Attacks, Technical Report TR-03-416. Department of Computer Science, Rice University (2003)

    Google Scholar 

  17. Debnath, N.C., Burgin, M.: Software metrics from the algorithmic perspective. In: Proceedings of the ISCA 18th International Conference “Computers and their Applications”, pp. 279–282. Honolulu, Hawaii (2003)

    Google Scholar 

  18. Dewey, T.G.: The algorithmic complexity of a protein. Phys. Rev. E 54, R39–R41 (1996)

    Article  Google Scholar 

  19. Dewey, T.G.: Algorithmic complexity and thermodynamics of sequence: structure relationships in proteins. Phys. Rev. E 56, 4545–4552 (1997)

    Google Scholar 

  20. Dzhunushaliev, V.D.: Kolmogorov’s algorithmic complexity and its probability interpretation in quantum gravity. Class. Quantum Gravity 15, 603–612 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dzhunushaliev, V.D., Singleton, D.: Algorithmic Complexity in Cosmology and Quantum Gravity, Preprint in Physics gr-qc/0108038, 2001 (electronic edition: http://arXiv.org)

  22. Hertel, J.: Inductive complexity of Goodstein’s theorem. Unconv. Comput. Nat. Comput. Lect. Notes Comput. Sci. 7445, 141–151 (2012)

    MathSciNet  MATH  Google Scholar 

  23. Kolmogorov, A.N.: Three approaches to the definition of the quantity of information. Probl. Inf. Trans. 1, 3–11 (1965)

    MathSciNet  MATH  Google Scholar 

  24. Kreinovich, V., Kunin, I.A.: Application of Kolmogorov Complexity to Advanced Problems in Mechanics, University of Texas at El Paso, Computer Science Department Reports, UTEP-CS-04-14 (2004)

    Google Scholar 

  25. Lewis, J.P.: Limits to software estimation. Softw. Eng. Notes 26, 54–59 (2001)

    Article  Google Scholar 

  26. Li, M., Vitanyi, P.: An Introduction to Kolmogorov Complexity and its Applications. Springer, New York (1997)

    Book  MATH  Google Scholar 

  27. Mansilla, R.: Algorithmic Complexity in Real Financial Markets, Preprint in Physics cond-mat/0104472 (2001) (electronic edition: http://arXiv.org)

  28. Shaw, F.Z., Chen, R.F., Tsao, H.W., Yen, C.T.: Algorithmic complexity as an index of cortical function in awake and pentobarbital-anesthetized rats. J. Neurosci. Methods 93(2), 101–110 (1999)

    Article  Google Scholar 

  29. Tegmark, M.: Does the universe in fact contain almost no information? Found. Phys. Lett. 9, 25–42 (1996)

    Article  MathSciNet  Google Scholar 

  30. Vitanyi, P.M.B.: Quantum Kolmogorov complexity based on classical descriptions. IEEE Trans. Inf. Theory 47(6), 2464–2479 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zimbardo, P.G., Weber, A.L., Jonson, R.L.: Psychology. Allyn and Bacon, San Francisco (2003)

    Google Scholar 

  32. Zurek, W.H.: Algorithmic randomness and physical entropy. Phys. Rev. A (3) 40(8), 4731–4751 (1989)

    Google Scholar 

  33. Zurek, W.H.: Algorithmic information content, Church-Turing thesis, physical entropy, and Maxwell’s demon. Information dynamics (Irsee, 1990): NATO Adv. Sci. Inst. Ser. B Phys., 256. Plenum, New York, 245–259 (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of California, Los Angeles, 405 Hilgard Ave., Los Angeles, CA, 90095, USA

    Mark Burgin

Authors
  1. Mark Burgin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark Burgin .

Editor information

Editors and Affiliations

  1. Unconventional Computing Centre, University of the West of England Unconventional Computing Centre, Bristol, United Kingdom

    Andrew Adamatzky

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Burgin, M. (2017). Decreasing Complexity in Inductive Computations. In: Adamatzky, A. (eds) Advances in Unconventional Computing. Emergence, Complexity and Computation, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-319-33924-5_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-33924-5_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-33923-8

  • Online ISBN: 978-3-319-33924-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation