Mathematical systems theory

, Volume 13, Issue 1, pp 95–104 | Cite as

A constructive generalization of the borel-cantelli lemma with application to the complexity of infinite strings

  • Richard A. DeMillo
  • Richard J. Lipton
Article

Abstract

This paper concerns a constructive adaptation of the classical Borel-Cantelli lemma which allows us to solve such decomposition problems as: when does there exist an infinite object that is decomposable into infinitely many parts that are maximally complex? A constructive proof is supplied of the key theorem and its degree is characterized. For completeness a classical (i.e., nonconstructive) proof is also provided.

Keywords

Computational Mathematic Constructive Proof Decomposition Problem Constructive Adaptation Constructive Generalization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. L. Chung and P. Erdös, On the applicability of the Borel-Cantelli lemma,Transactions of the American Mathematical Society 72, 179–186, 1952.Google Scholar
  2. 2.
    P. Erdös and J. Spencer,Probabilistic Methods in Combinatorics, Academic Press, 1974.Google Scholar
  3. 3.
    A. N. Kolmogorov, Three approaches to the quantitative definition of information,Promlemi Pederachi Informatsii 1, 3–11, 1965.Google Scholar
  4. 4.
    J. Lemperti, Wiener's test and Markov chains,J. Math. Anal. Appl. 6, 58–66, 1963.Google Scholar
  5. 5.
    R. Lipton, Polynomials with 0,1 coefficients that are hard to compute,Proceedings of the 16th IEEE Symposium on the Foundations of Computer Science, pp. 6–10, 1975.Google Scholar
  6. 6.
    H. Rogers,Theory of Recursive Functions and Effective Computability, McGraw-Hill, 1967.Google Scholar
  7. 7.
    V. Strassen, Polynomials with rational coefficients which are hard to compute,SIAM Journal of Computing 3, 128–148, 1974.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Richard A. DeMillo
    • 1
  • Richard J. Lipton
    • 2
  1. 1.School of Information and Computer ScienceGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of Computer ScienceYale UniversityNew HavenUSA

Personalised recommendations