Formalization of Shannon’s Theorems in SSReflect-Coq

  • Reynald Affeldt
  • Manabu Hagiwara
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7406)

Abstract

The most fundamental results of information theory are Shannon’s theorems. These theorems express the bounds for reliable data compression and transmission over a noisy channel. Their proofs are non-trivial but rarely detailed, even in the introductory literature. This lack of formal foundations makes it all the more unfortunate that crucial results in computer security rely solely on information theory (the so-called “unconditional security”). In this paper, we report on the formalization of a library for information theory in the SSReflect extension of the Coq proof-assistant. In particular, we produce the first formal proofs of the source coding theorem (that introduces the entropy as the bound for lossless compression), and the direct part of the more difficult channel coding theorem (that introduces the capacity as the bound for reliable communication over a noisy channel).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Shannon, C.E.: A Mathematical Theory of Communication. Bell System Technical Journal 27, 379–423, 623–656 (1948)MathSciNetMATHGoogle Scholar
  2. 2.
    Shannon, C.E.: Communication Theory of Secrecy Systems. Bell System Technical Journal 28, 656–715 (1949)MathSciNetMATHGoogle Scholar
  3. 3.
    Uyematsu, T.: Modern Shannon Theory, Information theory with types. Baifukan (1998) (in Japanese)Google Scholar
  4. 4.
    Hurd, J.: Formal Verification of Probabilistic Algorithms. PhD Thesis, Trinity College, University of Cambridge, UK (2001)Google Scholar
  5. 5.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley-Interscience (2006)Google Scholar
  6. 6.
    Hasan, O., Tahar, S.: Verification of Expectation Using Theorem Proving to Verify Expectation and Variance for Discrete Random Variables. J. Autom. Reasoning 41, 295–323 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bertot, Y., Gonthier, G., Ould Biha, S., Pasca, I.: Canonical Big Operators. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 86–101. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Audebaud, P., Paulin-Mohring, C.: Proofs of randomized algorithms in COQ. Sci. Comput. Program. 74(8), 568–589 (2009)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Coble, A.R.: Anonymity, Information, and Machine-Assisted Proof. PhD Thesis, King’s College, University of Cambridge, UK (2010)Google Scholar
  10. 10.
    The COQ Development Team. Reference Manual. Version 8.3. INRIA (2004-2010), http://coq.inria.fr
  11. 11.
    Mhamdi, T., Hasan, O., Tahar, S.: On the Formalization of the Lebesgue Integration Theory in HOL. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 387–402. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Gonthier, G., Mahboubi, A., Tassi, E.: A Small Scale Reflection Extension for the Coq system. Version 10. Technical report RR-6455. INRIA (2011)Google Scholar
  13. 13.
    Mhamdi, T., Hasan, O., Tahar, S.: Formalization of Entropy Measures in HOL. In: van Eekelen, M., Geuvers, H., Schmaltz, J., Wiedijk, F. (eds.) ITP 2011. LNCS, vol. 6898, pp. 233–248. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  14. 14.
    Affeldt, R., Hagiwara, M.: Formalization of Shannon’s Theorems in SSReflect-COQ. COQ scripts, http://staff.aist.go.jp/reynald.affeldt/shannon

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Reynald Affeldt
    • 1
  • Manabu Hagiwara
    • 1
  1. 1.Research Institute for Secure SystemsNational Institute of Advanced Industrial Science and TechnologyJapan

Personalised recommendations