Formalization of Shannon’s Theorems in SSReflect-Coq
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).
KeywordsMutual Information Typical Sequence Noisy Channel Direct Part Input Alphabet
Unable to display preview. Download preview PDF.
- 3.Uyematsu, T.: Modern Shannon Theory, Information theory with types. Baifukan (1998) (in Japanese)Google Scholar
- 4.Hurd, J.: Formal Verification of Probabilistic Algorithms. PhD Thesis, Trinity College, University of Cambridge, UK (2001)Google Scholar
- 5.Cover, T.M., Thomas, J.A.: Elements of Information Theory, 2nd edn. Wiley-Interscience (2006)Google Scholar
- 9.Coble, A.R.: Anonymity, Information, and Machine-Assisted Proof. PhD Thesis, King’s College, University of Cambridge, UK (2010)Google Scholar
- 10.The COQ Development Team. Reference Manual. Version 8.3. INRIA (2004-2010), http://coq.inria.fr
- 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
- 14.Affeldt, R., Hagiwara, M.: Formalization of Shannon’s Theorems in SSReflect-COQ. COQ scripts, http://staff.aist.go.jp/reynald.affeldt/shannon