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A Type-Theoretic Framework for Formal Reasoning with Different Logical Foundations

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Advances in Computer Science - ASIAN 2006. Secure Software and Related Issues (ASIAN 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4435))

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Abstract

A type-theoretic framework for formal reasoning with different logical foundations is introduced and studied. With logic-enriched type theories formulated in a logical framework, it allows various logical systems such as classical logic as well as intuitionistic logic to be used effectively alongside inductive data types and type universes. This provides an adequate basis for wider applications of type theory based theorem proving technology. Two notions of set are introduced in the framework and used in two case studies of classical reasoning: a predicative one in the formalisation of Weyl’s predicative mathematics and an impredicative one in the verification of security protocols.

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References

  1. Aczel, P., Gambino, N.: Collection principles in dependent type theory. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  2. Agda proof assistant (2000), http://www.cs.chalmers.se/~catarina/agda

  3. Adams, R., Luo, Z.: Weyl’s predicative classical mathematics as a logic-enriched type theory. In: TYPES 2006 (2006) (submitted)

    Google Scholar 

  4. Burrows, M., Abadi, M., Needham, R.: A logic of authentication. Proc. of the Royal Society of London 426, 233–271 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Constable, R., et al.: Implementing Mathematics with the NuPRL Proof Development System. Prentice-Hall, Englewood Cliffs (1986)

    Google Scholar 

  6. Coquand, T., Huet, G.: The calculus of constructions. Information and Computation 76(2/3) (1988)

    Google Scholar 

  7. Callaghan, P.C., Luo, Z.: An implementation of typed LF with coercive subtyping and universes. J. of Automated Reasoning 27(1), 3–27 (2001)

    Article  MATH  Google Scholar 

  8. The Coq Development Team. The Coq Proof Assistant Reference Manual (Version 8.0), INRIA (2004)

    Google Scholar 

  9. Feferman, S.: The significance of Hermann Weyl’s Das Kontinuum. In: Hendricks, V., et al. (eds.) Proof Theory (2000)

    Google Scholar 

  10. Feferman, S.: Predicativity. In: Shapiro, S. (ed.) The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford Univ Press, Oxford (2005)

    Google Scholar 

  11. Gambino, N., Aczel, P.: The generalised type-theoretic interpretation of constructive set theory. J. of Symbolic Logic 71(1), 67–103 (2006)

    Article  MATH  Google Scholar 

  12. Gonthier, G.: A computer checked proof of the four colour theorem (2005)

    Google Scholar 

  13. Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the Association for Computing Machinery 40(1), 143–184 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lowe, G.: Breaking and fixing the Needham-Schroeder public-key protocol using CSP and FDR. In: Margaria, T., Steffen, B. (eds.) TACAS 1996. LNCS, vol. 1055, Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  15. Luo, Z., Pollack, R.: LEGO Proof Development System: User’s Manual. LFCS Report ECS-LFCS-92-211, Dept of Computer Science, Univ of Edinburgh (1992)

    Google Scholar 

  16. Luo, Z.: Computation and Reasoning: A Type Theory for Computer Science. Oxford University Press, Oxford (1994)

    MATH  Google Scholar 

  17. Luo, Z.: PAL + : a lambda-free logical framework. Journal of Functional Programming 13(2), 317–338 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Martin-Löf, P.: Intuitionistic Type Theory. Bibliopolis (1984)

    Google Scholar 

  19. Magnusson, L., Nordström, B.: The ALF proof editor and its proof engine. In: Barendregt, H., Nipkow, T. (eds.) TYPES 1993. LNCS, vol. 806, Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  20. Nordström, B., Petersson, K., Smith, J.: Programming in Martin-Löf’s Type Theory: An Introduction. Oxford University Press, Oxford (1990)

    MATH  Google Scholar 

  21. Needham, R., Schroeder, M.: Using encryption for authentication in large networks of computers. Comm. of the ACM 21(12), 993–999 (1978)

    Article  MATH  Google Scholar 

  22. Paulson, L.: Isabelle: a generic theorem prover. LNCS, vol. 828. Springer, Heidelberg (1994)

    Book  MATH  Google Scholar 

  23. Paulson, L.: The inductive approach to verifying cryptographic protocols. Journal of Computer Security 6, 85–128 (1998)

    Article  Google Scholar 

  24. Paulson, L.: Proving security protocols correct. In: LICS (1999)

    Google Scholar 

  25. Simpson, S.: Subsystems of Second-Order Arithmetic. Springer, Heidelberg (1999)

    Book  MATH  Google Scholar 

  26. Smith, J.: The independence of Peano’s fourth axiom from Martin-Löf’s type theory without universes. Journal of Symbolic Logic 53(3) (1988)

    Google Scholar 

  27. Weyl, H.: The Continuum: a critical examination of the foundation of analysis. Dover Publ., (English translation of Das Kontinuum, 1918) (1994)

    Google Scholar 

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Authors and Affiliations

  1. Dept of Computer Science, Royal Holloway, Univ of London, Egham, Surrey, TW20 0EX, U.K.

    Zhaohui Luo

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  1. Zhaohui Luo
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Mitsu Okada Ichiro Satoh

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Luo, Z. (2007). A Type-Theoretic Framework for Formal Reasoning with Different Logical Foundations. In: Okada, M., Satoh, I. (eds) Advances in Computer Science - ASIAN 2006. Secure Software and Related Issues. ASIAN 2006. Lecture Notes in Computer Science, vol 4435. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77505-8_16

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  • DOI: https://doi.org/10.1007/978-3-540-77505-8_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-77504-1

  • Online ISBN: 978-3-540-77505-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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