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A Logical Interpretation of the λ-Calculus into the π-Calculus, Preserving Spine Reduction and Types

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CONCUR 2009 - Concurrency Theory (CONCUR 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5710))

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Abstract

We define a new, output-based encoding of the λ-calculus into the asynchronous π-calculus – enriched with pairing – that has its origin in mathematical logic, and show that this encoding respects one-step spine-reduction up to substitution, and that normal substitution is respected up to similarity. We will also show that it fully encodes lazy reduction of closed terms, in that term-substitution as well as each reduction step are modelled up to similarity. We then define a notion of type assignment for the π-calculus that uses the type constructor →, and show that all Curry-assignable types are preserved by the encoding.

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References

  1. Abadi, M., Gordon, A.: A Calculus for Cryptographic Protocols: The Spi Calculus. In: CC&CS 1997, pp. 36–47 (1997)

    Google Scholar 

  2. Abramsky, S.: The lazy lambda calculus. In: Research topics in functional programming, pp. 65–116 (1990)

    Google Scholar 

  3. Abramsky, S.: Proofs as Processes. TCS 135(1), 5–9 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. van Bakel, S., Cardelli, L., Vigliotti, M.G.: From X to π; Representing the Classical Sequent Calculus in π-calculus. In: CL&C 2008 (2008)

    Google Scholar 

  5. van Bakel, S., Lengrand, S., Lescanne, P.: The language formula_image: Circuits, computations and classical logic. In: Coppo, M., Lodi, E., Pinna, G.M. (eds.) ICTCS 2005. LNCS, vol. 3701, pp. 81–96. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  6. van Bakel, S., Lescanne, P.: Computation with Classical Sequents. MSCS 18, 555–609 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Barendregt, H.: The Lambda Calculus: its Syntax and Semantics. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  8. Barendregt, H.P., Kennaway, R., Klop, J.W., Sleep, M.R.: Needed Reduction and Spine Strategies for the Lambda Calculus. I&C 75(3), 191–231 (1987)

    MathSciNet  MATH  Google Scholar 

  9. Bellin, G., Scott, P.J.: On the pi-Calculus and Linear Logic. TCS 135(1), 11–65 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bloo, R., Rose, K.H.: Preservation of Strong Normalisation in Named Lambda Calculi with Explicit Substitution and Garbage Collection. In: CSN 1995 – Computer Science in the Netherlands, pp. 62–72 (1995)

    Google Scholar 

  11. Church, A.: A note on the entscheidungsproblem. JSL 1(1), 40–41 (1936)

    MATH  Google Scholar 

  12. Gentzen, G.: Investigations into logical deduction. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, p. 68, 1935. North Holland, Amsterdam (1969)

    Google Scholar 

  13. Gentzen, G.: Untersuchungen über das Logische Schliessen. Mathematische Zeitschrift 39, 176–210, 405–431 (1935)

    Article  MathSciNet  MATH  Google Scholar 

  14. Goubault-Larrecq, J.: A Few Remarks on SKInT. RR-3475, INRIA Rocquencourt (1998)

    Google Scholar 

  15. Honda, K., Tokoro, M.: An object calculus for asynchronous communication. In: America, P. (ed.) ECOOP 1991. LNCS, vol. 512, pp. 133–147. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  16. Honda, K., Yoshida, N.: On the Reduction-based Process Semantics. TCS 151, 437–486 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Honda, K., Yoshida, N., Berger, M.: Control in the π-Calculus. In: CW 2004 (2004)

    Google Scholar 

  18. Milner, R.: Function as processes. MSCS 2(2), 269–310 (1992)

    MathSciNet  Google Scholar 

  19. Parigot, M.: An algorithmic interpretation of classical natural deduction. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 190–201. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  20. Sangiorgi, D.: Expressing Mobility in Process Algebra: First Order and Higher order Paradigms. PhD thesis, Edinburgh University (1992)

    Google Scholar 

  21. Sangiorgi, D.: An Investigation into Functions as Processes. In: Main, M.G., Melton, A.C., Mislove, M.W., Schmidt, D., Brookes, S.D. (eds.) MFPS 1993. LNCS, vol. 802, pp. 143–159. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  22. Sangiorgi, D.: Lazy functions and processes. RR2515, INRIA, Sophia-Antipolis (1995)

    Google Scholar 

  23. Sangiorgi, D., Walker, D.: The Pi-Calculus. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  24. Sestoft, P.: Standard ML on the Web server. Department of Mathematics and Physics, Royal Veterinary and Agricultural University, Denmark (1996)

    Google Scholar 

  25. Thielecke, H.: Categorical Structure of Continuation Passing Style. PhD thesis, University of Edinburgh (1997)

    Google Scholar 

  26. Urban, C.: Classical Logic and Computation. PhD thesis, University of Cambridge (2000)

    Google Scholar 

  27. Urban, C., Bierman, G.M.: Strong normalisation of cut-elimination in classical logic. FI 45(1, 2), 123–155 (2001)

    MathSciNet  MATH  Google Scholar 

  28. de Vries, F.-J.: Böhm trees, bisimulations and observations in lambda calculus. In: FLP 1997, pp. 230–245 (1997)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computing, Imperial College, 180 Queen’s Gate, London, SW7 2BZ, UK

    Steffen van Bakel & Maria Grazia Vigliotti

Authors
  1. Steffen van Bakel
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  2. Maria Grazia Vigliotti
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Editors and Affiliations

  1. Department of Computer Science, University of Bologna, Mura A. Zamboni 7, 40127, Bologna, Italy

    Mario Bravetti  & Gianluigi Zavattaro  & 

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van Bakel, S., Vigliotti, M.G. (2009). A Logical Interpretation of the λ-Calculus into the π-Calculus, Preserving Spine Reduction and Types. In: Bravetti, M., Zavattaro, G. (eds) CONCUR 2009 - Concurrency Theory. CONCUR 2009. Lecture Notes in Computer Science, vol 5710. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04081-8_7

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  • DOI: https://doi.org/10.1007/978-3-642-04081-8_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04080-1

  • Online ISBN: 978-3-642-04081-8

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