Full Abstraction in a Subtyped pi-Calculus with Linear Types

  • Romain Demangeon
  • Kohei Honda
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6901)


We introduce a concise pi-calculus with directed choices and develop a theory of subtyping. Built on a simple behavioural intuition, the calculus offers exact semantic analysis of the extant notions of subtyping in functional programming languages and session-based programming languages. After illustrating the idea of subtyping through examples, we show type-directed embeddings of two known subtyped calculi, one for functions and another for session-based communications. In both cases, the behavioural content of the original subtyping is precisely captured in the fine-grained subtyping theory in the pi-calculus. We then establish full abstraction of these embeddings with respect to their standard semantics, Morris’s contextual congruence in the case of the functional calculus and testing equivalence for the concurrent calculus. For the full abstraction of the embedding of the session-based calculus, we introduce a new proof method centring on non-deterministic computational adequacy and definability. Partially suggested by a technique used by Quaglia and Walker for their full abstraction result, the new proof method extends the framework used in game-based semantics to the May/Must equivalences, giving a uniform proof method for both deterministic and non-deterministic languages.


Typing Rule Linear Type Proof Method Typing Context Session Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abadi, M., Cardelli, L.: A semantics of object types. In: LICS, pp. 332–341. IEEE Computer Society, Los Alamitos (1994)Google Scholar
  2. 2.
    Amadio, R.M., Cardelli, L.: Subtyping recursive types. ACM Trans. Program. Lang. Syst. 15(4), 575–631 (1993)CrossRefGoogle Scholar
  3. 3.
    Berger, M., Honda, K., Yoshida, N.: Genericity and the pi-calculus. Acta. Inf. 42(2-3), 83–141 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cardelli, L., Wegner, P.: On understanding types, data abstraction, and polymorphism. ACM Comput. Surv. 17(4), 471–522 (1985)CrossRefGoogle Scholar
  5. 5.
    Castagna, G., Nicola, R.D., Varacca, D.: Semantic subtyping for the pi-calculus. Theor. Comput. Sci. 398, 217–242 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Castellani, I., Hennessy, M.: Testing theories for asynchronous languages. In: Arvind, V., Ramanujam, R., (eds.) FST TCS 1998. LNCS, vol. 1530, pp. 90–102. Springer, Heidelberg (1998)Google Scholar
  7. 7.
    Demangeon, R., Honda, K.: Full abstraction in a subtyped pi-calculus with linear types, long version (2011) (in preparation),
  8. 8.
    Dezani-Ciancaglini, M., de’Liguoro, U.: Sessions and Session Types: An Overview. In: Laneve, C., Su, J. (eds.) WS-FM 2009. LNCS, vol. 6194, pp. 1–28. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Fiore, M.P., Honda, K.: Recursive types in games: Axiomatics and process representation. In: LICS, pp. 345–356 (1998)Google Scholar
  10. 10.
    Gay, S.J., Hole, M.: Subtyping for session types in the pi calculus. Acta. Inf. 42(2-3), 191–225 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Honda, K.: Composing processes. In: POPL, pp. 344–357 (1996)Google Scholar
  12. 12.
    Honda, K.: Processes and games. Electr. Notes Theor. Comput. Sci. 71 (2002)Google Scholar
  13. 13.
    Honda, K., Vasconcelos, V.T., Kubo, M.: Language primitives and type discipline for structured communication-based programming. In: Hankin, C. (ed.) ESOP 1998. LNCS, vol. 1381, pp. 122–138. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  14. 14.
    Honda, K., Yoshida, N.: Game-theoretic analysis of call-by-value computation. Theor. Comput. Sci. 221(1-2), 393–456 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Laird, J.: A game semantics of the asynchronous pi-calculus. In: Abadi, M., de Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 51–65. Springer, Heidelberg (2005)Google Scholar
  16. 16.
    Milner, R.: Functions as processes. Mathematical Structures in Computer Science 2(2), 119–141 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Morris, J.H.: Lambda-Calculus Models of Programming Languages. PhD Thesis, M.I.T (1968)Google Scholar
  18. 18.
    Padovani, L.: Session types at the mirror. In: ICE, pp. 71–86 (2009)Google Scholar
  19. 19.
    Padovani, L.: Fair subtyping for multi-party session types. In: De Meuter, W., Roman, G.-C. (eds.) COORDINATION 2011. LNCS, vol. 6721, pp. 127–141. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Pierce, B.C.: Types and programming languages. MIT Press, Cambridge (2002)zbMATHGoogle Scholar
  21. 21.
    Pierce, B.C., Sangiorgi, D.: Typing and subtyping for mobile processes. Mathematical Structures in Computer Science 6(5), 409–453 (1996)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Pierce, B.C., Steffen, M.: Higher-order subtyping. Theor. Comput. Sci. 176(1-2), 235–282 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Quaglia, P., Walker, D.: Types and full abstraction for polyadic pi-calculus. Inf. Comput. 200(2), 215–246 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sangiorgi, D., Walker, D.: The π-calculus: a Theory of Mobile Processes. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  25. 25.
    Takeuchi, K., Honda, K., Kubo, M.: An interaction-based language and its typing system. In: Halatsis, C., Philokyprou, G., Maritsas, D., Theodoridis, S. (eds.) PARLE 1994. LNCS, vol. 817, pp. 398–413. Springer, Heidelberg (1994)Google Scholar
  26. 26.
    Turner, N.: The polymorphic pi-calculus: Theory and Implementation. PhD thesis, Department of Computer Science, University of Edinburgh (1996)Google Scholar
  27. 27.
    Walker, D.: Objects in the pi-calculus. Inf. Comput. 116(2), 253–271 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Yoshida, N., Berger, M., Honda, K.: Strong Normalisation in the Pi-Calculus. Information and Computation 191(2), 145–202 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Romain Demangeon
    • 1
  • Kohei Honda
    • 1
  1. 1.Queen Mary, University of LondonUK

Personalised recommendations