A Truly Concurrent Game Model of the Asynchronous \(\pi \)-Calculus

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)


In game semantics, a computation is represented by a play, which is traditionally a sequence of messages exchanged by a program and an environment. Because of the sequentiality of plays, most game models for concurrent programs are a kind of interleaving semantics. Several frameworks for truly concurrent game models have been proposed, but no model has yet been applied to give a semantics of a complex concurrent calculus such as the \(\pi \)-calculus (with replication).

This paper proposes a truly concurrent version of the HO/N game model in which a play is not a sequence but a directed acyclic graph (DAG) with two kinds edges, justification pointers and causal edges. By using this model, we give the first truly concurrent game semantics for the asynchronous \(\pi \)-calculus. In order to illustrate a possible application, we propose an intersection type system for the asynchronous \(\pi \)-calculus by means of our game model, and discuss when a process can be completely characterised by the intersection type system.


HO/N game model True concurrency Asynchronous \(\pi \)-calculus 


  1. 1.
    Abramsky, S., Jagadeesan, R., Malacaria, P.: Full abstraction for PCF. Inf. Comput. 163(2), 409–470 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Abramsky, S., McCusker, G.: Linearity, sharing and state: a fully abstract game semantics for idealized algol with active expressions. Electr. Notes Theor. Comput. Sci. 3, 2–14 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Abramsky, S., Melliès, P.-A.: Concurrent games and full completeness. In: 14th Annual IEEE Symposium on Logic in Computer Science, pp. 431–442 (1999)Google Scholar
  4. 4.
    Baillot, P., Danos, V., Ehrhard, T., Regnier, L.: Timeless games. In: 11th International Workshop on Computer Science Logic, pp. 56–77 (1997)Google Scholar
  5. 5.
    Berger, M., Honda, K., Yoshida, N.: Sequentiality and the pi-calculus. In: TLCA, pp. 29–45 (2001)Google Scholar
  6. 6.
    Boreale, M.: On the expressiveness of internal mobility in name-passing calculi. Theor. Comput. Sci. 195(2), 205–226 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Boudes, P.: Thick subtrees, games and experiments. In: Curien, P.-L. (ed.) TLCA 2009. LNCS, vol. 5608, pp. 65–79. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02273-9_7 CrossRefGoogle Scholar
  8. 8.
    Castellan, S., Clairambault, P.: Causality vs. interleavings in concurrent game semantics. In: 27th International Conference on Concurrency Theory, CONCUR 2016, pp. 32:1–32:14 (2016)Google Scholar
  9. 9.
    Castellan, S., Clairambault, P., Winskel, G.: Symmetry in concurrent games. In: Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), CSL-LICS 2014, pp. 28:1–28:10 (2014)Google Scholar
  10. 10.
    Castellan, S., Clairambault, P., Winskel, G.: The parallel intensionally fully abstract games model of PCF. In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, pp. 232–243 (2015)Google Scholar
  11. 11.
    Crafa, S., Varacca, D., Yoshida, N.: Compositional event structure semantics for the internal \(pi\)-calculus. In: CONCUR 2007 - Concurrency Theory, 18th International Conference, CONCUR 2007, pp. 317–332 (2007)Google Scholar
  12. 12.
    Crafa, S., Varacca, D., Yoshida, N.: Event structure semantics of parallel extrusion in the pi-calculus. In: Foundations of Software Science and Computational Structures - 15th International Conference, FOSSACS 2012, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2012, pp. 225–239 (2012)Google Scholar
  13. 13.
    Curien, P.-L., Faggian, C.: An approach to innocent strategies as graphs. Inf. Comput. 214, 119–155 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Di Gianantonio, P., Lenisa, M.: Innocent game semantics via intersection type assignment systems. In: Computer Science Logic 2013, CSL 2013, pp. 231–247 (2013)Google Scholar
  15. 15.
    Ehrhard, T., Laurent, O.: Interpreting a finitary pi-calculus in differential interaction nets. Inf. Comput. 208(6), 606–633 (2010)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ehrhard, T., Regnier, L.: Uniformity and the taylor expansion of ordinary lambda-terms. Theor. Comput. Sci. 403(2–3), 347–372 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Faggian, C., Maurel, F.: Ludics nets, a game model of concurrent interaction. In: 20th IEEE Symposium on Logic in Computer Science (LICS 2005), pp. 376–385 (2005)Google Scholar
  18. 18.
    Ghica, D.R., Murawski, A.S.: Angelic semantics of fine-grained concurrency. Ann. Pure Appl. Logic 151(2–3), 89–114 (2008)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Honda, K., Laurent, O.: An exact correspondence between a typed pi-calculus and polarised proof-nets. Theor. Comput. Sci. 411(22–24), 2223–2238 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hyland, J.M.E., Ong, C.-H.L.: Pi-calculus, dialogue games and PCF. In: Proceedings of the Seventh International Conference on Functional Programming Languages and Computer Architecture, FPCA 1995, pp. 96–107 (1995)Google Scholar
  21. 21.
    Hyland, J.M.E., Ong, C.-H.L.: On full abstraction for PCF: I, II, and III. Inf. Comput. 163(2), 285–408 (2000)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jategaonkar Jagadeesan, L., Jagadeesan, R.: Causality and true concurrency: a data-flow analysis of the pi-calculus. In: Alagar, V.S., Nivat, M. (eds.) AMAST 1995. LNCS, vol. 936, pp. 277–291. Springer, Heidelberg (1995). doi:10.1007/3-540-60043-4_59 CrossRefGoogle Scholar
  23. 23.
    Laird, J.: A game semantics of idealized CSP. Electr. Notes Theor. Comput. Sci. 45, 232–257 (2001)CrossRefMATHGoogle Scholar
  24. 24.
    Laird, J.: A game semantics of the asynchronous \(\pi \)-calculus. In: 16th International Conference on CONCUR 2005 - Concurrency Theory, pp. 51–65 (2005)Google Scholar
  25. 25.
    Levy, P.B.: Morphisms between plays. In: Lecture Slides, GaLoP (2013)Google Scholar
  26. 26.
    Melliès, P.-A.: Asynchronous games 1: a group-theoretic formulation of uniformity (2003). (Unpublished manuscript)Google Scholar
  27. 27.
    Melliès, P.-A.: Asynchronous games 2: the true concurrency of innocence. Theor. Comput. Sci. 358(2–3), 200–228 (2006)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Melliès, P.-A.: Game semantics in string diagrams. In: Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science, LICS 2012, pp. 481–490 (2012)Google Scholar
  29. 29.
    Melliès, P.-A., Mimram, S.: Asynchronous games: innocence without alternation. In: Caires, L., Vasconcelos, V.T. (eds.) CONCUR 2007. LNCS, vol. 4703, pp. 395–411. Springer, Heidelberg (2007). doi:10.1007/978-3-540-74407-8_27 CrossRefGoogle Scholar
  30. 30.
    Melliès, P.-A., Mimram, S.: From asynchronous games to concurrent games (2008). (Unpublished manuscript)Google Scholar
  31. 31.
    Murawski, A.S., Tzevelekos, N.: Nominal game semantics. Found. Trends Program. Lang. 2(4), 191–269 (2016)CrossRefMATHGoogle Scholar
  32. 32.
    Nickau, H.: Hereditarily sequential functionals. In: Nerode, A., Matiyasevich, Y.V. (eds.) LFCS 1994. LNCS, vol. 813, pp. 253–264. Springer, Heidelberg (1994). doi:10.1007/3-540-58140-5_25 CrossRefGoogle Scholar
  33. 33.
    Power, J., Thielecke, H.: Closed Freyd-and kappa-categories. In: Automata, Languages and Programming. In: 26th International Colloquium, ICALP 1999, pp. 625–634 (1999)Google Scholar
  34. 34.
    Rideau, S., Winskel, G.: Concurrent strategies. In: Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science, LICS 2011, pp. 409–418 (2011)Google Scholar
  35. 35.
    Sangiorgi, D.: pi-calculus, internal mobility, and agent-passing calculi. Theor. Comput. Sci. 167(1&2), 235–274 (1996)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Tsukada, T., Ong, C.-H.L.: Nondeterminism in game semantics via sheaves. In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, pp. 220–231 (2015)Google Scholar
  37. 37.
    Tsukada, T., Ong, C.-H.L.: Plays as resource terms via non-idempotent intersection types. In: Grohe, M., Koskinen, E., Shankar, N. (eds.) Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2016, New York, USA, July 5–8, 2016, pp. 237–246. ACM (2016)Google Scholar
  38. 38.
    Varacca, D., Yoshida, N.: Typed event structures and the linear pi-calculus. Theor. Comput. Sci. 411(19), 1949–1973 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.The University of TokyoTokyoJapan

Personalised recommendations