Type Systems for Bigraphs

  • Ebbe Elsborg
  • Thomas T. Hildebrandt
  • Davide Sangiorgi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5474)


We propose a novel and uniform approach to type systems for (process) calculi, which roughly pushes the challenge of designing type systems and proving properties about them to the meta-model of bigraphs. Concretely, we propose to define type systems for the term language for bigraphs, which is based on a fixed set of elementary bigraphs and operators on these. An essential elementary bigraph is an ion, to which a control can be attached modelling its kind (its ordered number of channels and whether it is a guard), e.g. an input prefix of π-calculus. A model of a calculus is then a set of controls and a set of reaction rules, collectively a bigraphical reactive system (BRS). Possible advantages of developing bigraphical type systems include: a deeper understanding of a type system itself and its properties; transfer of the type systems to the concrete family of calculi that the BRS models; and the possibility of modularly adapting the type systems to extensions of the BRS (with new controls). As proof of concept we present a model of a π-calculus, develop an i/o-type system with subtyping on this model, prove crucial properties (including subject reduction) for this type system, and transfer these properties to the (typed) π-calculus.


Type System Main Lemma Typing Rule Type Soundness Type Judgment 
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  1. 1.
    Birkedal, L., Debois, S., Elsborg, E., Hildebrandt, T.T., Niss, H.: Bigraphical models of context-aware systems. In: Aceto, L., Ingólfsdóttir, A. (eds.) FOSSACS 2006. LNCS, vol. 3921, pp. 187–201. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  2. 2.
    Birkedal, L., Debois, S., Hildebrandt, T.T.: On the construction of sorted reactive systems. In: van Breugel, F., Chechik, M. (eds.) CONCUR 2008. LNCS, vol. 5201, pp. 218–232. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Bundgaard, M.N., Hildebrandt, T.T.: Bigraphical Semantics of Higher-Order Mobile Embedded Resources with Local Names. In: Proceedings of GT-VC 2005. ENTCS, vol. 154, pp. 7–29. Elsevier, Amsterdam (2006)Google Scholar
  4. 4.
    Bundgaard, M.N., Sassone, V.: Typed polyadic pi-calculus in bigraphs. In: Proceedings of PPDP 2006, pp. 1–12. ACM Press, New York (2006)Google Scholar
  5. 5.
    Damgaard, T.C., Birkedal, L.: Axiomatizing Binding Bigraphs. Nordic Journal of Computing 13(1-2), 58–77 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Igarashi, A., Kobayashi, N.: A generic type system for the Pi-calculus. TCS 311(1-3), 121–163 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jensen, O.H.: Mobile Processes in Bigraphs (Draft). PhD thesis, King’s College, University of Cambridge (submitted) (2007)Google Scholar
  8. 8.
    Jensen, O.H., Milner, R.: Bigraphs and Transitions. In: Proceedings of POPL 2003, pp. 38–49. ACM Press, New York (2003)Google Scholar
  9. 9.
    Jensen, O.H., Milner, R.: Bigraphs and mobile processes (revised). Technical Report UCAM-CL-TR-580, University of Cambridge (2004)Google Scholar
  10. 10.
    Kobayashi, N.: A type system for lock-free processes. Inf. & Comp. 177, 122–159 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kobayashi, N.: A new type system for deadlock-free processes. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 233–247. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    König, B.: A General Framework for Types in Graph Rewriting. Acta Inf. 42(4), 349–388 (2005); special issue: Types in concurrency, Part IIMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Leifer, J.J., Milner, R.: Transition systems, link graphs, and Petri nets. MSCS 16(6), 989–1047 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Milner, R.: Bigraphs for petri nets. In: Desel, J., Reisig, W., Rozenberg, G. (eds.) Lectures on Concurrency and Petri Nets. LNCS, vol. 3098, pp. 686–701. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Milner, R.: Axioms for bigraphical structure. MSCS 15(6), 1005–1032 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Milner, R.: Pure Bigraphs: Structure and dynamics. Inf. & Comp. 204(1) (2006)Google Scholar
  17. 17.
    Milner, R.: Local Bigraphs and Confluence: Two Conjectures. ENTCS 175(3) (June 2007)Google Scholar
  18. 18.
    Pierce, B.C., Sangiorgi, D.: Typing and Subtyping for Mobile processes. MSCS 6(5), 409–453 (1996)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Sangiorgi, D., Walker, D.: The Pi-calculus: a Theory of Mobile Processes. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  20. 20.
    Weiser, M.: Hot Topics – Ubiquitous Computing. IEEE Computer 26(10), 71–72 (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ebbe Elsborg
    • 1
  • Thomas T. Hildebrandt
    • 1
  • Davide Sangiorgi
    • 2
  1. 1.IT University of Copenhagen (ITU)Denmark
  2. 2.Università di BolognaItaly

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