Psi-calculi in Isabelle

  • Jesper Bengtson
  • Joachim Parrow
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5674)


Psi-calculi are extensions of the pi-calculus, accommodating arbitrary nominal datatypes to represent not only data but also communication channels, assertions and conditions, giving it an expressive power beyond the applied pi-calculus and the concurrent constraint pi-calculus.

We have formalised psi-calculi in the interactive theorem prover Isabelle using its nominal datatype package. One distinctive feature is that the framework needs to treat binding sequences, as opposed to single binders, in an efficient way. While different methods for formalising single binder calculi have been proposed over the last decades, representations for such binding sequences are not very well explored.

The main effort in the formalisation is to keep the machine checked proofs as close to their pen-and-paper counterparts as possible. We discuss two approaches to reasoning about binding sequences along with their strengths and weaknesses. We also cover custom induction rules to remove the bulk of manual alpha-conversions.


Operational Semantic Binding Sequence Induction Rule Freshness Condition Lambda Calculus 
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., Fournet, C.: Mobile values, new names, and secure communication. In: Proceedings of POPL 2001, pp. 104–115. ACM, New York (2001)Google Scholar
  2. 2.
    Aydemir, B., Charguéraud, A., Pierce, B.C., Pollack, R., Weirich, S.: Engineering formal metatheory. In: POPL 2008: Proceedings of the 35th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages, pp. 3–15. ACM, New York (2008)Google Scholar
  3. 3.
    Ballarin, C.: Locales and locale expressions in isabelle/isar. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 34–50. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Barendregt, H.P.: The Lambda Calculus – Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, vol. 103. North-Holland, Amsterdam (1984)zbMATHGoogle Scholar
  5. 5.
    Bengtson, J., Johansson, M., Parrow, J., Victor, B.: Psi-calculi: Mobile processes, nominal data, and logic. Technical report, Uppsala University (2009); (submitted),
  6. 6.
    Bengtson, J., Parrow, J.: Formalising the pi-calculus using nominal logic. In: Seidl, H. (ed.) FOSSACS 2007. LNCS, vol. 4423, pp. 63–77. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Berghofer, S., Urban, C.: Nominal Inversion Principles. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 71–85. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Buscemi, M.G., Montanari, U.: Open bisimulation for the concurrent constraint π-calculus. In: Drossopoulou, S. (ed.) ESOP 2008. LNCS, vol. 4960, pp. 254–268. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  9. 9.
    de Bruijn, N.G.: Lambda calculus notation with nameless dummies. a tool for automatic formula manipulation with application to the church-rosser theorem. Indagationes Mathematicae 34, 381–392 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hirschkoff, D.: A full formalisation of π-calculus theory in the calculus of constructions. In: Gunter, E.L., Felty, A.P. (eds.) TPHOLs 1997. LNCS, vol. 1275, pp. 153–169. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  11. 11.
    Honsell, F., Miculan, M., Scagnetto, I.: π-calculus in (co)inductive type theory. Theoretical Comput. Sci. 253(2), 239–285 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  13. 13.
    Pitts, A.M.: Nominal logic, a first order theory of names and binding. Information and Computation 186, 165–193 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Röckl, C., Hirschkoff, D.: A fully adequate shallow embedding of the π-calculus in Isabelle/HOL with mechanized syntax analysis. J. Funct. Program. 13(2), 415–451 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Urban, C.: Nominal techniques in Isabelle/HOL. Journal of Automated Reasoning 40(4), 327–356 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Urban, C., Berghofer, S., Norrish, M.: Barendregt’s variable convention in rule inductions. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603, pp. 35–50. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Jesper Bengtson
    • 1
  • Joachim Parrow
    • 1
  1. 1.Dept. of Information TechnologyUppsala UniversitySweden

Personalised recommendations