Edifices and Full Abstraction for the Symmetric Interaction Combinators

  • Damiano Mazza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4583)


The symmetric interaction combinators are a variant of Lafont’s interaction combinators. They are a graph-rewriting model of parallel deterministic computation. We define a notion analogous to that of head normal form in the λ-calculus, and make a semantical study of the corresponding observational equivalence. We associate with each net a compact metric space, called edifice, and prove that two nets are observationally equivalent iff they have the same edifice. Edifices may therefore be compared to Böhm trees in infinite η-normal form, or to Nakajima trees, and give a precise topological account of phenomena like infinite η-expansion.


Linear Logic Denotational Semantic Active Pair Binary Word Interaction Combinators 
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  1. Lafont, Y.: Interaction nets. In: Conference Record of POPL’90, pp. 95–108. ACM Press, New York (1990)Google Scholar
  2. Girard, J.Y.: Proof-nets: The parallel syntax for proof-theory. In: Ursini, A. (ed.) Logic and Algebra, Marcel Dekker (1996)Google Scholar
  3. Lafont, Y.: From proof nets to interaction nets. In: Girard, J.Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic, pp. 225–247. Cambridge University Press, Cambridge (1995)Google Scholar
  4. Gonthier, G., Abadi, M., Lévy, J.J.: The geometry of optimal lambda reduction. In: Conference Record of POPL 92, pp. 15–26. ACM Press, New York (1992)Google Scholar
  5. Mackie, I.: Efficient lambda evaluation with interaction nets. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 155–169. Springer, Heidelberg (2004)Google Scholar
  6. Mackie, I.: An interaction net implementation of additive and multiplicative structures. Journal of Logic and Computation 15(2), 219–237 (2005)zbMATHCrossRefGoogle Scholar
  7. Lafont, Y.: Interaction combinators. Information and Computation 137(1), 69–101 (1997)zbMATHCrossRefGoogle Scholar
  8. Fernández, M., Mackie, I.: Operational equivalence for interaction nets. Theoretical Computer Science 297(1-3), 157–181 (2003)zbMATHCrossRefGoogle Scholar
  9. Mazza, D.: A denotational semantics for the symmetric interaction combinators. Mathematical Structures in Computer Science, 17(3) (to appear, 2007)Google Scholar
  10. Wadsworth, C.: The relation between computational and denotational properties for Scott’s D  ∞  models. Siam J. Comput. 5(3), 488–521 (1976)zbMATHCrossRefGoogle Scholar
  11. Hyland, M.: A syntactic characterization of the equality in some models of the lambda calculus. J. London Math. Society 2(12), 361–370 (1976)CrossRefGoogle Scholar
  12. Nakajima, R.: Infinite normal forms for the λ-calculus. In: Böhm, C. (ed.) Lambda-Calculus and Computer Science Theory. LNCS, pp. 62–82. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  13. Danos, V., Regnier, L.: Proof nets and the Hilbert space. In: Girard, J.Y., Lafont, Y., Regnier, L. (eds.) Advances in Linear Logic, pp. 307–328. Cambridge University Press, Cambridge (1995)Google Scholar
  14. Mackie, I., Pinto, J.S.: Encoding linear logic with interaction combinators. Information and Computation 176(2), 153–186 (2002)zbMATHCrossRefGoogle Scholar
  15. Mazza, D.: Observational equivalence for the interaction combinators and internal separation. In: Mackie, I. (ed.) Proceedings of TERMGRAPH 2006. ENTCS, pp. 7–16. Elsevier, Amsterdam (2006)Google Scholar
  16. Kennaway, R., Klop, J.W., Sleep, R., de Vries, F.J.: Infinitary lambda calculus. Theoretical Computer Science 175(1), 93–125 (1997)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Damiano Mazza
    • 1
  1. 1.Laboratoire d’Informatique de Paris Nord 

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