On Isomorphisms of Intersection Types

  • Mariangiola Dezani-Ciancaglini
  • Roberto Di Cosmo
  • Elio Giovannetti
  • Makoto Tatsuta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5213)


The study of type isomorphisms for different λ-calculi started over twenty years ago, and a very wide body of knowledge has been established, both in terms of results and in terms of techniques. A notable missing piece of the puzzle was the characterization of type isomorphisms in the presence of intersection types. While at first thought this may seem to be a simple exercise, it turns out that not only finding the right characterization is not simple, but that the very notion of isomorphism in intersection types is an unexpectedly original element in the previously known landscape, breaking most of the known properties of isomorphisms of the typed λ-calculus. In particular, types that are equal in the standard models of intersection types may be non-isomorphic.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    van Bakel, S.: Complete restrictions of the intersection type discipline. Theoretical Computer Science 102(1), 135–163 (1992)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Barendregt, H., Coppo, M., Dezani-Ciancaglini, M.: A filter lambda model and the completeness of type assignment. The Journal of Symbolic Logic 48(4), 931–940 (1983)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bono, V., Venneri, B., Bettini, L.: A typed lambda calculus with intersection types. Theoretical Computer Science 398(1-3), 95–113 (2008)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bruce, K., Di Cosmo, R., Longo, G.: Provable isomorphisms of types. Mathematical Structures in Computer Science 2(2), 231–247 (1992)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bruce, K., Longo, G.: Provable isomorphisms and domain equations in models of typed languages. In: Sedgewick, R. (ed.) STOC 1985, pp. 263–272. ACM, New York (1985)CrossRefGoogle Scholar
  6. 6.
    Coppo, M., Dezani-Ciancaglini, M.: An extension of the basic functionality theory for the λ-calculus. Notre Dame Journal of Formal Logic 21(4), 685–693 (1980)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dezani-Ciancaglini, M.: Characterization of normal forms possessing an inverse in the λβ η-calculus. Theoretical Computer Science 2, 323–337 (1976)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Di Cosmo, R.: Second order isomorphic types. A proof theoretic study on second order λ-calculus with surjective pairing and terminal object. Information and Computation, pp. 176–201 (1995)Google Scholar
  9. 9.
    Di Cosmo, R.: A short survey of isomorphisms of types. Mathematical Structures in Computer Science 15, 825–838 (2005)MATHCrossRefGoogle Scholar
  10. 10.
    Fiore, M., Di Cosmo, R., Balat, V.: Remarks on isomorphisms in typed lambda calculi with empty and sum types. Annals of Pure and Applied Logic 141(1–2), 35–50 (2006)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Laurent, O.: Classical isomorphisms of types. Mathematical Structures in Computer Science 15, 969–1004 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Liquori, L., Ronchi Della Rocca, S.: Intersection types à la Church. Information and Computation 205(9), 1371–1386 (2007)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Martin, C.F.: Axiomatic bases for equational theories of natural numbers. Notices of the American Mathematical Society 19(7), 778 (1972)Google Scholar
  14. 14.
    Ronchi Della Rocca, S.: Principal type scheme and unification for intersection type discipline. Theoretical Computer Science 59, 1–29 (1988)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Soloviev, S.: A complete axiom system for isomorphism of types in closed categories. In: Voronkov, A. (ed.) LPAR 1993. LNCS, vol. 698, pp. 360–371. Springer, Heidelberg (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Mariangiola Dezani-Ciancaglini
    • 1
  • Roberto Di Cosmo
    • 2
  • Elio Giovannetti
    • 1
  • Makoto Tatsuta
    • 3
  1. 1.Dipartimento di InformaticaUniversità di TorinoTorinoItaly
  2. 2.Université Paris Diderot, PPS, UMR 7126ParisFrance
  3. 3.National Institute of InformaticsTokyoJapan

Personalised recommendations