A Linear Logical View of Linear Type Isomorphisms

  • Vincent Balat
  • Roberto Di Cosmo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1683)


The notion of isomorphisms of types has many theoretical as well as practical consequences, and isomorphisms of types have been investigated at length over the past years. Isomorphisms in weak system (like linear lambda calculus) have recently been investigated due to their practical interest in library search. In this paper we give a remarkably simple and elegant characterization of linear isomorphisms in the setting of Multiplicative Linear Logic (MLL), by making an essential use ofthe correctness criterion for Proof Nets due to Girard.


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  1. [1]
    A. Andreev and S. Soloviev. A deciding algorithm for linear isomorphism of types with complexity o(nlog 2(n)). In E. Moggi and G. Rossolini, editors, Category Theory and Computer Science, number 1290 in LNCS, pages 197–210CrossRefGoogle Scholar
  2. [2]
    M.-V. Aponte and R. Di Cosmo. Type isomorphisms for module signatures. In Programming Languages Implementation and Logic Programming (PLILP), volume 1140 of Lecture Notes in Computer Science, pages 334–346. Springer-Verlag, 1996.Google Scholar
  3. [3]
    G. Berry and P.-L. Curien. Theory and practice ofsequen tial algorithms: the kernel ofthe applicative language CDS. In M. Nivat and J. Reynolds, editors, Algebraic methods in semantics, pages 35–87. Cambridge University Press, 1985.Google Scholar
  4. [4]
    K. Bruce, R. Di Cosmo, and G. Longo. Provable isomorphisms oft ypes. Mathematical Structures in Computer Science, 2(2):231–247, 1992.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    K. Bruce and G. Longo. Provable isomorphisms and domain equations in models oft yped languages. ACM Symposium on Theory of Computing (STOC 85), May 1985.Google Scholar
  6. [6]
    D. Delahaye, R. Di Cosmo, and B. Werner. Recherche dans une bibliothèque de preuves Coq en utilisant le type et modulo isomorphismes. In PRC/GDR de programmation, Pôle Preuves et Spécifications Algébriques, November 1997.Google Scholar
  7. [7]
    R. Di Cosmo. Invertibility ofterms and valid isomorphisms. a prooftheoretic study on second order λ-calculus with surjective pairing and terminal object. Technical Report 91-10, LIENS-Ecole Normale Supérieure, 1991.Google Scholar
  8. [8]
    R. Di Cosmo. Type isomorphisms in a type assignment framework. In 19th Ann. ACM Symp. on Principles of Programming Languages (POPL), pages 200–210. ACM, 1992.Google Scholar
  9. [9]
    R. Di Cosmo. Deciding type isomorphisms in a type assignment framework. Journal of Functional Programming, 3(3):485–525, 1993. Special Issue on ML.MathSciNetGoogle Scholar
  10. [10]
    R. Di Cosmo. Isomorphisms of types: from λ-calculus to information retrieval and language design. Birkhauser, 1995. ISBN-0-8176-3763-X.Google Scholar
  11. [11]
    R. Di Cosmo and G. Longo. Constuctively equivalent propositions and isomorphisms of objects (or terms as natural transformations). In Moschovakis, editor, Logic from Computer Science, volume 21 of Mathematical Sciences Research Institute Publications, pages 73–94. Springer Verlag, Berkeley, 1991.Google Scholar
  12. [12]
    K. Dosen and Z. Petric. Isomorphic objects in symmetric monoidal closed categories. Technical Report 95-49-R, IRIT, 1995.Google Scholar
  13. [13]
    J.-Y. Girard. Linear logic. Theoretical Computer Science, 50, 1987.Google Scholar
  14. [14]
    S. Mac Lane and G. M. Kelly. Coherence in Closed Categories. Journal of Pure and Applied Algebra, 1(1):97–140, 1971.CrossRefMathSciNetGoogle Scholar
  15. [15]
    C. F. Martin. Axiomatic bases for equational theories of natural numbers. Notices of the Am. Math. Soc., 19(7):778, 1972.Google Scholar
  16. [16]
    L. Meertens and A. Siebes. Universal type isomorphisms in cartesian closed categories. Centrum voor Wiskunde en Informatica, Amsterdam, the Netherlands. E-mail: lambert,arno@cwi.nl, 1990.Google Scholar
  17. [17]
    M. Rittri. Retrieving library identifiers by equational matching oft ypes in 10th Int. Conf. on Automated Deduction. Lecture Notes in Computer Science, 449, July 1990.Google Scholar
  18. [18]
    M. Rittri. Searching program libraries by type and proving compiler correctness by bisimulation. PhD thesis, University ofGöteborg, Göteborg, Sweden, 1990.Google Scholar
  19. [19]
    M. Rittri. Using types as search keys in function libraries. Journal of Functional Programming, 1(1):71–89, 1991.MATHMathSciNetGoogle Scholar
  20. [20] C. Runciman and I. Toyn. Retrieving re-usable software components by polymorphic type. Journal of Functional Programming, 1(2):191–211, 1991.MATHMathSciNetGoogle Scholar
  21. [21]
    S. V. Soloviev. The category offinite sets and cartesian closed categories. Journal of Soviet Mathematics, 22(3):1387–1400, 1983.CrossRefGoogle Scholar
  22. [22]
    S. V. Soloviev. A complete axiom system for isomorphism of types in closed categories. In A. Voronkov, editor, Logic Programming and Automated Reasoning, 4th International Conference, volume 698 of Lecture Notes in Artificial Intelligence (subseries of LNCS), pages 360–371, St. Petersburg, Russia, 1993. Springer-Verlag.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Vincent Balat
    • 1
  • Roberto Di Cosmo
    • 1
  1. 1.LIENS École normale supérieureParisFrance

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