Abstract Syntax and Variable Binding for Linear Binders

  • Miki Tanaka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1893)


We apply the theory of binding algebra to syntax with linear binders. We construct a category of models for a linear binding signature. The initial model serves as abstract syntax for the signature. Moreover it contains structure for modelling simultaneous substitution. We use the presheaf category on the free strict symmetric monoidal category on 1 to construct models of each binding signature. This presheaf category has two monoidal structures, one of which models pairing of terms and the other simultaneous substitution.


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  1. 1.
    A. Barber, P. Gardner, M. Hasegawa, and G. Plotkin. From action calculi and linear logic. In Computer Science Logic’ 97 Selected Papers, volume 1414 of Lecture Notes in Computer Science, pages 78–97, 1998.Google Scholar
  2. 2.
    F. Bergeron, G. Labelle, and P. Leroux. Combinatorial species and tree-like structures, volume 67 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1998.Google Scholar
  3. 3.
    A. Barber, G. Plotkin. Dual intuitionistic linear logic. Submitted.Google Scholar
  4. 4.
    N. de Bruijn. Lambda calculus notations with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Indagationes Mathematicae, 34:381–391, 1972.Google Scholar
  5. 5.
    B.J. Day. On closed categories of functors. In Midwest Category Seminar Reports IV, volume 137 of Lecture Notes in Mathematics, pages 1–38, 1970.Google Scholar
  6. 6.
    M. Fiore, G. Plotkin, and D. Turi. Abstract syntax and variable binding. In Proceedings of 14th Symposium on Logic in Computer Science, pages 193–202, IEEE Computer Society Press, 1999.Google Scholar
  7. 7.
    M. Fiore, G. Plotkin. An axiomatisation of computationally adequate domain theoretic model of FPC. In Proceedings of 9th Symposium on Logic in Computer Science, pages 92–102, IEEE Computer Society Press, 1994.Google Scholar
  8. 8.
    M. Gabbay, A. Pitts. A new approach to abstract syntax involving binders. In Proceedings of 14th Symposium on Logic in Computer Science, pages 214–224, IEEE Computer Society Press, 1999.Google Scholar
  9. 9.
    R. Gordon, J.A. Power. Enrichment through variation. Journal of Pure and Applied Algebra, 120:167–185, 1997.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Hasegawa. Logical predicates for intuitionistic linear type theories. In Typed Lambda Calculi and its Applications, volume 1581 of Lecture Notes in Computer Science, pages 198–212, 1999.CrossRefGoogle Scholar
  11. 11.
    G.B. Im, G.M. Kelly. A universal property of the convolution monoidal structure. J. of Pure and Applied Algebra, 43:75–88, 1986.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    A. Joyal. Une théorie combinatoire des séries formelles. Advances in Mathematics, 42:1–82, 1981.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    A. Joyal. Foncteurs analytiques et espéces de structures. In Combinatoire enumerative, volume 1234 of Lecture Notes in Mathematics, pages 126–159, Springer-Verlag, 1987.Google Scholar
  14. 14.
    J. McCarthy. Towards a mathematical science of computation. In IFIP Congress 1962, North-Holland, 1963.Google Scholar
  15. 15.
    E. Moggi. Computational lambda-calculus and monads. In Proceedings of 4th Symposium on Logic in Computer Science, pages 14–23, IEEE Computer Society Press, 1989.Google Scholar
  16. 16.
    P.W. O’Hearn, J.C. Reynolds. From Algol to polymorphic linear lambda-calculus. Journal of the ACM, January 2000, Vol. 47 No. 1.Google Scholar
  17. 17.
    P. W. O’Hearn, R. D. Tennent. ed. Algol-like languages, In Progress in Theoretical Computer Science, Birkhauser, 1997.Google Scholar
  18. 18.
    P. W. O’Hearn. A model for Syntactic Control of Interference. Mathematical Structures in Computer Science, 3:435–465, 1993.MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    C. Strachey. The varieties of programming language. In Proceedings of the International Computing Symposium, pages 222–233. Cini Foundation, Venice, 1972.Google Scholar
  20. 20.
    M. Tanaka. Abstract syntax and variable binding for linear binders (extended version). Draft, 2000.Google Scholar
  21. 21.
    D. Turi, G. Plotkin. Towards a mathematical operational semantics. In Proceedings of 12th Symposium on Logic in Computer Science, pages 280–291, IEEE Computer Society Press, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Miki Tanaka
    • 1
  1. 1.Graduate School of InformaticsKyoto UniversityJapan

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