Partially Additive Categories and Fully Complete Models of Linear Logic

  • Esfandiar Haghverdi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2044)


We construct a new class of models for linear logic. These models are constructed on partially additive categories using the Int construction of Joyal, Street and Verity and double glueing construction of Hyland and Tan. We prove full completeness for MLL+MIX in these models.


Natural Transformation Monoidal Category Linear Logic Additive Category Symmetric Monoidal Category 
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.
    Abramsky, S.: Retracing Some Paths in Process Algebra. In CONCUR 96, SLNCS 1119 (1996) 1–17.Google Scholar
  2. 2.
    Abramsky, S. and Jagadeesan, R.: Games and full completeness for Multiplicative Linear Logic. J. of Symbolic Logic 59 (1994) 543–574.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Abramsky, S. and Mellies, P.: Concurrent Games and Full Completeness. In Proc. of 14th LICS (1999) 431–442.Google Scholar
  4. 4.
    Bainbridge, E.S., Freyd, P., Scedrov, A. and Scott, P.J.: Functorial Polymorphism. Theor. Comp. Science, 70 (1990) 35–64.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Blute, R.F.: Linear Logic, Coherence and Dinaturality. Theor. Computer Science 115 (1993) 3–41.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blute, R.F., Cockett, J.R.B., Seely, R.A.G. and Trimble, T.H.: Natural deduction and coherence for weakly distributive categories. Jour. Pure and Applied Algebra 113 (1996) 229–296.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Blute, R.F. and Scott, P.J.: The Shuffle Hopf Algebra and Noncommutative Full Completeness. Journal of Symbolic Logic 63(4) (1998) 1413–1436.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Blute, R.F. and Scott, P.J.: Linear Läuchli Semantics. Annals of Pure and Applied Logic 77 (1996) 101–142.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cockett, J.R.B. and Seely, R.A.G.: Weakly Distributive Categories. Journal of Pure and Applied Algebra 114 (1997) 133–173.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Costantini, F., Mascari, G. and Pedicini M.: Dynamics of Algorithms. Technical Report n.6 (1996).Google Scholar
  11. 11.
    Danos, V. and Regnier, L.: The Structure of Multiplicatives. Arch. Math. Logic 28 (1989) 181–203.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Devarajan, P., Hughes, D., Plotkin, G. and Pratt, V.: Full completeness of the multiplicative linear logic of Chu spaces. In Proc. of the 14th LICS, (1999) 234–243.Google Scholar
  13. 13.
    Fleury, A. and Rétoré, C.: The MIX Rule, Math. Struct. in Comp. Science 4 (1994) 273–285.zbMATHGoogle Scholar
  14. 14.
    Girard, J.Y., Scedrov, A. and Scott, P.J.: Normal forms and cut-free proofs as natural transformations. In Logic from Computer Science, (1991) 217–241.Google Scholar
  15. 15.
    Haghverdi, E.: A Categorical Approach to Linear Logic, Geometry of Interaction and Full Completeness, PhD Thesis, University of Ottawa, 2000. Available from http://
  16. 16.
    Haghverdi, E.: Unique decomposition categories, Geometry of Interaction and combinatory logic, Math. Struct. in Comp. Science, vol 10 (2000) 205–231.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Hamano, M.: Pontrajagin Duality and Full Completeness for MLL.Math. Struct. in Comp. Science, vol 10 (2000) 231–259.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hasegawa, M.: Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculus, TLCA’97, SLNCS 1210 (1997) 196–213.Google Scholar
  19. 19.
    Hasegawa, M.: Categorical glueing and logical predicates for models of linear logic, Manuscript (1998).Google Scholar
  20. 20.
    Hyland, M and Ong, L.: Full completeness for multiplicative linear logic without the mix-rule, electronically distributed manuscript, (1993).Google Scholar
  21. 21.
    Joyal, A., Street, R. and Verity, D.: Traced Monoidal Categories. Math. Proc. Camb. Phil. Soc. 119 (1996) 447–468.zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Kelly, G.M. and Mac Lane S. (1971), Coherence in closed categories. Jour. Pure and Applied Algebra, 1(1) (1971) 97–140.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lambek, J.: Deductive systems and categories II. Springer Lecture Notes in Mathematics 87 (1969).Google Scholar
  24. 24.
    Lambek, J. and Scott, P.J.: Introduction to higher order categorical logic. Cambridge University Press, (1986).Google Scholar
  25. 25.
    Loader, R.: Linear Logic, totality and full completeness. In Proc. LICS (1994).Google Scholar
  26. 26.
    Loader, R.: Models of Lambda Calculi and Linear Logic: Structural, equational and proof-theoretic characterisations. PhD Thesis, Oxford, (1994).Google Scholar
  27. 27.
    Manes, E.G. and Arbib, M.A.: Algebraic Approaches to Program Semantics. Springer-Verlag (1986).Google Scholar
  28. 28.
    Mascari, G. and Pedicini M.: Types and Dynamics in Partially Additive Categories. Idempotency, Ed. J. Gunawardena, (1995).Google Scholar
  29. 29.
    Panangaden, P.: Stochastic Techniques in Concurrency. Lecture Notes, BRICS, (1997).Google Scholar
  30. 30.
    Seely, R.A.G.: Linear logic, *-autonomous categories and cofree coalgebras. Categories in Computer Science and Logic, Contemp. Math. 92. AMS (1989).Google Scholar
  31. 31.
    Tan, A.M.: Full Completeness for Models of Linear Logic. PhD thesis, Cambridge, (1997).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Esfandiar Haghverdi
    • 1
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations