Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic

(Extended Abstract)
  • Marcelo P. Fiore
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4583)


In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigating differential structure in the context of linear logic. Consequently, within this setting, I introduce a notion of creation operator (as considered by physicists for bosonic Fock space in the context of quantum field theory), provide an equivalent description of creation operators in terms of creation maps, and show that they induce a differential operator satisfying all the basic laws of differentiation (the product and chain rules, the commutation relations, etc.).


Categorical Model Natural Transformation Annihilation Operator Creation Operator 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. Benton, N.: A mixed linear and non-linear logic: Proofs, terms and models. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, Springer, Heidelberg (1995)CrossRefGoogle Scholar
  2. Benton, N., Bierman, G., de Paiva, V., Hyland, M.: Linear λ-calculus and categorical models revisited. In: Martini, S., Börger, E., Kleine Büning, H., Jäger, G., Richter, M.M. (eds.) CSL 1992. LNCS, vol. 702, Springer, Heidelberg (1993)Google Scholar
  3. Bierman, G.: What is a categorical model of intuitionistic linear logic? In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 78–93. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  4. Blute, R., Cockett, J., Seely, R.: Differential categories. Mathematical Structures in Computer Science 16(6), 1049–1083 (2006)zbMATHCrossRefGoogle Scholar
  5. Blute, R., Panangaden, P., Seely, R.: Fock space: A model of linear exponential types (Corrected version of Holomorphic models of exponential types in linear logic). In: Main, M.G., Melton, A.C., Mislove, M.W., Schmidt, D., Brookes, S.D. (eds.) Mathematical Foundations of Programming Semantics. LNCS, vol. 802, Springer, Heidelberg (1993)Google Scholar
  6. Ehrhard, T.: On Köthe sequence spaces and linear logic. Mathematical Structures in Computer Science 12(5), 579–623 (2002)zbMATHCrossRefGoogle Scholar
  7. Ehrhard, T.: Finiteness spaces. Mathematical Structures in Computer Science 15(4), 615–646 (2005)zbMATHCrossRefGoogle Scholar
  8. Ehrhard, T., Regnier, L.: Differential interaction nets. Theoretical Computer Science 364(2), 166–195 (2006)zbMATHCrossRefGoogle Scholar
  9. Ehrhard, T., Reigner, L.: The differential lambda-calculus. Theoretical Computer Science 309(1-3), 1–41 (2003)zbMATHCrossRefGoogle Scholar
  10. Fiore, M.: Generalised species of structures: Cartesian closed and differential structure. Draft (2004)Google Scholar
  11. Fiore, M.: Mathematical models of computational and combinatorial structures (Invited address for). In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 25–46. Springer, Heidelberg (2005)Google Scholar
  12. Fiore, M., Gambino, N., Hyland, M., Winskel, G.: The cartesian closed bicategory of generalised species of structures. Preprint (2006)Google Scholar
  13. Frölicher, A., Kriegl, A.: Linear spaces and differentiation theory. Wiley Series in Pure and Applied Mathematics. Wiley-Interscience Publication, Chichester (1988)zbMATHGoogle Scholar
  14. Geroch, R.: Mathematical Physics. University of Chicago Press, Chicago (1985)zbMATHGoogle Scholar
  15. Hyvernat, P.: A logical investigation of interaction systems. PhD thesis, Université de la Mediterranée, Aix-Marseille 2 (2005)Google Scholar
  16. Kock, A.: Synthetic Differential Geometry. London Mathematical Society Lecture Notes Series, vol. 333. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  17. Lafont, Y.: Logiques, catégories et machines. PhD thesis, Université Paris, 7 (1988)Google Scholar
  18. Melliès, P.-A.: Categorical models of linear logic revisited. Theoretical Computer Science (to appear)Google Scholar
  19. Schalk, A.: What is a categorical model of linear logic? Notes for research students (2004)Google Scholar
  20. Seely, R.: Linear logic, *-autonomous categories and cofree coalgebras. In: Applications of Categories in Logic and Computer Science. Contemporary Mathematics, vol. 92 (1989)Google Scholar
  21. Sweedler, M.: Hopf algebras. Benjamin, NY (1969)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Marcelo P. Fiore
    • 1
  1. 1.Computer Laboratory, University of Cambridge 

Personalised recommendations