Abstract
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.).
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References
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)
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)
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)
Blute, R., Cockett, J., Seely, R.: Differential categories. Mathematical Structures in Computer Science 16(6), 1049–1083 (2006)
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)
Ehrhard, T.: On Köthe sequence spaces and linear logic. Mathematical Structures in Computer Science 12(5), 579–623 (2002)
Ehrhard, T.: Finiteness spaces. Mathematical Structures in Computer Science 15(4), 615–646 (2005)
Ehrhard, T., Regnier, L.: Differential interaction nets. Theoretical Computer Science 364(2), 166–195 (2006)
Ehrhard, T., Reigner, L.: The differential lambda-calculus. Theoretical Computer Science 309(1-3), 1–41 (2003)
Fiore, M.: Generalised species of structures: Cartesian closed and differential structure. Draft (2004)
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)
Fiore, M., Gambino, N., Hyland, M., Winskel, G.: The cartesian closed bicategory of generalised species of structures. Preprint (2006)
Frölicher, A., Kriegl, A.: Linear spaces and differentiation theory. Wiley Series in Pure and Applied Mathematics. Wiley-Interscience Publication, Chichester (1988)
Geroch, R.: Mathematical Physics. University of Chicago Press, Chicago (1985)
Hyvernat, P.: A logical investigation of interaction systems. PhD thesis, Université de la Mediterranée, Aix-Marseille 2 (2005)
Kock, A.: Synthetic Differential Geometry. London Mathematical Society Lecture Notes Series, vol. 333. Cambridge University Press, Cambridge (2006)
Lafont, Y.: Logiques, catégories et machines. PhD thesis, Université Paris, 7 (1988)
Melliès, P.-A.: Categorical models of linear logic revisited. Theoretical Computer Science (to appear)
Schalk, A.: What is a categorical model of linear logic? Notes for research students (2004)
Seely, R.: Linear logic, *-autonomous categories and cofree coalgebras. In: Applications of Categories in Logic and Computer Science. Contemporary Mathematics, vol. 92 (1989)
Sweedler, M.: Hopf algebras. Benjamin, NY (1969)
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Fiore, M.P. (2007). Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic. In: Della Rocca, S.R. (eds) Typed Lambda Calculi and Applications. TLCA 2007. Lecture Notes in Computer Science, vol 4583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73228-0_13
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DOI: https://doi.org/10.1007/978-3-540-73228-0_13
Publisher Name: Springer, Berlin, Heidelberg
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