Skip to main content

An explicit formula for the free exponential modality of linear logic


The exponential modality of linear logic associates to every formula A a commutative comonoid !A which can be duplicated in the course of reasoning. Here, we explain how to compute the free commutative comonoid !A as a sequential limit of equalizers in any symmetric monoidal category where this sequential limit exists and commutes with the tensor product. We apply this general recipe to a series of models of linear logic, typically based on coherence spaces, Conway games and finiteness spaces. This algebraic description unifies for the first time a number of apparently different constructions of the exponential modality in spaces and games. It also sheds light on the duplication policy of linear logic, and its interaction with classical duality and double negation completion.

Hide All
Ehrhard T. (2002). On Köthe sequence spaces and linear logic. Mathematical Structures in Computer Science 12 (05) 579623.
Ehrhard T. (2005). Finiteness spaces. Mathematical Structures in Computer Science 15 (4) 615646.
Girard J.-Y. (1987). Linear logic. Theoretical Computer Science 50 (1) 1102.
Girard J.-Y. (2006). Le point aveugle: Cours de logique: Tome 1, vers la perfection. Vision des Sciences, Hermann.
Hyland M. and Schalk A. (2002). Games on graphs and sequentially realizable functionals. In: Logic in Computer Science 02, IEEE Computer Society Press, Kopenhavn, 257264.
Joyal A. (1977). Remarques sur la théorie des jeux à deux personnes. Gazette des Sciences Mathématiques du Québec 1 (4) 4652.
Kelly M. (1982). Basic Concepts of Enriched Category Theory, Lecture Notes in Mathematics, vol. 64, Cambridge University Press.
Kelly M. and Laplaza M. (1980). Coherence for compact closed categories. Journal of Pure and Applied Algebra 19 193213.
Lafont Y. (1988). Logique, catégories et machines, Thèse de doctorat, Université de Paris 7, Denis Diderot.
Lawvere W. (1963). Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories. PhD thesis, Columbia University.
Melliès P. and Tabareau N. (2008). Free models of T-algebraic theories computed as Kan extensions. Available at:
Melliès P.-A. (2005). Asynchronous games 3: An innocent model of linear logic. Electronic Notes in Theoretical Computer Science 122 171192.
Melliès P.-A. (2009). Categorical Semantics of Linear Logic, Panoramas et Synthèses vol. 27, Société Mathématique de France.
Melliès P.-A., Tabareau N. and Tasson C. (2009). An explicit formula for the free exponential modality of linear logic. In: ICALP '09: Proceedings of the 36th Internatilonal Collogquium on Automata, Languages and Programming, Lecture Notes in Computer Science, Part II. vol. 5556, Springer-Verlag, 247260.
Seely R. (1989). Linear logic,*-autonomous categories and cofree coalgebras. In: Gray J. and Scedrov A. (eds.) Applications of Categories in Logic and Computer Science, Contemporary Mathematics, vol. 92. Americab Mathematis Society.
Tasson C. (2009). Sémantiques et syntaxes vectorielles de la logique linéaire. PhD thesis, Université Paris Diderot.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 15 *
Loading metrics...

Abstract views

Total abstract views: 106 *
Loading metrics...

* Views captured on Cambridge Core between 21st April 2017 - 18th December 2017. This data will be updated every 24 hours.