Categorical models of linear logic revisited

In this survey, we review the existing categorical axiomatizations of linear logic, with a special emphasis on Seely and Lafont presentations. In a first part, we explain why Benton, Bierman, de Paiva and Hyland had to replace Seely categories by a more complicated axiomatization, and how a while later, Benton managed to simplify this axiomatization. In a second part, we show how Lafont axiomatization may be relaxed, in order to admit exponential interpretations different from the free one. Finally, we illustrate with a few examples what categorical models can teach us about linear logic and its models.

[1]  Paul-André Melliès.  Comparing hierarchies of types in models of linear logic , 2004, Inf. Comput..

[2]  Pierre-Louis Curien.  On the Symmetry of Sequentiality , 1993, MFPS.

[3]  Michel Parigot,et al.  Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction , 1992, LPAR.

[4]  Roberto M. Amadio,et al.  Domains and lambda-calculi , 1998, Cambridge tracts in theoretical computer science.

[5]  S. Abramsky.  Game Semantics , 1999 .

[6]  Thomas Ehrhard.  A Relative PCF-Definability Result for Strongly Stable Functions and some Corollaries , 1999, Inf. Comput..

[7]  Olivier Laurent.  Polarized Proof-Nets and Lambda µ-Calculus , 1999 .

[8]  G. M. Kelly,et al.  A universal property of the convolution monoidal structure , 1986 .

[9]  Guy McCusker.  A Fully Abstract Relational Model of Syntactic Control of Interference , 2002, CSL.

[10]  Jean-Yves Girard,et al.  A new constructive logic: classic logic , 1991, Mathematical Structures in Computer Science.

[11]  Radha Jagadeesan,et al.  New foundations for the geometry of interaction , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[12]  Edmund Robinson,et al.  Premonoidal categories and notions of computation , 1997, Mathematical Structures in Computer Science.

[13]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[14]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[15]  A. Kock.  Strong functors and monoidal monads , 1972 .

[16]  Philippa Gardner,et al.  From Action Calculi to Linear Logic , 1997, CSL.

[17]  Thomas Streicher,et al.  Continuation Models Are Universal for -calculus , 1997 .

[18]  Michael Barr,et al.  *-Autonomous categories and linear logic , 1991, Mathematical Structures in Computer Science.

[19]  Jean-Yves Girard,et al.  The System F of Variable Types, Fifteen Years Later , 1986, Theor. Comput. Sci..

[20]  Andrew G. Barber,et al.  Linear type theories, semantics and action calculi , 1997 .

[21]  Peter Selinger,et al.  Control categories and duality: on the categorical semantics of the lambda-mu calculus , 2001, Mathematical Structures in Computer Science.

[22]  Thomas Ehrhard,et al.  Quantitative Semantics Revisited , 1999, TLCA.

[23]  Glynn Winskel,et al.  A Linear Metalanguage for Concurrency , 1998, AMAST.

[24]  Valeria C V de Paiva,et al.  Term Assignment for Intuitionistic Linear Logic , 1992 .

[25]  Nick Benton,et al.  A Mixed Linear and Non-Linear Logic: Proofs, Terms and Models (Extended Abstract) , 1994, CSL.

[26]  Gavin M. Bierman.  What is a Categorical Model of Intuitionistic Linear Logic? , 1995, TLCA.

[27]  Pierre-Louis Curien,et al.  Sequential Algorithms on Concrete Data Structures , 1982, Theor. Comput. Sci..

[28]  R. A. G. Seely,et al.  Linear Logic, -Autonomous Categories and Cofree Coalgebras , 1989 .

[29]  Gérard Berry,et al.  Stable Models of Typed lambda-Calculi , 1978, ICALP.

[30]  Masahito Hasegawa,et al.  Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi , 1997, TLCA.