Collapsing non-idempotent intersection types

We proved recently that the extensional collapse of the relational model of linear logic coincides with its Scott model, whose objects are preorders and morphisms are downwards closed relations. This result is obtained by the construction of a new model whose objects can be understood as preorders equipped with a realizability predicate. We present this model, which features a new duality, and explain how to use it for reducing normalization results in idempotent intersection types (usually proved by reducibility) to purely combinatorial methods. We illustrate this approach in the case of the call-by-value lambda-calculus, for which we introduce a new resource calculus, but it can be applied in the same way to many different calculi.

[1]  Klaus Keimel,et al.  Linear types and approximation , 2000, Math. Struct. Comput. Sci..

[2]  Jean-Yves Girard,et al.  Normal functors, power series and λ-calculus , 1988, Ann. Pure Appl. Log..

[3]  Thomas Ehrhard.  The Scott model of linear logic is the extensional collapse of its relational model , 2012, Theor. Comput. Sci..

[4]  Luca Roversi,et al.  The call-by-value [lambda]-calculus: a semantic investigation , 1999, Math. Struct. Comput. Sci..

[5]  Thomas Ehrhard,et al.  Uniformity and the Taylor expansion of ordinary lambda-terms , 2008, Theor. Comput. Sci..

[6]  Stéphane Lengrand,et al.  Filter Models: Non-idempotent Intersection Types, Orthogonality and Polymorphism , 2011, CSL.

[7]  Paolo Tranquilli,et al.  Intuitionistic differential nets and lambda-calculus , 2011, Theor. Comput. Sci..

[8]  Michele Pagani,et al.  A semantic measure of the execution time in linear logic , 2011, Theor. Comput. Sci..

[9]  Thomas Ehrhard,et al.  Differential Interaction Nets , 2005, WoLLIC.

[10]  Glynn Winskel.  A Linear Metalanguage for Concurrency , 1998 .

[11]  Martin Odersky,et al.  Call-by-name, call-by-value, call-by-need and the linear lambda calculus , 1995, MFPS.

[12]  Laurent Regnier,et al.  The differential lambda-calculus , 2003, Theor. Comput. Sci..

[13]  Eugenio Moggi,et al.  Notions of Computation and Monads , 1991, Inf. Comput..

[14]  Paul-André Melliès.  CATEGORICAL SEMANTICS OF LINEAR LOGIC , 2009 .

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

[16]  Marcelo P. Fiore,et al.  Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic , 2007, TLCA.

[17]  Michele Pagani,et al.  The Cut-Elimination Theorem for Differential Nets with Promotion , 2009, TLCA.

[18]  René David.  Every Unsolvable lambda Term has a Decoration , 1999, TLCA.

[19]  Frank Pfenning,et al.  Type Assignment for Intersections and Unions in Call-by-Value Languages , 2003, FoSSaCS.

[20]  Michael Huth,et al.  Linear Domains and Linear Maps , 1993, MFPS.

[21]  Glynn Winskel,et al.  Linear Logic in Computer Science: Linearity and Nonlinearity in Distributed Computation , 2004 .

[22]  Luca Roversi,et al.  Categorical semantics of the call-by-value lambda-calculus , 1995 .

[23]  Michele Pagani,et al.  Solvability in Resource Lambda-Calculus , 2010, FoSSaCS.