Call-by-Value Non-determinism in a Linear Logic Type Discipline

We consider the call-by-value λ-calculus extended with a may-convergent non-deterministic choice and a must-convergent parallel composition. Inspired by recent works on the relational semantics of linear logic and non-idempotent intersection types, we endow this calculus with a type system based on the so-called Girard’s second translation of intuitionistic logic into linear logic. We prove that a term is typable if and only if it is converging, and that its typing tree carries enough information to give a bound on the length of its lazy call-by-value reduction. Moreover, when the typing tree is minimal, such a bound becomes the exact length of the reduction.

[1]  M. Nivat.  Fiftieth volume of theoretical computer science , 1988 .

[2]  Stéphane Lengrand,et al.  Complexity of Strongly Normalising λ-Terms via Non-idempotent Intersection Types , 2011, FoSSaCS.

[3]  Patrick Sallé.  Une Généralisation de la Théorie des Types en \lambda-Calcul (Première partie) , 1980, RAIRO Theor. Informatics Appl..

[4]  Thomas Ehrhard,et al.  Collapsing non-idempotent intersection types , 2012, CSL.

[5]  Mariangiola Dezani-Ciancaglini,et al.  Filter models for conjunctive-disjunctive l-calculi , 1996 .

[6]  Gérard Boudol,et al.  Lambda-Calculi for (Strict) Parallel Functions , 1994, Inf. Comput..

[7]  Antonio Bucciarelli,et al.  A relational semantics for parallelism and non-determinism in a functional setting , 2012, Ann. Pure Appl. Log..

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

[9]  Giulio Manzonetto,et al.  A General Class of Models of H* , 2009, MFCS.

[10]  Roberto M. Amadio,et al.  Domains and Lambda-Calculi (Cambridge Tracts in Theoretical Computer Science) , 2008 .

[11]  Mariangiola Dezani-Ciancaglini,et al.  A new type assignment for λ-terms , 1978, Arch. Math. Log..

[12]  Mariangiola Dezani-Ciancaglini,et al.  A Filter Model for Concurrent lambda-Calculus , 1998, SIAM J. Comput..

[13]  Ian Stark,et al.  Free-Algebra Models for the pi-Calculus , 2005, FoSSaCS.

[14]  Rastislav Královič,et al.  Mathematical Foundations of Computer Science 2009, 34th International Symposium, MFCS 2009, Novy Smokovec, High Tatras, Slovakia, August 24-28, 2009. Proceedings , 2009, MFCS.

[15]  J. Krivine.  Lambda-calcul : types et modèles , 1990 .

[16]  Martin Odersky,et al.  Call-by-name, Call-by-value, Call-by-need and the Linear lambda Calculus , 1999, Theor. Comput. Sci..

[17]  Pablo Arrighi,et al.  A System F accounting for scalars , 2009, 0903.3741.

[18]  Gordon D. Plotkin,et al.  Call-by-Name, Call-by-Value and the lambda-Calculus , 1975, Theor. Comput. Sci..

[19]  Henk Barendregt,et al.  The Lambda Calculus: Its Syntax and Semantics , 1985 .

[20]  Lionel Vaux.  The algebraic lambda calculus , 2009, Math. Struct. Comput. Sci..

[21]  Olivier Laurent,et al.  Étude de la polarisation en logique , 2001 .

[22]  Gilles Dowek,et al.  Linear-algebraic lambda-calculus: higher-order, encodings, and confluence , 2008, RTA.

[23]  Michele Pagani,et al.  Linearity, Non-determinism and Solvability , 2010, Fundam. Informaticae.

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