Linearity in the Non-deterministic Call-by-Value Setting

We consider the non-deterministic extension of the call-by-value lambda calculus, which corresponds to the additive fragment of the linear-algebraic lambda-calculus. We define a fine-grained type system, capturing the right linearity present in such formalisms. After proving the subject reduction and the strong normalisation properties, we propose a translation of this calculus into the System F with pairs, which corresponds to a non linear fragment of linear logic. The translation provides a deeper understanding of the linearity in our setting.

[1]  Vincent Danos,et al.  Probabilistic game semantics , 2002, TOCL.

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

[3]  King,et al.  Demonstration of a fundamental quantum logic gate. , 1995, Physical review letters.

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

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

[6]  Gilles Dowek,et al.  Non determinism through type isomorphism , 2012, LSFA.

[7]  Thomas Ehrhard,et al.  Finiteness spaces , 2005, Mathematical Structures in Computer Science.

[8]  Chris Hankin,et al.  Probabilistic /lambda-calculus and Quantitative Program Analysis , 2005, J. Log. Comput..

[9]  Roberto Di Cosmo,et al.  Isomorphisms of Types , 1995, Progress in Theoretical Computer Science.

[10]  Richard Statman,et al.  Lambda Calculus with Types , 2013, Perspectives in logic.

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

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

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

[14]  Olivier Bournez,et al.  Rewriting Logic and Probabilities , 2003, RTA.

[15]  Catuscia Palamidessi,et al.  Probabilistic Asynchronous pi-Calculus , 2000, FoSSaCS.

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

[17]  Pablo Arrighi,et al.  Scalar System F for Linear-Algebraic λ-Calculus: Towards a Quantum Physical Logic , 2011, Electron. Notes Theor. Comput. Sci..

[18]  Richard Blute,et al.  Hopf algebras and linear logic , 1996, Mathematical Structures in Computer Science.

[19]  Simon Perdrix,et al.  Completeness of algebraic CPS simulations , 2011, DCM.

[20]  W. Wootters,et al.  A single quantum cannot be cloned , 1982, Nature.

[21]  Simon Perdrix,et al.  Equivalence of algebraic λ-calculi , 2010 .

[22]  Hélène Kirchner,et al.  Completion of a Set of Rules Modulo a Set of Equations , 1986, SIAM J. Comput..

[23]  J. Girard,et al.  Proofs and types , 1989 .

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

[25]  Christopher League.  Review of Isomorphisms of Types:: from λ-calculus to information retrieval and language design by Roberto Di Cosmo (Birkhauser, 1995) , 1997, SIGA.

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

[27]  Lionel Vaux.  On Linear Combinations of lambda -Terms , 2007, RTA.

[28]  Masako Takahashi.  Parallel Reductions in lambda-Calculus , 1995, Inf. Comput..

[29]  Michele Pagani,et al.  Parallel Reduction in Resource Lambda-Calculus , 2009, APLAS.

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

[31]  Matthew Hennessy,et al.  The Semantics of Call-By-Value and Call-By-Name in a Nondeterministic Environment , 1980, SIAM J. Comput..

[32]  Daniel J. Dougherty.  Adding Algebraic Rewriting to the Untyped Lambda Calculus , 1992, Inf. Comput..

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

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

[35]  Jean-Yves Girard.  Coherent Banach Spaces: A Continuous Denotational Semantics , 1999, Theor. Comput. Sci..

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