The Vectorial Lambda-Calculus

We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the linear-algebraic aspects of this extension of lambda-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We prove that the resulting typed lambda-calculus is strongly normalising and features a weak subject reduction. Finally, we show how to naturally encode matrices and vectors in this typed calculus.

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

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

[3]  Samson Abramsky,et al.  Handbook of logic in computer science. , 1992 .

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

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

[6]  Ugo de'Liguoro,et al.  Non deterministic extensions of untyped-calculus , 1995 .

[7]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[8]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[9]  Catuscia PalamidessiDept.  Probabilistic Asynchronous -calculus ? , 2000 .

[10]  Claude Kirchner,et al.  The Rho Cube , 2001, FoSSaCS.

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

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

[13]  André van Tonder,et al.  A Lambda Calculus for Quantum Computation , 2003, SIAM J. Comput..

[14]  Chris Hankin,et al.  Probabilistic λ-calculus and Quantitative Program Analysis , 2004 .

[15]  Jonathan Grattage.  A functional quantum programming language , 2005, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05).

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

[17]  Barbara Petit.  A Polymorphic Type System for the Lambda-Calculus with Constructors , 2009, TLCA.

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

[19]  Alejandro Ríos,et al.  The lambda-calculus with constructors: Syntax, confluence and separation , 2009, J. Funct. Program..

[20]  Christine Tasson.  Algebraic Totality, towards Completeness , 2009, TLCA.

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

[22]  Benoı̂t Valiron.  Orthogonality and Algebraic Lambda-Calculus , 2010 .

[23]  Thomas Ehrhard.  A Finiteness Structure on Resource Terms , 2010, 2010 25th Annual IEEE Symposium on Logic in Computer Science.

[24]  Alejandro D'iaz-Caro,et al.  A Type System for the Vectorial Aspect of the Linear-Algebraic Lambda-Calculus , 2010 .

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

[26]  Ugo Dal Lago,et al.  Probabilistic operational semantics for the lambda calculus , 2011, RAIRO Theor. Informatics Appl..

[27]  Alejandro Díaz-Caro,et al.  Linearity in the Non-deterministic Call-by-Value Setting , 2012, WoLLIC.

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

[29]  Gilles Dowek,et al.  The probability of non-confluent systems , 2013, DCM.

[30]  Giulio Manzonetto,et al.  Call-by-Value Non-determinism in a Linear Logic Type Discipline , 2013, LFCS.

[31]  Michele Alberti.  Normal Forms for the Algebraic Lambda-Calculus , 2013 .