Scalar System F for Linear-Algebraic λ-Calculus: Towards a Quantum Physical Logic

  • Authors:

  • Pablo Arrighi;Alejandro Díaz-Caro

  • Affiliations:

  • Université de Grenoble, Laboratoire d'Informatique de Grenoble, 220 rue de la Chimie, 38400 Saint Martin d'Hères, France;Université de Grenoble, Laboratoire d'Informatique de Grenoble, 220 rue de la Chimie, 38400 Saint Martin d'Hères, France

  • Venue:
  • Electronic Notes in Theoretical Computer Science (ENTCS)
  • Year:
  • 2011

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Abstract

The Linear-Algebraic @l-Calculus [Arrighi, P. and G. Dowek, Linear-algebraic @l-calculus: higher-order, encodings and confluence, Lecture Notes in Computer Science (RTA'08) 5117 (2008), pp. 17-31] extends the @l-calculus with the possibility of making arbitrary linear combinations of terms @a.t+@b.u. Since one can express fixed points over sums in this calculus, one has a notion of infinities arising, and hence indefinite forms. As a consequence, in order to guarantee the confluence, t-t does not always reduce to 0 - only if t is closed normal. In this paper we provide a System F like type system for the Linear-Algebraic @l-Calculus, which guarantees normalisation and hence no need for such restrictions, t-t always reduces to 0. Moreover this type system keeps track of 'the amount of a type'. As such it can be seen as probabilistic type system, guaranteeing that terms define correct probabilistic functions. It can also be seen as a step along the quest toward a quantum physical logic through the Curry-Howard isomorphism [Sorensen, M. H. and P. Urzyczyn, ''Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics),'' Elsevier Science Inc., New York, NY, USA, 2006].