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Conference Papers Year : 2011

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

Abstract

The Linear-Algebraic λ-Calculus extends the λ-calculus with the possibility of making arbitrary linear combinations of terms α.t+β.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 λ-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.

Dates and versions

hal-00924890 , version 1 (07-01-2014)

Identifiers

Cite

Pablo Arrighi, Alejandro Díaz-Caro. Scalar System F for Linear-Algebraic λ-Calculus: Towards a Quantum Physical Logic. QPL 2009 - International Workshop on Quantum Physics and Logic, Apr 2009, Oxford, United Kingdom. pp.219-229, ⟨10.1016/j.entcs.2011.01.033⟩. ⟨hal-00924890⟩
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