A System F accounting for scalars

The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend the lambda-calculus with the possibility of making arbitrary linear combinations of terms. In this paper we provide a fine-grained, System F-like type system for the linear-algebraic lambda-calculus. We show that this &quo...

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Bibliographic Details
Published in:Logical methods in computer science Vol. 8, Issue 1; no. 1
Main Authors: Arrighi, Pablo, Diaz-Caro, Alejandro
Format: Journal Article
Language:English
Published: Logical Methods in Computer Science Association 27.02.2012
Logical Methods in Computer Science e.V
Subjects:
Computer Science
computer science - logic in computer science
computer science - programming languages
f.4.1
Logic in Computer Science
quantum physics
barycentric calculus
linear-algebraic -calculus
type theory
ISSN:1860-5974, 1860-5974
Online Access:Get full text
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Summary:The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend the lambda-calculus with the possibility of making arbitrary linear combinations of terms. In this paper we provide a fine-grained, System F-like type system for the linear-algebraic lambda-calculus. We show that this "scalar" type system enjoys both the subject-reduction property and the strong-normalisation property, our main technical results. The latter yields a significant simplification of the linear-algebraic lambda-calculus itself, by removing the need for some restrictions in its reduction rules. But the more important, original feature of this scalar type system is that it keeps track of 'the amount of a type' that is present in each term. As an example of its use, we shown that it can serve as a guarantee that the normal form of a term is barycentric, i.e that its scalars are summing to one.
ISSN:1860-5974
1860-5974
DOI:10.2168/LMCS-8(1:11)2012