Lineal: A linear-algebraic lambda-calculus

Pablo Arrighi (1) , Gilles Dowek (2)

1 LIG - Laboratoire d'Informatique de Grenoble
2 DEDUCTEAM - Deduction modulo, interopérabilité et démonstration automatique

We provide a computational de nition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the lambda-calculus with the possibility to make arbitrary linear combinations of terms : alpha t + beta u. We describe how to \execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the eld of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus.

Domaines: Logique en informatique [cs.LO]

Pablo Arrighi, Gilles Dowek. Lineal: A linear-algebraic lambda-calculus. Logical Methods in Computer Science, 2013. ⟨hal-00919625⟩