Linear-algebraic λ-calculus: Higher-order, encodings, and confluence

Abstract

We introduce a minimal language combining higher-order computation and linear algebra. This language extends the λ-calculus with the possibility to make arbitrary linear combinations of terms α.t + β.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 field of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally we prove the confluence of the calculus, this is our main result. © 2008 Springer-Verlag.