Lineal: A linear-algebraic λ-calculus

We provide a computational definition 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 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 entire calculus. 1. Motivations Knuth and Bendix have described a method to transform an equational the-ory into a rewrite system [35]. In this paper, we show that this can be achieved for the theory of vector spaces. This yields a computational definition of the no