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

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

Linear-algebraic Λ-calculus

With a view towards models of quantum computation and/or the interpretation of linear logic, we define a functional language where all functions are linear operators by construction. A small step operational semantic (and hence an interpreter/simulator) is provided for this language in the form of a term rewrite system. The linear-algebraic λ-calculus hereby constructed is linear in a different...

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 higherorder 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 jus...

Focused Linear Logic and the λ-calculus

Linear logic enjoys strong symmetries inherited from classical logic while providing a constructive framework comparable to intuitionistic logic. However, the computational interpretation of sequent calculus presentations of linear logic remains problematic, mostly because of the many rule permutations allowed in the sequent calculus. We address this problem by providing a simple interpretation...

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

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...