The Vectorial Lambda Calculus Revisited

Abstract: We revisit the Vectorial Lambda Calculus, a typed version of Lineal. Vectorial (as well as Lineal) has been originally designed for quantum computing, as an extension to System F where linear combinations of lambda terms are also terms and linear combinations of types are also types. In its first presentation, Vectorial only provides a weakened version of the Subject Reduction property. We prove that our revised Vectorial Lambda Calculus supports the standard version of said property, answering a long standing issue. In addition we also introduce the concept of weight of types and terms, and prove a relation between the weight of terms and of its types.