Linear Explicit Substitutions

The-calculus adds explicit substitutions to the-calculus so as to provide a theoretical framework within which the implementation of functional programming languages can be studied. This paper gen-eralises the-calculus to provide a linear calculus of explicit substitutions , called xDILL, which analogously describes the implementation of linear functional programming languages. Our main observation is that there are non-trivial interactions between linearity and explicit substitutions and that xDILL is therefore best understood as a synthesis of its underlying logical structure and the technology of explicit substitutions. This is in contrast to the-calculus where the explicit substitutions are independent of the underlying logical structure.