A logical view of assignments

Imperative lambda calculus (ILC) is an abstract formal language obtained by extending the typed lambda calculus with imperative programming features, namely references and assignments. The language shares with typed lambda calculus important properties such as the Church-Rosser property and strong normalization. In this paper, we describe the logical symmetries that underlie ILC by exhibiting a constructive logic for which ILC forms the language of constructions. Central to this formulation is the view that references play a role similar to that of variables. References can be used to range over values and instantiated to specific values. Thus, we obtain a new form of universal quantification that uses references instead of variables. The essential term forms of ILC are then obtained as the constructions for the introduction and elimination of this quantifier. While references duplicate the role of variables, they also have important differences. References are semantic values whereas variables are syntactic entities and, secondly, references are reusable. These differences allow references to be used in a more flexible fashion leading to efficiency in constructions and algorithms.