System E: Expansion Variables for Flexible Typing with Linear and Non-linear Types and Intersection Types

Types are often used to control and analyze computer programs. Intersection types give great flexibility, but have been difficult to implement. The ! operator, used to distinguish between linear and non-linear types, has good potential for better resource-usage tracking, but has not been as flexible as one might want and has been difficult to use in compositional analysis. We introduce System E, a type system with expansion variables, linear intersection types, and the ! type constructor for creating non-linear types. System E is designed for maximum flexibility in automatic type inference and for ease of automatic manipulation of type information. Expansion variables allow postponing the choice of which typing rules to use until later constraint solving gives enough information to allow making a good choice. System E removes many difficulties that expansion variables had in the earlier System I and extends expansion variables to work with ! in addition to the intersection type constructor. We present subject reduction for call-by-need evaluation and discuss program analysis in System E.