Inferring Intersection Typings that are Equivalent to Call-by-Name and Call-by-Value Evaluations

We present a procedure to infer a typing for an arbitrary λ-term M in an intersection-type system that translates into exactly the call-by-name (resp., call-by-value) evaluation of M. Our framework is the recently developed System E which augments intersection types with expansion variables. The inferred typing for M is obtained by setting up a unification problem involving both type variables and expansion variables, which we solve with a confluent rewrite system. The inference procedure is compositional in the sense that typings for different program components can be inferred in any order, and without knowledge of the definition of other program components. Using expansion variables lets us achieve a compositional inference procedure easily. Termination of the procedure is generally undecidable. The procedure terminates and returns a typing if the input M is normalizing according to call-by-name (resp., call-by-value). The inferred typing is exact in the sense that the exact call-by-name (resp., call-by-value) behaviour of M can be obtained by a (polynomial) transformation of the typing. The inferred typing is also principal in the sense that any other typing that translates the call-by-name (resp., call-by-value) evaluation of M can be obtained from the inferred typing for M using a substitution-based transformation.