Principality and type inference for intersection types using expansion variables

Principality of typings is the property that for each typable term, there is a typing from which all other typings are obtained via some set of operations. Type inference is the problem of finding a typing for a given term, if possible. We define an intersection type system which has principal typings and types exactly the strongly normalizable λ-terms. More interestingly, every finite-rank restriction of this system (using Leivant’s first notion of rank) has principal typings and also has decidable type inference. This is in contrast to System F where the finite rank restriction for every finite rank at 3 and above has neither principal typings nor decidable type inference. Furthermore, the notion of principal typings for our system involves only one operation, substitution, rather than several operations (not all substitutionbased) as in earlier presentations of principality for intersection types (without rank restrictions). In our system the earlier notion of expansion is integrated in the form of expansion variables, which are subject to substitution as are ordinary variables. A unification-based type inference algorithm is presented using a new form of unification, β-unification. This is an expanded version of a report that appeared in the Proceedings of the 1999 ACM Symposium on Principles of Programming Languages, under the title “Principality and Decidable Type Inference for Finite-Rank Intersection Types” [KW99]. This work has been partly supported by EPSRC grants GR/L 36963 and GR/R 41545/01, by NATO grant CRG 971607, by NSF grants 9417382 (CCR), 9806745 (EIA), 9988529 (CCR), 0113193 (ITR), and by Sun Microsystems equipment grant EDUD-7826-990410-US.