Principality and decidable type inference for finite-rank intersection types

Abstract

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 X-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. This is also in contrast to earlier presentations of intersection types where the status (decidable or undecidable) of these properties is unknown for the finiterank restrictions at 3 and above. Furthermore, the notion of principal typings for our system involves only one operation, substitution, rather than several operations (not all substitution-based) as in earlier presentations of principality for intersection types (without rank restrictions). In our system the earlier notion of expunsion 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, P-unification.

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

“…(Do not confuse this with the much weaker property often (mis)named "principal types" associated with the Hindley/Milner type system [12,5] used by Haskell, OCaml, Standard ML, etc.) In contrast, intersection type systems often have principal typings (see [8] for a discussion), leading to our interest in them.…”

“…Several years ago, we developed a polymorphic type system for the λ-calculus called System I [8]. System I uses intersection types together with the new technology of expansion variables and has principal typings.…”

“…A "flat" substitution notion (as in System I [8]) which does not interleave expansions and substitutions can not express the composition S 5 ; S 6 . The substitution S 5 ; S 6 has an extra assignment (e 1 := ) at the end which has no effect (other than uglifying the example), because it follows the assignment (e 1 := (a 0 := τ )).…”

“…4 presents subtyping relations refl ("reflexive"), nlin ("non-linear"), and flex ("flexible") for use with System E. The refl subtyping relation only allows using subtyping in trivial ways that do not add typing power. When using refl , System E is similar to System I [8], although it types more terms because it has ω. We have implemented type inference using refl that always succeeds for any term M that has a β-normal form and that allows the β-normal form to be reconstructed from the typing.…”

Typing Local Definitions and Conditional Expressions with Rank 2 Intersection (Extended Abstract)

“…The fact that this type system is somewhat inflexible 1 has motivated the search for more expressive, but still decidable, type systems (see, for instance, [10,14,4,2,8,7,9]). The extensions based on intersection types are particular interesting since they generally have the principal typing property 2 , whose advantages w.r.t.…”