Polymorphic type inference

Abstract

The benefits of strong typing to disciplined programming, to cou:pile-time error detection and to program verification are well known. S!mng typing is especially natural for functional (applicative) mcnc... This gives rise to a hi,crarcby on polymorphic types, a notion that is equally natural for the type abstraction and type quantification disciplines. WC show that our algolithm V (of $3) cm be naturally cxtcndcd to the restriction of Lhc conjunctive discipline to types of rank 2 in the hierarchy, and we outline an argument showing that already for a low rank, scen)i1181Y ~, tYPc ill fcrcncc js not cffcctivcly decidable.

Beta-reduction as unification

“…Immediate from Corollary 23, using the fact that it is undecidable whether an arbitrary λ-term is β-SN. This is a "folk" result, which is not explicitly stated in [2], but here is a simple proof for it, based on a similar proof in [15]. The partial recursive functions are λI-definable [2, Theorem 9.2.16], which implies it is undecidable whether an arbitrary λI-term has a β-nf [2, Definition 9.2.3], by the undecidability of the Halting Problem.…”

“…Nevertheless, what is interesting is that there are several very different approaches to prove what is essentially the same result. For some of these approaches, we refer the reader to [13,15,18,21,22,23] as well as to some of the references cited therein. Yet another approach is based on the results of [10] after appropriate adjustments (mostly required by the fact that ∧ is an associative binary constructor in [10], which it is not here).…”

Intersection, Universally Quantified, and Reference Types

“…It is well know that the let constructor is syntactic sugar [10] [Section 11.5] and in presence of intersection or universally quantified types does not increase the typability of the language, since with either intersection or universally quantified types we can type the translation of let in pure λ -calculus [1], [6]. The only data types of Λ imp are the numerals, that is enough for discussing the typing problems shown in the introduction.…”

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.…”

“…the principal type property 3 of the ML type system have been described in [7]. In particular the system of rank 2 intersection types [10,14,15,7] is able to type all ML programs, has the principal typing property, decidable type inference, and complexity of type inference which is of the same order as in ML. The variant of the system of rank 2 intersection types considered by Jim [7] is particularly interesting since it includes a new rule for typing recursive definitions which allows to type some, but not all, examples of polymorphic recursion [12].…”