Refinement types and computational duality

Abstract

One lesson learned painfully over the past twenty years is the perilous interaction of Curry-style typing with evaluation order and side-effects. This led eventually to the value restriction on polymorphism in ML, as well as, more recently, to similar artifacts in type systems for ML with intersection and union refinement types. For example, some of the traditional subtyping laws for unions and intersections are unsound in the presence of effects, while union-elimination requires an evaluation context restriction in addition to the value restriction on intersection-introduction.Our aim is to show that rather than being ad hoc artifacts, phenomena such as the value and evaluation context restrictions arise naturally in type systems for effectful languages, out of principles of duality. Beginning with a review of recent work on the CurryHoward interpretation of focusing proofs as pattern-matching programs, we explain how to interpret intersection and union refinements on these programs, and how to logically derive the subtyping relationship via an identity coercion interpretation. The value restriction, etc., emerge out of this analysis. However, this abstract account does not immediately yield a decidable type system, essentially because the syntax is infinitary-both "infinitely wide" and "infinitely deep". We show how to mechanically construct a finitary syntax by applying two well-known PL techniques-patterncompilation and defunctionalization-and conclude by giving this finitary syntax an algorithmic refinement type system. Parallel to the text, we describe an embedding in the dependently-typed functional language Agda, both for the sake of clarifying these ideas, and also because formalization was an important guide in developing them. As one example, the Agda encoding split very naturally into an intrinsic ("Church") view of well-typed programs, and an extrinsic ("Curry") view of refinement typing for those programs.

Complexity of Model-Checking Call-by-Value Programs

“…This is a characteristic feature that the previous work for call-by-value calculi [3,19] does not have.…”

“…Zeilberger [19] proposed a principled design of the intersection type system based on the idea from focusing proofs [1]. Its connection to ours is currently unclear, mainly because of the difference of the target calculi.…”

Focalisation and Classical Realisability

“…Zeilberger motivated the use of focalisation in order to explain the "imperfections" of realistic typed programming languages such as the value restriction for intersection types in CBV [Zei09]. We show here how classical realisability concisely accounts for such imperfections.…”

Union, intersection and refinement types and reasoning about type disjointness for secure protocol implementations

“…Zeilberger [127] tries to explain why phenomena such as the value and evaluation context restrictions can arise synthetically from a logical view of refinement typing.…”