Branching Types

Although systems with intersection types have many unique capabilities, there has never been a fully satisfactory explicitly typed system with intersection types. We introduce and prove the basic properties of λ, a typed λ-calculus with branching types and types with quantification over type selection parameters. The new system λ is an explicitly typed system with the same expressiveness as a system with intersection types. Typing derivations in λ use branching types to squash together what would be separate parallel derivations in earlier systems with intersection types. Part I Informal Presentation This part presents our new system λ and its motivation and implications in a way that we hope is understandable yet without requiring too many technical details. The technical presentation and formal statements are deferred to part II. 1 Background and Motivation 1.1 Intersection Types Intersection types were independently invented near the end of the 1970s by Coppo and Dezani [CDC80] and Pottinger [Pot80]. Intersection types provide type polymorphism by listing type instances, differing from the more widely used ∀-quantified types [Gir72, Rey74], which provide type polymorphism by giving a type scheme that can be instantiated into various type instances via different substitutions of types for quantified type variables. The original motivation was for analyzing and/or synthesizing λ-models as well as in analyzing normalization properties, but over the last twenty years the scope of research on intersection types has broadened. Van Bakel has written a good summary of the early research [vB95]. Intersection types have many unique advantages over ∀-quantified types. First, they can characterize the behavior of λ-terms more precisely, and can be used to express exactly the results of many program analyses [PP01, AT00, WDMT97, WDMT02]. In particular, intersection types can give more detailed information about all the contexts in which a function is used. Type polymorphism with intersection types is also more flexible. For example, Urzyczyn [Urz97] proved the λ-term (λx.z(x(λfu.fu))(x(λvg.gv)))(λy.yyy) This work was partly supported by EPSRC grants GR/L 36963 and GR/R 41545/01, NATO grant CRG 971607, NSF grants CCR 9113196, 9417382, 9988529, and EIA 9806745, and Sun Microsystems equipment grant EDUD-7826-990410-US. Corresponding author. E-mail: [email protected]. Web: http://www.cee.hw.ac.uk/~jbw/ .