Higher-Order Subtyping with Type Intervals

Modern, statically typed programming languages provide various abstraction facilities at both the term- and type-level. Common abstraction mechanisms for types include parametric polymorphism -- a hallmark of functional languages -- and subtyping -- which is pervasive in object-oriented languages. Additionally, both kinds of languages may allow parametrized (or generic) datatype definitions in modules or classes. When several of these features are present in the same language, new and more expressive combinations arise, such as (1) bounded quantification, (2) bounded operator abstractions and (3) translucent type definitions. An example of such a language is Scala, which features all three of the aforementioned type-level constructs. This increases the expressivity of the language, but also the complexity of its type system. From a theoretical point of view, the various abstraction mechanisms have been studied through different extensions of Girard's higher-order polymorphic lambda-calculus F-omega. Higher-order subtyping and bounded polymorphism (1 and 2) have been formalized in F-omega-sub and its many variants; type definitions of various degrees of opacity (3) have been formalized through extensions of F-omega with singleton types. In this dissertation, I propose type intervals as a unifying concept for expressing (1--3) and other related constructs. In particular, I develop an extension of F-omega with interval kinds as a formal theory of higher-order subtyping with type intervals, and show how the familiar concepts of higher-order bounded quantification, bounded operator abstraction and singleton kinds can all be encoded in a semantics-preserving way using interval kinds. Going beyond the status quo, the theory is expressive enough to also cover less familiar constructs, such as lower-bounded operator abstractions and first-class, higher-order inequality constraints. I establish basic metatheoretic properties of the theory: I prove that subject reduction holds for well-kinded types w.r.t. full beta-reduction, that types and kinds are weakly normalizing, and that the theory is type safe w.r.t. its call-by-value operational reduction semantics. Key to this metatheoretic development is the use of hereditary substitution and the definition of an equivalent, canonical presentation of subtyping, which involves only normal types and kinds. The resulting metatheory is entirely syntactic, i.e. does not involve any model constructions, and has been fully mechanized in Agda. The extension of F-omega with interval kinds constitutes a stepping stone to the development of a higher-order version of the calculus of Dependent Object Types (DOT) -- the theoretical foundation of Scala's type system. In the last part of this dissertation, I briefly sketch a possible extension of the theory toward this goal and discuss some of the challenges involved in adapting the existing metatheory to that extension.