The significance of type theory to the theory of programming languages has long been recognized. Advances in programming languages have often derived from understanding that stems from type theory. However, these applications of type theory to practical programming languages have been indirect; the differences between practical languages and type theory have prevented direct connections between the two. This dissertation presents systematic techniques directly relating practical programming languages to type theory. These techniques allow programming languages to be interpreted in the rich mathematical domain of type theory. Such interpretations lead to semantics that are at once denotational and operational, combining the advantages of each, and they also lay the foundation for formal verification of computer programs in type theory. Previous type theories either have not provided adequate expressiveness to interpret practical languages, or have provided such expressiveness at the expense of essential features of the type theory. In particular, no previous type theory has supported a notion of partial functions (needed to interpret recursion in practical languages), and a notion of total functions and objects (needed to reason about data values), and an intrinsic notion of equality (needed for most interesting results). This dissertation presents the first type theory incorporating all three, and discusses issues arising in the design of that type theory. This type theory is used as the target of a type­theoretic semantics for a expressive programming calculus. This calculus may serve as an internal language for a variety of functional programming languages. The semantics is stated as a syntax­directed embedding of the programming calculus into type theory. A critical point arising in both the type theory and the type­theoretic semantics is the issue of admissibility. Admissibility governs what types it is legal to form recursive functions over. To build a useful type theory for partial functions it is necessary to have a wide class of admissible types. In particular, it is necessary for all the types arising in the type­theoretic semantics to be admissible. In this dissertation I present a class of admissible types that is considerably wider than any previously known class.

There are compelling benefits to using foundational type theory as a framework for programming language semantics. I give a semantics of an expressive programming calculus in the foundational type theory of Nuprl. Previous typetheoretic semantics have used less expressive type theories, or have sacrificed important programming constructs such as recursion and modules. The primary mechanisms of this semantics are partial types, for typing recursion, set types, for encoding power and singleton kinds, which are used for subtyping and module programming, and very dependent function types, for encoding signatures. Keywords Semantics, program verification, type theory, functional programming 1 INTRODUCTION Type theory has become a popular framework for formal reasoning in computer science and has formed the basis for a number of automated deduction systems, including Automath, Nuprl, HOL and Coq, among others. In addition to formalizing mathematics, these systems are widely used for the a...

We give a syntactic proof of decidability and consistency of equivalence for the singleton type calculus, which lies at the foundation of modern module systems such as that of ML. Unlike existing proofs, which work by constructing a model, our syntactic proof makes few demands on the underlying proof theory and mathematical foundation. Consequently, it can be — and has been — entirely formulated in the Twelf meta-logic, and provides an important piece of a Twelf-formalized type-safety proof for Standard ML. The proof works by translation of the singleton type calculus into a canonical presentation, adapted from work on logical frameworks, in which equivalent terms are written identically. Canonical forms are not preserved under standard substitution, so we employ an alternative definition of substitution called hereditary substitution, which contracts redices that arise during substitution. 1