Jason Hu

Over the past few decades, type theories as mathematical foundations have been extensively studied and are well understood. Many proof assistants implement type theories and have found important applications to provide critical security guarantees. In these applications, users often write meta-programs, programs that generate other programs, to implement proof search heuristics and improve their work efficiency. However, as opposed to the deep understanding of type theories, it remains unclear what foundation is suitable to support meta-programming in proof assistants. In this thesis, I investigate modal type theories, a specific approach to this problem. In modal type theories, modalities are a way to shallowly embed syntax into the systems, so users can write meta-programs that manipulate syntax through these modalities.
I explore two different styles of modal systems. In the first part, I investigate the Kripke-style systems, which faithfully model the familiar quasi-quoting style of meta-programming. I develop an explicit substitution calculus and scale it to dependent types, introducing Mint. I prove strong normalization of Mint, which implies its logical consistency, using an untyped domain model.
Nevertheless, the Kripke-style systems only support composition and execution of code, and they cannot easily support a general recursion principle on the structure of code. To support such a general recursion principle, I develop the layered style, where a system is divided into nested layers of sub-languages. The layered style scales quite naturally to dependent types, introducing DeLaM. DeLaM allows users to compose, execute and recurse on dependently typed code. I prove that DeLaM is weakly normalizing and its convertibility problem between types and terms is decidable. Hence, DeLaM provides a type-theoretic foundation to support type-safe meta-programming in proof assistants.

Dependent Object Types, or DOT, is a family of calculi developed to study the Scala programming language. These calculi have path dependent types as a feature, and potentially intersection types, union types and recursive types. So far, the study of DOT calculi mostly focuses on the soundness proof, which does not directly contribute to development of compilers. This thesis presents a detailed investigation of decidability and algorithmic properties of the family of DOT calculi.
In decidability analysis, the undecidability of subtyping of several calculi is formally established, including the D<: and D∧ calculi. Prior to this investigation, the undecidability of subtyping of all DOT calculi including D<: was open. Decidability analysis puts emphasis on a particular form of subtyping rules, called normal form. It turns out that a normal form definition is not only as expressive, but also more suggestive than the original definition. A conceptual device, called small-step analysis, is introduced to assist converting a usual definition of subtyping to its normal form definition. Moreover, decidability analysis gives direct contributions to the algorithmic analysis, by revealing two decidable fragments of D<: in declarative form, called the kernels. Decidability analysis also suggests a novel subtyping algorithm framework, stare-at subtyping. Stare-at subtyping and an existing algorithm are shown to be sound and complete w.r.t. their corresponding kernels.
In algorithmic analysis, stare-at subtyping is extended to other calculi, with more features than D<:, including D∧, μDART and jDOT. In μDART and jDOT, bi-directional type assignment algorithms are developed. The algorithms developed in this thesis are all shown to be sound with respect to their target calculi and terminating.
During the development of the algorithms, analysis shows a number of ways in which the Wadlerfest DOT calculus does not directly correspond to the Scala language, while substantially increases the difficulties of algorithmic design. jDOT, therefore, is developed as an alternative formalization of Scala.