Functional logic programming and probabilistic programming have demonstrated the broad benefits of combining laziness (non-strict evaluation with sharing of the results) with non-determinism. Yet these benefits are seldom enjoyed in functional programming, because the existing features for non-strictness, sharing, and nondeterminism in functional languages are tricky to combine. We present a practical way to write purely functional lazy non-deterministic programs that are efficient and perspicuous. We achieve this goal by embedding the programs into existing languages (such as Haskell, SML, and OCaml) with high-quality implementations, by making choices lazily and representing data with non-deterministic components, by working with custom monadic data types and search strategies, and by providing equational laws for the programmer to reason about their code.

Monads as an organizing principle for programming and semantics are notoriously difficult to grasp, yet they are a central and powerful abstraction in Haskell. This paper introduces a domain-specific language, MonadLab, that simplifies the construction of monads, and describes its implementation in Template Haskell. MonadLab makes monad construction truly first class, meaning that arcane theoretical issues with respect to monad transformers are completely hidden from the programmer. The motivation behind the design of MonadLab is to make monadic programming in Haskell simpler while providing a tool for non-Haskell experts that will assist them in understanding this powerful abstraction.

In the GADT (Generalized Algebraic Data Types) type system, a pattern-matching branch can draw type information from both the scrutinee type and the data constructor type. Even though the type system can handle complex interactions between the two types, most programs require only simple interactions in the form of parametric instantiation and type indexing. To explore the tradeoffs related to GADT patterns, we define the Pointwise GADT type system, which restricts GADTs to the common case of parametric instantiation and type indexing. The Pointwise GADT type system still accepts a wide range of GADT programs, while rejecting a pathological function whose pattern-matching branches can make arbitrarily different assumptions about the type environment. We also state and prove several properties of the type system, which we speculate might be useful in helping researchers design better type inference algorithms.

Generalized algebraic data types (GADTs) are a type system extension to algebraic data types that allows the type of an algebraic data value to vary with its shape. The GADT type system allows programmers to express detailed program properties as types (for example, that a function should return a list of the same length as its input), and a general-purpose type checker will automatically check those properties at compile time. Type inference for the GADT type system and the properties of the type system are both currently areas of active research. In this dissertation, I attack both problems simultaneously by exploiting the symbiosis between type system research and type inference research. Deficiencies of GADT type inference algorithms motivate research on specific aspects of the type system, and discoveries about the type system bring in new insights that lead to improved GADT type inference algorithms. The technical contributions of this dissertation are therefore twofold: in addition to new GADT type system properties (such as the prevalence of pointwise type information flow in GADT patterns, a generalized notion of existential types, and the effects of enforcing the GADT branch reachability requirement), I will also present a new GADT type inference algorithm that is significantly more powerful than existing algorithms. These contributions should help programmers use the GADT type system more effectively, and they should also enable language implementers to provide better support for the GADT type system. iii