Generalized algebraic data types (GADTs), sometimes known as “guarded recursive data types ” or “first-class phantom types”, are a simple but powerful generalization of the data types of Haskell and ML. Recent works have given compelling examples of the utility of GADTs, although type inference is known to be difficult. Our contribution is to show how to exploit programmer-supplied type annotations to make the type inference task almost embarrassingly easy. Our main technical innovation is wobbly types, which express in a declarative way the uncertainty caused by the incremental nature of typical type-inference algorithms.

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We study HMG(X), an extension of the constraint-based type system HM(X) with deep pattern matching, polymorphic recursion, and guarded algebraic data types. Guarded algebraic data types subsume the concepts known in the literature as indexed types, guarded recursive datatype constructors, (first-class) phantom types, and equality qualified types, and are closely related to inductive types. Their characteristic property is to allow every branch of a case construct to be typechecked under different assumptions about the type variables in scope. We prove that HMG(X) is sound and that, provided recursive definitions carry a type annotation, type inference can be reduced to constraint solving. Constraint solving is decidable, at least for some instances of X, but prohibitively expensive. Effective type inference for guarded algebraic data types is left as an issue for future research.

This paper addresses the question of how to extend OCaml’s Hindley-Milner type system with types indexed by logical propositions and proofs of the Coq theorem prover, thereby providing an expressive and extensible mechanism for ensuring fine-grained program invariants. We propose adopting the approached used by Shao et al. for certified binaries. This approach maintains a phase distinction between the computational and logical languages, thereby limiting effects and non-termination to the computational language, and maintaining the decidability of the type system. The extension subsumes language features such as impredicative first-class (higher-rank) polymorphism and type operators, that are notoriously difficult to integrate with the Hindley-Milner style of type inference that is used in OCaml. We make the observation that these features can be more easily integrated with type inference if the inference algorithm is free to adapt the order in which it solves typing constraints to each program. To this end we define a novel “order-free ” type inference algorithm. The key enabling technology is a graph representation of constraints and a constraint solver that performs Hindley-Milner inference with just three graph rewrite rules. 1

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