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Abstract. There are a number of extended forms of algebraic data types such as type classes with existential types and generalized algebraic data types. Such extensions are highly useful but their interaction has not been studied formally so far. Here, we present a unifying framework for these extensions. We show that the combination of type classes and generalized algebraic data types allows us to express a number of interesting properties which are desired by programmers. We support type checking based on a novel constraint solver. Our results show that our system is practical and greatly extends the expressive power of languages such as Haskell and ML. 1

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.

Tim Sheard sheard@cse.ogi.edu Emir Pasalic + pasalic@cse.ogi.edu ABSTRACT We report our experience with exploring a new point in the design space for formal reasoning systems: the development of the programming language##ngu .##209 is intended as both a practical programming language and a logic. The main goal of##102 is to allow programmers to describe and reason about semantic properties of programs from within the programming language itself, mainly by using a powerful type system.

Abstract The LR parser generators that are bundled with many functional programming language implementations produce code that is untyped, needlessly inefficient, or both. We show that, using generalized algebraic data types, it is possible to produce parsers that are well-typed (so they cannot unexpectedly crash or fail) and nevertheless efficient. This is a pleasing result as well as an illustration of the new expressiveness offered by generalized algebraic data types.

Abstract. The more expressive a type system, the more type information has to be provided in a program. Having to provide a type is sometimes a pain, but lacking expressivity is often even worse. There is a continuous struggle between expressivity and (type-)verbosity. However, even very expressive type systems allow type inference for parts of a program. Generic Haskell is an extension of Haskell that supports defining generic functions. Generic Haskell assumes that the type of a generic function is explicitly specified. This is often no problem, but sometimes it is rather painful to have to specify a type – in particular for generic functions with many dependencies – and sometimes the specified type can be generalized. In this paper, we identify three type inference problems specific to generic functions, and present (partial) solutions to each of them. 1

Abstract. We observe that the combination of multi-parameter type classes with existential types and type annotations leads to a loss of principal types. As a consequence type inference in implementations such as GHC is incomplete. This may be a surprising fact for users of these standard features. We conduct a concise investigation of the problem and are able to precisely explain why we lose principality and completeness of inference. As remedy we propose several novel inference methods to retain completeness and principality. 1

Type inference for Hindley/Milner and variants is well understood as a constraint solving problem. Recent extensions to Hindley/Milner such as generalized algebraic data types (GADTs) force us to go beyond this approach to inference. In this paper we show how to perform type inference for GADTs using Herbrand constraint abduction, a solving method to infer missing facts in terms of Herbrand constraints, i.e. conjunctions of type equations. But typeinference for GADTs is very hard, we are the first to give example programs with an infinite number of maximal types. We propose to rule out several kinds of “non-intuitive ” solutions and show that we can construct in this way a complete and decidable type inference approach for GADTs and sketch how to support type error diagnosis. Our results point out new direction how to perform type inference for expressive type systems.

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