introduc e a notion of guarded rec ursive (g.r.) datatype c#w struc tors, generalizing the notion ofrec# rsive datatypes in func tional programming languages suc h as ML and Haskell. We address both theoret ic#t and prac# ic## issues resulted from this generalization. On one hand, we design a type system to formalize the notion of g.r. datatypec onstruc - tors and then prove the soundness of the type system. On the other hand, we present some signific ant applic ations (e.g., implementing objec ts, implementing stagedc omputation, etc# ) of g.r. datatype c# nstruc#S rs, arguing that g.r. datatypec onstruc torsc an have far-reac hingc onsequenc es in programming. The mainc ontribution of the paper lies in the rec#I0 ition and then the formalization of a programming notion that is of both theoretic# l interest and prac tic# l use.

Generic Haskell is an extension of Haskell that supports the construction of generic programs. During the development of several applications, such as an XML editor and compressor, we encountered a number of limitations with the existing (Classic) Generic Haskell language, as implemented by the current Generic Haskell compiler. Specifically,

In this paper, we explore an extension to Haskell type classes that allows a type class declaration to define data types as well as values (or methods). Similarly, an instance declaration gives a witness for such data types, as well as a witness for each method. It turns out that this extension directly supports the idea of a type-indexed type, and is useful in many applications, especially for self-optimising libraries that adapt their data representations and algorithms in a type-directed manner.

A polytypic function is a function that can be instantiated on many data types to obtain data type specific functionality. Examples of polytypic functions are the functions that can be derived in Haskell, such as show , read , and ` '. More advanced examples are functions for digital searching, pattern matching, unification, rewriting, and structure editing. For each of these problems, we not only have to define polytypic functionality, but also a type-indexed data type: a data type that is constructed in a generic way from an argument data type. For example, in the case of digital searching we have to define a search tree type by induction on the structure of the type of search keys. This paper shows how to define type-indexed data types, discusses several examples of type-indexed data types, and shows how to specialize type-indexed data types. The approach has been implemented in Generic Haskell, a generic programming extension of the functional language Haskell.

Type abstraction is a key feature of ML-like languages for writing large programs. Marshalling is necessary for writing distributed programs, exchanging values via network byte-streams or persistent stores. In this paper we combine the two, developing compile-time and run-time semantics for marshalling, that guarantee abstraction-safety between separately-built programs. We obtain a namespace for abstract types that is global, i.e. meaningful between programs, by hashing module declarations. We examine the scenarios in which values of abstract types are communicated from one program to another, and ensure, by constructing hashes appropriately, that the dynamic and static notions of type equality mirror each other. We use singleton kinds to express abstraction in the static semantics; abstraction is tracked in the dynamic semantics by coloured brackets. These allow us to prove preservation, erasure, and coincidence results. We argue that our proposal is a good basis for extensions to existing ML-like languages, pragmatically straightforward for language users and for implementors.

Two different ways of defining ad-hoc polymorphic operations commonly occur in programming languages. With the first form polymorphic operations are defined inductively on the structure of types while with the second form polymorphic operations are defined for specific sets of types. In intensional type analysis operations are defined by induction on the structure of types. Therefore no new cases are necessary for user-defined types, because these types are equivalent to their underlying structure. However, intensional type analysis is “closed ” to extension, as the behavior of the operations cannot be differentiated for the new types, thus destroying the distinctions that these types are designed to express. Haskell type classes on the other hand define polymorphic operations for sets of types. Operations defined by class instances are considered “open”—the programmer can add instances for new types without modifying existing code. However, the operations must be extended with specialized code for each new type, and it may be tedious or even impossible to add extensions that apply to a large universe of new types. Both approaches have their benefits, so it is important to let programmers decide which is most appropriate for their needs. In this paper, we define a language that supports both forms of ad-hoc polymorphism, using the same basic constructs.

Functional logic overloading is a novel approach to userdefined overloading that extends Haskell’s concept of type classes in significant ways. Whereas type classes are conceptually predicates on types in standard Haskell, they are type functions in our approach. Thus, we can base type inference on the evaluation of functional logic programs. Functional logic programming provides a solid theoretical foundation for type functions and, at the same time, allows for programmable overloading resolution strategies by choosing different evaluation strategies for functional logic programs. Type inference with type functions is an instance of type inference with constrained types, where the underlying constraint system is defined by a functional logic program. We have designed a variant of Haskell which supports our approach to overloading, and implemented a prototype frontend for the language.

Run-time type analysis is an increasingly important linguistic mechanism in modern programming languages. Language runtime systems use it to implement services such as accurate garbage collection, serialization, cloning and structural equality. Component frameworks rely on it to provide reflection mechanisms so they may discover and interact with program interfaces dynamically. Run-time type analysis is also crucial for large, distributed systems that must be dynamically extended, because it allows those systems to check program invariants when new code and new forms of data are added. Finally, many generic user-level algorithms for iteration, pattern matching, and unification can be defined through type analysis mechanisms. However, existing frameworks for run-time type analysis were designed for simple type systems. They do not scale well to the sophisticated type systems of modern and next-generation programming languages that include complex constructs such as first-class abstract types, recursive types, objects, and type parameterization. In addition, facilities to support type analysis often require complicated

Abstract. Reynolds ’ abstraction theorem [21], often referred to as the parametricity theorem, can be used to derive properties about functional programs solely from their types. Unfortunately, in the presence of runtime type analysis, the abstraction properties of polymorphic programs are no longer valid. However, runtime type analysis can be implemented with term-level representations of types, as in the λR language of Crary et al. [10], where case analysis on these runtime representations introduces type refinement. In this paper, we show that representation-based analysis is consistent with type abstraction by extending the abstraction theorem to such a language. We also discuss the “free theorems” that result. This work provides a foundation for the more general problem of extending the abstraction theorem to languages with generalized algebraic datatypes (gadts). 1

Type-directed programming is an important and widely used paradigm in the design of software. With this form of programming, a program may analyze type information to determine its behavior. By analyzing the structure of data, many operations, such as serialization, cloning, structural equality, and iterators, may be defined once, for all types of data. The benefit of type-directed programming is that as software evolves, operations need not be updated---they will automatically adapt to new data forms. Otherwise, each of these operations must be individually redefined for each type of data, forcing programmers to revisit the same program logic many times during a program's lifetime.