Ralf Hinze

A new approach to generic functional programming

This paper describes a new approach to generic functional programming, which allows us to define functions generically for all datatypes expressible in Haskell. A generic function is one that is defined by induction on the structure of types. Typical examples include pretty printers, parsers, and comparison functions. The advanced type system of Haskell presents a real challenge: datatypes may be parameterized not only by types but also by type constructors, type definitions may involve mutual recursion, and recursive calls of type constructors can be arbitrarily nested. We show that—despite this complexity—a generic function is uniquely defined by giving cases for primitive types and type constructors (such as disjoint unions and cartesian products). Given this information a generic function can be specialized to arbitrary Haskell datatypes. The key idea of the approach is to model types by terms of the simply typed λ-calculus augmented by a family of recursion operators. While conceptually simple, our approach places high demands on the type system: it requires polymorphic recursion, rank-n types, and a strong form of type constructor polymorphism. Finally, we point out connections to Haskells class system and show that our approach generalizes type classes in some respects.

Derivable Type Classes

Abstract Generic programming allows you to write a function once, and use it many times at different types. A lot of good foundational work on generic programming has been done. The goal of this paper is to propose a practical way of supporting generic programming within the Haskell language, without radically changing the language or its type system. The key idea is to present generic programming as a richer language in which to write default method definitions in a class declaration. On the way, we came across a separate issue, concerning type-class overloading where higher kinds are involved. We propose a simple type-class system extension to allow the programmer to write richer contexts than is currently possible.

Polytypic values possess polykinded types

Abstract A polytypic value is one that is defined by induction on the structure of types. In Haskell types are assigned so-called kinds that distinguish between manifest types like the type of integers and functions on types like the list type constructor. Previous approaches to polytypic programming were restricted in that they only allowed to parameterize values by types of one fixed kind. In this paper, we show how to define values that are indexed by types of arbitrary kinds. It turns out that these polytypic values possess types that are indexed by kinds. We present several examples that demonstrate that the additional flexibility is useful in practice. One paradigmatic example is the mapping function, which describes the functorial action on arrows. A single polytypic definition yields mapping functions for data types of arbitrary kinds including first- and higher-order functors. Haskells type system essentially corresponds to the simply typed lambda calculus with kinds playing the role of types. We show that the specialization of a polytypic value to concrete instances of data types can be phrased as an interpretation of the simply typed lambda calculus. This allows us to employ logical relations to prove properties of polytypic values. Among other things, we show that the polytypic mapping function satisfies suitable generalizations of the functorial laws.

A lightweight implementation of generics and dynamics

The recent years have seen a number of proposals for extending statically typed languages by dynamics or generics. Most proposals --- if not all --- require significant extensions to the underlying language. In this paper we show that this need not be the case. We propose a particularly lightweight extension that supports both dynamics and generics. Furthermore, the two features are smoothly integrated: dynamic values, for instance, can be passed to generic functions. Our proposal makes do with a standard Hindley-Milner type system augmented by existential types. Building upon these ideas we have implemented a small library that is readily usable both with Hugs and with the Glasgow Haskell compiler.

Generics for the masses

A generic function is a function that can be instantiated on many data types to obtain data type specific functionality. Examples of generic functions are the functions that can be derived in Haskell, such as show, read, and ‘==’. The recent years have seen a number of proposals that support the definition of generic functions. Some of the proposals define new languages, some define extensions to existing languages. As a common characteristic none of the proposals can be made to work within Haskell 98: they all require something extra, either a more sophisticated type system or an additional language construct. The purpose of this paper is to show that one can, in fact, program generically within Haskell 98 obviating to some extent the need for fancy type systems or separate tools. Haskells type classes are at the heart of this approach: they ensure that generic functions can be defined succinctly and, in particular, that they can be used painlessly. We detail three different implementations of generics both from a practical and from a theoretical perspective.

“Scrap your boilerplate” reloaded

The paper “Scrap your boilerplate” (SYB) introduces a combinator library for generic programming that offers generic traversals and queries. Classically, support for generic programming consists of two essential ingredients: a way to write (type-)overloaded functions, and independently, a way to access the structure of data types. SYB seems to lack the second. As a consequence, it is difficult to compare with other approaches such as PolyP or Generic Haskell. In this paper we reveal the structural view that SYB builds upon. This allows us to define the combinators as generic functions in the classical sense. We explain the SYB approach in this changed setting from ground up, and use the understanding gained to relate it to other generic programming approaches. Furthermore, we show that the SYB view is applicable to a very large class of data types, including generalized algebraic data types.

Deriving backtracking monad transformers

In a paper about pretty printing J. Hughes introduced two fundamental techniques for deriving programs from their specification, where a specification consists of a signature and properties that the operations of the signature are required to satisfy. Briefly, the first technique, the term implementation, represents the operations by terms and works by defining a mapping from operations to observations --- this mapping can be seen as defining a simple interpreter. The second, the context-passing implementation, represents operations as functions from their calling context to observations. We apply both techniques to derive a backtracking monad transformer that adds backtracking to an arbitrary monad. In addition to the usual backtracking operations --- failure and nondeterministic choice --- the prolog cut and an operation for delimiting the effect of a cut are supported.

Finger trees: a simple general-purpose data structure

We introduce 2-3 finger trees, a functional representation of persistent sequences supporting access to the ends in amortized constant time, and concatenation and splitting in time logarithmic in the size of the smaller piece. Representations achieving these bounds have appeared previously, but 2-3 finger trees are much simpler, as are the operations on them. Further, by defining the split operation in a general form, we obtain a general purpose data structure that can serve as a sequence, priority queue, search tree, priority search queue and more.

Generalizing generalized tries

A trie is a search tree scheme that employs the structure of search keys to organize information. Tries were originally devised as a means to represent a collection of records indexed by strings over a fixed alphabet. Based on work by C. P. Wadsworth and others, R. H. Connelly and F. L. Morris generalized the concept to permit indexing by elements built according to an arbitrary signature. Here we go one step further, and define tries and operations on tries generically for arbitrary datatypes of first-order kind, including parameterized and nested datatypes. The derivation employs techniques recently developed in the context of polytypic programming and can be regarded as a comprehensive case study in this new programming paradigm. It is well known that for the implementation of generalized tries, nested datatypes and polymorphic recursion are needed. Implementing tries for first-order kinded datatypes places even greater demands on the type system: it requires rank-2 type signatures and second-order nested datatypes. Despite these requirements, the definition of tries is surprisingly simple, which is mostly due to the framework of polytypic programming.

Open data types and open functions

The problem of supporting the modular extensibility of both data and functions in one programming language at the same time is known as the expression problem. Functional languages traditionally make it easy to add new functions, but extending data (adding new data constructors) requires modifying existing code. We present a semantically and syntactically lightweight variant of open data types and open functions as a solution to the expression problem in the Haskell language. Constructors of open data types and equations of open functions may appear scattered throughout a program with several modules. The intended semantics is as follows: the program should behave as if the data types and functions were closed, defined in one place. The order of function equations is determined by best-fit pattern matching, where a specific pattern takes precedence over an unspecific one. We show that our solution is applicable to the expression problem, generic programming, and exceptions. We sketch two implementations: a direct implementation of the semantics, and a scheme based on mutually recursive modules that permits separate compilation