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The differential lambda-calculus
, 2003
"... We present an extension of the lambda-calculus with differential constructions. We state and prove some basic results (confluence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus. ..."
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Cited by 70 (11 self)
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We present an extension of the lambda-calculus with differential constructions. We state and prove some basic results (confluence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus.
Generic programming within dependently typed programming
- In Generic Programming, 2003. Proceedings of the IFIP TC2 Working Conference on Generic Programming, Schloss Dagstuhl
, 2003
"... Abstract We show how higher kinded generic programming can be represented faithfully within a dependently typed programming system. This development has been implemented using the Oleg system. The present work can be seen as evidence for our thesis that extensions of type systems can be done by prog ..."
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Cited by 66 (8 self)
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Abstract We show how higher kinded generic programming can be represented faithfully within a dependently typed programming system. This development has been implemented using the Oleg system. The present work can be seen as evidence for our thesis that extensions of type systems can be done by programming within a dependently typed language, using data as codes for types. 1.
Type-Indexed Data Types
- SCIENCE OF COMPUTER PROGRAMMING
, 2004
"... 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, ..."
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Cited by 62 (23 self)
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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.
Faking It: Simulating Dependent Types in Haskell
, 2001
"... Dependent types reflect the fact that validity of data is often a relative notion by allowing prior data to affect the types of subsequent data. Not only does this make for a precise type system, but also a highly generic one: both the type and the program for each instance of a family of operations ..."
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Cited by 37 (6 self)
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Dependent types reflect the fact that validity of data is often a relative notion by allowing prior data to affect the types of subsequent data. Not only does this make for a precise type system, but also a highly generic one: both the type and the program for each instance of a family of operations can be computed from the data which codes for that instance. Recent experimental extensions to the Haskell type class mechanism give us strong tools to relativize types to other types. We may simulate some aspects of dependent typing by making counterfeit type-level copies of data, with type constructors simulating data constructors and type classes simulating datatypes. This paper gives examples of the technique and discusses its potential. 1
Indexed Containers
"... Abstract. The search for an expressive calculus of datatypes in which canonical algorithms can be easily written and proven correct has proved to be an enduring challenge to the theoretical computer science community. Approaches such as polynomial types, strictly positive types and inductive types h ..."
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Cited by 36 (5 self)
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Abstract. The search for an expressive calculus of datatypes in which canonical algorithms can be easily written and proven correct has proved to be an enduring challenge to the theoretical computer science community. Approaches such as polynomial types, strictly positive types and inductive types have all met with some success but they tend not to cover important examples such as types with variable binding, types with constraints, nested types, dependent types etc. In order to compute with such types, we generalise from the traditional treatment of types as free standing entities to families of types which have some form of indexing. The hallmark of such indexed types is that one must usually compute not with an individual type in the family, but rather with the whole family simultaneously. We implement this simple idea by generalising our previous work on containers to what we call indexed containers and show that they cover a number of sophisticated datatypes and, indeed, other computationally interesting structures such as the refinement calculus and interaction structures. Finally, and rather surprisingly, the extra structure inherent in indexed containers simplifies the theory of containers and thereby allows for a much richer and more expressive calculus. 1
Generic Haskell: applications
- In Generic Programming, Advanced Lectures, volume 2793 of LNCS
, 2003
"... Generic Haskell is an extension of Haskell that supports the construction of generic programs. These lecture notes discuss three advanced generic programming applications: generic dictionaries, compressing XML documents, and the zipper: a data structure used to represent a tree together with a s ..."
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Cited by 32 (17 self)
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Generic Haskell is an extension of Haskell that supports the construction of generic programs. These lecture notes discuss three advanced generic programming applications: generic dictionaries, compressing XML documents, and the zipper: a data structure used to represent a tree together with a subtree that is the focus of attention, where that focus may move left, right, up or down the tree. When describing and implementing these examples, we will encounter some advanced features of Generic Haskell, such as type-indexed data types, dependencies between and generic abstractions of generic functions, adjusting a generic function using a default case, and generic functions with a special case for a particular constructor.
Comparing Libraries for Generic Programming in Haskell
, 2008
"... Datatype-generic programming is defining functions that depend on the structure, or “shape”, of datatypes. It has been around for more than 10 years, and a lot of progress has been made, in particular in the lazy functional programming language Haskell. There are more than 10 proposals for generic p ..."
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Cited by 25 (10 self)
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Datatype-generic programming is defining functions that depend on the structure, or “shape”, of datatypes. It has been around for more than 10 years, and a lot of progress has been made, in particular in the lazy functional programming language Haskell. There are more than 10 proposals for generic programming libraries or language extensions for Haskell. To compare and characterize the many generic programming libraries in a typed functional language, we introduce a set of criteria and develop a generic programming benchmark: a set of characteristic examples testing various facets of datatype-generic programming. We have implemented the benchmark for nine existing Haskell generic programming libraries and present the evaluation of the libraries. The comparison is useful for reaching a common standard for generic programming, but also for a programmer who has to choose a particular approach for datatype-generic programming.
Exploring the regular tree types
- In Types for Proofs and Programs
, 2004
"... Abstract. In this paper we use the Epigram language to define the universe of regular tree types—closed under empty, unit, sum, product and least fixpoint. We then present a generic decision procedure for Epigram’s in-built equality at each type, taking a complementary approach to that of Benke, Dyb ..."
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Cited by 23 (4 self)
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Abstract. In this paper we use the Epigram language to define the universe of regular tree types—closed under empty, unit, sum, product and least fixpoint. We then present a generic decision procedure for Epigram’s in-built equality at each type, taking a complementary approach to that of Benke, Dybjer and Jansson [7]. We also give a generic definition of map, taking our inspiration from Jansson and Jeuring [21]. Finally, we equip the regular universe with the partial derivative which can be interpreted functionally as Huet’s notion of ‘zipper’, as suggested by McBride in [27] and implemented (without the fixpoint case) in Generic Haskell by Hinze, Jeuring and Löh [18]. We aim to show through these examples that generic programming can be ordinary programming in a dependently typed language. 1
