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52
Containers  Constructing Strictly Positive Types
, 2004
"... ... with disjoint coproducts and initial algebras of container functors (the categorical analogue of Wtypes) — and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any MartinLöf category. Central to our development are t ..."
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Cited by 83 (28 self)
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... with disjoint coproducts and initial algebras of container functors (the categorical analogue of Wtypes) — and then establish that nested strictly positive inductive and coinductive types, which we call strictly positive types, exist in any MartinLöf category. Central to our development are the notions of containers and container functors, introduced in Abbott, Altenkirch, and Ghani (2003a). These provide a new conceptual analysis of data structures and polymorphic functions by exploiting dependent type theory as a convenient way to define constructions in MartinLöf categories. We also show that morphisms between containers can be full and faithfully interpreted as polymorphic functions (i.e. natural transformations) and that, in the presence of Wtypes, all strictly positive types (including nested inductive and coinductive types) give rise to containers.
Wellfounded Trees and Dependent Polynomial Functors
 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class ..."
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Cited by 39 (6 self)
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We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.
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
Symmetric Lenses
"... Lenses—bidirectional transformations between pairs of connected structures—have been extensively studied and are beginning to find their way into industrial practice. However, some aspects of their foundations remain poorly understood. In particular, most previous work has focused on the special cas ..."
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Cited by 34 (2 self)
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Lenses—bidirectional transformations between pairs of connected structures—have been extensively studied and are beginning to find their way into industrial practice. However, some aspects of their foundations remain poorly understood. In particular, most previous work has focused on the special case of asymmetric lenses, where one of the structures is taken as primary and the other is thought of as a projection, or view. A few studies have considered symmetric variants, where each structure contains information not present in the other, but these all lack the basic operation of composition. Moreover, while many domainspecific languages based on lenses have been designed, lenses have not been thoroughly studied from a more fundamental algebraic perspective. We offer two contributions to the theory of lenses. First, we present a new symmetric formulation, based on complements, an old idea from the database literature. This formulation generalizes the familiar structure of asymmetric lenses, and it admits a good notion of composition. Second, we explore the algebraic structure of the space of symmetric lenses. We present generalizations of a number of known constructions on asymmetric lenses and settle some longstanding questions about their properties—in particular, we prove the existence of (symmetric monoidal) tensor products and sums and the nonexistence of full categorical products or sums in the category of symmetric lenses. We then show how the methods of universal algebra can be applied to build iterator lenses for structured data such as lists and trees, yielding lenses for operations like mapping, filtering, and concatenation from first principles. Finally, we investigate an even more general technique for constructing mapping combinators, based on the theory of containers. 1.
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 inbuilt 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 inbuilt 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
Constructing polymorphic programs with quotient types
 In 7th International Conference on Mathematics of Program Construction (MPC
, 2004
"... Abstract. The efficient representation and manipulation of data is one of the fundamental tasks in the construction of large software systems. Parametric polymorphism has been one of the most successful approaches to date but, as of yet, has not been applicable to programming with quotient datatypes ..."
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Cited by 15 (5 self)
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Abstract. The efficient representation and manipulation of data is one of the fundamental tasks in the construction of large software systems. Parametric polymorphism has been one of the most successful approaches to date but, as of yet, has not been applicable to programming with quotient datatypes such as unordered pairs, cyclic lists, bags etc. This paper provides the basis for writing polymorphic programs over quotient datatypes by extending our recently developed theory of containers. 1
Polynomial functors and polynomial monads
, 2009
"... Abstract. We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship wi ..."
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Cited by 15 (5 self)
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Abstract. We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.
Continuous functions on final coalgebras
, 2007
"... In a previous paper we have given a representation of continuous functions on streams, both discretevalued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We ga ..."
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Cited by 12 (1 self)
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In a previous paper we have given a representation of continuous functions on streams, both discretevalued functions, and functions between streams. the topology on streams is the ‘Baire ’ topology induced by taking as a basic neighbourhood the set of streams that share a given finite prefix. We gave also a combinator on the representations of stream processing functions that reflects composition. Streams are the simplest example of a nontrivial final coalgebras, playing in the coalgebraic realm the same role as do the natural numbers in the algebraic realm. Here we extend our previous results to cover the case of final coalgebras for a broad class of functors generalising (×A). The functors we deal with are those that arise from countable signatures of finiteplace untyped operators. These have many applications. The topology we put on the final coalgebra for such a functor is that induced by taking for basic neighbourhoods the set of infinite objects which share a common prefix, according to the usual definition of the final coalgebra as the limit of a certain inverse chain starting at �. 1
Generic programming with dependent types
 Spring School on Datatype Generic Programming
, 2006
"... In these lecture notes we give an overview of recent research on the relationship ..."
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Cited by 11 (1 self)
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In these lecture notes we give an overview of recent research on the relationship
∂ for Data: Differentiating Data Structures
"... This paper and our conference paper (Abbott, Altenkirch, Ghani, and McBride, 2003b) explain and analyse the notion of the derivative of a data structure as the type of its onehole contexts based on the central observation made by McBride (2001). To make the idea precise we need a generic notion of ..."
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Cited by 10 (1 self)
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This paper and our conference paper (Abbott, Altenkirch, Ghani, and McBride, 2003b) explain and analyse the notion of the derivative of a data structure as the type of its onehole contexts based on the central observation made by McBride (2001). To make the idea precise we need a generic notion of a data type, which leads to the notion of a container, introduced in (Abbott, Altenkirch, and Ghani, 2003a) and investigated extensively in (Abbott, 2003). Using containers we can provide a notion of linear map which is the concept missing from McBride’s first analysis. We verify the usual laws of differential calculus including the chain rule and establish laws for initial algebras and terminal coalgebras.