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40
Nested datatypes
 In MPC’98, volume 1422 of LNCS
, 1998
"... Abstract. A nested datatype, also known as a nonregular datatype, is a parametrised datatype whose declaration involves different instances of the accompanying type parameters. Nested datatypes have been mostly ignored in functional programming until recently, but they are turning out to be both th ..."
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Cited by 79 (5 self)
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Abstract. A nested datatype, also known as a nonregular datatype, is a parametrised datatype whose declaration involves different instances of the accompanying type parameters. Nested datatypes have been mostly ignored in functional programming until recently, but they are turning out to be both theoretically important and useful in practice. The aim of this paper is to suggest a functorial semantics for such datatypes, with an associated calculational theory that mirrors and extends the standard theory for regular datatypes. Though elegant and generic, the proposed approach appears more limited than one would like, and some of the limitations are discussed. 1
Inductive and Coinductive types with Iteration and Recursion
 Proceedings of the 1992 Workshop on Types for Proofs and Programs, Bastad
, 1992
"... We study (extensions of) simply and polymorphically typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in ter ..."
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Cited by 51 (0 self)
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We study (extensions of) simply and polymorphically typed lambda calculus from a point of view of how iterative and recursive functions on inductive types are represented. The inductive types can usually be understood as initial algebras in a certain category and then recursion can be defined in terms of iteration. However, in the syntax we often have only weak initiality, which makes the definition of recursion in terms of iteration inefficient or just impossible. We propose a categorical notion of (primitive) recursion which can easily be added as computation rule to a typed lambda calculus and gives us a clear view on what the dual of recursion, corecursion, on coinductive types is. (The same notion has, independently, been proposed by [Mendler 1991].) We look at how these syntactic notions work out in the simply typed lambda calculus and the polymorphic lambda calculus. It will turn out that in the syntax, recursion can be defined in terms of corecursion and vice versa using polymo...
Presheaf Models for Concurrency
, 1999
"... In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their wo ..."
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Cited by 45 (19 self)
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In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a builtin notion of bisimulation. We show how
Wellfounded Trees in Categories
, 1999
"... this paper, we give an abstract 2 categorical characterization of Wtypes. We calculate these Wtypes explicitly in some categories of presheaves and sheaves on a site, and in the gluing category or Freyd cover. (We also have an explicit description in the case of Hyland's realizability topos, whic ..."
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Cited by 37 (6 self)
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this paper, we give an abstract 2 categorical characterization of Wtypes. We calculate these Wtypes explicitly in some categories of presheaves and sheaves on a site, and in the gluing category or Freyd cover. (We also have an explicit description in the case of Hyland's realizability topos, which will be presented in [17].) These explicit calculations can be formalized in a weak predicative metatheory, and lead to the result that if E is any suitably filtered pretopos with dependent products and Wtypes, then so is the category of internal sheaves on a site in E (Remark 5.9). Our paper is organized as follows. In Section 2 we review some standard definitions concerning pretoposes and dependent products. In Section 3 we present the categorical definition of the Wconstruction, and in Section 4 we prove some of its basic functoriality properties; e.g., that it turns coequalizers into equalizers. In Section 5, a construction is presented which to each map between (pre)sheaves of sets associates a sheaf of wellfounded trees, and it is proved that this is in fact the Wtype in the category (pre)sheaves of sets (Theorem 5.6). In Section 6, we discuss the Wconstruction for the Freyd cover. Finally, in Section 7 it is shown how these categorical constructions are not only analogous to but explicitly related to MartinLof type theory. 2 Pretoposes and dependent products
From Dynamic Programming to Greedy Algorithms
 Formal Program Development, volume 755 of Lecture Notes in Computer Science
, 1992
"... A calculus of relations is used to reason about specifications and algorithms for optimisation problems. It is shown how certain greedy algorithms can be seen as refinements of dynamic programming. Throughout, the maximum lateness problem is used as a motivating example. 1 Introduction An optimisat ..."
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Cited by 14 (3 self)
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A calculus of relations is used to reason about specifications and algorithms for optimisation problems. It is shown how certain greedy algorithms can be seen as refinements of dynamic programming. Throughout, the maximum lateness problem is used as a motivating example. 1 Introduction An optimisation problem can be solved by dynamic programming if an optimal solution is composed of optimal solutions to subproblems. This property, which is known as the principle of optimality, can be formalised as a monotonicity condition. If the principle of optimality is satisfied, one can compute a solution by decomposing the input in all possible ways, recursively solving the subproblems, and then combining optimal solutions to subproblems into an optimal solution for the whole problem. By contrast, a greedy algorithm considers only one decomposition of the argument. This decomposition is usually unbalanced, and greedy in the sense that at each step the algorithm reduces the input as much as poss...
Primitive (co)recursion and courseofvalues (co)iteration, categorically
 Informatica
, 1999
"... Abstract. In the mainstream categorical approach to typed (total) functional programming, datatypes are modelled as initial algebras and codatatypes as terminal coalgebras. The basic function definition schemes of iteration and coiteration are modelled by constructions known as catamorphisms and ana ..."
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Cited by 13 (6 self)
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Abstract. In the mainstream categorical approach to typed (total) functional programming, datatypes are modelled as initial algebras and codatatypes as terminal coalgebras. The basic function definition schemes of iteration and coiteration are modelled by constructions known as catamorphisms and anamorphisms. Primitive recursion has been captured by a construction called paramorphisms. We draw attention to the dual construction of apomorphisms, and show on examples that primitive corecursion is a useful function definition scheme. We also put forward and study two novel constructions, viz., histomorphisms and futumorphisms, that capture the powerful schemes of courseofvalue iteration and its dual, respectively, and argue that even these are helpful.
Generic Programming With Relations and Functors
 Journal of Functional Programming
, 1999
"... This paper explores the idea of generic programming in which programs are parameterised by data types. Part of the constructive theory of lists, specically the part dealing with properties of segments, is generalised in two ways: from lists to arbitrary inductive data types, and from functions to ..."
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Cited by 9 (6 self)
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This paper explores the idea of generic programming in which programs are parameterised by data types. Part of the constructive theory of lists, specically the part dealing with properties of segments, is generalised in two ways: from lists to arbitrary inductive data types, and from functions to relations. The new theory is used to solve a generic problem about segments. 1 Introduction To what extent is it possible to construct programs without knowing exactly what data types are involved? At rst sight this may seem a strange question, but consider the case of pattern matching. Over lists, this problem can be formulated in terms of two strings, a pattern and a text; the object is to determine if and where the pattern occurs as a segment of the text. Now, pattern matching can be generalised to other data types, including arrays and trees of various kinds; the essential step is to be able to dene the notion of `segment' in these types. So the intriguing question arises: can one...
Functional programming with apomorphisms (corecursion
 Proceedings of the Estonian Academy of Sciences: Physics, Mathematics
, 1998
"... Abstract. In the mainstream categorical approach to typed (total) functional programming, functions with inductive source types defined by primitive recursion are called paramorphisms; the utility of primitive recursion as a scheme for defining functions in programming is well known. We draw attenti ..."
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Cited by 9 (1 self)
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Abstract. In the mainstream categorical approach to typed (total) functional programming, functions with inductive source types defined by primitive recursion are called paramorphisms; the utility of primitive recursion as a scheme for defining functions in programming is well known. We draw attention to the dual notion of apomorphisms — with coinductive target types defined by primitive corecursion and show on examples that primitive corecursion is useful in programming. Key words: typed (total) functional programming, categorical program calculation, (co)datatypes, (co)recursion forms. 1.