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Programming with bananas, lenses, envelopes and barbed wire
 In FPCA
, 1991
"... We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example Functions in Bird and Wadler's &qu ..."
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Cited by 334 (12 self)
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We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example Functions in Bird and Wadler's "Introduction to Functional Programming " can be expressed using these operators. 1
Polytypic Data Conversion Programs
 Science of Computer Programming
, 2001
"... Several generic programs for converting values from regular datatypes to some other format, together with their corresponding inverses, are constructed. Among the formats considered are shape plus contents, compact bit streams and pretty printed strings. The different data conversion programs are co ..."
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Cited by 25 (9 self)
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Several generic programs for converting values from regular datatypes to some other format, together with their corresponding inverses, are constructed. Among the formats considered are shape plus contents, compact bit streams and pretty printed strings. The different data conversion programs are constructed using John Hughes' arrow combinators along with a proof that printing (from a regular datatype to another format) followed by parsing (from that format back to the regular datatype) is the identity. The printers and parsers are described in PolyP, a polytypic extension of the functional language Haskell.
Proof Methods for Structured Corecursive Programs
, 1999
"... Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. Ho ..."
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Cited by 12 (3 self)
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Corecursive programs produce values of greatest fixpoint types, in contrast to recursive programs, which consume values of least fixpoint types. There are a number of widely used methods for proving properties of corecursive programs, including fixpoint induction, the take lemma, and coinduction. However, these methods are all rather lowlevel, in the sense that they do not exploit the common structure that is often present in corecursive definitions. We argue for a more structured approach to proving properties of corecursive programs. In particular, we show that by writing corecursive programs using an operator called unfold that encapsulates a common pattern of corecursive de nition, we can then use highlevel algebraic properties of this operator to conduct proofs in a purely calculational style that avoids the use of either induction or coinduction.
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...
Abstract From Parity Games to Circular Proofs
"... We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. µbicomplete categories. We argue that parity games with a given starting position ..."
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We survey on the ongoing research that relates the combinatorics of parity games to the algebra of categories with finite products, finite coproducts, initial algebras and final coalgebras of definable functors, i.e. µbicomplete categories. We argue that parity games with a given starting position play the role of terms for the theory of µbicomplete categories. We show that the interpretation of a parity game in the category of sets and functions is the set of deterministic winning strategies for one player in the game. We discuss bounded memory communication strategies between two parity games and their computational significance. We describe how an attempt to formalize them within the algebra of µbicomplete categories leads to develop a calculus of proofs that are allowed to contain cycles. This paper is a survey on our recent work lifting results on free µlattices [1,2] to a categorical setting. A µlattice is a lattice with enough least and greatest fixed points to interpret formal µterms. A generalization of this notion leads to consider categories with finite products, finite coproducts, and enough initial algebras and final coalgebras of functors. We call these categories µbicomplete. The outcome of this research is so far described in [3,4,5]. A main goal for us is to understand how the algebra of µbicomplete categories describes a computational situation through the combinatorics of games; when attempting to achieve this goal, computational logic and prooftheory become unavoidable ingredients. It is the aim of this note to give insights on how these four worlds – categories, games, computation and logic – relate in this context. As the need of a mathematical formalization has too often hidden these relationships, we shall present here only informal arguments. The reader will find formal proofs of the statements in the references cited above.