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Representations of First Order Function Types as Terminal Coalgebras
 In Typed Lambda Calculi and Applications, TLCA 2001, number 2044 in Lecture Notes in Computer Science
, 2001
"... terminal coalgebras ..."
Generic Accumulations
, 2002
"... which are eventually used in later stages of the computation. We present a generic definition of accumulations, achieved by the introduction of a new recursive operator on inductive types. We also show that the notion of downwards accumulation developed by Gibbons is subsumed by our notion of acc ..."
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which are eventually used in later stages of the computation. We present a generic definition of accumulations, achieved by the introduction of a new recursive operator on inductive types. We also show that the notion of downwards accumulation developed by Gibbons is subsumed by our notion of accumulation.
Functional Transformation Systems
, 2000
"... . We demonstrate how functional programming can be used as a platform for developing integrated transformation systems that are typed, scalable, easy to use, and adaptable. The kernel of such a functional transformation system for a certain language is a transformation framework generated from t ..."
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. We demonstrate how functional programming can be used as a platform for developing integrated transformation systems that are typed, scalable, easy to use, and adaptable. The kernel of such a functional transformation system for a certain language is a transformation framework generated from the language's grammar. The framework provides a fold traversal scheme and adaptable basic traversals. From the grammar a parser and prettyprinter is also generated. There is also support for storage and interchange of intermediate results. A software renovation case study illustrates our approach and its practical value. 1 Introduction Transformation systems, i.e., systems which implement program transformations, are essential in many application areas, including software renovation [5, 2], reverse engineering, (domainspecific) language implementation, and program optimization. Transformations and the accompanying analyses are usually implemented as traversals over syntax trees apply...
An Algebraic View of Structural Induction
 Computer Science Logic 1994, number 933 in Lect. Notes Comp. Sci
, 1994
"... We propose a uniform, categorytheoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial algebras for certain kind of endofunctors T : B !B on a bicartesian/distributive category B . Regarding ..."
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Cited by 9 (5 self)
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We propose a uniform, categorytheoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial algebras for certain kind of endofunctors T : B !B on a bicartesian/distributive category B . Regarding a predicate logic as a fibration p : P!B over B , we consider a logical predicate lifting of T to the total category P. Then, a predicate is inductive precisely when it carries an algebra structure for such lifted endofunctor. The validity of the induction principle is formulated by requiring that the `truth' predicate functor ? : B !Ppreserve initial algebras. We then show that when the fibration admits a comprehension principle, analogous to the one in set theory, it satisfies the induction principle. We also consider the appropriate extensions of the above formulation to deal with initiality (and induction) in arbitrary contexts, i.e. the `stability' property of the induction principle. 1...
Towards Merging Recursion and Comonads
, 2000
"... Comonads are mathematical structures that account naturally for effects that derive from the context in which a program is executed. This paper reports ongoing work on the interaction between recursion and comonads. Two applications are shown that naturally lead to versions of a comonadic fold op ..."
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Comonads are mathematical structures that account naturally for effects that derive from the context in which a program is executed. This paper reports ongoing work on the interaction between recursion and comonads. Two applications are shown that naturally lead to versions of a comonadic fold operator on the product comonad. Both versions capture functions that require extra arguments for their computation and are related with the notion of strong datatype. 1 Introduction One of the main features of recursive operators derivable from datatype definitions is that they impose a structure upon programs which can be exploited for program transformation. Recursive operators structure functional programs according to the data structures they traverse or generate and come equipped with a battery of algebraic laws, also derivable from type definitions, which are used in program calculations [24, 11, 5, 15]. Some of these laws, the socalled fusion laws, are particularly interesting in p...
Inductive Data Types for Predicate Transformers
 Information Processing Letters
, 1995
"... Introduction Modern functional programming languages [5, 6] and specification formalisms [3] are built around the notion of inductive data types and homomorphisms on these data types. Such homomorphisms, which correspond to the familiar fold or reduce operators in functional programming, are calle ..."
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Introduction Modern functional programming languages [5, 6] and specification formalisms [3] are built around the notion of inductive data types and homomorphisms on these data types. Such homomorphisms, which correspond to the familiar fold or reduce operators in functional programming, are called catamorphisms. In this note, it is shown how catamorphisms can be generalised from functions to relations, and from relations to predicate transformers. The first step of this generalisation (from functions to relations) was already achieved in a slightly different setting by Backhouse et al. [2]; the generalisation to predicate transformers is new. In practical terms, the main result presented here says that a calculus based on predicate transformers (like the refinement calculus studied by Back, Morgan and others [1, 14]) can be enriched with program constructors for iterating over inductive data types. The refineme
Programming with Algebras
 Advanced Functional Programming, number 925 in Lecture Notes in Computer Science
, 1995
"... Introduction From the early days of computing, many individuals have recognized that algebras provide interesting mathematical models for at least some aspects of programs. In mathematics, an algebra consists of a set (called the carrier of the algebra), together with a finite set of total function ..."
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Introduction From the early days of computing, many individuals have recognized that algebras provide interesting mathematical models for at least some aspects of programs. In mathematics, an algebra consists of a set (called the carrier of the algebra), together with a finite set of total functions that have the carrier set as their common codomain. The algebras we learn in school, however, are usually those derived from number theory and programs are more diverse, if not richer, than operations on numbers. A somewhat more abstract notion, called signature algebras, has been used for some time to to model abstract data types [GTW78]. A signature defines a set of typed operator symbols without specifying functions that would be the actual operators. Thus a signature defines a class of algebras, namely the algebras whose operators conform to the typing constraints imposed by the signature. Signature algebras have been helpful in understanding the issues involved in abstract dat
Decomposing Typed Lambda Calculus Into a Couple of Categorical Programming Languages
 Proc. CTCS , Lect. Notes in Computer Science 953
, 1995
"... . We give two categorical programming languages with variable arrows and associated abstraction/reduction mechanisms, which extend the possibility of categorical programming [Hag87, CF92] in practice. These languages are complementary to each other  one of them provides a firstorder programming s ..."
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Cited by 8 (0 self)
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. We give two categorical programming languages with variable arrows and associated abstraction/reduction mechanisms, which extend the possibility of categorical programming [Hag87, CF92] in practice. These languages are complementary to each other  one of them provides a firstorder programming style whereas the other does higherorder  and are "children" of the simply typed lambda calculus in the sense that we can decompose typed lambda calculus into them and, conversely, the combination of them is equivalent to typed lambda calculus. This decomposition is a consequence of a semantic analysis on typed lambda calculus due to C. Hermida and B. Jacobs [HJ94]. 1 Introduction There have been several attempts applying category theory to designing programming languages directly, especially to typed functional programming languages, since category theory itself has been a typed functional language for various mathematics. If one (possibly a programmer or a mathematician) can regard a cat...
A Calculational Approach to Strong Datatypes
 Department of Informatics, University of Oslo
, 1997
"... This paper describes calculational properties of fold, a recursive functional on inductive types that extends the standard catamorphism [MFP91] to global parameters. As described by Cockett and Spencer [CS91] fold is definable for those inductive types that are strong in the sense of being given by ..."
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Cited by 5 (2 self)
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This paper describes calculational properties of fold, a recursive functional on inductive types that extends the standard catamorphism [MFP91] to global parameters. As described by Cockett and Spencer [CS91] fold is definable for those inductive types that are strong in the sense of being given by an algebra that happens to be initial with parameters. The laws for fold are presented in two groups. The first one corresponds to standard laws for catamorphisms now adapted to folds when these are regarded as catamorphisms in a category of socalled Xactions (for X an object of parameters). The second group contains laws that describe the combination of folds and the interaction between folds and catamorphisms. 1 Introduction In programming semantics, a recursive type is understood as a solution of a recursive type equation. Least fixpoints of covariant recursive type equations correspond to the socalled inductive types which contain only finite elements, like natural numbers, lists o...
What is a Data Type?
, 1996
"... A program derivation is said to be polytypic if some of its parameters are data types. Polytypic program derivations necessitate a general, noninductive definition of `data type'. Here we propose such a definition: a data type is a relator that has membership. It is shown how this definition implie ..."
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Cited by 4 (1 self)
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A program derivation is said to be polytypic if some of its parameters are data types. Polytypic program derivations necessitate a general, noninductive definition of `data type'. Here we propose such a definition: a data type is a relator that has membership. It is shown how this definition implies various other properties that are shared by all data types. In particular, all data types have a unique strength, and all natural transformations between data types are strong. 1 Introduction What is a data type? It is easy to list a number of examples: pairs, lists, bags, finite sets, possibly infinite sets, function spaces . . . but such a list of examples hardly makes a definition. The obvious formalisation is a definition that builds up the class of data types inductively; such an inductive definition, however, leads to cumbersome proofs if we want to prove a property of all data types. Here we aim to give a noninductive characterisation, defining a data type as a mathematical object...