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63
Bananas in Space: Extending Fold and Unfold to Exponential Types
, 1995
"... Fold and unfold are general purpose functionals for processing and constructing lists. By using the categorical approach of modelling recursive datatypes as fixed points of functors, these functionals and their algebraic properties were generalised from lists to polynomial (sumofproduct) datatypes ..."
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Cited by 95 (6 self)
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Fold and unfold are general purpose functionals for processing and constructing lists. By using the categorical approach of modelling recursive datatypes as fixed points of functors, these functionals and their algebraic properties were generalised from lists to polynomial (sumofproduct) datatypes. However, the restriction to polynomial datatypes is a serious limitation: it precludes the use of exponentials (functionspaces) , whereas it is central to functional programming that functions are firstclass values, and so exponentials should be able to be used freely in datatype definitions. In this paper we explain how Freyd's work on modelling recursive datatypes as fixed points of difunctors shows how to generalise fold and unfold from polynomial datatypes to those involving exponentials. Knowledge of category theory is not required; we use Gofer throughout as our metalanguage, making extensive use of constructor classes. 1 Introduction During the 1980s, Bird and Meertens [6, 22] d...
Datatypegeneric programming
 Spring School on DatatypeGeneric Programming, volume 4719 of Lecture Notes in Computer Science
"... Abstract. Generic programming aims to increase the flexibility of programming languages, by expanding the possibilities for parametrization — ideally, without also expanding the possibilities for uncaught errors. The term means different things to different people: parametric polymorphism, data abst ..."
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Cited by 48 (12 self)
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Abstract. Generic programming aims to increase the flexibility of programming languages, by expanding the possibilities for parametrization — ideally, without also expanding the possibilities for uncaught errors. The term means different things to different people: parametric polymorphism, data abstraction, metaprogramming, and so on. We use it to mean polytypism, that is, parametrization by the shape of data structures rather than their contents. To avoid confusion with other uses, we have coined the qualified term datatypegeneric programming for this purpose. In these lecture notes, we expand on the definition of datatypegeneric programming, and present some examples of datatypegeneric programs. We also explore the connection with design patterns in objectoriented programming; in particular, we argue that certain design patterns are just higherorder datatypegeneric programs. 1
Calculate Polytypically!
 In PLILP'96, volume 1140 of LNCS
, 1996
"... A polytypic function definition is a function definition that is parametrised with a datatype. It embraces a class of algorithms. As an example we define a simple polytypic "crush" combinator that can be used to calculate polytypically. The ability to define functions polytypically adds another leve ..."
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Cited by 41 (3 self)
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A polytypic function definition is a function definition that is parametrised with a datatype. It embraces a class of algorithms. As an example we define a simple polytypic "crush" combinator that can be used to calculate polytypically. The ability to define functions polytypically adds another level of flexibility in the reusability of programming idioms and in the design of libraries of interoperable components.
Calculating Functional Programs
 Algebraic and Coalgebraic Methods in the Mathematics of Program Construction, volume 2297 of LNCS, chapter 5
, 2000
"... A good way of developing a correct program is to calculate it from its specification. Functional programming languages are especially suitable for this, because their referential transparency greatly helps calculation. We discuss the ideas behind program calculation, and illustrate with an examp ..."
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Cited by 26 (8 self)
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A good way of developing a correct program is to calculate it from its specification. Functional programming languages are especially suitable for this, because their referential transparency greatly helps calculation. We discuss the ideas behind program calculation, and illustrate with an example (the maximum segment sum problem). We show that calculations are driven by promotion, and that promotion properties arise from universal properties of the data types involved. 1 Context The history of computing is a story of two contrasting trends. On the one hand, the cost and cost/performance ratio of computer hardware plummets; on the other, computer software is overcomplex, unreliable and almost inevitably over budget. Clearly, we have learnt how to build computers, but not yet how to program them. It is now widely accepted that adhoc approaches to constructing software break down as projects get more ambitious. A more formal approach, based on sound mathematical foundations, i...
Manufacturing Datatypes
, 1999
"... This paper describes a general framework for designing purely functional datatypes that automatically satisfy given size or structural constraints. Using the framework we develop implementations of different matrix types (eg square matrices) and implementations of several tree types (eg Braun trees, ..."
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Cited by 24 (3 self)
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This paper describes a general framework for designing purely functional datatypes that automatically satisfy given size or structural constraints. Using the framework we develop implementations of different matrix types (eg square matrices) and implementations of several tree types (eg Braun trees, 23 trees). Consider, for instance, representing square n \Theta n matrices. The usual representation using lists of lists fails to meet the structural constraints: there is no way to ensure that the outer list and the inner lists have the same length. The main idea of our approach is to solve in a first step a related, but simpler problem, namely to generate the multiset of all square numbers. In order to describe this multiset we employ recursion equations involving finite multisets, multiset union, addition and multiplication lifted to multisets. In a second step we mechanically derive datatype definitions from these recursion equations which enforce the `squareness' constraint. The tra...
Fast and Loose Reasoning is Morally Correct
, 2006
"... Functional programmers often reason about programs as if they were written in a total language, expecting the results to carry over to nontotal (partial) languages. We justify such reasoning. ..."
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Cited by 24 (0 self)
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Functional programmers often reason about programs as if they were written in a total language, expecting the results to carry over to nontotal (partial) languages. We justify such reasoning.
Fold and Unfold for Program Semantics
 In Proc. 3rd ACM SIGPLAN International Conference on Functional Programming
, 1998
"... In this paper we explain how recursion operators can be used to structure and reason about program semantics within a functional language. In particular, we show how the recursion operator fold can be used to structure denotational semantics, how the dual recursion operator unfold can be used to str ..."
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Cited by 22 (4 self)
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In this paper we explain how recursion operators can be used to structure and reason about program semantics within a functional language. In particular, we show how the recursion operator fold can be used to structure denotational semantics, how the dual recursion operator unfold can be used to structure operational semantics, and how algebraic properties of these operators can be used to reason about program semantics. The techniques are explained with the aid of two main examples, the first concerning arithmetic expressions, and the second concerning Milner's concurrent language CCS. The aim of the paper is to give functional programmers new insights into recursion operators, program semantics, and the relationships between them. 1 Introduction Many computations are naturally expressed as recursive programs defined in terms of themselves, and properties proved of such programs using some form of inductive argument. Not surprisingly, many programs will have a similar recursive stru...
Proof Methods for Corecursive Programs
 Fundamenta Informaticae Special Issue on Program Transformation
, 1999
"... This article is a tutorial on four methods for proving properties of corecursive programs: fixpoint induction, the approximation lemma, coinduction, and fusion. ..."
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Cited by 21 (6 self)
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This article is a tutorial on four methods for proving properties of corecursive programs: fixpoint induction, the approximation lemma, coinduction, and fusion.
Comparing Libraries for Generic Programming in Haskell
, 2008
"... Datatypegeneric 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 20 (10 self)
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Datatypegeneric 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 datatypegeneric 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 datatypegeneric programming.
A Refinement Algebra for ObjectOriented Programming
, 2003
"... In this article we introduce a comprehensive set of algebraic laws for rool, a language similar to sequential Java but with a copy semantics. We present a few laws of commands, but focus on the objectoriented features of the language. We show that this set of laws is complete in the sense that ..."
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Cited by 16 (3 self)
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In this article we introduce a comprehensive set of algebraic laws for rool, a language similar to sequential Java but with a copy semantics. We present a few laws of commands, but focus on the objectoriented features of the language. We show that this set of laws is complete in the sense that it is sufficient to reduce an arbitrary rool program to a normal form expressed in a restricted subset of the rool operators. We also