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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...
Galois Connections Presented Calculationally.
, 1992
"... properties of Galois connections 29 4.1 Preorders : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 4.1.1 Calculating in preorders : : : : : : : : : : : : : : : : : : : : : : : : 30 4.1.2 Alternative definitions : : : : : : : : : : : : : : : : : : : : : : : : : 33 4.1.3 Un ..."
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Cited by 16 (0 self)
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properties of Galois connections 29 4.1 Preorders : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 4.1.1 Calculating in preorders : : : : : : : : : : : : : : : : : : : : : : : : 30 4.1.2 Alternative definitions : : : : : : : : : : : : : : : : : : : : : : : : : 33 4.1.3 Uniqueness of adjoints in a preorder : : : : : : : : : : : : : : : : : 34 4.2 Partial orders : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 35 4.2.1 More cancellation laws : : : : : : : : : : : : : : : : : : : : : : : : : 35 4.2.2 Existence of adjoints : : : : : : : : : : : : : : : : : : : : : : : : : : 36 4.2.3 The closure connection : : : : : : : : : : : : : : : : : : : : : : : : : 40 4.2.4 "Perfect" connections : : : : : : : : : : : : : : : : : : : : : : : : : : 41 4.3 Complete lattices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 i 5 Application: The Domain Operator 47 5.1 Monotypes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :...
Between Functions and Relations in Calculating Programs
, 1992
"... This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made ..."
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Cited by 15 (4 self)
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This thesis is about the calculational approach to programming, in which one derives programs from specifications. One such calculational paradigm is Ruby, the relational calculus developed by Jones and Sheeran for describing and designing circuits. We identify two shortcomings with derivations made using Ruby. The first is that the notion of a program being an implementation of a specification has never been made precise. The second is to do with types. Fundamental to the use of type information in deriving programs is the idea of having types as special kinds of programs. In Ruby, types are partial equivalence relations (pers). Unfortunately, manipulating some formulae involving types has proved difficult within Ruby. In particular, the preconditions of the `induction' laws that are much used within program derivation often work out to be assertions about types; such assertions have typically been verified either by informal arguments or by using predicate calculus, rather than by ap...
Transposing relations: from Maybe functions to hash tables
 In MPC’04, volume 3125 of LNCS
, 2004
"... Abstract. Functional transposition is a technique for converting relations into functions aimed at developing the relational algebra via the algebra of functions. This paper attempts to develop a basis for generic transposition. Two instances of this construction are considered, one applicable to an ..."
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Cited by 15 (11 self)
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Abstract. Functional transposition is a technique for converting relations into functions aimed at developing the relational algebra via the algebra of functions. This paper attempts to develop a basis for generic transposition. Two instances of this construction are considered, one applicable to any relation and the other applicable to simple relations only. Our illustration of the usefulness of the generic transpose takes advantage of the free theorem of a polymorphic function. We show how to derive laws of relational combinators as free theorems of their transposes. Finally, we relate the topic of functional transposition with the hashing technique for efficient data representation. 1
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...
Pointwise Relational Programming
 In Algebraic Methodology and Software Technology, volume 1816 of LNCS
, 2000
"... The pointfree relational calculus has been very successful as a language for discussing general programming principles. However, when it comes to specific applications, the calculus can be rather awkward to use: some things are more clearly and simply expressed using variables. The combination of v ..."
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Cited by 6 (0 self)
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The pointfree relational calculus has been very successful as a language for discussing general programming principles. However, when it comes to specific applications, the calculus can be rather awkward to use: some things are more clearly and simply expressed using variables. The combination of variables and relational combinators such as converse and choice yields a kind of nondeterministic functional programming language. We give a semantics for such a language, and illustrate with an example application.
Between Dynamic Programming and Greedy: Data Compression
 Programming Research Group, 11 Keble Road, Oxford OX1 3QD
, 1995
"... The derivation of certain algorithms can be seen as a hybrid form of dynamic programming and the greedy paradigm. We present a generic theorem about such algorithms, and show how it can be applied to the derivation of an algorithm for data compression. 1 Introduction Dynamic programming is a techni ..."
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Cited by 5 (0 self)
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The derivation of certain algorithms can be seen as a hybrid form of dynamic programming and the greedy paradigm. We present a generic theorem about such algorithms, and show how it can be applied to the derivation of an algorithm for data compression. 1 Introduction Dynamic programming is a technique for solving optimisation problems. A typical dynamic programming algorithm proceeds by decomposing the input in all possible ways, recursively solving the subproblems, and combining optimal solutions to subproblems into an optimal solution for the whole problem. The greedy paradigm is also a technique for solving optimisation problems and differs from dynamic programming in that only one decomposition of the input is considered. Such a decomposition is usually chosen to maximise some objective function, and this explains the term `greedy'. In our earlier work, we have characterised the use of dynamic programming and the greedy paradigm, using the categorical calculus of relations to der...
Transforming Data by Calculation
 IN GTTSE’07, VOLUME 5235 OF LNCS
, 2008
"... This paper addresses the foundations of datamodel transformation. A catalog of data mappings is presented which includes abstraction and representation relations and associated constraints. These are justified in an algebraic style via the pointfreetransform, a technique whereby predicates are lif ..."
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Cited by 5 (3 self)
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This paper addresses the foundations of datamodel transformation. A catalog of data mappings is presented which includes abstraction and representation relations and associated constraints. These are justified in an algebraic style via the pointfreetransform, a technique whereby predicates are lifted to binary relation terms (of the algebra of programming) in a twolevel style encompassing both data and operations. This approach to data calculation, which also includes transformation of recursive data models into “flat ” database schemes, is offered as alternative to standard database design from abstract models. The calculus is also used to establish a link between the proposed transformational style and bidirectional lenses developed in the context of the classical viewupdate problem.
Pointfree factorization of operation refinement
 In FM’06, volume 4085 of LNCS
, 2006
"... Abstract. The standard operation refinement ordering is a kind of “meet of opposites”: nondeterminism reduction suggests “smaller ” behaviour while increase of definition suggests “larger ” behaviour. Groves ’ factorization of this ordering into two simpler relations, one per refinement concern, ma ..."
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Cited by 4 (4 self)
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Abstract. The standard operation refinement ordering is a kind of “meet of opposites”: nondeterminism reduction suggests “smaller ” behaviour while increase of definition suggests “larger ” behaviour. Groves ’ factorization of this ordering into two simpler relations, one per refinement concern, makes it more mathematically tractable but is far from fully exploited in the literature. We present a pointfree theory for this factorization which is more agile and calculational than the standard settheoretic approach. In particular, we show that factorization leads to a simple proof of structural refinement for arbitrary parametric types and exploit factor instantiation across different subclasses of (relational) operation. The prospect of generalizing the factorization to coalgebraic refinement is discussed.
Inductive Datatypes with Laws and Subtyping – A Relational Model
 Faculty of Mathematics and Computing
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