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Generic programming: An introduction
 3rd International Summer School on Advanced Functional Programming
, 1999
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When Do Datatypes Commute?
 Category Theory and Computer Science, 7th International Conference, volume 1290 of LNCS
, 1997
"... Polytypic programs are programs that are parameterised by type constructors (like List), unlike polymorphic programs which are parameterised by types (like Int). In this paper we formulate precisely the polytypic programming problem of "commuting " two datatypes. The precise formulation involves ..."
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Cited by 15 (3 self)
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Polytypic programs are programs that are parameterised by type constructors (like List), unlike polymorphic programs which are parameterised by types (like Int). In this paper we formulate precisely the polytypic programming problem of "commuting " two datatypes. The precise formulation involves a novel notion of higher order polymorphism. We demonstrate via a number of examples the relevance and interest of the problem, and we show that all "regular datatypes" (the sort of datatypes that one can define in a functional programming language) do indeed commute according to our specification. The framework we use is the theory of allegories, a combination of category theory with the pointfree relation calculus. 1 Polytypism The ability to abstract is vital to success in computer programming. At the macro level of requirements engineering the successful designer is the one able to abstract from the particular wishes of a few clients a general purpose product that can capture a l...
Universal regular path queries
 HigherOrder and Symbolic Computation
, 2003
"... Given are a directed edgelabelled graph G with a distinguished node n0, and a regular expression P which may contain variables. We wish to compute all substitutions φ (of symbols for variables), together with all nodes n such that all paths n0 → n are in φ(P). We derive an algorithm for this proble ..."
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Cited by 12 (1 self)
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Given are a directed edgelabelled graph G with a distinguished node n0, and a regular expression P which may contain variables. We wish to compute all substitutions φ (of symbols for variables), together with all nodes n such that all paths n0 → n are in φ(P). We derive an algorithm for this problem using relational algebra, and show how it may be implemented in Prolog. The motivation for the problem derives from a declarative framework for specifying compiler optimisations. 1 Bob Paige and IFIP WG 2.1 Bob Paige was a longstanding member of IFIP Working Group 2.1 on Algorithmic Languages and Calculi. In recent years, the main aim of this group has been to investigate the derivation of algorithms from specifications by program transformation. Already in the mideighties, Bob was way ahead of the pack: instead of applying transformational techniques to wellworn examples, he was applying his theories of program transformation to new problems, and discovering new algorithms [16, 48, 52]. The secret of his success lay partly in his insistence on the study of general algorithm design strategies (in particular
Container Types Categorically
, 2000
"... A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a ..."
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Cited by 12 (0 self)
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A program derivation is said to be polytypic if some of its parameters are data types. Often these data types are container types, whose elements store data. Polytypic program derivations necessitate a general, noninductive definition of `container (data) type'. Here we propose such a definition: a container type is a relator that has membership. It is shown how this definition implies various other properties that are shared by all container types. In particular, all container types have a unique strength, and all natural transformations between container types are strong. Capsule Review Progress in a scientific dicipline is readily equated with an increase in the volume of knowledge, but the true milestones are formed by the introduction of solid, precise and usable definitions. Here you will find the first generic (`polytypic') definition of the notion of `container type', a definition that is remarkably simple and suitable for formal generic proofs (as is amply illustrated in t...
Characterizing Determinacy in Kleene Algebras
 INFORMATION SCIENCES
, 2000
"... Elements of Kleene algebras can be used, among others, as abstractions of the inputoutput semantics of nondeterministic programs or as models for the association of pointers with their target objects. In the first case, one seeks to distinguish the subclass of elements that correspond to determinist ..."
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Cited by 11 (5 self)
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Elements of Kleene algebras can be used, among others, as abstractions of the inputoutput semantics of nondeterministic programs or as models for the association of pointers with their target objects. In the first case, one seeks to distinguish the subclass of elements that correspond to deterministic programs. In the second case one is only interested in functional correspondences, since it does not make sense for a pointer to point to two di#erent objects. We discuss several candidate notions of determinacy and clarify their relationship. Some characterizations that are equivalent in the case where the underlying Kleene algebra is an (abstract) relation algebra are not equivalent for general Kleene algebras.
Final Dialgebras: From Categories to Allegories
 Workshop on Fixed Points in Computer Science
, 1999
"... The study of inductive and coinductive types (like finite lists and streams, respectively) is usually conducted within the framework of category theory, which to all intents and purposes is a theory of sets and functions between sets. Allegory theory, an extension of category theory due to Freyd, is ..."
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Cited by 6 (3 self)
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The study of inductive and coinductive types (like finite lists and streams, respectively) is usually conducted within the framework of category theory, which to all intents and purposes is a theory of sets and functions between sets. Allegory theory, an extension of category theory due to Freyd, is better suited to modelling relations between sets as opposed to functions between sets. The question thus arises of how to extend the standard categorical results on the existence of final objects in categories (for example, coalgebras and products) to their existence in allegories. The motivation is to streamline current work on generic programming, in which the use of a relational theory rather than a functional theory has proved to be desirable. In this paper, we define the notion of a relational final dialgebra and prove, for an important class of dialgebras, that a relational final dialgebra exists in an allegory if and only if a final dialgebra exists in the underlying category of map...
Categorical Fixed Point Calculus
, 1995
"... A number of latticetheoretic fixed point rules are generalised to category theory and applied to the construction of isomorphisms between list structures. 1 Introduction Category theoreticians view a preordered set as a particular sort of category in which there is at most one arrow between any pa ..."
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Cited by 6 (0 self)
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A number of latticetheoretic fixed point rules are generalised to category theory and applied to the construction of isomorphisms between list structures. 1 Introduction Category theoreticians view a preordered set as a particular sort of category in which there is at most one arrow between any pair of objects. According to this view, several concepts of lattice theory are instances of concepts of category theory as shown in table 1. An alternative viewpoint, advocated by Lambek [9], is that lattice theory is a valuable source of inspiration for novel results in category theory. Indeed, it is our view that for the purposes of advancing programming methodology category theory may profitably be regarded as "coherently constructive lattice theory 1 ". That is to say, arrows between objects of a category may be seen as "witnesses" to a preordering between the objects. Category theory is thus "constructive" because it is a theory about how to construct such witnesses rather than a theor...
Mathematics of Recursive Program Construction
, 2001
"... A discipline for the design of recursive programs is presented. The core concept ..."
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Cited by 5 (1 self)
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A discipline for the design of recursive programs is presented. The core concept
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...
Inductive Datatypes with Laws and Subtyping – A Relational Model
 Faculty of Mathematics and Computing
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